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Signal Processing
Vol. 1, Nos. 1–2 (2007) 1–194
c 2007 L. R. Rabiner and R. W. Schafer
Introduction to Digital Speech Processing
Lawrence R. Rabiner
and Ronald W. Schafer
Rutgers University and University of California, Santa Barbara, USA,
Hewlett-Packard Laboratories, Palo Alto, CA, USA
Since even before the time of Alexander Graham Bell's revolution-
ary invention, engineers and scientists have studied the phenomenon
of speech communication with an eye on creating more efficient and
effective systems of human-to-human and human-to-machine communi-
cation. Starting in the 1960s, digital signal processing (DSP), assumed
a central role in speech studies, and today DSP is the key to realizing
the fruits of the knowledge that has been gained through decades of
research. Concomitant advances in integrated circuit technology and
computer architecture have aligned to create a technological environ-
ment with virtually limitless opportunities for innovation in speech
communication applications. In this text, we highlight the central role
of DSP techniques in modern speech communication research and appli-
cations. We present a comprehensive overview of digital speech process-
ing that ranges from the basic nature of the speech signal, through a
variety of methods of representing speech in digital form, to applica-
tions in voice communication and automatic synthesis and recognition
of speech. The breadth of this subject does not allow us to discuss any
aspect of speech processing to great depth; hence our goal is to pro-
vide a useful introduction to the wide range of important concepts that
comprise the field of digital speech processing. A more comprehensive
treatment will appear in the forthcoming book, Theory and Application
of Digital Speech Processing [101].
The fundamental purpose of speech is communication, i.e., the trans-
mission of messages. According to Shannon's information theory [116],
a message represented as a sequence of discrete symbols can be quanti-
fied by its information content in bits, and the rate of transmission of
information is measured in bits/second (bps). In speech production, as
well as in many human-engineered electronic communication systems,
the information to be transmitted is encoded in the form of a contin-
uously varying (analog) waveform that can be transmitted, recorded,
manipulated, and ultimately decoded by a human listener. In the case
of speech, the fundamental analog form of the message is an acous-
tic waveform, which we call the speech signal. Speech signals, as illus-
trated in Figure 1.1, can be converted to an electrical waveform by
a microphone, further manipulated by both analog and digital signal
processing, and then converted back to acoustic form by a loudspeaker,
a telephone handset or headphone, as desired. This form of speech pro-
cessing is, of course, the basis for Bell's telephone invention as well as
today's multitude of devices for recording, transmitting, and manip-
ulating speech and audio signals. Although Bell made his invention
without knowing the fundamentals of information theory, these ideas
4 Introduction
0 0.02 0.04 0.06 0.08 0.1 0.12
time in seconds
Fig. 1.1 A speech waveform with phonetic labels for the text message "Should we chase."
have assumed great importance in the design of sophisticated modern
communications systems. Therefore, even though our main focus will
be mostly on the speech waveform and its representation in the form of
parametric models, it is nevertheless useful to begin with a discussion
of how information is encoded in the speech waveform.
1.1 The Speech Chain
Figure 1.2 shows the complete process of producing and perceiving
speech from the formulation of a message in the brain of a talker, to
the creation of the speech signal, and finally to the understanding of
the message by a listener. In their classic introduction to speech sci-
ence, Denes and Pinson aptly referred to this process as the "speech
chain" [29]. The process starts in the upper left as a message repre-
sented somehow in the brain of the speaker. The message information
can be thought of as having a number of different representations dur-
ing the process of speech production (the upper path in Figure 1.2).
1.1 The Speech Chain 5
Vocal Tract
discrete input continuous input
Speech Production
Speech Perception
continuous output discrete output
articulatory motions
phonemes, words
50 bps 200 bps 2000 bps 64-700 Kbps
information rate
Fig. 1.2 The Speech Chain: from message, to speech signal, to understanding.
For example the message could be represented initially as English text.
In order to "speak" the message, the talker implicitly converts the text
into a symbolic representation of the sequence of sounds corresponding
to the spoken version of the text. This step, called the language code
generator in Figure 1.2, converts text symbols to phonetic symbols
(along with stress and durational information) that describe the basic
sounds of a spoken version of the message and the manner (i.e., the
speed and emphasis) in which the sounds are intended to be produced.
As an example, the segments of the waveform of Figure 1.1 are labeled
with phonetic symbols using a computer-keyboard-friendly code called
Thus, the text "should we chase" is represented phoneti-
cally (in ARPAbet symbols) as [SH UH D — W IY — CH EY S]. (See
Chapter 2 for more discussion of phonetic transcription.) The third step
in the speech production process is the conversion to "neuro-muscular
controls," i.e., the set of control signals that direct the neuro-muscular
system to move the speech articulators, namely the tongue, lips, teeth,
The International Phonetic Association (IPA) provides a set of rules for phonetic tran-
scription using an equivalent set of specialized symbols. The ARPAbet code does not
require special fonts and is thus more convenient for computer applications.
6 Introduction
jaw and velum, in a manner that is consistent with the sounds of the
desired spoken message and with the desired degree of emphasis. The
end result of the neuro-muscular controls step is a set of articulatory
motions (continuous control) that cause the vocal tract articulators to
move in a prescribed manner in order to create the desired sounds.
Finally the last step in the Speech Production process is the "vocal
tract system" that physically creates the necessary sound sources and
the appropriate vocal tract shapes over time so as to create an acoustic
waveform, such as the one shown in Figure 1.1, that encodes the infor-
mation in the desired message into the speech signal.
To determine the rate of information flow during speech produc-
tion, assume that there are about 32 symbols (letters) in the language
(in English there are 26 letters, but if we include simple punctuation
we get a count closer to 32 = 2
symbols). Furthermore, the rate of
speaking for most people is about 10 symbols per second (somewhat
on the high side, but still acceptable for a rough information rate esti-
mate). Hence, assuming independent letters as a simple approximation,
we estimate the base information rate of the text message as about
50 bps (5 bits per symbol times 10 symbols per second). At the second
stage of the process, where the text representation is converted into
phonemes and prosody (e.g., pitch and stress) markers, the informa-
tion rate is estimated to increase by a factor of 4 to about 200 bps. For
example, the ARBAbet phonetic symbol set used to label the speech
sounds in Figure 1.1 contains approximately 64 = 2
symbols, or about
6 bits/phoneme (again a rough approximation assuming independence
of phonemes). In Figure 1.1, there are 8 phonemes in approximately
600 ms. This leads to an estimate of 8 6/0.6 = 80 bps. Additional
information required to describe prosodic features of the signal (e.g.,
duration, pitch, loudness) could easily add 100 bps to the total infor-
mation rate for a message encoded as a speech signal.
The information representations for the first two stages in the speech
chain are discrete so we can readily estimate the rate of information
flow with some simple assumptions. For the next stage in the speech
production part of the speech chain, the representation becomes con-
tinuous (in the form of control signals for articulatory motion). If they
could be measured, we could estimate the spectral bandwidth of these
1.1 The Speech Chain 7
control signals and appropriately sample and quantize these signals to
obtain equivalent digital signals for which the data rate could be esti-
mated. The articulators move relatively slowly compared to the time
variation of the resulting acoustic waveform. Estimates of bandwidth
and required accuracy suggest that the total data rate of the sampled
articulatory control signals is about 2000 bps [34]. Thus, the original
text message is represented by a set of continuously varying signals
whose digital representation requires a much higher data rate than
the information rate that we estimated for transmission of the mes-
sage as a speech signal.
Finally, as we will see later, the data rate
of the digitized speech waveform at the end of the speech production
part of the speech chain can be anywhere from 64,000 to more than
700,000 bps. We arrive at such numbers by examining the sampling
rate and quantization required to represent the speech signal with a
desired perceptual fidelity. For example, "telephone quality" requires
that a bandwidth of 0–4 kHz be preserved, implying a sampling rate of
8000 samples/s. Each sample can be quantized with 8 bits on a log scale,
resulting in a bit rate of 64,000 bps. This representation is highly intelli-
gible (i.e., humans can readily extract the message from it) but to most
listeners, it will sound different from the original speech signal uttered
by the talker. On the other hand, the speech waveform can be repre-
sented with "CD quality" using a sampling rate of 44,100 samples/s
with 16 bit samples, or a data rate of 705,600 bps. In this case, the
reproduced acoustic signal will be virtually indistinguishable from the
original speech signal.
As we move from text to speech waveform through the speech chain,
the result is an encoding of the message that can be effectively transmit-
ted by acoustic wave propagation and robustly decoded by the hear-
ing mechanism of a listener. The above analysis of data rates shows
that as we move from text to sampled speech waveform, the data rate
can increase by a factor of 10,000. Part of this extra information rep-
resents characteristics of the talker such as emotional state, speech
mannerisms, accent, etc., but much of it is due to the inefficiency
Note that we introduce the term data rate for digital representations to distinguish from
the inherent information content of the message represented by the speech signal.
8 Introduction
of simply sampling and finely quantizing analog signals. Thus, moti-
vated by an awareness of the low intrinsic information rate of speech,
a central theme of much of digital speech processing is to obtain a
digital representation with lower data rate than that of the sampled
The complete speech chain consists of a speech production/
generation model, of the type discussed above, as well as a speech
perception/recognition model, as shown progressing to the left in the
bottom half of Figure 1.2. The speech perception model shows the series
of steps from capturing speech at the ear to understanding the mes-
sage encoded in the speech signal. The first step is the effective con-
version of the acoustic waveform to a spectral representation. This is
done within the inner ear by the basilar membrane, which acts as a
non-uniform spectrum analyzer by spatially separating the spectral
components of the incoming speech signal and thereby analyzing them
by what amounts to a non-uniform filter bank. The next step in the
speech perception process is a neural transduction of the spectral fea-
tures into a set of sound features (or distinctive features as they are
referred to in the area of linguistics) that can be decoded and processed
by the brain. The next step in the process is a conversion of the sound
features into the set of phonemes, words, and sentences associated with
the in-coming message by a language translation process in the human
brain. Finally, the last step in the speech perception model is the con-
version of the phonemes, words and sentences of the message into an
understanding of the meaning of the basic message in order to be able
to respond to or take some appropriate action. Our fundamental under-
standing of the processes in most of the speech perception modules in
Figure 1.2 is rudimentary at best, but it is generally agreed that some
physical correlate of each of the steps in the speech perception model
occur within the human brain, and thus the entire model is useful for
thinking about the processes that occur.
There is one additional process shown in the diagram of the com-
plete speech chain in Figure 1.2 that we have not discussed — namely
the transmission channel between the speech generation and speech
perception parts of the model. In its simplest embodiment, this trans-
mission channel consists of just the acoustic wave connection between
1.2 Applications of Digital Speech Processing 9
a speaker and a listener who are in a common space. It is essen-
tial to include this transmission channel in our model for the speech
chain since it includes real world noise and channel distortions that
make speech and message understanding more difficult in real com-
munication environments. More interestingly for our purpose here —
it is in this domain that we find the applications of digital speech
1.2 Applications of Digital Speech Processing
The first step in most applications of digital speech processing is to
convert the acoustic waveform to a sequence of numbers. Most modern
A-to-D converters operate by sampling at a very high rate, applying a
digital lowpass filter with cutoff set to preserve a prescribed bandwidth,
and then reducing the sampling rate to the desired sampling rate, which
can be as low as twice the cutoff frequency of the sharp-cutoff digital
filter. This discrete-time representation is the starting point for most
applications. From this point, other representations are obtained by
digital processing. For the most part, these alternative representations
are based on incorporating knowledge about the workings of the speech
chain as depicted in Figure 1.2. As we will see, it is possible to incor-
porate aspects of both the speech production and speech perception
process into the digital representation and processing. It is not an over-
simplification to assert that digital speech processing is grounded in a
set of techniques that have the goal of pushing the data rate of the
speech representation to the left along either the upper or lower path
in Figure 1.2.
The remainder of this chapter is devoted to a brief summary of the
applications of digital speech processing, i.e., the systems that people
interact with daily. Our discussion will confirm the importance of the
digital representation in all application areas.
1.2.1 Speech Coding
Perhaps the most widespread applications of digital speech process-
ing technology occur in the areas of digital transmission and storage
10 Introduction
Channel or
samples data
) (t x
] [n x ] [n y
] [n y ] [n x ) (t x
Fig. 1.3 Speech coding block diagram — encoder and decoder.
of speech signals. In these areas the centrality of the digital repre-
sentation is obvious, since the goal is to compress the digital wave-
form representation of speech into a lower bit-rate representation. It
is common to refer to this activity as "speech coding" or "speech
Figure 1.3 shows a block diagram of a generic speech encod-
ing/decoding (or compression) system. In the upper part of the figure,
the A-to-D converter converts the analog speech signal x
(t) to a sam-
pled waveform representation x[n]. The digital signal x[n] is analyzed
and coded by digital computation algorithms to produce a new digital
signal y[n] that can be transmitted over a digital communication chan-
nel or stored in a digital storage medium as ˆ y[n]. As we will see, there
are a myriad of ways to do the encoding so as to reduce the data rate
over that of the sampled and quantized speech waveform x[n]. Because
the digital representation at this point is often not directly related to
the sampled speech waveform, y[n] and ˆ y[n] are appropriately referred
to as data signals that represent the speech signal. The lower path in
Figure 1.3 shows the decoder associated with the speech coder. The
received data signal ˆ y[n] is decoded using the inverse of the analysis
processing, giving the sequence of samples ˆ x[n] which is then converted
(using a D-to-A Converter) back to an analog signal ˆ x
(t) for human
listening. The decoder is often called a synthesizer because it must
reconstitute the speech waveform from data that may bear no direct
relationship to the waveform.
1.2 Applications of Digital Speech Processing 11
With carefully designed error protection coding of the digital
representation, the transmitted (y[n]) and received (ˆ y[n]) data can be
essentially identical. This is the quintessential feature of digital coding.
In theory, perfect transmission of the coded digital representation is
possible even under very noisy channel conditions, and in the case of
digital storage, it is possible to store a perfect copy of the digital repre-
sentation in perpetuity if sufficient care is taken to update the storage
medium as storage technology advances. This means that the speech
signal can be reconstructed to within the accuracy of the original cod-
ing for as long as the digital representation is retained. In either case,
the goal of the speech coder is to start with samples of the speech signal
and reduce (compress) the data rate required to represent the speech
signal while maintaining a desired perceptual fidelity. The compressed
representation can be more efficiently transmitted or stored, or the bits
saved can be devoted to error protection.
Speech coders enable a broad range of applications including nar-
rowband and broadband wired telephony, cellular communications,
voice over internet protocol (VoIP) (which utilizes the internet as
a real-time communications medium), secure voice for privacy and
encryption (for national security applications), extremely narrowband
communications channels (such as battlefield applications using high
frequency (HF) radio), and for storage of speech for telephone answer-
ing machines, interactive voice response (IVR) systems, and pre-
recorded messages. Speech coders often utilize many aspects of both
the speech production and speech perception processes, and hence may
not be useful for more general audio signals such as music. Coders that
are based on incorporating only aspects of sound perception generally
do not achieve as much compression as those based on speech produc-
tion, but they are more general and can be used for all types of audio
signals. These coders are widely deployed in MP3 and AAC players and
for audio in digital television systems [120].
1.2.2 Text-to-Speech Synthesis
For many years, scientists and engineers have studied the speech pro-
duction process with the goal of building a system that can start with
12 Introduction
Fig. 1.4 Text-to-speech synthesis system block diagram.
text and produce speech automatically. In a sense, a text-to-speech
synthesizer such as depicted in Figure 1.4 is a digital simulation of the
entire upper part of the speech chain diagram. The input to the system
is ordinary text such as an email message or an article from a newspa-
per or magazine. The first block in the text-to-speech synthesis system,
labeled linguistic rules, has the job of converting the printed text input
into a set of sounds that the machine must synthesize. The conversion
from text to sounds involves a set of linguistic rules that must determine
the appropriate set of sounds (perhaps including things like emphasis,
pauses, rates of speaking, etc.) so that the resulting synthetic speech
will express the words and intent of the text message in what passes
for a natural voice that can be decoded accurately by human speech
perception. This is more difficult than simply looking up the words in
a pronouncing dictionary because the linguistic rules must determine
how to pronounce acronyms, how to pronounce ambiguous words like
read, bass, object, how to pronounce abbreviations like St. (street or
Saint), Dr. (Doctor or drive), and how to properly pronounce proper
names, specialized terms, etc. Once the proper pronunciation of the
text has been determined, the role of the synthesis algorithm is to cre-
ate the appropriate sound sequence to represent the text message in
the form of speech. In essence, the synthesis algorithm must simulate
the action of the vocal tract system in creating the sounds of speech.
There are many procedures for assembling the speech sounds and com-
piling them into a proper sentence, but the most promising one today is
called "unit selection and concatenation." In this method, the computer
stores multiple versions of each of the basic units of speech (phones, half
phones, syllables, etc.), and then decides which sequence of speech units
sounds best for the particular text message that is being produced. The
basic digital representation is not generally the sampled speech wave.
Instead, some sort of compressed representation is normally used to
1.2 Applications of Digital Speech Processing 13
save memory and, more importantly, to allow convenient manipulation
of durations and blending of adjacent sounds. Thus, the speech syn-
thesis algorithm would include an appropriate decoder, as discussed in
Section 1.2.1, whose output is converted to an analog representation
via the D-to-A converter.
Text-to-speech synthesis systems are an essential component of
modern human–machine communications systems and are used to do
things like read email messages over a telephone, provide voice out-
put from GPS systems in automobiles, provide the voices for talking
agents for completion of transactions over the internet, handle call cen-
ter help desks and customer care applications, serve as the voice for
providing information from handheld devices such as foreign language
phrasebooks, dictionaries, crossword puzzle helpers, and as the voice of
announcement machines that provide information such as stock quotes,
airline schedules, updates on arrivals and departures of flights, etc.
Another important application is in reading machines for the blind,
where an optical character recognition system provides the text input
to a speech synthesis system.
1.2.3 Speech Recognition and Other Pattern
Matching Problems
Another large class of digital speech processing applications is con-
cerned with the automatic extraction of information from the speech
signal. Most such systems involve some sort of pattern matching.
Figure 1.5 shows a block diagram of a generic approach to pattern
matching problems in speech processing. Such problems include the
following: speech recognition, where the object is to extract the mes-
sage from the speech signal; speaker recognition, where the goal is
to identify who is speaking; speaker verification, where the goal is
to verify a speaker's claimed identity from analysis of their speech
Fig. 1.5 Block diagram of general pattern matching system for speech signals.
14 Introduction
signal; word spotting, which involves monitoring a speech signal for
the occurrence of specified words or phrases; and automatic index-
ing of speech recordings based on recognition (or spotting) of spoken
The first block in the pattern matching system converts the ana-
log speech waveform to digital form using an A-to-D converter. The
feature analysis module converts the sampled speech signal to a set
of feature vectors. Often, the same analysis techniques that are used
in speech coding are also used to derive the feature vectors. The final
block in the system, namely the pattern matching block, dynamically
time aligns the set of feature vectors representing the speech signal with
a concatenated set of stored patterns, and chooses the identity associ-
ated with the pattern which is the closest match to the time-aligned set
of feature vectors of the speech signal. The symbolic output consists
of a set of recognized words, in the case of speech recognition, or the
identity of the best matching talker, in the case of speaker recognition,
or a decision as to whether to accept or reject the identity claim of a
speaker in the case of speaker verification.
Although the block diagram of Figure 1.5 represents a wide range
of speech pattern matching problems, the biggest use has been in the
area of recognition and understanding of speech in support of human–
machine communication by voice. The major areas where such a system
finds applications include command and control of computer software,
voice dictation to create letters, memos, and other documents, natu-
ral language voice dialogues with machines to enable help desks and
call centers, and for agent services such as calendar entry and update,
address list modification and entry, etc.
Pattern recognition applications often occur in conjunction with
other digital speech processing applications. For example, one of the
preeminent uses of speech technology is in portable communication
devices. Speech coding at bit rates on the order of 8 Kbps enables nor-
mal voice conversations in cell phones. Spoken name speech recognition
in cellphones enables voice dialing capability that can automatically
dial the number associated with the recognized name. Names from
directories with upwards of several hundred names can readily be rec-
ognized and dialed using simple speech recognition technology.
1.2 Applications of Digital Speech Processing 15
Another major speech application that has long been a dream
of speech researchers is automatic language translation. The goal of
language translation systems is to convert spoken words in one lan-
guage to spoken words in another language so as to facilitate natural
language voice dialogues between people speaking different languages.
Language translation technology requires speech synthesis systems that
work in both languages, along with speech recognition (and generally
natural language understanding) that also works for both languages;
hence it is a very difficult task and one for which only limited progress
has been made. When such systems exist, it will be possible for people
speaking different languages to communicate at data rates on the order
of that of printed text reading!
1.2.4 Other Speech Applications
The range of speech communication applications is illustrated in
Figure 1.6. As seen in this figure, the techniques of digital speech
processing are a key ingredient of a wide range of applications that
include the three areas of transmission/storage, speech synthesis, and
speech recognition as well as many others such as speaker identification,
speech signal quality enhancement, and aids for the hearing- or visually-
The block diagram in Figure 1.7 represents any system where time
signals such as speech are processed by the techniques of DSP. This
figure simply depicts the notion that once the speech signal is sampled,
it can be manipulated in virtually limitless ways by DSP techniques.
Here again, manipulations and modifications of the speech signal are
& Storage
Digital Speech Processing
of Speech
Aids for the
Fig. 1.6 Range of speech communication applications.
16 Introduction
Fig. 1.7 General block diagram for application of digital signal processing to speech signals.
usually achieved by transforming the speech signal into an alternative
representation (that is motivated by our understanding of speech pro-
duction and speech perception), operating on that representation by
further digital computation, and then transforming back to the wave-
form domain, using a D-to-A converter.
One important application area is speech enhancement, where the
goal is to remove or suppress noise or echo or reverberation picked up by
a microphone along with the desired speech signal. In human-to-human
communication, the goal of speech enhancement systems is to make the
speech more intelligible and more natural; however, in reality the best
that has been achieved so far is less perceptually annoying speech that
essentially maintains, but does not improve, the intelligibility of the
noisy speech. Success has been achieved, however, in making distorted
speech signals more useful for further processing as part of a speech
coder, synthesizer, or recognizer. An excellent reference in this area is
the recent textbook by Loizou [72].
Other examples of manipulation of the speech signal include
timescale modification to align voices with video segments, to mod-
ify voice qualities, and to speed-up or slow-down prerecorded speech
(e.g., for talking books, rapid review of voice mail messages, or careful
scrutinizing of spoken material).
1.3 Our Goal for this Text
We have discussed the speech signal and how it encodes information
for human communication. We have given a brief overview of the way
in which digital speech processing is being applied today, and we have
hinted at some of the possibilities that exist for the future. These and
many more examples all rely on the basic principles of digital speech
processing, which we will discuss in the remainder of this text. We
make no pretense of exhaustive coverage. The subject is too broad and
1.3 Our Goal for this Text 17
too deep. Our goal is only to provide an up-to-date introduction to this
fascinating field. We will not be able to go into great depth, and we will
not be able to cover all the possible applications of digital speech pro-
cessing techniques. Instead our focus is on the fundamentals of digital
speech processing and their application to coding, synthesis, and recog-
nition. This means that some of the latest algorithmic innovations and
applications will not be discussed — not because they are not interest-
ing, but simply because there are so many fundamental tried-and-true
techniques that remain at the core of digital speech processing. We
hope that this text will stimulate readers to investigate the subject in
greater depth using the extensive set of references provided.
The Speech Signal
As the discussion in Chapter 1 shows, the goal in many applications of
digital speech processing techniques is to move the digital representa-
tion of the speech signal from the waveform samples back up the speech
chain toward the message. To gain a better idea of what this means,
this chapter provides a brief overview of the phonetic representation of
speech and an introduction to models for the production of the speech
2.1 Phonetic Representation of Speech
Speech can be represented phonetically by a finite set of symbols called
the phonemes of the language, the number of which depends upon the
language and the refinement of the analysis. For most languages the
number of phonemes is between 32 and 64. A condensed inventory of
the sounds of speech in the English language is given in Table 2.1,
where the phonemes are denoted by a set of ASCII symbols called the
ARPAbet. Table 2.1 also includes some simple examples of ARPAbet
transcriptions of words containing each of the phonemes of English.
Additional phonemes can be added to Table 2.1 to account for allo-
phonic variations and events such as glottal stops and pauses.
2.1 Phonetic Representation of Speech 19
Table 2.1 Condensed list of ARPAbet phonetic symbols for North American English.
Class ARPAbet Example Transcription
Vowels and IY beet [B IY T]
diphthongs IH bit [B IH T]
EY bait [B EY T]
EH bet [B EH T]
AE bat [B AE T]
AA bob [B AA B]
AO born [B AO R N]
UH book [B UH K]
OW boat [B OW T]
UW boot [B UW T]
AH but [B AH T ]
ER bird [B ER D]
AY buy [B AY]
AW down [D AW N]
OY boy [B OY]
Glides Y you [Y UH]
R rent [R EH N T]
Liquids W wit [W IH T]
L l et [L EH T]
Nasals M met [M EH T]
N net [N EH T]
NG sing [S IH NG]
Stops P pat [P AE T]
B bet [B EH T]
T ten [T EH N]
D debt [D EH T]
K kit [K IH T]
G get [G EH T]
Fricatives HH hat [HH AE T]
F f at [F AE T]
V vat [V AE T]
TH thing [TH IH NG]
DH that [DH AE T]
S sat [S AE T]
Z zoo [Z UW]
SH shut [SH AH T]
ZH azure [AE ZH ER]
Affricates CH chase [CH EY S]
JH j udge [JH AH JH ]
This set of 39 phonemes is used in the CMU Pronouncing Dictionary available on-line at
Figure 1.1 on p. 4 shows how the sounds corresponding to the text
"should we chase" are encoded into a speech waveform. We see that, for
the most part, phonemes have a distinctive appearance in the speech
waveform. Thus sounds like /SH/ and /S/ look like (spectrally shaped)
20 The Speech Signal
random noise, while the vowel sounds /UH/, /IY/, and /EY/ are highly
structured and quasi-periodic. These differences result from the distinc-
tively different ways that these sounds are produced.
2.2 Models for Speech Production
A schematic longitudinal cross-sectional drawing of the human vocal
tract mechanism is given in Figure 2.1 [35]. This diagram highlights
the essential physical features of human anatomy that enter into the
final stages of the speech production process. It shows the vocal tract
as a tube of nonuniform cross-sectional area that is bounded at one end
by the vocal cords and at the other by the mouth opening. This tube
serves as an acoustic transmission system for sounds generated inside
the vocal tract. For creating nasal sounds like /M/, /N/, or /NG/,
a side-branch tube, called the nasal tract, is connected to the main
acoustic branch by the trapdoor action of the velum. This branch path
radiates sound at the nostrils. The shape (variation of cross-section
along the axis) of the vocal tract varies with time due to motions of
the lips, jaw, tongue, and velum. Although the actual human vocal
tract is not laid out along a straight line as in Figure 2.1, this type of
model is a reasonable approximation for wavelengths of the sounds in
The sounds of speech are generated in the system of Figure 2.1 in
several ways. Voiced sounds (vowels, liquids, glides, nasals in Table 2.1)
Fig. 2.1 Schematic model of the vocal tract system. (After Flanagan et al. [35].)
2.2 Models for Speech Production 21
are produced when the vocal tract tube is excited by pulses of air pres-
sure resulting from quasi-periodic opening and closing of the glottal
orifice (opening between the vocal cords). Examples in Figure 1.1 are
the vowels /UH/, /IY/, and /EY/, and the liquid consonant /W/.
Unvoiced sounds are produced by creating a constriction somewhere in
the vocal tract tube and forcing air through that constriction, thereby
creating turbulent air flow, which acts as a random noise excitation of
the vocal tract tube. Examples are the unvoiced fricative sounds such as
/SH/ and /S/. A third sound production mechanism is when the vocal
tract is partially closed off causing turbulent flow due to the constric-
tion, at the same time allowing quasi-periodic flow due to vocal cord
vibrations. Sounds produced in this manner include the voiced frica-
tives /V/, /DH/, /Z/, and /ZH/. Finally, plosive sounds such as /P/,
/T/, and /K/ and affricates such as /CH/ are formed by momentarily
closing off air flow, allowing pressure to build up behind the closure, and
then abruptly releasing the pressure. All these excitation sources cre-
ate a wide-band excitation signal to the vocal tract tube, which acts as
an acoustic transmission line with certain vocal tract shape-dependent
resonances that tend to emphasize some frequencies of the excitation
relative to others.
As discussed in Chapter 1 and illustrated by the waveform in
Figure 1.1, the general character of the speech signal varies at the
phoneme rate, which is on the order of 10 phonemes per second, while
the detailed time variations of the speech waveform are at a much higher
rate. That is, the changes in vocal tract configuration occur relatively
slowly compared to the detailed time variation of the speech signal. The
sounds created in the vocal tract are shaped in the frequency domain
by the frequency response of the vocal tract. The resonance frequencies
resulting from a particular configuration of the articulators are instru-
mental in forming the sound corresponding to a given phoneme. These
resonance frequencies are called the formant frequencies of the sound
[32, 34]. In summary, the fine structure of the time waveform is created
by the sound sources in the vocal tract, and the resonances of the vocal
tract tube shape these sound sources into the phonemes.
The system of Figure 2.1 can be described by acoustic theory,
and numerical techniques can be used to create a complete physical
22 The Speech Signal
excitation signal speech signal
Vocal Tract
Fig. 2.2 Source/system model for a speech signal.
simulation of sound generation and transmission in the vocal tract
[36, 93], but, for the most part, it is sufficient to model the produc-
tion of a sampled speech signal by a discrete-time system model such
as the one depicted in Figure 2.2. The discrete-time time-varying linear
system on the right in Figure 2.2 simulates the frequency shaping of
the vocal tract tube. The excitation generator on the left simulates the
different modes of sound generation in the vocal tract. Samples of a
speech signal are assumed to be the output of the time-varying linear
In general such a model is called a source/system model of speech
production. The short-time frequency response of the linear system
simulates the frequency shaping of the vocal tract system, and since the
vocal tract changes shape relatively slowly, it is reasonable to assume
that the linear system response does not vary over time intervals on the
order of 10 ms or so. Thus, it is common to characterize the discrete-
time linear system by a system function of the form:
H(z) =
1 −
(1 − d
(1 − c
, (2.1)
where the filter coefficients a
and b
(labeled as vocal tract parameters
in Figure 2.2) change at a rate on the order of 50–100 times/s. Some
of the poles (c
) of the system function lie close to the unit circle
and create resonances to model the formant frequencies. In detailed
modeling of speech production [32, 34, 64], it is sometimes useful to
employ zeros (d
) of the system function to model nasal and fricative
2.2 Models for Speech Production 23
sounds. However, as we discuss further in Chapter 4, many applications
of the source/system model only include poles in the model because this
simplifies the analysis required to estimate the parameters of the model
from the speech signal.
The box labeled excitation generator in Figure 2.2 creates an appro-
priate excitation for the type of sound being produced. For voiced
speech the excitation to the linear system is a quasi-periodic sequence
of discrete (glottal) pulses that look very much like those shown in the
righthand half of the excitation signal waveform in Figure 2.2. The fun-
damental frequency of the glottal excitation determines the perceived
pitch of the voice. The individual finite-duration glottal pulses have a
lowpass spectrum that depends on a number of factors [105]. There-
fore, the periodic sequence of smooth glottal pulses has a harmonic line
spectrum with components that decrease in amplitude with increasing
frequency. Often it is convenient to absorb the glottal pulse spectrum
contribution into the vocal tract system model of (2.1). This can be
achieved by a small increase in the order of the denominator over what
would be needed to represent the formant resonances. For unvoiced
speech, the linear system is excited by a random number generator
that produces a discrete-time noise signal with flat spectrum as shown
in the left-hand half of the excitation signal. The excitation in Fig-
ure 2.2 switches from unvoiced to voiced leading to the speech signal
output as shown in the figure. In either case, the linear system imposes
its frequency response on the spectrum to create the speech sounds.
This model of speech as the output of a slowly time-varying digital
filter with an excitation that captures the nature of the voiced/unvoiced
distinction in speech production is the basis for thinking about the
speech signal, and a wide variety of digital representations of the speech
signal are based on it. That is, the speech signal is represented by the
parameters of the model instead of the sampled waveform. By assuming
that the properties of the speech signal (and the model) are constant
over short time intervals, it is possible to compute/measure/estimate
the parameters of the model by analyzing short blocks of samples of
the speech signal. It is through such models and analysis techniques
that we are able to build properties of the speech production process
into digital representations of the speech signal.
24 The Speech Signal
2.3 More Refined Models
Source/system models as shown in Figure 2.2 with the system char-
acterized by a time-sequence of time-invariant systems are quite suffi-
cient for most applications in speech processing, and we shall rely on
such models throughout this text. However, such models are based on
many approximations including the assumption that the source and the
system do not interact, the assumption of linearity, and the assump-
tion that the distributed continuous-time vocal tract transmission sys-
tem can be modeled by a discrete linear time-invariant system. Fluid
mechanics and acoustic wave propagation theory are fundamental phys-
ical principles that must be applied for detailed modeling of speech
production. Since the early work of Flanagan and Ishizaka [34, 36, 51]
much work has been devoted to creating detailed simulations of glottal
flow, the interaction of the glottal source and the vocal tract in speech
production, and the nonlinearities that enter into sound generation and
transmission in the vocal tract. Stevens [121] and Quatieri [94] provide
useful discussions of these effects. For many years, researchers have
sought to measure the physical dimensions of the human vocal tract
during speech production. This information is essential for detailed sim-
ulations based on acoustic theory. Early efforts to measure vocal tract
area functions involved hand tracing on X-ray pictures [32]. Recent
advances in MRI imaging and computer image analysis have provided
significant advances in this area of speech science [17].
Hearing and Auditory Perception
In Chapter 2, we introduced the speech production process and showed
how we could model speech production using discrete-time systems.
In this chapter we turn to the perception side of the speech chain to
discuss properties of human sound perception that can be employed to
create digital representations of the speech signal that are perceptually
3.1 The Human Ear
Figure 3.1 shows a schematic view of the human ear showing the three
distinct sound processing sections, namely: the outer ear consisting of
the pinna, which gathers sound and conducts it through the external
canal to the middle ear; the middle ear beginning at the tympanic
membrane, or eardrum, and including three small bones, the malleus
(also called the hammer), the incus (also called the anvil) and the stapes
(also called the stirrup), which perform a transduction from acoustic
waves to mechanical pressure waves; and finally, the inner ear, which
consists of the cochlea and the set of neural connections to the auditory
nerve, which conducts the neural signals to the brain.
26 Hearing and Auditory Perception
Fig. 3.1 Schematic view of the human ear (inner and middle structures enlarged). (After
Flanagan [34].)
Figure 3.2 [107] depicts a block diagram abstraction of the auditory
system. The acoustic wave is transmitted from the outer ear to the
inner ear where the ear drum and bone structures convert the sound
wave to mechanical vibrations which ultimately are transferred to the
basilar membrane inside the cochlea. The basilar membrane vibrates in
a frequency-selective manner along its extent and thereby performs a
rough (non-uniform) spectral analysis of the sound. Distributed along
Fig. 3.2 Schematic model of the auditory mechanism. (After Sachs et al. [107].)
3.2 Perception of Loudness 27
the basilar membrane are a set of inner hair cells that serve to convert
motion along the basilar membrane to neural activity. This produces
an auditory nerve representation in both time and frequency. The pro-
cessing at higher levels in the brain, shown in Figure 3.2 as a sequence
of central processing with multiple representations followed by some
type of pattern recognition, is not well understood and we can only
postulate the mechanisms used by the human brain to perceive sound
or speech. Even so, a wealth of knowledge about how sounds are per-
ceived has been discovered by careful experiments that use tones and
noise signals to stimulate the auditory system of human observers in
very specific and controlled ways. These experiments have yielded much
valuable knowledge about the sensitivity of the human auditory system
to acoustic properties such as intensity and frequency.
3.2 Perception of Loudness
A key factor in the perception of speech and other sounds is loudness.
Loudness is a perceptual quality that is related to the physical property
of sound pressure level. Loudness is quantified by relating the actual
sound pressure level of a pure tone (in dB relative to a standard refer-
ence level) to the perceived loudness of the same tone (in a unit called
phons) over the range of human hearing (20 Hz–20 kHz). This relation-
ship is shown in Figure 3.3 [37, 103]. These loudness curves show that
the perception of loudness is frequency-dependent. Specifically, the dot-
ted curve at the bottom of the figure labeled "threshold of audibility"
shows the sound pressure level that is required for a sound of a given
frequency to be just audible (by a person with normal hearing). It can
be seen that low frequencies must be significantly more intense than
frequencies in the mid-range in order that they be perceived at all. The
solid curves are equal-loudness-level contours measured by comparing
sounds at various frequencies with a pure tone of frequency 1000 Hz and
known sound pressure level. For example, the point at frequency 100 Hz
on the curve labeled 50 (phons) is obtained by adjusting the power of
the 100 Hz tone until it sounds as loud as a 1000 Hz tone having a sound
pressure level of 50 dB. Careful measurements of this kind show that a
100 Hz tone must have a sound pressure level of about 60 dB in order
28 Hearing and Auditory Perception
Fig. 3.3 Loudness level for human hearing. (After Fletcher and Munson [37].)
to be perceived to be equal in loudness to the 1000 Hz tone of sound
pressure level 50 dB. By convention, both the 50 dB 1000 Hz tone and
the 60 dB 100 Hz tone are said to have a loudness level of 50 phons
(pronounced as /F OW N Z/).
The equal-loudness-level curves show that the auditory system is
most sensitive for frequencies ranging from about 100 Hz up to about
6 kHz with the greatest sensitivity at around 3 to 4 kHz. This is almost
precisely the range of frequencies occupied by most of the sounds of
3.3 Critical Bands
The non-uniform frequency analysis performed by the basilar mem-
brane can be thought of as equivalent to that of a set of bandpass filters
whose frequency responses become increasingly broad with increas-
ing frequency. An idealized version of such a filter bank is depicted
schematically in Figure 3.4. In reality, the bandpass filters are not ideal
3.4 Pitch Perception 29
Fig. 3.4 Schematic representation of bandpass filters according to the critical band theory
of hearing.
as shown in Figure 3.4, but their frequency responses overlap signifi-
cantly since points on the basilar membrane cannot vibrate indepen-
dently of each other. Even so, the concept of bandpass filter analysis
in the cochlea is well established, and the critical bandwidths have
been defined and measured using a variety of methods, showing that
the effective bandwidths are constant at about 100 Hz for center fre-
quencies below 500 Hz, and with a relative bandwidth of about 20%
of the center frequency above 500 Hz. An equation that fits empirical
measurements over the auditory range is
= 25 + 75[1 + 1.4(f
, (3.1)
where ∆f
is the critical bandwidth associated with center frequency
[134]. Approximately 25 critical band filters span the range from
0 to 20 kHz. The concept of critical bands is very important in under-
standing such phenomena as loudness perception, pitch perception, and
masking, and it therefore provides motivation for digital representations
of the speech signal that are based on a frequency decomposition.
3.4 Pitch Perception
Most musical sounds as well as voiced speech sounds have a periodic
structure when viewed over short time intervals, and such sounds are
perceived by the auditory system as having a quality known as pitch.
Like loudness, pitch is a subjective attribute of sound that is related to
the fundamental frequency of the sound, which is a physical attribute of
the acoustic waveform [122]. The relationship between pitch (measured
30 Hearing and Auditory Perception
Fig. 3.5 Relation between subjective pitch and frequency of a pure tone.
on a nonlinear frequency scale called the mel-scale) and frequency of a
pure tone is approximated by the equation [122]:
Pitch in mels = 1127log
(1 + f/700), (3.2)
which is plotted in Figure 3.5. This expression is calibrated so that a
frequency of 1000 Hz corresponds to a pitch of 1000 mels. This empirical
scale describes the results of experiments where subjects were asked to
adjust the pitch of a measurement tone to half the pitch of a reference
tone. To calibrate the scale, a tone of frequency 1000 Hz is given a
pitch of 1000 mels. Below 1000 Hz, the relationship between pitch and
frequency is nearly proportional. For higher frequencies, however, the
relationship is nonlinear. For example, (3.2) shows that a frequency of
f = 5000 Hz corresponds to a pitch of 2364 mels.
The psychophysical phenomenon of pitch, as quantified by the mel-
scale, can be related to the concept of critical bands [134]. It turns out
that more or less independently of the center frequency of the band, one
critical bandwidth corresponds to about 100 mels on the pitch scale.
This is shown in Figure 3.5, where a critical band of width ∆f
= 160 Hz
centered on f
= 1000 Hz maps into a band of width 106 mels and a
critical band of width 100 Hz centered on 350 Hz maps into a band of
width 107 mels. Thus, what we know about pitch perception reinforces
the notion that the auditory system performs a frequency analysis that
can be simulated with a bank of bandpass filters whose bandwidths
increase as center frequency increases.
3.5 Auditory Masking 31
Voiced speech is quasi-periodic, but contains many frequencies. Nev-
ertheless, many of the results obtained with pure tones are relevant to
the perception of voice pitch as well. Often the term pitch period is used
for the fundamental period of the voiced speech signal even though its
usage in this way is somewhat imprecise.
3.5 Auditory Masking
The phenomenon of critical band auditory analysis can be explained
intuitively in terms of vibrations of the basilar membrane. A related
phenomenon, called masking, is also attributable to the mechanical
vibrations of the basilar membrane. Masking occurs when one sound
makes a second superimposed sound inaudible. Loud tones causing
strong vibrations at a point on the basilar membrane can swamp out
vibrations that occur nearby. Pure tones can mask other pure tones,
and noise can mask pure tones as well. A detailed discussion of masking
can be found in [134].
Figure 3.6 illustrates masking of tones by tones. The notion that a
sound becomes inaudible can be quantified with respect to the thresh-
old of audibility. As shown in Figure 3.6, an intense tone (called the
masker) tends to raise the threshold of audibility around its location on
the frequency axis as shown by the solid line. All spectral components
whose level is below this raised threshold are masked and therefore do
of hearing
log frequency
Fig. 3.6 Illustration of effects of masking.
32 Hearing and Auditory Perception
not need to be reproduced in a speech (or audio) processing system
because they would not be heard. Similarly, any spectral component
whose level is above the raised threshold is not masked, and therefore
will be heard. It has been shown that the masking effect is greater for
frequencies above the masking frequency than below. This is shown in
Figure 3.6 where the falloff of the shifted threshold is less abrupt above
than below the masker.
Masking is widely employed in digital representations of speech (and
audio) signals by "hiding" errors in the representation in areas where
the threshold of hearing is elevated by strong frequency components in
the signal [120]. In this way, it is possible to achieve lower data rate
representations while maintaining a high degree of perceptual fidelity.
3.6 Complete Model of Auditory Processing
In Chapter 2, we described elements of a generative model of speech
production which could, in theory, completely describe and model the
ways in which speech is produced by humans. In this chapter, we
described elements of a model of speech perception. However, the
problem that arises is that our detailed knowledge of how speech is
perceived and understood, beyond the basilar membrane processing
of the inner ear, is rudimentary at best, and thus we rely on psy-
chophysical experimentation to understand the role of loudness, critical
bands, pitch perception, and auditory masking in speech perception in
humans. Although some excellent auditory models have been proposed
[41, 48, 73, 115] and used in a range of speech processing systems, all
such models are incomplete representations of our knowledge about
how speech is understood.
Short-Time Analysis of Speech
In Figure 2.2 of Chapter 2, we presented a model for speech produc-
tion in which an excitation source provides the basic temporal fine
structure while a slowly varying filter provides spectral shaping (often
referred to as the spectrum envelope) to create the various sounds of
speech. In Figure 2.2, the source/system separation was presented at
an abstract level, and few details of the excitation or the linear sys-
tem were given. Both the excitation and the linear system were defined
implicitly by the assertion that the sampled speech signal was the out-
put of the overall system. Clearly, this is not sufficient to uniquely
specify either the excitation or the system. Since our goal is to extract
parameters of the model by analysis of the speech signal, it is com-
mon to assume structures (or representations) for both the excitation
generator and the linear system. One such model uses a more detailed
representation of the excitation in terms of separate source generators
for voiced and unvoiced speech as shown in Figure 4.1. In this model
the unvoiced excitation is assumed to be a random noise sequence,
and the voiced excitation is assumed to be a periodic impulse train
with impulses spaced by the pitch period (P
) rounded to the nearest
34 Short-Time Analysis of Speech
Fig. 4.1 Voiced/unvoiced/system model for a speech signal.
The pulses needed to model the glottal flow waveform dur-
ing voiced speech are assumed to be combined (by convolution) with
the impulse response of the linear system, which is assumed to be
slowly-time-varying (changing every 50–100 ms or so). By this we mean
that over the timescale of phonemes, the impulse response, frequency
response, and system function of the system remains relatively con-
stant. For example over time intervals of tens of milliseconds, the sys-
tem can be described by the convolution expression
ˆ n
[n] =
ˆ n
ˆ n
[n − m], (4.1)
where the subscript ˆ n denotes the time index pointing to the block of
samples of the entire speech signal s[n] wherein the impulse response
ˆ n
[m] applies. We use n for the time index within that interval, and m is
the index of summation in the convolution sum. In this model, the gain
ˆ n
is absorbed into h
ˆ n
[m] for convenience. As discussed in Chapter 2,
the most general linear time-invariant system would be characterized
by a rational system function as given in (2.1). However, to simplify
analysis, it is often assumed that the system is an all-pole system with
As mentioned in Chapter 3, the period of voiced speech is related to the fundamental
frequency (perceived as pitch) of the voice.
system function of the form:
H(z) =
1 −
. (4.2)
The coefficients G and a
in (4.2) change with time, and should there-
fore be indexed with the subscript ˆ n as in (4.1) and Figure 4.1, but
this complication of notation is not generally necessary since it is usu-
ally clear that the system function or difference equation only applies
over a short time interval.
Although the linear system is assumed to
model the composite spectrum effects of radiation, vocal tract tube,
and glottal excitation pulse shape (for voiced speech only) over a short
time interval, the linear system in the model is commonly referred
to as simply the "vocal tract" system and the corresponding impulse
response is called the "vocal tract impulse response." For all-pole linear
systems, as represented by (4.2), the input and output are related by
a difference equation of the form:
s[n] =
s[n − k] + Ge[n], (4.3)
where as discussed above, we have suppressed the indication of the time
at which the difference equation applies.
With such a model as the basis, common practice is to parti-
tion the analysis of the speech signal into techniques for extracting
the parameters of the excitation model, such as the pitch period and
voiced/unvoiced classification, and techniques for extracting the lin-
ear system model (which imparts the spectrum envelope or spectrum
Because of the slowly varying nature of the speech signal, it is com-
mon to process speech in blocks (also called "frames") over which the
properties of the speech waveform can be assumed to remain relatively
constant. This leads to the basic principle of short-time analysis, which
In general, we use n and m for discrete indices for sequences, but whenever we want to
indicate a specific analysis time, we use ˆ n.
36 Short-Time Analysis of Speech
is represented in a general form by the equation:
ˆ n
T ¦x[m]w[ˆ n − m]¦, (4.4)
where X
ˆ n
represents the short-time analysis parameter (or vector of
parameters) at analysis time ˆ n.
The operator T¦ ¦ defines the nature
of the short-time analysis function, and w[ˆ n − m] represents a time-
shifted window sequence, whose purpose is to select a segment of the
sequence x[m] in the neighborhood of sample m = ˆ n. We will see several
examples of such operators that are designed to extract or highlight
certain features of the speech signal. The infinite limits in (4.4) imply
summation over all nonzero values of the windowed segment x
ˆ n
[m] =
x[m]w[ˆ n − m]; i.e., for all m in the region of support of the window.
For example, a finite-duration
window might be a Hamming window
defined by
[m] =
0.54 + 0.46cos(πm/M) −M ≤ m≤ M
0 otherwise.
Figure 4.2 shows a discrete-time Hamming window and its discrete-time
Fourier transform, as used throughout this chapter.
It can be shown
that a (2M + 1) sample Hamming window has a frequency main lobe
(full) bandwidth of 4π/M. Other windows will have similar properties,
i.e., they will be concentrated in time and frequency, and the frequency
width will be inversely proportional to the time width [89].
Figure 4.3 shows a 125 ms segment of a speech waveform that
includes both unvoiced (0–50 ms) and voiced speech (50–125 ms). Also
shown is a sequence of data windows of duration 40 ms and shifted by
15 ms (320 samples at 16 kHz sampling rate) between windows. This
illustrates how short-time analysis is implemented.
Sometimes (4.4) is normalized by dividing by the effective window length, i.e.,
w[m], so that X
ˆ n
is a weighted average.
An example of an infinite-duration window is w[n] = na
for n ≥ 0. Such a window can
lead to recursive implementations of short-time analysis functions [9].
We have assumed the time origin at the center of a symmetric interval of 2M + 1 samples.
A causal window would be shifted to the right by M samples.
4.1 Short-Time Energy and Zero-Crossing Rate 37
Fig. 4.2 Hamming window (a) and its discrete-time Fourier transform (b).
0 20 40 60 80 100 120
time in ms
Fig. 4.3 Section of speech waveform with short-time analysis windows.
4.1 Short-Time Energy and Zero-Crossing Rate
Two basic short-time analysis functions useful for speech signals are the
short-time energy and the short-time zero-crossing rate. These func-
tions are simple to compute, and they are useful for estimating prop-
erties of the excitation function in the model.
The short-time energy is defined as
ˆ n
(x[m]w[ˆ n − m])
[ˆ n − m]. (4.6)
38 Short-Time Analysis of Speech
In this case the operator T¦ ¦ is simply squaring the windowed samples.
As shown in (4.6), it is often possible to express short-time analysis
operators as a convolution or linear filtering operation. In this case,
ˆ n
= x
[n] ∗ h
n=ˆ n
, where the impulse response of the linear filter
is h
[n] = w
Similarly, the short-time zero crossing rate is defined as the weighted
average of the number of times the speech signal changes sign within
the time window. Representing this operator in terms of linear filtering
leads to
ˆ n
0.5[sgn¦x[m]¦ − sgn¦x[m − 1]¦[ w[ˆ n − m], (4.7)
sgn¦x¦ =
1 x ≥ 0
−1 x < 0.
Since 0.5[sgn¦x[m]¦ − sgn¦x[m − 1]¦[ is equal to 1 if x[m] and x[m − 1]
have different algebraic signs and 0 if they have the same sign, it follows
that Z
ˆ n
in (4.7) is a weighted sum of all the instances of alternating sign
(zero-crossing) that fall within the support region of the shifted window
w[ˆ n − m]. While this is a convenient representation that fits the general
framework of (4.4), the computation of Z
ˆ n
could be implemented in
other ways.
Figure 4.4 shows an example of the short-time energy and zero-
crossing rate for a segment of speech with a transition from unvoiced
to voiced speech. In both cases, the window is a Hamming window
(two examples shown) of duration 25 ms (equivalent to 401 samples
at a 16 kHz sampling rate).
Thus, both the short-time energy and
the short-time zero-crossing rate are output of a lowpass filter whose
frequency response is as shown in Figure 4.2(b). For the 401-point
Hamming window used in Figure 4.4, the frequency response is very
small for discrete-time frequencies above 2π/200 rad/s (equivalent to
16000/200 = 80 Hz analog frequency). This means that the short-time
In the case of the short-time energy, the window applied to the signal samples was the
square-root of the Hamming window, so that he[n] = w
[n] is the Hamming window
defined by (4.5).
4.1 Short-Time Energy and Zero-Crossing Rate 39
0 20 40 60 80 100 120
time in ms
Fig. 4.4 Section of speech waveform with short-time energy and zero-crossing rate
energy and zero-crossing rate functions are slowly varying compared
to the time variations of the speech signal, and therefore, they can be
sampled at a much lower rate than that of the original speech signal.
For finite-length windows like the Hamming window, this reduction of
the sampling rate is accomplished by moving the window position ˆ n in
jumps of more than one sample as shown in Figure 4.3.
Note that during the unvoiced interval, the zero-crossing rate is rel-
atively high compared to the zero-crossing rate in the voiced interval.
Conversely, the energy is relatively low in the unvoiced region com-
pared to the energy in the voiced region. Note also that there is a small
shift of the two curves relative to events in the time waveform. This is
due to the time delay of M samples (equivalent to 12.5 ms) added to
make the analysis window filter causal.
The short-time energy and short-time zero-crossing rate are impor-
tant because they abstract valuable information about the speech sig-
nal, and they are simple to compute. The short-time energy is an indi-
cation of the amplitude of the signal in the interval around time ˆ n. From
our model, we expect unvoiced regions to have lower short-time energy
than voiced regions. Similarly, the short-time zero-crossing rate is a
crude frequency analyzer. Voiced signals have a high frequency (HF)
falloff due to the lowpass nature of the glottal pulses, while unvoiced
sounds have much more HF energy. Thus, the short-time energy and
short-time zero-crossing rate can be the basis for an algorithm for mak-
ing a decision as to whether the speech signal is voiced or unvoiced at
40 Short-Time Analysis of Speech
any particular time ˆ n. A complete algorithm would involve measure-
ments of the statistical distributions of the energy and zero-crossing
rate for both voiced and unvoiced speech segments (and also back-
ground noise distributions). These distributions can be used to derive
thresholds used in voiced/unvoiced decision [100].
4.2 Short-Time Autocorrelation Function (STACF)
The autocorrelation function is often used as a means of detecting
periodicity in signals, and it is also the basis for many spectrum analysis
methods. This makes it a useful tool for short-time speech analysis. The
STACF is defined as the deterministic autocorrelation function of the
sequence x
ˆ n
[m] = x[m]w[ˆ n − m] that is selected by the window shifted
to time ˆ n, i.e.,
ˆ n
[] =
ˆ n
ˆ n
[m + ]
x[m]w[ˆ n − m]x[m + ]w[ˆ n − m − ]. (4.9)
Using the familiar even-symmetric property of the autocorrelation,
ˆ n
[−] = φ
ˆ n
[], (4.9) can be expressed in terms of linear time-invariant
(LTI) filtering as
ˆ n
[] =
x[m]x[m − ] ˜ w
[ˆ n − m], (4.10)
where ˜ w
[m] = w[m]w[m + ]. Note that the STACF is a two-
dimensional function of the discrete-time index ˆ n (the window position)
and the discrete-lag index . If the window has finite duration, (4.9) can
be evaluated directly or using FFT techniques (see Section 4.6). For
infinite-duration decaying exponential windows, the short-time auto-
correlation of (4.10) can be computed recursively at time ˆ n by using a
different filter ˜ w
[m] for each lag value [8, 9].
To see how the short-time autocorrelation can be used in speech
analysis, assume that a segment of the sampled speech signal is a seg-
ment of the output of the discrete-time model shown in Figure 4.1 where
4.2 Short-Time Autocorrelation Function (STACF) 41
the system is characterized at a particular analysis time by an impulse
response h[n], and the input is either a periodic impulse train or random
white noise. (Note that we have suppressed the indication of analysis
time.) Different segments of the speech signal will have the same form
of model with different excitation and system impulse response. That
is, assume that s[n] = e[n] ∗ h[n], where e[n] is the excitation to the
linear system with impulse response h[n]. A well known, and easily
proved, property of the autocorrelation function is that
[] = φ
[] ∗ φ
[], (4.11)
i.e., the autocorrelation function of s[n] = e[n] ∗ h[n] is the convolution
of the autocorrelation functions of e[n] and h[n]. In the case of the
speech signal, h[n] represents the combined (by convolution) effects
of the glottal pulse shape (for voiced speech), vocal tract shape, and
radiation at the lips. For voiced speech, the autocorrelation of a peri-
odic impulse train excitation with period P
is a periodic impulse train
sequence with the same period. In this case, therefore, the autocorre-
lation of voiced speech is the periodic autocorrelation function
[] =
[ − mP
]. (4.12)
In the case of unvoiced speech, the excitation can be assumed to be ran-
dom white noise, whose stochastic autocorrelation function would be
an impulse sequence at = 0. Therefore, the autocorrelation function
of unvoiced speech computed using averaging would be simply
[] = φ
[]. (4.13)
Equation (4.12) assumes periodic computation of an infinite periodic
signal, and (4.13) assumes probability average or averaging over an
infinite time interval (for stationary random signals). However, the
deterministic autocorrelation function of a finite-length segment of the
speech waveform will have properties similar to those of (4.12) and
(4.13) except that the correlation values will taper off with lag due
to the tapering of the window and the fact that less and less data is
involved in the computation of the short-time autocorrelation for longer
42 Short-Time Analysis of Speech
0 10 20 30 40
0 10 20 30 40
0 10 20 30 40
0 20 40
(a) Voiced Segment (b) Voiced Autocorrelation
(c) Unvoiced Segment (d) Unvoiced Autocorrelation
time in ms time in ms
Fig. 4.5 Voiced and unvoiced segments of speech and their corresponding STACF.
lag values. This tapering off in level is depicted in Figure 4.5 for both a
voiced and an unvoiced speech segment. Note the peak in the autocor-
relation function for the voiced segment at the pitch period and twice
the pitch period, and note the absence of such peaks in the autocorrela-
tion function for the unvoiced segment. This suggests that the STACF
could be the basis for an algorithm for estimating/detecting the pitch
period of speech. Usually such algorithms involve the autocorrelation
function and other short-time measurements such as zero-crossings and
energy to aid in making the voiced/unvoiced decision.
Finally, observe that the STACF implicitly contains the short-time
energy since
ˆ n
(x[m]w[ˆ n − m])
= φ
ˆ n
[0]. (4.14)
4.3 Short-Time Fourier Transform (STFT)
The short-time analysis functions discussed so far are examples of
the general short-time analysis principle that is the basis for most
4.3 Short-Time Fourier Transform (STFT) 43
algorithms for speech processing. We now turn our attention to what is
perhaps the most important basic concept in digital speech processing.
In subsequent chapters, we will find that the STFT defined as
ˆ n
j ˆ ω
) =
x[m]w[ˆ n − m]e
−j ˆ ωm
, (4.15)
is the basis for a wide range of speech analysis, coding and synthesis sys-
tems. By definition, for fixed analysis time ˆ n, the STFT is the discrete-
time Fourier transform (DTFT) of the signal x
ˆ n
[m] = x[m]w[ˆ n − m],
i.e., the DTFT of the (usually finite-duration) signal selected and
amplitude-weighted by the sliding window w[ˆ n − m] [24, 89, 129]. Thus,
the STFT is a function of two variables; ˆ n the discrete-time index denot-
ing the window position, and ˆ ω representing the analysis frequency.
Since (4.15) is a sequence of DTFTs, the two-dimensional function
ˆ n
j ˆ ω
) at discrete-time ˆ n is a periodic function of continuous radian
frequency ˆ ω with period 2π [89].
As in the case of the other short-time analysis functions discussed
in this chapter, the STFT can be expressed in terms of a linear fil-
tering operation. For example, (4.15) can be expressed as the discrete
ˆ n
j ˆ ω
) = (x[n]e
−j ˆ ωn
) ∗ w[n]
n=ˆ n
, (4.16)
or, alternatively,
ˆ n
j ˆ ω
) =
x[n] ∗ (w[n]e
j ˆ ωn
−j ˆ ωn
n=ˆ n
. (4.17)
Recall that a typical window like a Hamming window, when viewed
as a linear filter impulse response, has a lowpass frequency response
with the cutoff frequency varying inversely with the window length.
(See Figure 4.2(b).) This means that for a fixed value of ˆ ω, X
ˆ n
j ˆ ω
is slowly varying as ˆ n varies. Equation (4.16) can be interpreted as
follows: the amplitude modulation x[n]e
−j ˆ ωn
shifts the spectrum of
As before, we use ˆ n to specify the analysis time, and ˆ ω is used to distinguish the STFT
analysis frequency from the frequency variable of the non-time-dependent Fourier trans-
form frequency variable ω.
44 Short-Time Analysis of Speech
x[n] down by ˆ ω, and the window (lowpass) filter selects the resulting
band of frequencies around zero frequency. This is, of course, the
band of frequencies of x[n] that were originally centered on analysis
frequency ˆ ω. An identical conclusion follows from (4.17): x[n] is the
input to a bandpass filter with impulse response w[n]e
j ˆ ωn
, which selects
the band of frequencies centered on ˆ ω. Then that band of frequencies is
shifted down by the amplitude modulation with e
−j ˆ ωn
, resulting again
in the same lowpass signal [89].
In summary, the STFT has three interpretations: (1) It is a sequence
of discrete-time Fourier transforms of windowed signal segments, i.e., a
periodic function of ˆ ω at each window position ˆ n. (2) For each frequency
ˆ ω with ˆ n varying, it is the time sequence output of a lowpass filter that
follows frequency down-shifting by ˆ ω. (3) For each frequency ˆ ω, it is
the time sequence output resulting from frequency down-shifting the
output of a bandpass filter.
4.4 Sampling the STFT in Time and Frequency
As defined in (4.15), the STFT is a function of a continuous analysis
frequency ˆ ω. The STFT becomes a practical tool for both analysis
and applications when it is implemented with a finite-duration window
moved in steps of R > 1 samples in time and computed at a discrete
set of frequencies as in
[k] =
x[m]w[rR − m]e
k = 0, 1, . . . , N − 1,
where N is the number of uniformly spaced frequencies across the
interval 0 ≤ ˆ ω < 2π, and L is the window length (in samples). Note
that we have assumed that w[m] is causal and nonzero only in the
range 0 ≤ m≤ L − 1 so that the windowed segment x[m]w[rR − m]
is nonzero over rR − L + 1 ≤ m≤ rR. To aid in interpretation, it is
helpful to write (4.18) in the equivalent form:
[k] =
k = 0, 1, . . . , N − 1, (4.19)
4.4 Sampling the STFT in Time and Frequency 45
[k] =
x[rR − m]w[m]e
k = 0, 1, . . . , N − 1. (4.20)
Since we have assumed, for specificity, that w[m] = 0 only in the range
0 ≤ m≤ L − 1, the alternative form,
[k], has the interpretation of
an N-point DFT of the sequence x[rR − m]w[m], which, due to the
definition of the window, is nonzero in the interval 0 ≤ m≤ L − 1.
(4.20), the analysis time rR is shifted to the time origin of the DFT
computation, and the segment of the speech signal is the time-reversed
sequence of L samples that precedes the analysis time. The complex
exponential factor e
in (4.19) results from the shift of the
time origin.
For a discrete-time Fourier transform interpretation, it follows that
[k] can be computed by the following process:
(1) Form the sequence x
[m] = x[rR − m]w[m], for m =
(2) Compute the complex conjugate of the N-point DFT of
the sequence x
[m]. (This can be done efficiently with an
N-point FFT algorithm.)
(3) The multiplication by e
can be done if necessary,
but often can be omitted (as in computing a sound spectro-
gram or spectrographic display).
(4) Move the time origin by R samples (i.e., r →r + 1) and
repeat steps (1), (2), and (3), etc.
The remaining issue for complete specification of the sampled STFT
is specification of the temporal sampling period, R, and the number of
uniform frequencies, N. It can easily be shown that both R and N
are determined entirely by the time width and frequency bandwidth of
the lowpass window, w[m], used to compute the STFT [1], giving the
The DFT is normally defined with a negative exponent. Thus, since x[rR − m]w[m] is
real, (4.20) is the complex conjugate of the DFT of the windowed sequence [89].
46 Short-Time Analysis of Speech
following constraints on R and N:
(1) R ≤ L/(2C) where C is a constant that is dependent on the
window frequency bandwidth; C = 2 for a Hamming window,
C = 1 for a rectangular window.
(2) N ≥ L, where L is the window length in samples.
Constraint (1) above is related to sampling the STFT in time at a
rate of twice the window bandwidth in frequency in order to eliminate
aliasing in frequency of the STFT, and constraint (2) above is related
to sampling in frequency at a rate of twice the equivalent time width
of the window to ensure that there is no aliasing in time of the STFT.
4.5 The Speech Spectrogram
Since the 1940s, the sound spectrogram has been a basic tool for gain-
ing understanding of how the sounds of speech are produced and how
phonetic information is encoded in the speech signal. Up until the
1970s, spectrograms were made by an ingenious device comprised of
an audio tape loop, variable analog bandpass filter, and electrically
sensitive paper [66]. Today spectrograms like those in Figure 4.6 are
made by DSP techniques [87] and displayed as either pseudo-color or
gray-scale images on computer screens.
Sound spectrograms like those in Figure 4.6 are simply a display of
the magnitude of the STFT. Specifically, the images in Figure 4.6 are
plots of
, f
) = 20log
[k][ = 20log
[k][, (4.21)
where the plot axes are labeled in terms of analog time and fre-
quency through the relations t
= rRT and f
= k/(NT), where T is
the sampling period of the discrete-time signal x[n] = x
(nT). In order
to make smooth looking plots like those in Figure 4.6, R is usually
quite small compared to both the window length L and the num-
ber of samples in the frequency dimension, N, which may be much
larger than the window length L. Such a function of two variables
can be plotted on a two dimensional surface (such as this text) as
either a gray-scale or a color-mapped image. Figure 4.6 shows the time
4.5 The Speech Spectrogram 47
0 100 200 300 400 500 600
time in ms
0 100 200 300 400 500 600
Fig. 4.6 Spectrogram for speech signal of Figure 1.1.
waveform at the top and two spectrograms computed with different
length analysis windows. The bars on the right calibrate the color
map (in dB).
A careful interpretation of (4.20) and the corresponding spectro-
gram images leads to valuable insight into the nature of the speech
signal. First note that the window sequence w[m] is nonzero only over
the interval 0 ≤ m≤ L − 1. The length of the window has a major
effect on the spectrogram image. The upper spectrogram in Figure 4.6
was computed with a window length of 101 samples, corresponding to
10 ms time duration. This window length is on the order of the length
of a pitch period of the waveform during voiced intervals. As a result,
in voiced intervals, the spectrogram displays vertically oriented stri-
ations corresponding to the fact that the sliding window sometimes
includes mostly large amplitude samples, then mostly small amplitude
48 Short-Time Analysis of Speech
samples, etc. As a result of the short analysis window, each individ-
ual pitch period is resolved in the time dimension, but the resolution
in the frequency dimension is poor. For this reason, if the analysis
window is short, the spectrogram is called a wide-band spectrogram.
This is consistent with the linear filtering interpretation of the STFT,
since a short analysis filter has a wide passband. Conversely, when the
window length is long, the spectrogram is a narrow-band spectrogram,
which is characterized by good frequency resolution and poor time
The upper plot in Figure 4.7, for example, shows S(t
, f
) as a
function of f
at time t
= 430 ms. This vertical slice through the
spectrogram is at the position of the black vertical line in the upper
spectrogram of Figure 4.6. Note the three broad peaks in the spec-
trum slice at time t
= 430 ms, and observe that similar slices would
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
frequency in kHz
Fig. 4.7 Short-time spectrum at time 430 ms (dark vertical line in Figure 4.6) with Hamming
window of length M = 101 in upper plot and M = 401 in lower plot.
4.6 Relation of STFT to STACF 49
be obtained at other times around t
= 430 ms. These large peaks are
representative of the underlying resonances of the vocal tract at the
corresponding time in the production of the speech signal.
The lower spectrogram in Figure 4.6 was computed with a win-
dow length of 401 samples, corresponding to 40 ms time duration. This
window length is on the order of several pitch periods of the waveform
during voiced intervals. As a result, the spectrogram no longer displays
vertically oriented striations since several periods are included in the
window no matter where it is placed on the waveform in the vicinity
of the analysis time t
. As a result, the spectrogram is not as sensitive
to rapid time variations, but the resolution in the frequency dimension
is much better. Therefore, the striations tend to be horizontally ori-
ented in the narrow-band case since the fundamental frequency and its
harmonics are all resolved. The lower plot in Figure 4.7, for example,
shows S(t
, f
) as a function of f
at time t
= 430 ms. In this case,
the signal is voiced at the position of the window, so over the analy-
sis window interval it acts very much like a periodic signal. Periodic
signals have Fourier spectra that are composed of impulses at the fun-
damental frequency, f
, and at integer multiples (harmonics) of the
fundamental frequency [89]. Multiplying by the analysis window in the
time-domain results in convolution of the Fourier transform of the win-
dow with the impulses in the spectrum of the periodic signal [89]. This
is evident in the lower plot of Figure 4.7, where the local maxima of the
curve are spaced at multiples of the fundamental frequency f
where T
is the fundamental period (pitch period) of the signal.
4.6 Relation of STFT to STACF
A basic property of Fourier transforms is that the inverse Fourier trans-
form of the magnitude-squared of the Fourier transform of a signal is the
autocorrelation function for that signal [89]. Since the STFT defined in
(4.15) is a discrete-time Fourier transform for fixed window position,
it follows that φ
ˆ n
[] given by (4.9) is related to the STFT given by
Each "ripple" in the lower plot is essentially a frequency-shifted copy of the Fourier trans-
form of the Hamming window used in the analysis.
50 Short-Time Analysis of Speech
(4.15) as
ˆ n
[] =
ˆ n
j ˆ ω
j ˆ ω
dˆ ω. (4.22)
The STACF can also be computed from the sampled STFT. In partic-
ular, an inverse DFT can be used to compute
[] =
[] = φ
[] for = 0, 1, . . . , L − 1 if N ≥ 2L. If L < N < 2L
time aliasing will occur, but
[] = φ
[] for = 0, 1, . . . , N − L. Note
that (4.14) shows that the short-time energy can also be obtained from
either (4.22) or (4.23) by setting = 0.
Figure 4.8 illustrates the equivalence between the STACF and the
STFT. Figures 4.8(a) and 4.8(c) show the voiced and unvoiced autocor-
relation functions that were shown in Figures 4.5(b) and 4.5(d). On the
right are the corresponding STFTs. The peaks around 9 and 18 ms in
0 10 20 30 40
0 10 20 30 40
0 2000 4000 6000 8000
0 2000 4000 6000 8000
(b) Voiced STFT (a) Voiced Autocorrelation
(d) Unvoiced STFT (c) Unvoiced Autocorrelation
time in msec frequency in kHz
Fig. 4.8 STACF and corresponding STFT.
4.7 Short-Time Fourier Synthesis 51
the autocorrelation function of the voiced segment imply a fundamen-
tal frequency at this time of approximately 1000/9 = 111 Hz. Similarly,
note in the STFT on the right in Figure 4.8(b) that there are approx-
imately 18 local regularly spaced peaks in the range 0–2000 Hz. Thus,
we can estimate the fundamental frequency to be 2000/18 = 111 Hz as
4.7 Short-Time Fourier Synthesis
From the linear filtering point of view of the STFT, (4.19) and (4.20)
represent the downsampled (by the factor R) output of the process
of bandpass filtering with impulse response w[n]e
by frequency-down-shifting by (2πk/N). Alternatively, (4.18) shows
that X
[k] is a downsampled output of the lowpass window filter
with frequency-down-shifted input x[n]e
. The left half of Fig-
ure 4.9 shows a block diagram representation of the STFT as a com-
bination of modulation, followed by lowpass filtering, followed by a
downsampler by R. This structure is often called a filter bank, and
the outputs of the individual filters are called the channel signals. The
x x
f [n]
f [n]
f [n]
] [k X
] [n x
n N j
) 1 (
− −
n N j
) 1 (
n j
) 1 (
n j
) 0 (
n j
) 1 (
n j
) 0 (
] [n y
] [k X
] [k Y
Fig. 4.9 Filterbank interpretation of short-time Fourier analysis and synthesis.
52 Short-Time Analysis of Speech
spectrogram is comprised of the set of outputs of the filter bank with
each channel signal corresponding to a horizontal line in the spectro-
gram. If we want to use the STFT for other types of speech process-
ing, we may need to consider if and how it is possible to reconstruct
the speech signal from the STFT, i.e., from the channel signals. The
remainder of Figure 4.9 shows how this can be done.
First, the diagram shows the possibility that the STFT might be
modified by some sort of processing. An example might be quantization
of the channel signals for data compression. The modified STFT is
denoted as Y
[k]. The remaining parts of the structure on the right in
Figure 4.9 implement the synthesis of a new time sequence y[n] from
the STFT. This part of the diagram represents the synthesis equation
y[n] =
[k]f[n − rR]
. (4.24)
The steps represented by (4.24) and the right half of Figure 4.9 involve
first upsampling by R followed by linear filtering with f[n] (called
the synthesis window). This operation, defined by the part within the
parenthesis in (4.24), interpolates the STFT Y
[k] to the time sam-
pling rate of the speech signal for each channel k.
Then, synthesis
is achieved by modulation with e
, which up-shifts the lowpass
interpolated channel signals back to their original frequency bands cen-
tered on frequencies (2πk/N). The sum of the up-shifted signals is the
synthesized output.
There are many variations on the short-time Fourier analy-
sis/synthesis paradigm. The FFT algorithm can be used to implement
both analysis and synthesis; special efficiencies result when L > R = N,
and the analysis and synthesis can be accomplished using only real fil-
tering and with modulation operations being implicitly achieved by
the downsampling [24, 129]. However, it is most important to note
that with careful choice of the parameters L, R, and N and careful
design of the analysis window w[m] together with the synthesis win-
dow f[n], it is possible to reconstruct the signal x[n] with negligible
The sequence f[n] is the impulse response of a LTI filter with gain R/N and normalized
cutoff frequency π/R. It is often a scaled version of w[n].
4.8 Short-Time Analysis is Fundamental to our Thinking 53
error (y[n] = x[n]) from the unmodified STFT Y
[k] = X
[k]. One
condition that guarantees exact reconstruction is [24, 129].
w[rR − n + qN]f[n − rR] =
1 q = 0
0 q = 0.
Short-time Fourier analysis and synthesis is generally formulated
with equally spaced channels with equal bandwidths. However, we
have seen that models for auditory processing involve nonuniform filter
banks. Such filter banks can be implemented as tree structures where
frequency bands are successively divided into low- and HF bands. Such
structures are essentially the same as wavelet decompositions, a topic
beyond the scope of this text, but one for which a large body of liter-
ature exists. See, for example [18, 125, 129].
4.8 Short-Time Analysis is Fundamental to our Thinking
It can be argued that the short-time analysis principle, and particu-
larly the short-time Fourier representation of speech, is fundamental to
our thinking about the speech signal and it leads us to a wide variety
of techniques for achieving our goal of moving from the sampled time
waveform back along the speech chain toward the implicit message.
The fact that almost perfect reconstruction can be achieved from the
filter bank channel signals gives the short-time Fourier representation
major credibility in the digital speech processing tool kit. This impor-
tance is strengthened by the fact that, as discussed briefly in Chapter 3,
models for auditory processing are based on a filter bank as the first
stage of processing. Much of our knowledge of perceptual effects is
framed in terms of frequency analysis, and thus, the STFT representa-
tion provides a natural framework within which this knowledge can be
represented and exploited to obtain efficient representations of speech
and more general audio signals. We will have more to say about this
in Chapter 7, but first we will consider (in Chapter 5) the technique
of cepstrum analysis, which is based directly on the STFT, and the
technique of linear predictive analysis (in Chapter 6), which is based
on the STACF and thus equivalently on the STFT.
Homomorphic Speech Analysis
The STFT provides a useful framework for thinking about almost all
the important techniques for analysis of speech that have been devel-
oped so far. An important concept that flows directly from the STFT
is the cepstrum, more specifically, the short-time cepstrum, of speech.
In this chapter, we explore the use of the short-time cepstrum as a
representation of speech and as a basis for estimating the parameters
of the speech generation model. A more detailed discussion of the uses
of the cepstrum in speech processing can be found in [110].
5.1 Definition of the Cepstrum and Complex Cepstrum
The cepstrum was defined by Bogert, Healy, and Tukey to be the inverse
Fourier transform of the log magnitude spectrum of a signal [16]. Their
original definition, loosely framed in terms of spectrum analysis of ana-
log signals, was motivated by the fact that the logarithm of the Fourier
spectrum of a signal containing an echo has an additive periodic com-
ponent depending only on the echo size and delay, and that further
Fourier analysis of the log spectrum can aid in detecting the presence
5.1 Definition of the Cepstrum and Complex Cepstrum 55
of that echo. Oppenheim, Schafer, and Stockham showed that the cep-
strum is related to the more general concept of homomorphic filtering
of signals that are combined by convolution [85, 90, 109]. They gave a
definition of the cepstrum of a discrete-time signal as
c[n] =
dω, (5.1a)
where log[X(e
)[ is the logarithm of the magnitude of the DTFT
of the signal, and they extended the concept by defining the complex
cepstrum as
ˆ x[n] =
dω, (5.1b)
where log¦X(e
)¦ is the complex logarithm of X(e
) defined as
) = log¦X(e
)¦ = log[X(e
)[ + j arg[X(e
)]. (5.1c)
The transformation implied by (5.1b) is depicted as the block diagram
in Figure 5.1. The same diagram represents the cepstrum if the complex
logarithm is replaced by the logarithm of the magnitude of the DTFT.
Since we restrict our attention to real sequences, x[n], it follows from
the symmetry properties of Fourier transforms that the cepstrum is the
even part of the complex cepstrum, i.e., c[n] = (ˆ x[n] + ˆ x[−n])/2 [89].
As shown in Figure 5.1, the operation of computing the complex cep-
strum from the input can be denoted as ˆ x[n] = D
¦x[n]¦. In the theory
of homomorphic systems D
¦ ¦ is called the characteristic system for
convolution. The connection between the cepstrum concept and homo-
morphic filtering of convolved signals is that the complex cepstrum has
the property that if x[n] = x
[n] ∗ x
[n], then
ˆ x[n] = D
[n] ∗ x
[n]¦ = ˆ x
[n] + ˆ x
[n]. (5.2)
Fig. 5.1 Computing the complex cepstrum using the DTFT.
56 Homomorphic Speech Analysis
Fig. 5.2 The inverse of the characteristic system for convolution (inverse complex cepstrum).
That is, the complex cepstrum operator transforms convolution into
addition. This property, which is true for both the cepstrum and the
complex cepstrum, is what makes the cepstrum and the complex cep-
strum useful for speech analysis, since our model for speech produc-
tion involves convolution of the excitation with the vocal tract impulse
response, and our goal is often to separate the excitation signal from
the vocal tract signal. In the case of the complex cepstrum, the inverse
of the characteristic system exists as in Figure 5.2, which shows the
reverse cascade of the inverses of the operators in Figure 5.1. Homo-
morphic filtering of convolved signals is achieved by forming a modified
complex cepstrum
ˆ y[n] = g[n]ˆ x[n], (5.3)
where g[n] is a window (a "lifter" in the terminology of Bogert et al.)
which selects a portion of the complex cepstrum for inverse processing.
A modified output signal y[n] can then be obtained as the output of
Figure 5.2 with ˆ y[n] given by (5.3) as input. Observe that (5.3) defines
a linear operator in the conventional sense, i.e., if ˆ x[n] = ˆ x
[n] + ˆ x
then ˆ y[n] = g[n]ˆ x
[n] + g[n]ˆ x
[n]. Therefore, the output of the inverse
characteristic system will have the form y[n] = y
[n] ∗ y
[n], where
ˆ y
[n] = g[n]ˆ x
[n] is the complex cepstrum of y
[n], etc. Examples of
lifters used in homomorphic filtering of speech are given in Sections 5.4
and 5.6.2.
The key issue in the definition and computation of the complex cep-
strum is the computation of the complex logarithm; more specifically,
the computation of the phase angle arg[X(e
)], which must be done
so as to preserve an additive combination of phases for two signals
combined by convolution [90, 109].
5.2 The Short-Time Cepstrum 57
The independent variable of the cepstrum and complex cepstrum
is nominally time. The crucial observation leading to the cepstrum
concept is that the log spectrum can be treated as a waveform to be
subjected to further Fourier analysis. To emphasize this interchanging
of domains of reference, Bogert et al. [16] coined the word cepstrum
by transposing some of the letters in the word spectrum. They created
many other special terms in this way including quefrency as the name
for the independent variable of the cepstrum and liftering for the oper-
ation of linear filtering the log magnitude spectrum by the operation
of (5.3). Only the terms cepstrum, quefrency, and liftering are widely
used today.
5.2 The Short-Time Cepstrum
The application of these definitions to speech requires that the
DTFT be replaced by the STFT. Thus the short-time cepstrum is
defined as
ˆ n
[m] =
ˆ n
j ˆ ω
j ˆ ωm
dˆ ω, (5.4)
where X
ˆ n
j ˆ ω
) is the STFT defined in (4.15), and the short-time
complex cepstrum is likewise defined by replacing X(e
j ˆ ω
) by X
ˆ n
j ˆ ω
in (5.1b).
The similarity to the STACF defined by (4.9) should be
clear. The short-time cepstrum is a sequence of cepstra of windowed
finite-duration segments of the speech waveform. By analogy, a "cep-
strogram" would be an image obtained by plotting the magnitude of
the short-time cepstrum as a function of quefrency m and analysis
time ˆ n.
5.3 Computation of the Cepstrum
In (5.1a) and (5.1b), the cepstrum and complex cepstrum are defined
in terms of the DTFT. This is useful in the basic definitions, but not
In cases where we wish to explicitly indicate analysis-time dependence of the short-time
cepstrum, we will use ˆ n for the analysis time and m for quefrency as in (5.4), but as in
other instances, we often suppress the subscript ˆ n.
58 Homomorphic Speech Analysis
for use in processing sampled speech signals. Fortunately, several com-
putational options exist.
5.3.1 Computation Using the DFT
Since the DFT (computed with an FFT algorithm) is a sampled
(in frequency) version of the DTFT of a finite-length sequence (i.e.,
X[k] = X(e
) [89]), the DFT and inverse DFT can be substi-
tuted for the DTFT and its inverse in (5.1a) and (5.1b) as shown in
Figure 5.3, which shows that the complex cepstrum can be computed
approximately using the equations
X[k] =
X[k] = log[X[k][ + j arg¦X[k]¦ (5.5b)
ˆ x[n] =
. (5.5c)
Note the "tilde" ˜ symbol above ˆ x[n] in (5.5c) and in Figure 5.3. Its
purpose is to emphasize that using the DFT instead of the DTFT
results in an approximation to (5.1b) due to the time-domain aliasing
resulting from the sampling of the log of the DTFT [89]. That is,
ˆ x[n] =
ˆ x[n + rN], (5.6)
where ˆ x[n] is the complex cepstrum defined by (5.1b). An identical
equation holds for the time-aliased cepstrum ˜ c[n].
The effect of time-domain aliasing can be made negligible by using a
large value for N. A more serious problem in computation of the com-
plex cepstrum is the computation of the complex logarithm. This is
Fig. 5.3 Computing the cepstrum or complex cepstrum using the DFT.
5.3 Computation of the Cepstrum 59
because the angle of a complex number is usually specified modulo
2π, i.e., by the principal value. In order for the complex cepstrum
to be evaluated properly, the phase of the sampled DTFT must be
evaluated as samples of a continuous function of frequency. If the princi-
pal value phase is first computed at discrete frequencies, then it must be
"unwrapped" modulo 2π in order to ensure that convolutions are trans-
formed into additions. A variety of algorithms have been developed for
phase unwrapping [90, 109, 127]. While accurate phase unwrapping
presents a challenge in computing the complex cepstrum, it is not a
problem in computing the cepstrum, since the phase is not used. Fur-
thermore, phase unwrapping can be avoided by using numerical com-
putation of the poles and zeros of the z-transform.
5.3.2 z-Transform Analysis
The characteristic system for convolution can also be represented by
the two-sided z-transform as depicted in Figure 5.4. This is very useful
for theoretical investigations, and recent developments in polynomial
root finding [117] have made the z-transform representation a viable
computational basis as well. For this purpose, we assume that the input
signal x[n] has a rational z-transform of the form:
X(z) = X
(z), (5.7)
(z) = z
(1 − a
) =
(1 − a
z), (5.8a)
(z) =
(1 − e
), (5.8b)
(z) = A
(1 − b
(1 − c
. (5.8c)
The zeros of X
(z), i.e., z
= a
, are zeros of X(z) outside of the
unit circle ([a
(z) is thus the maximum-phase part of X(z).
60 Homomorphic Speech Analysis
Fig. 5.4 z-transform representation of characteristic system for convolution.
(z) contains all the zeros (with angles θ
) on the unit circle. The
minimum-phase part is X
(z), where b
and c
are zeros and poles,
respectively, that are inside the unit circle ([b
[ < 1 and [c
[ < 1). The
factor z
implies a shift of M
samples to the left. It is included to
simplify the results in (5.11).
The complex cepstrum of x[n] is determined by assuming that the
complex logarithm log¦X(z)¦ results in the sum of logarithms of each
of the product terms, i.e.,
X(z) = log
log(1 − a
z) +
log(1 − e
+ log[A[ +
log(1 − b
) −
log(1 − c
). (5.9)
Applying the power series expansion
log(1 − a) = −
[a[ < 1 (5.10)
to each of the terms in (5.9) and collecting the coefficients of the positive
and negative powers of z gives
ˆ x[n] =
n < 0
log[A[ + log
n = 0
n > 0.
Given all the poles and zeros of a z-transform X(z), (5.11) allows
us to compute the complex cepstrum with no approximation. This is
5.3 Computation of the Cepstrum 61
the case in theoretical analysis where the poles and zeros are specified.
However, (5.11) is also useful as the basis for computation. All that is
needed is a process for obtaining the z-transform as a rational function
and a process for finding the zeros of the numerator and denominator.
This has become more feasible with increasing computational power
and with new advances in finding roots of large polynomials [117].
One method of obtaining a z-transform is simply to select a finite-
length sequence of samples of a signal. The z-transform is then simply
a polynomial with the samples x[n] as coefficients, i.e.,
X(z) =
(1 − a
(1 − b
). (5.12)
A second method that yields a z-transform is the method of linear
predictive analysis, to be discussed in Chapter 6.
5.3.3 Recursive Computation of the Complex Cepstrum
Another approach to computing the complex cepstrum applies only to
minimum-phase signals, i.e., signals having a z-transform whose poles
and zeros are inside the unit circle. An example would be the impulse
response of an all-pole vocal tract model with system function
H(z) =
1 −
(1 − c
. (5.13)
Such models are implicit in the use of linear predictive analysis of speech
(Chapter 6). In this case, all the poles c
must be inside the unit circle
for stability of the system. From (5.11) it follows that the complex
cepstrum of the impulse response h[n] corresponding to H(z) is
h[n] =
0 n < 0
log[G[ n = 0
n > 0.
62 Homomorphic Speech Analysis
It can be shown [90] that the impulse response and its complex cep-
strum are related by the recursion formula:
h[n] =
0 n < 0
logG n = 0
h[k]h[n − k]
n ≥ 1.
Furthermore, working with the reciprocal (negative logarithm) of
(5.13), it can be shown that there is a direct recursive relationship
between the coefficients of the denominator polynomial in (5.13) and
the complex cepstrum of the impulse response of the model filter, i.e.,
h[n] =
0 n < 0
logG n = 0
n > 0.
From (5.16) it follows that the coefficients of the denominator polyno-
mial can be obtained from the complex cepstrum through
h[n] −
1 ≤ n ≤ p. (5.17)
From (5.17), it follows that p + 1 values of the complex cepstrum are
sufficient to fully determine the speech model system in (5.13) since all
the denominator coefficients and G can be computed from
h[n] for n =
0, 1, . . . , p using (5.17). This fact is the basis for the use of the cepstrum
in speech coding and speech recognition as a vector representation of
the vocal tract properties of a frame of speech.
5.4 Short-Time Homomorphic Filtering of Speech
Figure 5.5 shows an example of the short-time cepstrum of speech for
the segments of voiced and unvoiced speech in Figures 4.5(a) and 4.5(c).
The low quefrency part of the cepstrum is expected to be representative
of the slow variations (with frequency) in the log spectrum, while the
5.4 Short-Time Homomorphic Filtering of Speech 63
0 10 20 30 40
0 10 20 30 40
0 2000 4000 6000 8000
0 2000 4000 6000 8000
(b) Voiced log STFT (a) Voiced Cepstrum
(d) Unvoiced log STFT (c) Unvoiced Cepstrum
time in ms frequency in kHz
Fig. 5.5 Short-time cepstra and corresponding STFTs and homomorphically-smoothed
high quefrency components would correspond to the more rapid fluctu-
ations of the log spectrum. This is illustrated by the plots in Figure 5.5.
The log magnitudes of the corresponding short-time spectra are shown
on the right as Figures 5.5(b) and 5.5(d). Note that the spectrum for
the voiced segment in Figure 5.5(b) has a structure of periodic ripples
due to the harmonic structure of the quasi-periodic segment of voiced
speech. This periodic structure in the log spectrum of Figure 5.5(b),
manifests itself in the cepstrum peak at a quefrency of about 9 ms
in Figure 5.5(a). The existence of this peak in the quefrency range of
expected pitch periods strongly signals voiced speech. Furthermore, the
quefrency of the peak is an accurate estimate of the pitch period dur-
ing the corresponding speech interval. As shown in Figure 4.5(b), the
autocorrelation function also displays an indication of periodicity, but
not nearly as unambiguously as does the cepstrum. On the other hand,
note that the rapid variations of the unvoiced spectra appear random
with no periodic structure. This is typical of Fourier transforms (peri-
odograms) of short segments of random signals. As a result, there is no
strong peak indicating periodicity as in the voiced case.
64 Homomorphic Speech Analysis
To illustrate the effect of liftering, the quefrencies above 5 ms are
multiplied by zero and the quefrencies below 5 ms are multiplied by 1
(with a short transition taper as shown in Figures 5.5(a) and 5.5(c)).
The DFT of the resulting modified cepstrum is plotted as the smooth
curve that is superimposed on the short-time spectra in Figures 5.5(b)
and 5.5(d), respectively. These slowly varying log spectra clearly retain
the general spectral shape with peaks corresponding to the formant
resonance structure for the segment of speech under analysis. Therefore,
a useful perspective is that by liftering the cepstrum, it is possible
to separate information about the vocal tract contributions from the
short-time speech spectrum [88].
5.5 Application to Pitch Detection
The cepstrum was first applied in speech processing to determine the
excitation parameters for the discrete-time speech model of Figure 7.10.
Noll [83] applied the short-time cepstrum to detect local periodicity
(voiced speech) or the lack thereof (unvoiced speech). This is illus-
trated in Figure 5.6, which shows a plot that is very similar to the plot
first published by Noll [83]. On the left is a sequence of log short-time
spectra (rapidly varying curves) and on the right is the correspond-
ing sequence of cepstra computed from the log spectra on the left.
The successive spectra and cepstra are for 50 ms segments obtained by
moving the window in steps of 12.5 ms (100 samples at a sampling rate
of 8000 samples/sec). From the discussion of Section 5.4, it is apparent
that for the positions 1 through 5, the window includes only unvoiced
speech, while for positions 6 and 7 the signal within the window is
partly voiced and partly unvoiced. For positions 8 through 15 the win-
dow only includes voiced speech. Note again that the rapid variations
of the unvoiced spectra appear random with no periodic structure. On
the other hand, the spectra for voiced segments have a structure of
periodic ripples due to the harmonic structure of the quasi-periodic
segment of voiced speech. As can be seen from the plots on the right,
the cepstrum peak at a quefrency of about 11–12 ms strongly signals
voiced speech, and the quefrency of the peak is an accurate estimate of
the pitch period during the corresponding speech interval.
5.5 Application to Pitch Detection 65
0 1 2 3 4
0 5 10 15 20 25
Short-Time Log Spectra in Cepstrum Analysis
frequency in kHz
Short-Time Cepstra
time in ms
Fig. 5.6 Short-time cepstra and corresponding STFTs and homomorphically-smoothed
The essence of the pitch detection algorithm proposed by Noll is
to compute a sequence of short-time cepstra and search each succes-
sive cepstrum for a peak in the quefrency region of the expected pitch
period. Presence of a strong peak implies voiced speech, and the que-
frency location of the peak gives the estimate of the pitch period. As
in most model-based signal processing applications of concepts such
as the cepstrum, the pitch detection algorithm includes many fea-
tures designed to handle cases that do not fit the underlying model
very well. For example, for frames 6 and 7 the cepstrum peak is weak
66 Homomorphic Speech Analysis
corresponding to the transition from unvoiced to voiced speech. In other
problematic cases, the peak at twice the pitch period may be stronger
than the peak at the quefrency of the pitch period. Noll applied tem-
poral continuity constraints to prevent such errors.
5.6 Applications to Pattern Recognition
Perhaps the most pervasive application of the cepstrum in speech pro-
cessing is its use in pattern recognition systems such as design of vec-
tor quantizers (VQ) and automatic speech recognizers (ASR). In such
applications, a speech signal is represented on a frame-by-frame basis
by a sequence of short-time cepstra. As we have shown, cepstra can be
computed either by z-transform analysis or by DFT implementation
of the characteristic system. In either case, we can assume that the
cepstrum vector corresponds to a gain-normalized (c[0] = 0) minimum-
phase vocal tract impulse response that is defined by the complex
h[n] =
2c[n] 1 ≤ n ≤ n
0 n < 0.
In problems such as VQ or ASR, a test pattern c[n] (vector of cep-
strum values n = 1, 2, . . . , n
) is compared against a similarly defined
reference pattern ¯ c[n]. Such comparisons require a distance measure.
For example, the Euclidean distance applied to the cepstrum would give
[c[n] − ¯ c[n][
. (5.19a)
Equivalently in the frequency domain,
j ˆ ω
)[ − log[
j ˆ ω
dˆ ω, (5.19b)
where log[H(e
j ˆ ω
)[ is the log magnitude of the DTFT of h[n] corre-
sponding to the complex cepstrum in (5.18) or the real part of the
For minimum-phase signals, the complex cepstrum satisfies h[n] = 0 for n < 0. Since the
cepstrum is always the even part of the complex cepstrum, it follows that
h[n] = 2c[n] for
n > 0.
5.6 Applications to Pattern Recognition 67
DTFT of
h[n] in (5.18). Thus, cepstrum-based comparisons are strongly
related to comparisons of smoothed short-time spectra. Therefore, the
cepstrum offers an effective and flexible representation of speech for
pattern recognition problems, and its interpretation as a difference of
log spectra suggests a match to auditory perception mechanisms.
5.6.1 Compensation for Linear Filtering
Suppose that we have only a linearly filtered version of the speech
signal, y[n] = h[n] ∗ x[n], instead of x[n]. If the analysis window is long
compared to the length of h[n], the short-time cepstrum of one frame
of the filtered speech signal y[n] will be
ˆ n
[m] = c
ˆ n
[m] + c
[m], (5.20)
where c
[m] will appear essentially the same in each frame. Therefore,
if we can estimate c
which we assume is non-time-varying, we can
obtain c
ˆ n
[m] at each frame from c
ˆ n
[m] by subtraction, i.e., c
[m] =
ˆ n
[m] − c
[m]. This property is extremely attractive in situations
where the reference pattern ¯ c[m] has been obtained under different
recording or transmission conditions from those used to acquire the
test vector. In these circumstances, the test vector can be compensated
for the effects of the linear filtering prior to computing the distance
measures used for comparison of patterns.
Another approach to removing the effects of linear distortions is to
observe that the cepstrum component due to the distortion is the same
in each frame. Therefore, it can be removed by a simple first difference
operation of the form:
ˆ n
[m] = c
ˆ n
[m] − c
ˆ n−1
[m]. (5.21)
It is clear that if c
ˆ n
[m] = c
ˆ n
[m] + c
[m] with c
[m] being indepen-
dent of ˆ n, then ∆c
ˆ n
[m] = ∆c
ˆ n
[m], i.e., the linear distortion effects
are removed.
In this section, it will be useful to use somewhat more complicated notation. Specifically,
we denote the cepstrum at analysis time ˆ n of a signal x[n] as c
ˆ n
[m], where m denotes
the quefrency index of the cepstrum.
Stockham [124] showed how c
[m] for such linear distortions can be estimated from the
signal y[n] by time averaging the log of the STFT.
68 Homomorphic Speech Analysis
Furui [39, 40] first noted that the sequence of cepstrum values has
temporal information that could be of value for a speaker verification
system. He used polynomial fits to cepstrum sequences to extract sim-
ple representations of the temporal variation. The delta cepstrum as
defined in (5.21) is simply the slope of a first-order polynomial fit to
the cepstrum time evolution.
5.6.2 Liftered Cepstrum Distance Measures
In using linear predictive analysis (discussed in Chapter 6) to obtain
cepstrum feature vectors for pattern recognition problems, it is
observed that there is significant statistical variability due to a variety
of factors including short-time analysis window position, bias toward
harmonic peaks, and additive noise [63, 126]. A solution to this problem
is to use weighted distance measures of the form:
[n] [c[n] − ¯ c[n][
, (5.22a)
which can be written as the Euclidean distance of liftered cepstra
[g[n]c[n] − g[n]¯ c[n][
. (5.22b)
Tohkura [126] found, for example, that when averaged over many
frames of speech and speakers, cepstrum values c[n] have zero means
and variances on the order of 1/n
. This suggests that g[n] = n for
n = 1, 2, . . . , n
could be used to equalize the contributions for each
term to the cepstrum difference.
Juang et al. [63] observed that the variability due to the vagaries of
LPC analysis could be lessened by using a lifter of the form:
g[n] = 1 + 0.5n
) n = 1, 2, . . . , n
. (5.23)
Tests of weighted distance measures showed consistent improvements
in automatic speech recognition tasks.
Itakura and Umezaki [55] used the group delay function to derive a
different cepstrum weighting function. Instead of g[n] = n for all n, or
5.6 Applications to Pattern Recognition 69
the lifter of (5.23), Itakura proposed the lifter
g[n] = n
. (5.24)
This lifter has great flexibility. For example, if s = 0 we have simply
lowpass liftering of the cepstrum. If s = 1 and τ is large, we have essen-
tially g[n] = n for small n with high quefrency tapering. Itakura and
Umezaki [55] tested the group delay spectrum distance measure in an
automatic speech recognition system. They found that for clean test
utterances, the difference in recognition rate was small for different
values of s when τ ≈ 5 although performance suffered with increas-
ing s for larger values of τ. This was attributed to the fact that for
larger s the group delay spectrum becomes very sharply peaked and
thus more sensitive to small differences in formant locations. However,
in test conditions with additive white noise and also with linear filter-
ing distortions, recognition rates improved significantly with τ = 5 and
increasing values of the parameter s.
5.6.3 Mel-Frequency Cepstrum Coefficients
As we have seen, weighted cepstrum distance measures have a directly
equivalent interpretation in terms of distance in the frequency domain.
This is significant in light of models for human perception of sound,
which, as noted in Chapter 3, are based upon a frequency analysis
performed in the inner ear. With this in mind, Davis and Mermelstein
[27] formulated a new type of cepstrum representation that has come to
be widely used and known as the mel-frequency cepstrum coefficients
The basic idea is to compute a frequency analysis based upon a
filter bank with approximately critical band spacing of the filters and
bandwidths. For 4 kHz bandwidth, approximately 20 filters are used.
In most implementations, a short-time Fourier analysis is done first,
resulting in a DFT X
ˆ n
[k] for analysis time ˆ n. Then the DFT values
are grouped together in critical bands and weighted by a triangular
weighting function as depicted in Figure 5.7. Note that the bandwidths
in Figure 5.7 are constant for center frequencies below 1 kHz and then
increase exponentially up to half the sampling rate of 4 kHz resulting
70 Homomorphic Speech Analysis
0 1000 2000 3000 4000
frequency in Hz
Fig. 5.7 Weighting functions for Mel-frequency filter bank.
in a total of 22 "filters." The mel-frequency spectrum at analysis time
ˆ n is defined for r = 1, 2, . . . , R as
ˆ n
[r] =
ˆ n
, (5.25a)
where V
[k] is the triangular weighting function for the rth filter ranging
from DFT index L
to U
, where
is a normalizing factor for the rth mel-filter. This normalization is built
into the weighting functions of Figure 5.7. It is needed so that a per-
fectly flat input Fourier spectrum will produce a flat mel-spectrum. For
each frame, a discrete cosine transform of the log of the magnitude of
the filter outputs is computed to form the function mfcc
ˆ n
[m], i.e.,
ˆ n
[m] =
ˆ n
r +
. (5.26)
Typically, mfcc
ˆ n
[m] is evaluated for a number of coefficients, N
that is less than the number of mel-filters, e.g., N
= 13 and R = 22.
Figure 5.8 shows the result of mfcc analysis of a frame of voiced speech
in comparison with the short-time Fourier spectrum, LPC spectrum
(discussed in Chapter 6), and a homomorphically smoothed spectrum.
The large dots are the values of log(MF
ˆ n
[r]) and the line interpolated
5.7 The Role of the Cepstrum 71
0 500 1000 1500 2000 2500 3000 3500 4000
frequency in Hz
Short-time Fourier Transform
Homomorphic smoothing,
LPC smoothing,
Mel cepstrum smoothing,
Fig. 5.8 Comparison of spectral smoothing methods.
between them is a spectrum reconstructed at the original DFT frequen-
cies. Note that all these spectra are different, but they have in common
that they have peaks at the formant resonances. At higher frequencies,
the reconstructed mel-spectrum of course has more smoothing due to
the structure of the filter bank.
Note that the delta cepstrum idea expressed by (5.21) can be applied
to mfcc to remove the effects of linear filtering as long as the frequency
response of the distorting linear filter does not vary much across each
of the mel-frequency bands.
5.7 The Role of the Cepstrum
As discussed in this chapter, the cepstrum entered the realm of speech
processing as a basis for pitch detection. It remains one of the most
effective indicators of voice pitch that have been devised. Because the
72 Homomorphic Speech Analysis
vocal tract and excitation components are well separated in the cep-
strum, it was natural to consider analysis techniques for estimation of
the vocal tract system as well [86, 88, 111]. While separation techniques
based on the cepstrum can be very effective, the linear predictive anal-
ysis methods to be discussed in the next chapter have proven to be
more effective for a variety of reasons. Nevertheless, the cepstrum con-
cept has demonstrated its value when applied to vocal tract system
estimates obtained by linear predictive analysis.
Linear Predictive Analysis
Linear predictive analysis is one of the most powerful and widely
used speech analysis techniques. The importance of this method lies
both in its ability to provide accurate estimates of the speech param-
eters and in its relative speed of computation. In this chapter, we
present a formulation of the ideas behind linear prediction, and dis-
cuss some of the issues that are involved in using it in practical speech
6.1 Linear Prediction and the Speech Model
We come to the idea of linear prediction of speech by recalling the
source/system model that was introduced in Chapter 2, where the
sampled speech signal was modeled as the output of a linear, slowly
time-varying system excited by either quasi-periodic impulses (during
voiced speech), or random noise (during unvoiced speech). The par-
ticular form of the source/system model implied by linear predictive
analysis is depicted in Figure 6.1, where the speech model is the part
inside the dashed box. Over short time intervals, the linear system is
74 Linear Predictive Analysis
Fig. 6.1 Model for linear predictive analysis of speech signals.
described by an all-pole system function of the form:
H(z) =
1 −
. (6.1)
In linear predictive analysis, the excitation is defined implicitly by the
vocal tract system model, i.e., the excitation is whatever is needed
to produce s[n] at the output of the system. The major advantage of
this model is that the gain parameter, G, and the filter coefficients
¦ can be estimated in a very straightforward and computationally
efficient manner by the method of linear predictive analysis.
For the system of Figure 6.1 with the vocal tract model of (6.1), the
speech samples s[n] are related to the excitation e[n] by the difference
s[n] =
s[n − k] + Ge[n]. (6.2)
A linear predictor with prediction coefficients, α
, is defined as a system
whose output is
˜ s[n] =
s[n − k], (6.3)
and the prediction error, defined as the amount by which ˜ s[n] fails to
exactly predict sample s[n], is
d[n] = s[n] − ˜ s[n] = s[n] −
s[n − k]. (6.4)
6.1 Linear Prediction and the Speech Model 75
From (6.4) it follows that the prediction error sequence is the output
of an FIR linear system whose system function is
A(z) = 1 −
. (6.5)
It can be seen by comparing (6.2) and (6.4) that if the speech signal
obeys the model of (6.2) exactly, and if α
= a
, then d[n] = Ge[n].
Thus, the prediction error filter, A(z), will be an inverse filter for the
system, H(z), of (6.1), i.e.,
H(z) =
. (6.6)
The basic problem of linear prediction analysis is to determine the
set of predictor coefficients ¦α
¦ directly from the speech signal in
order to obtain a useful estimate of the time-varying vocal tract system
through the use of (6.6). The basic approach is to find a set of predic-
tor coefficients that will minimize the mean-squared prediction error
over a short segment of the speech waveform. The resulting param-
eters are then assumed to be the parameters of the system function
H(z) in the model for production of the given segment of the speech
waveform. This process is repeated periodically at a rate appropriate
to track the phonetic variation of speech (i.e., order of 50–100 times
per second).
That this approach will lead to useful results may not be immedi-
ately obvious, but it can be justified in several ways. First, recall that
if α
= a
, then d[n] = Ge[n]. For voiced speech this means that d[n]
would consist of a train of impulses, i.e., d[n] would be small except at
isolated samples spaced by the current pitch period, P
. Thus, finding
s that minimize the mean-squared prediction error seems consistent
with this observation. A second motivation for this approach follows
from the fact that if a signal is generated by (6.2) with non-time-varying
coefficients and excited either by a single impulse or by a stationary
white noise input, then it can be shown that the predictor coefficients
that result from minimizing the mean-squared prediction error (over
all time) are identical to the coefficients of (6.2). A third pragmatic
justification for using the minimum mean-squared prediction error as a
76 Linear Predictive Analysis
basis for estimating the model parameters is that this approach leads to
an exceedingly useful and accurate representation of the speech signal
that can be obtained by efficient solution of a set of linear equations.
The short-time average prediction error is defined as
ˆ n
ˆ n
ˆ n
[m] −
ˆ n
[m − k]
, (6.7)
where s
ˆ n
[m] is a segment of speech that has been selected in a neigh-
borhood of the analysis time ˆ n, i.e.,
ˆ n
[m] = s[m + ˆ n] − M
≤ m≤ M
. (6.8)
That is, the time origin of the analysis segment is shifted to sample
ˆ n of the entire signal. The notation ' ` denotes averaging over a finite
number of samples. The details of specific definitions of the averaging
operation will be discussed below.
We can find the values of α
that minimize E
ˆ n
in (6.7) by setting
ˆ n
= 0, for i = 1, 2, . . . , p, thereby obtaining the equations
˜ α
ˆ n
[m − i]s
ˆ n
[m − k]` = 's
ˆ n
[m − i]s
ˆ n
[m]` 1 ≤ i ≤ p, (6.9)
where the ˜ α
are the values of α
that minimize E
ˆ n
in (6.7). (Since the
˜ α
are unique, we will drop the tilde and use the notation α
to denote
the values that minimize E
ˆ n
.) If we define
ˆ n
[i, k] = 's
ˆ n
[m − i]s
ˆ n
[m − k]`, (6.10)
then (6.9) can be written more compactly as
ˆ n
[i, k] = ϕ
ˆ n
[i, 0] i = 1, 2, . . . , p. (6.11)
More accurately, the solutions of (6.7) only provide a stationary point which can be shown
to be a minimum of E
ˆ n
since E
ˆ n
is a convex function of the α
The quantities ϕ
ˆ n
[i, k] are in the form of a correlation function for the speech segment
ˆ n
[m]. The details of the definition of the averaging operation used in (6.10) have a
significant effect on the properties of the prediction coefficients that are obtained by solving
6.2 Computing the Prediction Coefficients 77
If we know ϕ
ˆ n
[i, k] for 1 ≤ i ≤ p and 0 ≤ k ≤ p, this set of p equations
in p unknowns, which can be represented by the matrix equation:
Φα = ψ, (6.12)
can be solved for the vector α = ¦α
¦ of unknown predictor coefficients
that minimize the average squared prediction error for the segment
ˆ n
Using (6.7) and (6.9), the minimum mean-squared prediction
error can be shown to be [3, 5]
ˆ n
= ϕ
ˆ n
[0, 0] −
ˆ n
[0, k]. (6.13)
Thus, the total minimum mean-squared error consists of a fixed com-
ponent equal to the mean-squared value of the signal segment minus a
term that depends on the predictor coefficients that satisfy (6.11), i.e.,
the optimum coefficients reduce E
ˆ n
in (6.13) the most.
To solve for the optimum predictor coefficients, we must first com-
pute the quantities ϕ
ˆ n
[i, k] for 1 ≤ i ≤ p and 0 ≤ k ≤ p. Once this
is done we only have to solve (6.11) to obtain the α
s. Thus, in
principle, linear prediction analysis is very straightforward. However,
the details of the computation of ϕ
ˆ n
[i, k] and the subsequent solu-
tion of the equations are somewhat intricate and further discussion
is required.
6.2 Computing the Prediction Coefficients
So far we have not been explicit about the meaning of the averaging
notation ' ` used to define the mean-squared prediction error in (6.7).
As we have stated, in a short-time analysis procedure, the averaging
must be over a finite interval. We shall see below that two methods for
linear predictive analysis emerge out of a consideration of the limits of
summation and the definition of the waveform segment s
ˆ n
Although the α
s are functions of ˆ n (the time index at which they are estimated) this
dependence will not be explicitly shown. We shall also find it advantageous to drop the
subscripts ˆ n on E
ˆ n
, s
ˆ n
[m], and ϕ
ˆ n
[i, k] when no confusion will result.
These two methods, applied to the same speech signal yield slightly different optimum
predictor coefficients.
78 Linear Predictive Analysis
6.2.1 The Covariance Method
One approach to computing the prediction coefficients is based on the
ˆ n
ˆ n
ˆ n
[m] −
ˆ n
[m − k]
, (6.14)
where −M
≤ n ≤ M
. The quantities ϕ
ˆ n
[i, k] needed in (6.11) inherit
the same definition of the averaging operator, i.e.,
ˆ n
[i, k] =
ˆ n
[m − i]s
ˆ n
[m − k]
1 ≤ i ≤ p
0 ≤ k ≤ p.
Both (6.14) and (6.15) require values of s
ˆ n
[m] = s[m + ˆ n] over the
range −M
− p ≤ m≤ M
. By changes of index of summation, (6.15)
can be expressed in the equivalent forms:
ˆ n
[i, k] =
ˆ n
ˆ n
[m + i − k] (6.16a)
ˆ n
ˆ n
[m + k − i], (6.16b)
from which it follows that ϕ
ˆ n
[i, k] = ϕ
ˆ n
[k, i].
Figure 6.2 shows the sequences that are involved in computing the
mean-squared prediction error as defined by (6.14). The top part of
this figure shows a sampled speech signal s[m], and the box denotes
a segment of that waveform selected around some time index ˆ n. The
second plot shows that segment extracted as a finite-length sequence
ˆ n
[m] = s[m + ˆ n] for −M
− p ≤ m≤ M
. Note the p "extra" samples
(light shading) at the beginning that are needed to start the prediction
error filter at time −M
. This method does not require any assump-
tion about the signal outside the interval −M
− p ≤ m≤ M
the samples s
ˆ n
[m] for −M
− p ≤ m≤ M
are sufficient to evaluate
ϕ[i, k] for all required values of i and k. The third plot shows the pre-
diction error computed with the optimum predictor coefficients. Note
that in solving for the optimum prediction coefficients using (6.11), the
6.2 Computing the Prediction Coefficients 79
Fig. 6.2 Illustration of windowing and prediction error for the covariance method.
prediction error is implicitly computed over the range −M
≤ m≤ M
as required in (6.14) and (6.15). The minimum mean-squared predic-
tion error would be simply the sum of squares of all the samples shown.
In most cases of linear prediction, the prediction error sequence is not
explicitly computed since solution of (6.11) does not require it.
The mathematical structure that defines the covariance method of
linear predictive analysis implies a number of useful properties of the
solution. It is worthwhile to summarize them as follows:
(C.1) The mean-squared prediction error satisfies E
ˆ n
≥ 0. With
this method, it is theoretically possible for the average
error to be exactly zero.
(C.2) The matrix Φ in (6.12) is a symmetric positive-semi-
definite matrix.
(C.3) The roots of the prediction error filter A(z) in (6.5) are
not guaranteed to lie within the unit circle of the z-plane.
This implies that the vocal tract model filter (6.6) is not
guaranteed to be stable.
80 Linear Predictive Analysis
(C.4) As a result of (C.2), the equations (6.11) can be solved
efficiently using, for example, the well known Cholesky
decomposition of the covariance matrix Φ [3].
6.2.2 The Autocorrelation Method
Perhaps the most widely used method of linear predictive analysis
is called the autocorrelation method because the covariance function
ˆ n
[i, k] needed in (6.11) reduces to the STACF φ
ˆ n
[[i − k[] that we dis-
cussed in Chapter 4 [53, 54, 74, 78]. In the autocorrelation method, the
analysis segment s
ˆ n
[m] is defined as
ˆ n
[m] =
s[n + m]w[m] −M
≤ m≤ M
0 otherwise,
where the analysis window w[m] is used to taper the edges of the seg-
ment to zero. Since the analysis segment is defined by the windowing
of (6.17) to be zero outside the interval −M
≤ m≤ M
, it follows that
the prediction error sequence d
ˆ n
[m] can be nonzero only in the range
≤ m≤ M
+ p. Therefore, E
ˆ n
is defined as
ˆ n
ˆ n
ˆ n
. (6.18)
The windowing of (6.17) allows us to use the infinite limits to signify
that the sum is over all nonzero values of d
ˆ n
[m]. Applying this notion
to (6.16a) and (6.16b) leads to the conclusion that
ˆ n
[i, k] =
ˆ n
ˆ n
[m + [i − k[] = φ
ˆ n
[[i − k[]. (6.19)
Thus, ϕ[i, k] is a function only of [i − k[. Therefore, we can replace
ˆ n
[i, k] by φ
ˆ n
[[i − k[], which is the STACF defined in Chapter 4 as
ˆ n
[k] =
ˆ n
ˆ n
[m + k] = φ
ˆ n
[−k]. (6.20)
The resulting set of equations for the optimum predictor coefficients is
ˆ n
[[i − k[] = φ
ˆ n
[i] i = 1, 2, . . . , p. (6.21)
6.2 Computing the Prediction Coefficients 81
Fig. 6.3 Illustration of windowing and prediction error for the autocorrelation method.
Figure 6.3 shows the sequences that are involved in computing the
optimum prediction coefficients using the autocorrelation method. The
upper plot shows the same sampled speech signal s[m] as in Figure 6.2
with a Hamming window centered at time index ˆ n. The middle plot
shows the result of multiplying the signal s[ˆ n + m] by the window w[m]
and redefining the time origin to obtain s
ˆ n
[m]. Note that the zero-
valued samples outside the window are shown with light shading. The
third plot shows the prediction error computed using the optimum
coefficients. Note that for this segment, the prediction error (which is
implicit in the solution of (6.21)) is nonzero over the range −M
≤ m≤
+ p. Also note the lightly shaded p samples at the beginning. These
samples can be large due to the fact that the predictor must predict
these samples from the zero-valued samples that precede the windowed
segment s
ˆ n
[m]. It is easy to see that at least one of these first p samples
of the prediction error must be nonzero. Similarly, the last p samples of
the prediction error can be large due to the fact that the predictor must
predict zero-valued samples from windowed speech samples. It can also
be seen that at least one of these last p samples of the prediction error
must be nonzero. For this reason, it follows that E
ˆ n
, being the sum of
82 Linear Predictive Analysis
squares of the prediction error samples, must always be strictly greater
than zero.
As in the case of the covariance method, the mathematical structure
of the autocorrelation method implies a number of properties of the
solution, including the following:
(A.1) The mean-squared prediction error satisfies E
ˆ n
> 0. With
this method, it is theoretically impossible for the error
to be exactly zero because there will always be at least
one sample at the beginning and one at the end of the
prediction error sequence that will be nonzero.
(A.2) The matrix Φ in (6.12) is a symmetric positive-definite
Toeplitz matrix [46].
(A.3) The roots of the prediction error filter A(z) in (6.5) are
guaranteed to lie within the unit circle of the z-plane so
that the vocal tract model filter of (6.6) is guaranteed to
be stable.
(A.4) As a result of (A.2), the equations (6.11) can be solved
efficiently using the Levinson–Durbin algorithm, which,
because of its many implications, we discuss in more
detail in Section 6.3.
6.3 The Levinson–Durbin Recursion
As stated in (A.2) above, the matrix Φin (6.12) is a symmetric positive-
definite Toeplitz matrix, which means that all the elements on a given
diagonal in the matrix are equal. Equation (6.22) shows the detailed
structure of the matrix equation Φα = ψ for the autocorrelation
φ[0] φ[1] φ[p − 1]
φ[1] φ[0] φ[p − 2]
φ[p − 1] φ[p − 2] φ[0]
For this reason, a tapering window is generally used in the autocorrelation method.
6.3 The Levinson–Durbin Recursion 83
Note that the vector ψ is composed of almost the same autocorrelation
values as comprise Φ. Because of the special structure of (6.22) it is
possible to derive a recursive algorithm for inverting the matrix Φ.
That algorithm, known as the Levinson–Durbin algorithm, is specified
by the following steps:
Levinson–Durbin Algorithm
= φ[0] (D.1)
for i = 1, 2, . . . , p
φ[i] −
φ[i − j]
= k
if i > 1 then for j = 1, 2, . . . , i − 1
= α
− k
= (1 − k
= α
j = 1, 2, . . . , p (D.6)
An important feature of the Levinson–Durbin algorithm is that it
determines by recursion the optimum ith-order predictor from the opti-
mum (i − 1)th-order predictor, and as part of the process, all predic-
tors from order 0 (no prediction) to order p are computed along with
the corresponding mean-squared prediction errors E
. Specifically, the
equations labeled (D.3) and (D.4) can be used to show that the pre-
diction error system function satisfies
(z) = A
(z) − k
). (6.23)
Defining the ith-order forward prediction error e
[n] as the out-
put of the prediction error filter with system function A
(z) and
[n] as the output of the ith-order backward prediction error filter
(z) = z
), (6.23) leads (after some manipulations) to an
interpretation of the Levinson–Durbin algorithm in terms of a lattice
filter structure as in Figure 6.4(a).
84 Linear Predictive Analysis
Fig. 6.4 Lattice structures derived from the Levinson–Durbin recursion. (a) Prediction error
filter A(z). (b) Vocal tract filter H(z) = 1/A(z).
Also, by solving two equations in two unknowns recursively, it is
possible to start at the output of Figure 6.4(a) and work to the left
eventually computing s[n] in terms of e[n]. The lattice structure corre-
sponding to this is shown in Figure 6.4(b). Lattice structures like this
can be derived from acoustic principles applied to a physical model
composed of concatenated lossless tubes [101]. If the input and output
signals in such physical models are sampled at just the right sampling
rate, the sampled signals are related by a transfer function identical to
(6.6). In this case, the coefficients k
behave as reflection coefficients at
the tube boundaries [3, 78, 101].
Note that the parameters k
for i = 1, 2, . . . , p play a key role in
the Levinson–Durbin recursion and also in the lattice filter interpreta-
tion. Itakura and Saito [53, 54], showed that the parameters k
in the
Levinson–Durbin recursion and the lattice filter interpretation obtained
from it also could be derived by looking at linear predictive analysis
from a statistical perspective. They called the k
parameters, PARCOR
(for partial correlation) coefficients [54], because they can be computed
directly as a ratio of cross-correlation values between the forward and
6.4 LPC Spectrum 85
backward prediction errors at the output of the (i − 1)th stage of pre-
diction in Figure 6.4(a), i.e.,
[m − 1]
[m − 1]
. (6.24)
In the PARCOR interpretation, each stage of Figure 6.4(a) removes
part of the correlation in the input signal. The PARCOR coefficients
computed using (6.24) are identical to the k
obtained as a result of the
Levinson–Durbin algorithm. Indeed, Equation (D.2) in the Levinson–
Durbin algorithm can be replaced by (6.24), and the result is an algo-
rithm for transforming the PARCOR representation into the linear pre-
dictor coefficient representation.
The Levinson–Durbin formulation provides one more piece of useful
information about the PARCOR coefficients. Specifically, from Equa-
tion (D.5) of the algorithm, it follows that, since E
= (1 − k
is strictly greater than zero for predictors of all orders, it must be true
that −1 < k
< 1 for all i. It can be shown that this condition on the
PARCORs also guarantees that all the zeros of a prediction error filter
(z) of any order must be strictly inside the unit circle of the z-plane
[74, 78].
6.4 LPC Spectrum
The frequency-domain interpretation of linear predictive analysis pro-
vides an informative link to our earlier discussions of the STFT and cep-
strum analysis. The autocorrelation method is based on the short-time
autocorrelation function, φ
ˆ n
[m], which is the inverse discrete Fourier
transform of the magnitude-squared of the STFT, [S
ˆ n
j ˆ ω
, of the
windowed speech signal s
ˆ n
[m] = s[n + m]w[m]. The values φ
ˆ n
[m] for
m = 0, 1, . . . , p are used to compute the prediction coefficients and gain,
which in turn define the vocal tract system function H(z) in (6.6).
Therefore, the magnitude-squared of the frequency response of this sys-
tem, obtained by evaluating H(z) on the unit circle at angles 2πf/f
86 Linear Predictive Analysis
is of the form:
1 −
, (6.25)
and can be thought of as an alternative short-time spectral representa-
tion. Figure 6.5 shows a comparison between short-time Fourier analy-
sis and linear predictive spectrum analysis for segments of voiced and
unvoiced speech. Figures 6.5(a) and 6.5(c) show the STACF, with the
first 23 values plotted with a heavy line. These values are used to
compute the predictor coefficients and gain for an LPC model with
p = 22. The frequency responses of the corresponding vocal tract sys-
tem models are computed using (6.25) where the sampling frequency is
= 16 kHz. These frequency responses are superimposed on the corre-
sponding STFTs (shown in gray). As we have observed before, the rapid
variations with frequency in the STFT are due primarily to the exci-
tation, while the overall shape is assumed to be determined by the
effects of glottal pulse, vocal tract transfer function, and radiation. In
0 5 10 15
0 5 10 15
0 2000 4000 6000 8000
0 2000 4000 6000 8000
(b) Voiced LPC Spectrum (a) Voiced Autocorrelation
(d) Unvoiced LPC Spectrum (c) Unvoiced Autocorrelation
time in ms frequency in kHz
Fig. 6.5 Comparison of short-time Fourier analysis with linear predictive analysis.
6.4 LPC Spectrum 87
cepstrum analysis, the excitation effects are removed by lowpass lifter-
ing the cepstrum. In linear predictive spectrum analysis, the excitation
effects are removed by focusing on the low-time autocorrelation coef-
ficients. The amount of smoothing of the spectrum is controlled by
the choice of p. Figure 6.5 shows that a linear prediction model with
p = 22 matches the general shape of the short-time spectrum, but does
not represent all its local peaks and valleys, and this is exactly what is
The question naturally arises as to how p should be chosen.
Figure 6.6 offers a suggestion. In Figure 6.6(a) are shown the STFT
(in gray) and the frequency responses of a 12th-order model (heavy
dark line) and a 40th-order model (thin dark line). Evidently, the lin-
ear predictive spectra tend to favor the peaks of the short-time Fourier
transform. That this is true in general can be argued using the Parseval
theorem of Fourier analysis [74]. This is in contrast to homomorphic
Fig. 6.6 Effect of predictor order on: (a) estimating the spectral envelope of a 4-kHz band-
width signal; and (b) the normalized mean-squared prediction error for the same signal.
88 Linear Predictive Analysis
smoothing of the STFT, which tends toward an average of the peaks
and valleys of the STFT. As an example, see Figure 5.8, which shows
a comparison between the short-time Fourier spectrum, the LPC spec-
trum with p = 12, a homomorphically smoothed spectrum using 13
cepstrum values, and a mel-frequency spectrum.
Figure 6.6(b) shows the normalized mean-squared prediction error
/φ[0] as a function of p for the segment of speech used to
produce Figure 6.6(a). Note the sharp decrease from p = 0 (no pre-
diction implies V
= 1) to p = 1 and the less abrupt decrease there-
after. Furthermore, notice that the mean-squared error curve flattens
out above about p = 12 and then decreases modestly thereafter. The
choice p = 12 gives a good match to the general shape of the STFT,
highlighting the formant structure imposed by the vocal tract filter
while ignoring the periodic pitch structure. Observe that p = 40 gives
a much different result. In this case, the vocal tract model is highly
influenced by the pitch harmonics in the short-time spectrum. It can
be shown that if p is increased to the length of the windowed speech
segment, that [H(e
ˆ n
, but as pointed out in
(A.1) above, E
does not go to zero [75].
In a particular application, the prediction order is generally fixed
at a value that captures the general spectral shape due to the glot-
tal pulse, vocal tract resonances, and radiation. From the acoustic
theory of speech production, it follows that the glottal pulse spec-
trum is lowpass, the radiation filtering is highpass, and the vocal tract
imposes a resonance structure that, for adult speakers, is comprised
of about one resonance per kilohertz of frequency [101]. For the sam-
pled speech signal, the combination of the lowpass glottal pulse spec-
trum and the highpass filtering of radiation are usually adequately
represented by one or two additional complex pole pairs. When cou-
pled with an estimate of one resonance per kilohertz, this leads to a
rule of thumb of p = 4 + f
/1000. For example, for a sampling rate of
= 8000 Hz, it is common to use a predictor order of 10–12. Note that
in Figure 6.5 where the sampling rate was f
= 16000 Hz, a predictor
order of p = 22 gave a good representation of the overall shape and
resonance structure of the speech segment over the band from 0 to
8000 Hz.
6.5 Equivalent Representations 89
The example of Figure 6.6 illustrates an important point about
linear predictive analysis. In that example, the speaker was a high-
pitched female, and the wide spacing between harmonics causes the
peaks of the linear predictive vocal tract model to be biased toward
those harmonics. This effect is a serious limitation for high-pitched
voices, but is less problematic for male voices where the spacing between
harmonics is generally much smaller.
6.5 Equivalent Representations
The basic parameters obtained by linear predictive analysis are the gain
G and the prediction coefficients ¦α
¦. From these, a variety of different
equivalent representations can be obtained. These different representa-
tions are important, particularly when they are used in speech coding,
because we often wish to quantize the model parameters for efficiency
in storage or transmission of coded speech.
In this section, we give only a brief summary of the most important
equivalent representations.
6.5.1 Roots of Prediction Error System Function
Equation (6.5) shows that the system function of the prediction error
filter is a polynomial in z
and therefore it can be represented in terms
of its zeros as
A(z) = 1 −
(1 − z
). (6.26)
According to (6.6), the zeros of A(z) are the poles of H(z). Therefore,
if the model order is chosen judiciously as discussed in the previous
section, then it can be expected that roughly f
/1000 of the roots will
be close in frequency (angle in the z-plane) to the formant frequencies.
Figure 6.7 shows an example of the roots (marked with ) of a 12th-
order predictor. Note that eight (four complex conjugate pairs) of the
roots are close to the unit circle. These are the poles of H(z) that model
the formant resonances. The remaining four roots lie well within the
unit circle, which means that they only provide for the overall spectral
shaping resulting from the glottal and radiation spectral shaping.
90 Linear Predictive Analysis
Fig. 6.7 Poles of H(z) (zeros of A(z)) marked with × and LSP roots marked with ∗ and o.
The prediction coefficients are perhaps the most susceptible to quan-
tization errors of all the equivalent representations. It is well-known
that the roots of a polynomial are highly sensitive to errors in its coef-
ficients — all the roots being a function of all the coefficients [89].
Since the pole locations are crucial to accurate representation of the
spectrum, it is important to maintain high accuracy in the location of
the zeros of A(z). One possibility, not often invoked, would be to factor
the polynomial as in (6.26) and then quantize each root (magnitude
and angle) individually.
6.5.2 LSP Coefficients
A much more desirable alternative to quantization of the roots of A(z)
was introduced by Itakura [52], who defined the line spectrum pair
(LSP) polynomials
P(z) = A(z) + z
) (6.27a)
Q(z) = A(z) − z
). (6.27b)
Note the similarity to (6.23), from which it follows that P(z) and Q(z) are system functions
of lattice filters obtained by extending Figure 6.4(a) with an additional section with k
∓1, respectively.
6.5 Equivalent Representations 91
This transformation of the linear prediction parameters is invertible:
to recover A(z) from P(z) and Q(z) simply add the two equations to
obtain A(z) = (P(z) + Q(z))/2.
An illustration of the LSP representation is given in Figure 6.7,
which shows the roots of A(z), P(z), and Q(z) for a 12th-order pre-
dictor. This new representation has some very interesting and useful
properties that are confirmed by Figure 6.7 and summarized below
[52, 119].
(LSP.1) All the roots of P(z) and Q(z) are on the unit circle,
i.e., they can be represented as
P(z) =
(1 − e
) (6.28a)
Q(z) =
(1 − e
). (6.28b)
The normalized discrete-time frequencies (angles in the
z-plane), Ω
and Θ
, are called the line spectrum fre-
quencies or LSFs. Knowing the p LSFs that lie in the
range 0 < ω < π is sufficient to completely define the
LSP polynomials and therefore A(z).
(LSP.2) If p is an even integer, then P(−1) = 0 and Q(1) = 0.
(LSP.3) The PARCOR coefficients corresponding to A(z) sat-
isfy [k
[ < 1 if and only if the roots of P(z) and Q(z)
alternate on the unit circle, i.e., the LSFs are interlaced
over the range of angles 0 to π.
(LSP.4) The LSFs are close together when the roots of A(z)
are close to the unit circle.
These properties make it possible to represent the linear predictor
by quantized differences between the successive LSFs [119]. If prop-
erty (LSP.3) is maintained through the quantization, the reconstructed
polynomial A(z) will still have its zeros inside the unit circle.
92 Linear Predictive Analysis
6.5.3 Cepstrum of Vocal Tract Impulse Response
One of the most useful alternative representations of the linear predic-
tor is the cepstrum of the impulse response, h[n], of the vocal tract
filter. The impulse response can be computed recursively through the
difference equation:
h[n] =
h[n − k] + Gδ[n], (6.29)
or a closed form expression for the impulse response can be obtained
by making a partial fraction expansion of the system function H(z).
As discussed in Section 5.3.3, since H(z) is a minimum-phase sys-
tem (all its poles inside the unit circle), it follows that the complex cep-
strum of h[n] can be computed using (5.14), which would require that
the poles of H(z) be found by polynomial rooting. Then, any desired
number of cepstrum values can be computed. However, because of the
minimum phase condition, a more direct recursive computation is pos-
sible [101]. Equation (5.16) gives the recursion for computing
h[n] from
the predictor coefficients and gain, and (5.17) gives the inverse recursion
for computing the predictor coefficients from the complex cepstrum of
the vocal tract impulse response. These relationships are particularly
useful in speech recognition applications where a small number of cep-
strum values is used as a feature vector.
6.5.4 PARCOR Coefficients
We have seen that the PARCOR coefficients are bounded by ±1. This
makes them an attractive parameter for quantization.
We have mentioned that if we have a set of PARCOR coefficients,
we can simply use them at step (D.2) of the Levinson–Durbin algo-
rithm thereby obtaining an algorithm for converting PARCORs to
predictor coefficients. By working backward through the Levinson–
Durbin algorithm, we can compute the PARCOR coefficients from
a given set of predictor coefficients. The resulting algorithm is
given below.
6.5 Equivalent Representations 93
Predictor-to-PARCOR Algorithm
= α
j = 1, 2, . . . , p
= α
for i = p, p − 1, . . . , 2
for j = 1, 2, . . . , i − 1
+ k
1 − k
= α
6.5.5 Log Area Coefficients
As mentioned above, the lattice filter representation of the vocal tract
system is strongly related to concatenated acoustic tube models of the
physics of sound propagation in the vocal tract. Such models are char-
acterized by a set of tube cross-sectional areas denoted A
. These "area
function" parameters are useful alternative representations of the vocal
tract model obtained from linear predictive analysis. Specifically, the
log area ratio parameters are defined as
= log
= log
1 − k
1 + k
, (6.30)
where A
and A
are the areas of two successive tubes and the PAR-
COR coefficient −k
is the reflection coefficient for sound waves imping-
ing on the junction between the two tubes [3, 78, 101]. The inverse
transformation (from g
to k
) is
1 − e
1 + e
. (6.31)
The PARCOR coefficients can be converted to predictor coefficients if
required using the technique discussed in Section 6.5.4.
Viswanathan and Makhoul [131] showed that the frequency response
of a vocal tract filter represented by quantized log area ratio coefficients
is relatively insensitive to quantization errors.
94 Linear Predictive Analysis
6.6 The Role of Linear Prediction
In this chapter, we have discussed many of the fundamentals of linear
predictive analysis of speech. The value of linear predictive analysis
stems from its ability to represent in a compact form the part of the
speech model associated with the vocal tract, which in turn is closely
related to the phonemic representation of the speech signal. Over 40
years of research on linear predictive methods have yielded a wealth of
knowledge that has been widely and effectively applied in almost every
area of speech processing, but especially in speech coding, speech syn-
thesis, and speech recognition. The present chapter and Chapters 3–5
provide the basis for the remaining chapters of this text, which focus
on these particular application areas.
Digital Speech Coding
This chapter, the first of three on digital speech processing applications,
focuses on specific techniques that are used in digital speech coding.
We begin by describing the basic operation of sampling a speech signal
and directly quantizing and encoding of the samples. The remainder
of the chapter discusses a wide variety of techniques that represent the
speech signal in terms of parametric models of speech production and
7.1 Sampling and Quantization of Speech (PCM)
In any application of digital signal processing (DSP), the first step is
sampling and quantization of the resulting samples into digital form.
These operations, which comprise the process of A-to-D conversion,
are depicted for convenience in analysis and discussion in the block
diagram of Figure 7.1.
Current A-to-D and D-to-A converters use oversampling techniques to implement the
functions depicted in Figure 7.1.
96 Digital Speech Coding
) (t x
] [n x ] [ ˆ n x
] [n c
A-to-D Converter
] [n c′
] [ ˆ n x′
D-to-A Converter
) ( ˆ t x′
Fig. 7.1 The operations of sampling and quantization.
The sampler produces a sequence of numbers x[n] = x
(nT), where
T is the sampling period and f
= 1/T is the sampling frequency. The-
oretically, the samples x[n] are real numbers that are useful for theo-
retical analysis, but never available for computations. The well known
Shannon sampling theorem states that a bandlimited signal can be
reconstructed exactly from samples taken at twice the highest frequency
in the input signal spectrum (typically between 7 and 20 kHz for speech
and audio signals) [89]. Often, we use a lowpass filter to remove spectral
information above some frequency of interest (e.g., 4 kHz) and then use
a sampling rate such as 8000 samples/s to avoid aliasing distortion [89].
Figure 7.2 depicts a typical quantizer definition. The quantizer sim-
ply takes the real number inputs x[n] and assigns an output ˆ x[n] accord-
ing to the nonlinear discrete-output mapping Q¦ ¦.
In the case of
the example of Figure 7.2, the output samples are mapped to one
of eight possible values, with samples within the peak-to-peak range
being rounded and samples outside the range being "clipped" to either
the maximum positive or negative level. For samples within range, the
quantization error, defined as
e[n] = ˆ x[n] − x[n] (7.1)
Note that in this chapter, the notation ˆ x[n] denotes the quantized version of x[n], not the
complex cepstrum of x[n] as in Chapter 5.
7.1 Sampling and Quantization of Speech (PCM) 97
Fig. 7.2 8-level mid-tread quantizer.
satisfies the condition
−∆/2 < e[n] ≤ ∆/2, (7.2)
where ∆ is the quantizer step size. A B-bit quantizer such as the one
shown in Figure 7.2 has 2
levels (1 bit usually signals the sign).
Therefore, if the peak-to-peak range is 2X
, the step size will be
. Because the quantization levels are uniformly spaced by
∆, such quantizers are called uniform quantizers. Traditionally, repre-
sentation of a speech signal by binary-coded quantized samples is called
pulse-code modulation (or just PCM) because binary numbers can be
represented for transmission as on/off pulse amplitude modulation.
The block marked "encoder" in Figure 7.1 represents the assigning
of a binary code word to each quantization level. These code words rep-
resent the quantized signal amplitudes, and generally, as in Figure 7.2,
these code words are chosen to correspond to some convenient binary
number system such that arithmetic can be done on the code words as
98 Digital Speech Coding
if they were proportional to the signal samples.
In cases where accu-
rate amplitude calibration is desired, the step size could be applied as
depicted in the block labeled "Decoder" at the bottom of Figure 7.1.
Generally, we do not need to be concerned with the fine distinction
between the quantized samples and the coded samples when we have
a fixed step size, but this is not the case if the step size is adapted
from sample-to-sample. Note that the combination of the decoder
and lowpass reconstruction filter represents the operation of D-to-A
conversion [89].
The operation of quantization confronts us with a dilemma. Most
signals, speech especially, have a wide dynamic range, i.e., their ampli-
tudes vary greatly between voiced and unvoiced sounds and between
speakers. This means that we need a large peak-to-peak range so as to
avoid clipping of the loudest sounds. However, for a given number of
levels (bits), the step size ∆ = 2X
is proportional to the peak-to-
peak range. Therefore, as the peak-to-peak range increases, (7.2) states
that the size of the maximum quantization error grows. Furthermore,
for a uniform quantizer, the maximum size of the quantization error
is the same whether the signal sample is large or small. For a given
peak-to-peak range for a quantizer such as the one in Figure 7.2 we
can decrease the quantization error only by adding more levels (bits).
This is the fundamental problem of quantization.
The data rate (measured in bits/second) of sampled and quantized
speech signals is I = B f
. The standard values for sampling and quan-
tizing sound signals (speech, singing, instrumental music) are B = 16
and f
= 44.1 or 48 kHz. This leads to a bitrate of I = 16 44100 =
705, 600 bits/s.
This value is more than adequate and much more than
desired for most speech communication applications. The bit rate can
be lowered by using fewer bits/sample and/or using a lower sampling
rate; however, both these simple approaches degrade the perceived
quality of the speech signal. This chapter deals with a wide range of
In a real A-to-D converter, the sampler, quantizer, and coder are all integrated into one
system, and only the binary code words c[n] are available.
denotes the possibility of errors in the codewords. Symbol errors would cause addi-
tional error in the reconstructed samples.
Even higher rates (24 bits and 96 kHz sampling rate) are used for high-quality audio.
7.1 Sampling and Quantization of Speech (PCM) 99
techniques for significantly reducing the bit rate while maintaining an
adequate level of speech quality.
7.1.1 Uniform Quantization Noise Analysis
A more quantitative description of the effect of quantization can be
obtained using random signal analysis applied to the quantization error.
If the number of bits in the quantizer is reasonably high and no clip-
ping occurs, the quantization error sequence, although it is completely
determined by the signal amplitudes, nevertheless behaves as if it is a
random signal with the following properties [12]:
(Q.1) The noise samples appear to be
uncorrelated with the
signal samples.
(Q.2) Under certain assumptions, notably smooth input prob-
ability density functions and high rate quantizers, the
noise samples appear to be uncorrelated from sample-to-
sample, i.e., e[n] acts like a white noise sequence.
(Q.3) The amplitudes of noise samples are uniformly dis-
tributed across the range −∆/2 < e[n] ≤ ∆/2, resulting
in average power σ
= ∆
These simplifying assumptions allow a linear analysis that yields accu-
rate results if the signal is not too coarsely quantized. Under these con-
ditions, it can be shown that if the output levels of the quantizer are
optimized, then the quantizer error will be uncorrelated with the quan-
tizer output (however not the quantizer input, as commonly stated).
These results can be easily shown to hold in the simple case of a uni-
formly distributed memoryless input and Bennett has shown how the
result can be extended to inputs with smooth densities if the bit rate
is assumed high [12].
With these assumptions, it is possible to derive the following
formula for the signal-to-quantizing-noise ratio (in dB) of a B-bit
By "appear to be" we mean that measured correlations are small. Bennett has shown that
the correlations are only small because the error is small and, under suitable conditions,
the correlations are equal to the negative of the error variance [12]. The condition "uncor-
related" often implies independence, but in this case the error is a deterministic function
of the input and hence it cannot be independent of the input.
100 Digital Speech Coding
uniform quantizer:
= 10log
= 6.02B + 4.78 − 20log
, (7.3)
where σ
and σ
are the rms values of the input signal and quantization
noise samples, respectively. The formula of Equation (7.3) is an increas-
ingly good approximation as the bit rate (or the number of quantization
levels) gets large. It can be way off, however, if the bit rate is not large
[12]. Figure 7.3 shows a comparison of (7.3) with signal-to-quantization-
noise ratios measured for speech signals. The measurements were done
by quantizing 16-bit samples to 8, 9, and 10 bits. The faint dashed
lines are from (7.3) and the dark dashed lines are measured values for
uniform quantization. There is good agreement between these graphs
indicating that (7.3) is a reasonable estimate of SNR.
Note that X
is a fixed parameter of the quantizer, while σ
depends on the input signal level. As signal level increases, the ratio
decreases moving to the left in Figure 7.3. When σ
gets close
to X
, many samples are clipped, and the assumptions underlying
(7.3) no longer hold. This accounts for the precipitous fall in SNR for
< 8.
Comparison of -Law and Uniform Quantizers
Fig. 7.3 Comparison of µ-law and linear quantization for B = 8, 9, 10: Equation (7.3) — light
dashed lines. Measured uniform quantization SNR — dark dashed lines. µ-law (µ = 100)
compression — solid lines.
7.1 Sampling and Quantization of Speech (PCM) 101
Also, it should be noted that (7.3) and Figure 7.3 show that with
all other parameters being fixed, increasing B by 1 bit (doubling the
number of quantization levels) increases the SNR by 6 dB. On the other
hand, it is also important to note that halving σ
decreases the SNR
by 6 dB. In other words, cutting the signal level in half is like throwing
away one bit (half of the levels) of the quantizer. Thus, it is exceed-
ingly important to keep input signal levels as high as possible without
7.1.2 µ-Law Quantization
Also note that the SNR curves in Figure 7.3 decrease linearly with
increasing values of log
). This is because the size of the quan-
tization noise remains the same as the signal level decreases. If the
quantization error were proportional to the signal amplitude, the SNR
would be constant regardless of the size of the signal. This could be
achieved if log(x[n]) were quantized instead of x[n], but this is not pos-
sible because the log function is ill-behaved for small values of x[n]. As
a compromise, µ-law compression (of the dynamic range) of the signal,
defined as
y[n] = X
1 + µ
log(1 + µ)
sign(x[n]) (7.4)
can be used prior to uniform quantization [118]. Quantization of µ-law
compressed signals is often termed log-PCM. If µ is large (> 100) this
nonlinear transformation has the effect of distributing the effective
quantization levels uniformly for small samples and logarithmically over
the remaining range of the input samples. The flat curves in Figure 7.3
show that the signal-to-noise ratio with µ-law compression remains rel-
atively constant over a wide range of input levels.
µ-law quantization for speech is standardized in the CCITT
G.711 standard. In particular, 8-bit, µ = 255, log-PCM with f
8000 samples/s is widely used for digital telephony. This is an exam-
ple of where speech quality is deliberately compromised in order to
achieve a much lower bit rate. Once the bandwidth restriction has been
imposed, 8-bit log-PCM introduces little or no perceptible distortion.
102 Digital Speech Coding
This configuration is often referred to as "toll quality" because when
it was introduced at the beginning of the digital telephony era, it was
perceived to render speech equivalent to speech transmitted over the
best long distance lines. Nowadays, readily available hardware devices
convert analog signals directly into binary-coded µ-law samples and
also expand compressed samples back to uniform scale for conver-
sion back to analog signals. Such an A-to-D/D-to-A device is called
a encoder/decoder or "codec." If the digital output of a codec is used
as input to a speech processing algorithm, it is generally necessary to
restore the linearity of the amplitudes through the inverse of (7.4).
7.1.3 Non-Uniform and Adaptive Quantization
µ-law compression is an example of non-uniform quantization. It is
based on the intuitive notion of constant percentage error. A more rig-
orous approach is to design a non-uniform quantizer that minimizes the
mean-squared quantization error. To do this analytically, it is necessary
to know the probability distribution of the signal sample values so that
the most probable samples, which for speech are the low amplitude
samples, will incur less error than the least probable samples. To apply
this idea to the design of a non-uniform quantizer for speech requires
an assumption of an analytical form for the probability distribution or
some algorithmic approach based on measured distributions. The fun-
damentals of optimum quantization were established by Lloyd [71] and
Max [79]. Paez and Glisson [91] gave an algorithm for designing opti-
mum quantizers for assumed Laplace and Gamma probability densities,
which are useful approximations to measured distributions for speech.
Lloyd [71] gave an algorithm for designing optimum non-uniform quan-
tizers based on sampled speech signals.
Optimum non-uniform quan-
tizers can improve the signal-to-noise ratio by as much as 6 dB over
µ-law quantizers with the same number of bits, but with little or no
improvement in perceived quality of reproduction, however.
Lloyd's work was initially published in a Bell Laboratories Technical Note with portions
of the material having been presented at the Institute of Mathematical Statistics Meeting
in Atlantic City, New Jersey in September 1957. Subsequently this pioneering work was
published in the open literature in March 1982 [71].
7.2 Digital Speech Coding 103
Another way to deal with the wide dynamic range of speech is to let
the quantizer step size vary with time. When the quantizer in a PCM
system is adaptive, the system is called an adaptive PCM (APCM)
system. For example, the step size could satisfy
∆[n] = ∆
ˆ n
, (7.5)
where ∆
is a constant and E
ˆ n
is the short-time energy of the speech
signal defined, for example, as in (4.6). Such adaption of the step size
is equivalent to fixed quantization of the signal after division by the
rms signal amplitude (E
ˆ n
. With this definition, the step size will
go up and down with the local rms amplitude of the speech signal.
The adaptation speed can be adjusted by varying the analysis win-
dow size. If the step size control is based on the unquantized signal
samples, the quantizer is called a feed-forward adaptive quantizer, and
the step size information must be transmitted or stored along with the
quantized sample, thereby adding to the information rate. To amortize
this overhead, feedforward quantization is usually applied to blocks of
speech samples. On the other hand, if the step size control is based on
past quantized samples, the quantizer is a feedback adaptive quantizer.
Jayant [57] studied a class of feedback adaptive quantizers where the
step size ∆[n] is a function of the step size for the previous sample,
∆[n − 1]. In this approach, if the previous sample is quantized using
step size ∆[n − 1] to one of the low quantization levels, then the step
size is decreased for the next sample. On the other hand, the step size
is increased if the previous sample was quantized in one of the highest
levels. By basing the adaptation on the previous sample, it is not nec-
essary to transmit the step size. It can be derived at the decoder by
the same algorithm as used for encoding.
Adaptive quantizers can achieve about 6 dB improvement (equiva-
lent to adding one bit) over fixed quantizers [57, 58].
7.2 Digital Speech Coding
As we have seen, the data rate of sampled and quantized speech is
B f
. Virtually perfect perceptual quality is achievable with high data
104 Digital Speech Coding
By reducing B or f
, the bit rate can be reduced, but percep-
tual quality may suffer. Reducing f
requires bandwidth reduction,
and reducing B too much will introduce audible distortion that may
resemble random noise. The main objective in digital speech coding
(sometimes called speech compression) is to lower the bit rate while
maintaining an adequate level of perceptual fidelity. In addition to the
two dimensions of quality and bit rate, the complexity (in terms of
digital computation) is often of equal concern.
Since the 1930s, engineers and speech scientists have worked toward
the ultimate goal of achieving more efficient representations. This work
has ledtoabasic approachinspeechcodingthat is basedonthe use of DSP
techniques and the incorporation of knowledge of speech production and
perception into the quantization process. There are many ways that this
can be done. Traditionally, speech coding methods have been classified
according to whether they attempt to preserve the waveform of the
speech signal or whether they only seek to maintain an acceptable level
of perceptual quality and intelligibility. The former are generally called
waveform coders, and the latter model-based coders. Model-based coders
are designed for obtaining efficient digital representations of speech and
only speech. Straightforward sampling and uniformquantization (PCM)
is perhaps the only pure waveform coder. Non-uniform quantization
and adaptive quantization are simple attempts to build in properties
of the speech signal (time-varying amplitude distribution) and speech
perception (quantization noise is masked by loud sounds), but these
extensions still are aimed at preserving the waveform.
Most modern model-based coding methods are based on the notion
that the speech signal can be represented in terms of an excitation sig-
nal and a time-varying vocal tract system. These two components are
quantized, and then a "quantized" speech signal can be reconstructed
by exciting the quantized filter with the quantized excitation. Figure 7.4
illustrates why linear predictive analysis can be fruitfully applied for
model-based coding. The upper plot is a segment of a speech signal.
If the signal that is reconstructed from the digital representation is not perceptibly different
from the original analog speech signal, the digital representation is often referred to as
being "transparent."
7.2 Digital Speech Coding 105
0 100 200 300 400
0 100 200 300 400
Speech Waveform
Prediction Error Sequence
sample index
Fig. 7.4 Speech signal and corresponding prediction error signal for a 12th-order predictor.
The lower plot is the output of a linear prediction error filter A(z)
(with p = 12) that was derived from the given segment by the tech-
niques of Chapter 6. Note that the prediction error sequence amplitude
of Figure 7.4 is a factor of five lower than that of the signal itself, which
means that for a fixed number of bits, this segment of the prediction
error could be quantized with a smaller step size than the waveform
itself. If the quantized prediction error (residual) segment is used as
input to the corresponding vocal tract filter H(z) = 1/A(z), an approx-
imation to the original segment of speech would be obtained.
direct implementation of this idea does not lead to a practical method
of speech coding, it is nevertheless suggestive of more practical schemes
which we will discuss.
Today, the line between waveform coders and model-based coders
is not distinct. A more useful classification of speech coders focusses
on how the speech production and perception models are incorporated
into the quantization process. One class of systems simply attempts to
extract an excitation signal and a vocal tract system from the speech
signal without any attempt to preserve a relationship between the
waveforms of the original and the quantized speech. Such systems are
attractive because they can be implemented with modest computation.
To implement this time-varying inverse filtering/reconstruction requires care in fitting the
segments of the signals together at the block boundaries. Overlap-add methods [89] can
be used effectively for this purpose.
106 Digital Speech Coding
These coders, which we designate as open-loop coders, are also called
vocoders (voice coder) since they are based on the principles estab-
lished by H. Dudley early in the history of speech processing research
[30]. A second class of coders employs the source/system model for
speech production inside a feedback loop, and thus are called closed-
loop coders. These compare the quantized output to the original input
and attempt to minimize the difference between the two in some pre-
scribed sense. Differential PCM systems are simple examples of this
class, but increased availability of computational power has made it
possible to implement much more sophisticated closed-loop systems
called analysis-by-synthesis coders. Closed loop systems, since they
explicitly attempt to minimize a time-domain distortion measure, often
do a good job of preserving the speech waveform while employing many
of the same techniques used in open-loop systems.
7.3 Closed-Loop Coders
7.3.1 Predictive Coding
The essential features of predictive coding of speech were set forth in a
classic paper by Atal and Schroeder [5], although the basic principles of
predictive quantization were introduced by Cutler [26]. Figure 7.5 shows
a general block diagram of a large class of speech coding systems that
are called adaptive differential PCM (ADPCM) systems. These systems
are generally classified as waveform coders, but we prefer to emphasize
that they are closed-loop, model-based systems that also preserve the
waveform. The reader should ignore initially the blocks concerned with
adaptation and all the dotted control lines and focus instead on the core
DPCM system, which is comprised of a feedback structure that includes
the blocks labeled Q and P. The quantizer Q can have a number of
levels ranging from 2 to much higher, it can be uniform or non-uniform,
and it can be adaptive or not, but irrespective of the type of quantizer,
the quantized output can be expressed as
d[n] = d[n] + e[n], where e[n]
is the quantization error. The block labeled P is a linear predictor, so
˜ x[n] =
ˆ x[n − k], (7.6)
7.3 Closed-Loop Coders 107
Fig. 7.5 General block diagram for adaptive differential PCM (ADPCM).
i.e., the signal ˜ x[n] is predicted based on p past samples of the signal
ˆ x[n].
The signal d[n] = x[n] − ˜ x[n], the difference between the input
and the predicted signal, is the input to the quantizer. Finally, the
In a feed-forward adaptive predictor, the predictor parameters are estimated from the
input signal x[n], although, as shown in Figure 7.5, they can also be estimated from past
samples of the reconstructed signal ˆ x[n]. The latter would be termed a feedback adaptive
108 Digital Speech Coding
relationship between ˆ x[n] and
d[n] is
ˆ x[n] =
ˆ x[n − k] +
d[n], (7.7)
i.e., the signal ˆ x[n] is the output of what we have called the "vocal
tract filter," and
d[n] is the quantized excitation signal. Finally, since
ˆ x[n] = ˜ x[n] +
d[n] it follows that
ˆ x[n] = x[n] + e[n]. (7.8)
This is the key result for DPCM systems. It says that no matter what
predictor is used in Figure 7.5, the quantization error in ˆ x[n] is iden-
tical to the quantization error in the quantized excitation signal
If the prediction is good, then the variance of d[n] will be less than
the variance of x[n] so it will be possible to use a smaller step size and
therefore to reconstruct ˆ x[n] with less error than if x[n] were quan-
tized directly. The feedback structure of Figure 7.5 contains within it a
source/system speech model, which, as shown in Figure 7.5(b), is the
system needed to reconstruct the quantized speech signal ˆ x[n] from the
coded difference signal input.
From (7.8), the signal-to-noise ratio of the ADPCM system is
SNR = σ
. A simple, but informative modification leads to
, (7.9)
where the obvious definitions apply.
is the signal-to-noise ratio of the quantizer, which, for fine
quantization (i.e., large number of quantization levels with small step
size), is given by (7.3). The number of bits B determines the bit-rate
of the coded difference signal. As shown in Figure 7.5, the quantizer
can be either feedforward- or feedback-adaptive. As indicated, if the
quantizer is feedforward adaptive, step size information must be part
of the digital representation.
The quantity G
is called the prediction gain, i.e., if G
> 1 it
represents an improvement gained by placing a predictor based on the
speech model inside the feedback loop around the quantizer. Neglecting
7.3 Closed-Loop Coders 109
the correlation between the signal and the quantization error, it can be
shown that
1 −
, (7.10)
where ρ[k] = φ[k]/φ[0] is the normalized autocorrelation function used
to compute the optimum predictor coefficients. The predictor can be
either fixed or adaptive, and adaptive predictors can be either feed-
forward or feedback (derived by analyzing past samples of the quan-
tized speech signal reconstructed as part of the coder). Fixed predictors
can be designed based on long-term average correlation functions. The
lower figure in Figure 7.6 shows an estimated long-term autocorrela-
tion function for speech sampled at an 8 kHz sampling rate. The upper
plot shows 10log
computed from (7.10) as a function of predictor
order p. Note that a first-order fixed predictor can yield about 6 dB
prediction gain so that either the quantizer can have 1 bit less or the
0 2 4 6 8 10 12
0 2 4 6 8 10 12
predictor order p
autocorrelation lag
Fig. 7.6 Long-term autocorrelation function ρ[m] (lower plot) and corresponding prediction
gain G
110 Digital Speech Coding
reconstructed signal can have a 6 dB higher SNR. Higher-order fixed
predictors can achieve about 4 dB more prediction gain, and adapting
the predictor at a phoneme rate can produce an additional 4 dB of gain
[84]. Great flexibility is inherent in the block diagram of Figure 7.5, and
not surprisingly, many systems based upon the basic principle of dif-
ferential quantization have been proposed, studied, and implemented
as standard systems. Here we can only mention a few of the most
7.3.2 Delta Modulation
Delta modulation (DM) systems are the simplest differential coding
systems since they use a 1-bit quantizer and usually only a first-order
predictor. DM systems originated in the classic work of Cutler [26]
and deJager [28]. While DM systems evolved somewhat independently
of the more general differential coding methods, they nevertheless fit
neatly into the general theory of predictive coding. To see how such
systems can work, note that the optimum predictor coefficient for a
first-order predictor is α
= ρ[1], so from (7.10) it follows that the pre-
diction gain for a first-order predictor is
1 − ρ
. (7.11)
Furthermore, note the nature of the long-term autocorrelation function
in Figure 7.6. This correlation function is for f
= 8000 samples/s. If
the same bandwidth speech signal were over-sampled at a sampling
rate higher than 8000 samples/s, we could expect that the correlation
value ρ[1] would lie on a smooth curve interpolating between samples
0 and 1 in Figure 7.6, i.e., as f
gets large, both ρ[1] and α
unity and G
gets large. Thus, even though a 1-bit quantizer has a
very low SNR, this can be compensated by the prediction gain due to
Delta modulation systems can be implemented with very simple
hardware, and in contrast to more general predictive coding systems,
they are not generally implemented with block processing. Instead,
adaptation algorithms are usually based on the bit stream at the output
of the 1-bit quantizer. The simplest delta modulators use a very high
7.3 Closed-Loop Coders 111
sampling rate with a fixed predictor. Their bit rate, being equal to f
must be very high to achieve good quality reproduction. This limitation
can be mitigated with only modest increase in complexity by using an
adaptive 1-bit quantizer [47, 56]. For example, Jayant [56] showed that
for bandlimited (3 kHz) speech, adaptive delta modulation (ADM) with
bit rate (sampling rate) 40 kbits/s has about the same SNR as 6-bit
log PCM sampled at 6.6 kHz. Furthermore, he showed that doubling
the sampling rate of his ADM system improved the SNR by about
10 dB.
Adaptive delta modulation has the following advantages: it is very
simple to implement, only bit synchronization is required in transmis-
sion, it has very low coding delay, and it can be adjusted to be robust to
channel errors. For these reasons, adaptive delta modulation (ADM) is
still used for terminal-cost-sensitive digital transmission applications.
7.3.3 Adaptive Differential PCM Systems
Differential quantization with a multi-bit quantizer is called differen-
tial PCM (DPCM). As depicted in Figure 7.5, adaptive differential
PCM systems can have any combination of adaptive or fixed quantiz-
ers and/or predictors. Generally, the operations depicted in Figure 7.5
are implemented on short blocks of input signal samples, which intro-
duces delay. A careful comparison of a variety of ADPCM systems by
Noll [84] gives a valuable perspective on the relative contributions of
the quantizer and predictor. Noll compared log PCM, adaptive PCM,
fixed DPCM, and three ADPCM systems all using quantizers of 2, 3,
4, and 5-bits with 8 kHz sampling rate. For all bit rates (16, 24, 32, and
40 kbps), the results were as follows
(1) Log PCM had the lowest SNR.
(2) Adapting the quantizer (APCM) improved SNR by 6 dB.
(3) Adding first-order fixed or adaptive prediction improved the
SNR by about 4 dB over APCM.
The bit rates mentioned do not include overhead information for the quantized prediction
coefficients and quantizer step sizes. See Section 220.127.116.11 for a discussion of this issue.
112 Digital Speech Coding
(4) A fourth-order adaptive predictor added about 4 dB, and
increasing the predictor order to 12 added only 2 dB more.
The superior performance of ADPCM relative to log-PCM and its
relatively low computational demands have led to several standard ver-
sions (ITU G.721, G.726, G.727), which are often operated at 32 kbps
where quality is superior to log-PCM (ITU G.711) at 64 kbps.
The classic paper by Atal and Schroeder [5] contained a number
of ideas that have since been applied with great success in adaptive
predictive coding of speech. One of these concerns the quantization
noise introduced by ADPCM, which we have seen is simply added to
the input signal in the reconstruction process. If we invoke the white
noise approximation for quantization error, it is clear that the noise will
be most prominent in the speech at high frequencies where speech spec-
trum amplitudes are low. A simple solution to this problem is to pre-
emphasize the speech signal before coding, i.e., filter the speech with a
linear filter that boosts the high-frequency (HF) part of the spectrum.
After reconstruction of the pre-emphasized speech, it can be filtered
with the inverse system to restore the spectral balance, and in the
process, the noise spectrum will take on the shape of the de-emphasis
filter spectrum. A simple pre-emphasis system of this type has system
function (1 − γz
) where γ is less than one.
A more sophisticated
application of this idea is to replace the simple fixed pre-emphasis fil-
ter with a time-varying filter designed to shape the quantization noise
spectrum. An equivalent approach is to define a new feedback cod-
ing system where the quantization noise is computed as the difference
between the input and output of the quantizer and then shaped before
adding to the prediction residual. All these approaches raise the noise
spectrum at low frequencies and lower it at high frequencies, thus tak-
ing advantage of the masking effects of the prominent low frequencies
in speech [2].
Another idea that has been applied effectively in ADPCM systems
as well as analysis-by-synthesis systems is to include a long-delay pre-
dictor to capture the periodicity as well as the short-delay correlation
Values around γ = 0.4 for fs = 8 kHz work best. If γ is too close to one, the low frequency
noise is emphasized too much.
7.3 Closed-Loop Coders 113
inherent in voiced speech. Including the simplest long-delay predictor
would change (7.6) to
˜ x[n] = βˆ x[n − M] +
(ˆ x[n] − βˆ x[n − M]). (7.12)
The parameter M is essentially the pitch period (in samples) of voiced
speech, and the parameter β accounts for amplitude variations between
periods. The predictor parameters in (7.12) cannot be jointly opti-
mized. One sub-optimal approach is to estimate β and M first by
determining M that maximizes the correlation around the expected
pitch period and setting β equal to the normalized correlation at M.
Then the short-delay predictor parameters α
are estimated from the
output of the long-delay prediction error filter.
When this type of
predictor is used, the "vocal tract system" used in the decoder would
have system function
H(z) =
1 −
1 − βz
. (7.13)
Adding long-delay prediction takes more information out of the speech
signal and encodes it in the predictor. Even more complicated predic-
tors with multiple delays and gains have been found to improve the
performance of ADPCM systems significantly [2].
220.127.116.11 Coding the ADPCM Parameters
A glance at Figure 7.5 shows that if the quantizer and predictor are
fixed or feedback-adaptive, then only the coded difference signal is
needed to reconstruct the speech signal (assuming that the decoder
has the same coefficients and feedback-adaptation algorithms). Other-
wise, the predictor coefficients and quantizer step size must be included
as auxiliary data (side information) along with the quantized difference
Normally, this analysis would be performed on the input speech signal with the resulting
predictor coefficients applied to the reconstructed signal as in (7.12).
114 Digital Speech Coding
signal. The difference signal will have the same sampling rate as the
input signal. The step size and predictor parameters will be estimated
and changed at a much lower sampling rate, e.g., 50–100 times/s. The
total bit rate will be the sum of all the bit rates.
Predictor Quantization The predictor parameters must be quan-
tized for efficient digital representation of the speech signal. As dis-
cussed in Chapter 6, the short-delay predictors can be represented in
many equivalent ways, almost all of which are preferable to direct quan-
tization of the predictor coefficients themselves. The PARCOR coeffi-
cients can be quantized with a non-uniform quantizer or transformed
with an inverse sine or hyperbolic tangent function to flatten their sta-
tistical distribution and then quantized with a fixed uniform quantizer.
Each coefficient can be allocated a number of bits contingent on its
importance in accurately representing the speech spectrum [131].
Atal [2] reported a system which used a 20th-order predictor and
quantized the resulting set of PARCOR coefficients after transforma-
tion with an inverse sine function. The number of bits per coefficient
ranged from 5 for each of the lowest two PARCORs down to 1 each for
the six highest-order PARCORs, yielding a total of 40 bits per frame.
If the parameters are up-dated 100 times/s, then a total of 4000 bps for
the short-delay predictor information is required. To save bit rate, it
is possible to up-date the short-delay predictor 50 times/s for a total
bit rate contribution of 2000 bps [2]. The long delay predictor has a
delay parameter M, which requires 7 bits to cover the range of pitch
periods to be expected with 8 kHz sampling rate of the input. The
long-delay predictor in the system mentioned above used delays of
M − 1, M, M + 1 and three gains each quantized to 4 or 5 bit accu-
racy. This gave a total of 20 bits/frame and added 2000 bps to the
overall bit rate.
Vector Quantization for Predictor Quantization Another
approach to quantizing the predictor parameters is called vector quan-
tization, or VQ [45]. The basic principle of vector quantization is
depicted in Figure 7.7. VQ can be applied in any context where a set
of parameters naturally groups together into a vector (in the sense of
7.3 Closed-Loop Coders 115
Code Book
Code Book
Encoder Decoder
Fig. 7.7 Vector quantization.
linear algebra).
For example the predictor coefficients can be repre-
sented by vectors of predictor coefficients, PARCOR coefficients, cep-
strum values, log area ratios, or line spectrum frequencies. In VQ, a
vector, v, to be quantized is compared exhaustively to a "codebook"
populated with representative vectors of that type. The index i of the
closest vector ˆ v
to v, according to a prescribed distortion measure,
is returned from the exhaustive search. This index i then represents
the quantized vector in the sense that if the codebook is known, the
corresponding quantized vector ˆ v
can be looked up. If the codebook
has 2
entries, the index i can be represented by a B bit number.
The design of a vector quantizer requires a training set consisting
of a large number (L) of examples of vectors that are drawn from the
same distribution as vectors to be quantized later, along with a dis-
tortion measure for comparing two vectors. Using an iterative process,
codebook vectors are formed from the training set of vectors. The
"training phase" is computationally intense, but need only be done
once, and it is facilitated by the LBG algorithm [70]. One advantage of
VQ is that the distortion measure can be designed to sort the vectors
into perceptually relevant prototype vectors. The resulting codebook is
then used to quantize general test vectors. Typically a VQ training set
consists of at least 10 times the ultimate number of codebook vectors,
i.e., L ≥ 10 2
Independent quantization of individual speech samples or individual predictor coefficients
is called scalar quantization.
116 Digital Speech Coding
The quantization process is also time consuming because a given
vector must be systematically compared with all the codebook vectors
to determine which one it is closest to. This has led to numerous inno-
vations in codebook structure that speed up the look up process [45].
Difference Signal Quantization It is necessary to code the differ-
ence signal in ADPCM. Since the difference signal has the same sam-
pling rate as the input signal, it is desirable to use as few bits as possible
for the quantizer. If straightforward B-bit uniform quantization is used,
the additional bit rate would be B f
. This was the approach used in
earlier studies by Noll [84] as mentioned in Section 7.3.3. In order to
reduce the bit rate for coding the difference samples, some sort of block
coding must be applied. One approach is to design a multi-bit quantizer
that only operates on the largest samples of the difference signal. Atal
and Schroeder [2, 7] proposed to precede the quantizer by a "center
clipper," which is a system that sets to zero value all samples whose
amplitudes are within a threshold band and passes those samples above
threshold unchanged. By adjusting the threshold, the number of zero
samples can be controlled. For high thresholds, long runs of zero sam-
ples result, and the entropy of the quantized difference signal becomes
less than one. The blocks of zero samples can be encoded efficiently
using variable-length-to-block codes yielding average bits/sample on
the order of 0.7 [2]. For a sampling rate of 8 kHz, this coding method
needs a total of 5600 bps for the difference signal. When this bit rate
of the differential coder is added to the bit rate for quantizing the
predictor information (≈4000 bps) the total comes to approximately
10,000 bps.
184.108.40.206 Quality vs. Bit Rate for ADPCM Coders
ADPCM systems normally operate with an 8 kHz sampling rate, which
is a compromise aimed at providing adequate intelligibility and mod-
erate bit rate for voice communication. With this sampling rate, the
speech signal must be bandlimited to somewhat less than 4 kHz by a
lowpass filter. Adaptive prediction and quantization can easily lower
the bit rate to 32 kbps with no degradation in perceived quality (with
7.3 Closed-Loop Coders 117
64 kbps log PCM toll quality as a reference). Of course, raising the
bit rate for ADPCM above 64 kbps cannot improve the quality over
log-PCM because of the inherent frequency limitation. However, the
bit rate can be lowered below 32 kbps with only modest distortion
until about 10 kbps. Below this value, quality degrades significantly.
In order to achieve near-toll-quality at low bit rates, all the techniques
discussed above and more must be brought into play. This increases
the system complexity, although not unreasonably so for current DSP
hardware. Thus, ADPCM is an attractive coding method when toll
quality is required at modest cost, and where adequate transmission
and/or storage capacity is available to support bit rates on the order
of 10 kbps.
7.3.4 Analysis-by-Synthesis Coding
While ADPCM coding can produce excellent results at moderate bit
rates, its performance is fundamentally constrained by the fact that the
difference signal has the same sampling rate as the input signal. The
center clipping quantizer produces a difference signal that can be coded
efficiently. However, this approach is clearly not optimal since the center
clipper throws away information in order to obtain a sparse sequence.
What is needed is a way of creating an excitation signal for the vocal
tract filter that is both efficient to code and also produces decoded
speech of high quality. This can be done within the same closed-loop
framework as ADPCM, but with some significant modifications.
220.127.116.11 Basic Analysis-by-Synthesis Coding System
Figure 7.8 shows the block diagram of another class of closed-loop
digital speech coders. These systems are called "analysis-by-synthesis
coders" because the excitation is built up using an iterative process to
produce a "synthetic" vocal tract filter output ˆ x[n] that matches the
input speech signal according to a perceptually weighted error crite-
rion. As in the case of the most sophisticated ADPCM systems, the
operations of Figure 7.8 are carried out on blocks of speech samples. In
particular, the difference, d[n], between the input, x[n], and the output
of the vocal tract filter, ˆ x[n], is filtered with a linear filter called the
118 Digital Speech Coding
Vocal Tract
] [n x ] [n d
] [
n d
] [ ˆ n x
] [n d′
Fig. 7.8 Structure of analysis-by-synthesis speech coders.
perceptual weighting filter, W(z). As the first step in coding a block of
speech samples, both the vocal tract filter and the perceptual weighting
filter are derived from a linear predictive analysis of the block. Then,
the excitation signal is determined from the perceptually weighted dif-
ference signal d
[n] by an algorithm represented by the block labeled
"Excitation Generator."
Note the similarity of Figure 7.8 to the core ADPCM diagram of
Figure 7.5. The perceptual weighting and excitation generator inside
the dotted box play the role played by the quantizer in ADPCM, where
an adaptive quantization algorithm operates on d[n] to produce a quan-
tized difference signal
d[n], which is the input to the vocal tract system.
In ADPCM, the vocal tract model is in the same position in the closed-
loop system, but instead of the synthetic output ˆ x[n], a signal ˜ x[n]
predicted from ˆ x[n] is subtracted from the input to form the difference
signal. This is a key difference. In ADPCM, the synthetic output differs
from the input x[n] by the quantization error. In analysis-by-synthesis,
ˆ x[n] = x[n] − d[n], i.e., the reconstruction error is −d[n], and a percep-
tually weighted version of that error is minimized in the mean-squared
sense by the selection of the excitation
220.127.116.11 Perceptual Weighting of the Difference Signal
Since −d[n] is the error in the reconstructed signal, it is desirable to
shape its spectrum to take advantage of perceptual masking effects. In
ADPCM, this is accomplished by quantization noise feedback or preem-
phasis/deemphasis filtering. In analysis-by-synthesis coding, the input
7.3 Closed-Loop Coders 119
to the vocal tract filter is determined so as to minimize the perceptually-
weighted error d
[n]. The weighting is implemented by linear filtering,
i.e., d
[n] = d[n] ∗ w[n] with the weighting filter usually defined in terms
of the vocal tract filter as the linear system with system function
W(z) =
. (7.14)
The poles of W(z) lie at the same angle but at α
times the radii of
the poles of H(z), and the zeros of W(z) are at the same angles but at
radii of α
times the radii of the poles of H(z). If α
> α
the frequency
response is like a "controlled" inverse filter for H(z), which is the shape
desired. Figure 7.9 shows the frequency response of such a filter, where
typical values of α
= 0.9 and α
= 0.4 are used in (7.14). Clearly, this
filter tends to emphasize the high frequencies (where the vocal tract
filter gain is low) and it deemphasizes the low frequencies in the error
signal. Thus, it follows that the error will be distributed in frequency so
that relatively more error occurs at low frequencies, where, in this case,
such errors would be masked by the high amplitude low frequencies.
By varying α
and α
in (7.14) the relative distribution of error can be
0 1000 2000 3000 4000
frequency in Hz,
Fig. 7.9 Comparison of frequency responses of vocal tract filter and perceptual weighting
filter in analysis-by-synthesis coding.
120 Digital Speech Coding
188.8.131.52 Generating the Excitation Signal
Most analysis-by-synthesis systems generate the excitation from a finite
fixed collection of input components, which we designate here as f
for 0 ≤ n ≤ L − 1, where L is the excitation frame length and γ ranges
over the finite set of components. The input is composed of a finite sum
of scaled components selected from the given collection, i.e.,
d[n] =
[n]. (7.15)
The β
s and the sequences f
[n] are chosen to minimize
((x[n] − h[n] ∗
d[n]) ∗ w[n])
, (7.16)
where h[n] is the vocal tract impulse response and w[n] is the impulse
response of the perceptual weighting filter with system function (7.14).
Since the component sequences are assumed to be known at both the
coder and decoder, the βs and γs are all that is needed to represent the
d[n]. It is very difficult to solve simultaneously for the optimumβs
and γs that minimize (7.16). However, satisfactory results are obtained
by solving for the component signals one at a time.
Starting with the assumption that the excitation signal is zero dur-
ing the current excitation
analysis frame (indexed 0 ≤ n ≤ L − 1), the
output in the current frame due to the excitation determined for previ-
ous frames is computed and denoted ˆ x
[n] for 0 ≤ n ≤ L − 1. Normally
this would be a decaying signal that could be truncated after L samples.
The error signal in the current frame at this initial stage of the iterative
process would be d
[n] = x[n] − ˆ x
[n], and the perceptually weighted
error would be d
[n] = d
[n] ∗ w[n]. Now at the first iteration stage,
assume that we have determined which of the collection of input com-
ponents f
[n] will reduce the weighted mean-squared error the most
and we have also determined the required gain β
. By superposition,
ˆ x
[n] = ˆ x
[n] + β
[n] ∗ h[n]. The weighted error at the first itera-
tion is d
[n] = (d
[n] − β
[n] ∗ h[n] ∗ w[n]), or if we define the per-
ceptually weighted vocal tract impulse response as h
[n] = h[n] ∗ w[n]
Several analysis frames are usually included in one linear predictive analysis frame.
7.3 Closed-Loop Coders 121
and invoke the distributive property of convolution, then the weighted
error sequence is d
[n] = d
[n] − β
[n] ∗ h
[n]. This process is contin-
ued by finding the next component, subtracting it from the previously
computed residual error, and so forth. Generalizing to the kth iteration,
the corresponding equation takes the form:
[n] = d
[n] − β
[n], (7.17)
where y
[n] = f
[n] ∗ h
[n] is the output of the perceptually weighted
vocal tract impulse response due to the input component f
[n]. The
mean-squared error at stage k of the iteration is defined as
[n] − β
, (7.18)
where it is assumed that d
[n] is the weighted difference signal that
remains after k − 1 steps of the process.
Assuming that γ
is known, the value of β
that minimizes E
(7.18) is
, (7.19)
and the corresponding minimum mean-squared error is
− β
. (7.20)
While we have assumed in the above discussion that f
[n] and
are known, we have not discussed how they can be found. Equa-
tion (7.20) suggests that the mean-squared error can be minimized by
, (7.21)
122 Digital Speech Coding
which is essentially the normalized cross-correlation between the new
input component and the weighted residual error d
[n]. An exhaus-
tive search through the collection of input components will determine
which component f
[n] will maximize the quantity in (7.21) and thus
reduce the mean-squared error the most. Once this component is found,
the corresponding β
can be found from (7.19), and the new error
sequence computed from (7.17).
After N iterations of this process, the complete set of component
input sequences and corresponding coefficients will have been deter-
If desired, the set of input sequences so determined can be
assumed and a new set of βs can be found jointly by solving a set of
N linear equations.
126.96.36.199 Multi-Pulse Excitation Linear Prediction (MPLP)
The procedure described in Section 18.104.22.168 is quite general since the
only constraint was that the collection of component input sequences be
finite. The first analysis-by-synthesis coder was called multi-pulse linear
predictive coding [4]. In this system, the component input sequences are
simply isolated unit impulse sequences, i.e., f
[n] = δ[n − γ], where γ
is an integer such that 0 ≤ γ ≤ L − 1. The excitation sequence derived
by the process described in Section 22.214.171.124 is therefore of the form:
d[n] =
δ[n − γ
], (7.22)
where N is usually on the order of 4–5 impulses in a 5 ms (40 samples
for f
= 8 kHz) excitation analysis frame.
In this case, an exhaustive
search at each stage can be achieved by computing just one cross-
correlation function and locating the impulse at the maximum of the
cross-correlation. This is because the components y
[n] are all just
shifted versions of h
Alternatively, N need not be fixed in advance. The error reduction process can be stopped
when E
falls below a prescribed threshold.
It is advantageous to include two or more excitation analysis frames in one linear predic-
tive analysis frame, which normally is of 20 ms (160 samples) duration.
7.3 Closed-Loop Coders 123
In a speech coder based on multi-pulse analysis-by-synthesis, the
speech signal is represented by quantized versions of the prediction
coefficients, the pulse locations, and the pulse amplitudes. The predic-
tor coefficients can be coded as discussed for ADPCM by encoding any
of the alternative representations either by scalar or vector quantiza-
tion. A number of different approaches have been devised for coding
the excitation parameters. These usually involve some sort of differ-
ential coding scheme. At bit rates on the order of 10–16 kbps multi-
pulse coding can approach toll quality; however, below about 10 kbps,
quality degrades rapidly due to the many parameters that must be
coded [13].
Regular-pulse excitation (RPE) is a special case of multipulse exci-
tation where it is easier to encode the impulse locations. Specifically
after determining the location γ
of the first impulse, all other impulses
are located at integer multiples of a fixed spacing on either side of γ
Thus, only γ
, the integer multiples, and the gain constants must be
encoded. Using a pre-set spacing of 4 and an excitation analysis win-
dow of 40 samples yields high quality at about 10 kbps [67]. Because the
impulse locations are fixed after the first iteration, RPE requires signifi-
cantly less computation than full-blown multipulse excitation analysis.
One version of RPE is the basis for the 13 kbps digital coder in the
GSM mobile communications system.
18.104.22.168 Code-Excited Linear Prediction (CELP)
Since the major portion of the bit rate of an analysis-by-synthesis coder
lies in the excitation signal, it is not surprising that a great deal of
effort has gone into schemes for finding excitation signals that are eas-
ier to encode than multipulse excitation, but yet maintain high qual-
ity of reproduction of the speech signal. The next major innovation
in the history of analysis-by-synthesis systems was called code-excited
linear predictive coding or CELP [112].
In this scheme, the excita-
tion signal components are Gaussian random sequences stored in a
Earlier work by Stewart [123] used a residual codebook (populated by the Lloyd algorithm
[71]) with a low complexity trellis search.
124 Digital Speech Coding
If the codebook contains 2
sequences, then a given
sequence can be specified by an M-bit number. Typical codebook sizes
are 256, 512 or 1024. Within the codebook, the sequence lengths are
typically L = 40–60 samples (5–7.5 ms at 8 kHz sampling rate). If N
sequences are used to form the excitation, then a total of M N bits
will be required to code the sequences. Additional bits are still required
to code the β
s. With this method, the decoder must also have a copy
of the analysis codebook so that the excitation can be regenerated at
the decoder.
The main disadvantage of CELP is the high computational cost of
exhaustively searching the codebook at each stage of the error min-
imization. This is because each codebook sequence must be filtered
with the perceptually weighted impulse response before computing the
cross correlation with the residual error sequence. Efficient searching
schemes and structured codebooks have eased the computational bur-
den, and modern DSP hardware can easily implement the computations
in real time.
The basic CELP framework has been applied in the development of
numerous speech coders that operate with bit rates in the range from
4800 to 16000 bps. The Federal Standard 1016 (FS1016) was adopted
by the Department of Defense for use in secure voice transmission
at 4800 bps. Another system called VSELP (vector sum excited linear
prediction), which uses multiple codebooks to achieve a bit rate of
8000 bps, was adopted in 1989 for North American Cellular Systems.
The GSM half rate coder operating at 5600 bps is based on the IS-54
VSELP coder. Due to the block processing structure, CELP coders
can introduce more than 40 ms delay into communication systems. The
ITU G.728 low-delay CELP coder uses very short excitation analysis
frames and backward linear prediction to achieve high quality at a bit
rate of 16,000 bps with delays less than 2 ms. Still another standardized
CELP system is the ITU G.729 conjugate-structure algebraic CELP
(CS-ACELP) coder that is used in some mobile communication systems.
While it may seem counter-intuitive that the excitation could be comprised of white noise
sequence, remember that, in fact, the object of linear prediction is to produce just such
a signal.
7.4 Open-Loop Coders 125
22.214.171.124 Long-Delay Predictors in Analysis-by-Synthesis
Long-delay predictors have an interesting interpretation in analysis-by-
synthesis coding. If we maintain a memory of one or more frames of
previous values of the excitation signal, we can add a term β
d[n − γ
to (7.15). The long segment of the past excitation acts as a sort of
codebook where the L-sample codebook sequences overlap by L − 1
samples. The first step of the excitation computation would be to com-
pute γ
using (7.21) and then β
using (7.19). For each value of γ
to be
tested, this requires that the weighted vocal tract impulse response be
convolved with L-sample segments of the past excitation starting with
sample γ
. This can be done recursively to save computation. Then the
component β
d[n − γ
] can be subtracted from the initial error to start
the iteration process in either the multipulse or the CELP framework.
The additional bit rate for coding β
and γ
is often well worthwhile,
and long-delay predictors are used in many of the standardized coders
mentioned above. This incorporation of components of the past history
of the excitation has been referred to as "self-excitation" [104] or the
use of an "adaptive codebook" [19].
7.4 Open-Loop Coders
ADPCM coding has not been useful below about 9600 bps, multipulse
coding has similar limitations, and CELP coding has not been used at
bit rates below about 4800 bits/s. While these closed-loop systems have
many attractive features, it has not been possible to generate excitation
signals that can be coded at low bit rates and also produce high quality
synthetic speech. For bit rates below 4800 bps, engineers have turned
to the vocoder principles that were established decades ago. We call
these systems open-loop systems because they do not determine the
excitation by a feedback process.
7.4.1 The Two-State Excitation Model
Figure 7.10 shows a source/system model for speech that is closely
related to physical models for speech production. As we have discussed
126 Digital Speech Coding
Fig. 7.10 Two-state-excitation model for speech synthesis.
in the previous chapters, the excitation model can be very simple.
Unvoiced sounds are produced by exciting the system with white noise,
and voiced sounds are produced by a periodic impulse train excitation,
where the spacing between impulses is the pitch period, P
. We shall
refer to this as the two-state excitation model. The slowly-time-varying
linear system models the combined effects of vocal tract transmission,
radiation at the lips, and, in the case of voiced speech, the lowpass
frequency shaping of the glottal pulse. The V/UV (voiced/unvoiced
excitation) switch produces the alternating voiced and unvoiced seg-
ments of speech, and the gain parameter G controls the level of the
filter output. When values for V/UV decision, G, P
, and the param-
eters of the linear system are supplied at periodic intervals (frames)
then the model becomes a speech synthesizer, as discussed in Chap-
ter 8. When the parameters of the model are estimated directly
from a speech signal, the combination of estimator and synthesizer
becomes a vocoder or, as we prefer, an open-loop analysis/synthesis
speech coder.
220.127.116.11 Pitch, Gain, and V/UV Detection
The fundamental frequency of voiced speech can range from well below
100 Hz for low-pitched male speakers to over 250 Hz for high-pitched
voices of women and children. The fundamental frequency varies slowly,
with time, more or less at the same rate as the vocal tract motions. It
is common to estimate the fundamental frequency F
or equivalently,
7.4 Open-Loop Coders 127
the pitch period P
at a frame rate of about 50–100 times/s.
To do this, short segments of speech are analyzed to detect periodicity
(signaling voiced speech) or aperiodicity (signaling unvoiced speech).
One of the simplest, yet most effective approaches to pitch detection,
operates directly on the time waveform by locating corresponding peaks
and valleys in the waveform, and measuring the times between the
peaks [43]. The STACF is also useful for this purpose as illustrated
in Figure 4.8, which shows autocorrelation functions for both voiced
speech, evincing a peak at the pitch period, and unvoiced speech, which
shows no evidence of periodicity. In pitch detection applications of the
short-time autocorrelation, it is common to preprocess the speech by
a spectrum flattening operation such as center clipping [96] or inverse
filtering [77]. This preprocessing tends to enhance the peak at the pitch
period for voiced speech while suppressing the local correlation due to
formant resonances.
Another approach to pitch detection is suggested by Figure 5.5,
which shows the cepstra of segments of voiced and unvoiced speech. In
this case, a strong peak in the expected pitch period range signals voiced
speech, and the location of the peak is the pitch period. Similarly, lack
of a peak in the expected range signals unvoiced speech [83]. Figure 5.6
shows a sequence of short-time cepstra which moves from unvoiced to
voiced speech in going from the bottom to the top of the figure. The
time variation of the pitch period is evident in the upper plots.
The gain parameter G is also found by analysis of short segments
of speech. It should be chosen so that the short-time energy of the syn-
thetic output matches the short-time energy of the input speech signal.
For this purpose, the autocorrelation function value φ
ˆ n
[0] at lag 0 or
the cepstrum value c
ˆ n
[0] at quefrency 0 can be used to determine the
energy of the segment of the input signal.
For digital coding applications, the pitch period, V/UV, and gain
must be quantized. Typical values are 7 bits for pitch period (P
= 0
signals UV) and 5 bits for G. For a frame rate of 50 frames/s, this totals
600 bps, which is well below the bit rate used to encode the excitation
signal in closed-loop coders such as ADPCM or CELP. Since the vocal
tract filter can be coded as in ADPCM or CELP, much lower total bit
128 Digital Speech Coding
rates are common in open-loop systems. This comes at a large cost in
quality of the synthetic speech output, however.
220.127.116.11 Vocal Tract System Estimation
The vocal tract system in the synthesizer of Figure 7.10 can take many
forms. The primary methods that have been used have been homomor-
phic filtering and linear predictive analysis as discussed in Chapters 5
and 6, respectively.
The Homomorphic Vocoder Homomorphic filtering can be used
to extract a sequence of impulse responses from the sequence of cep-
stra that result from short-time cepstrum analysis. Thus, one cepstrum
computation can yield both an estimate of pitch and the vocal tract
impulse response. In the original homomorphic vocoder, the impulse
response was digitally coded by quantizing each cepstrum value indi-
vidually (scalar quantization) [86]. The impulse response, reconstructed
from the quantized cepstrum at the synthesizer, is simply convolved
with the excitation created from the quantized pitch, voicing, and gain
information, i.e., s[n] = Ge[n] ∗ h[n].
In a more recent application of
homomorphic analysis in an analysis-by-synthesis framework [21], the
cepstrum values were coded using vector quantization, and the excita-
tion derived by analysis-by-synthesis as described in Section 188.8.131.52. In
still another approach to digital coding, homomorphic filtering was used
to remove excitation effects in the short-time spectrum and then three
formant frequencies were estimated from the smoothed spectra. Fig-
ure 5.6 shows an example of the formants estimated for voiced speech.
These formant frequencies were used to control the resonance frequen-
cies of a synthesizer comprised of a cascade of second-order section IIR
digital filters [111]. Such a speech coder is called a formant vocoder.
LPC Vocoder Linear predictive analysis can also be used to esti-
mate the vocal tract system for an open-loop coder with two-state
Care must be taken at frame boundaries. For example, the impulse response can be
changed at the time a new pitch impulse occurs, and the resulting output can overlap
into the next frame.
7.4 Open-Loop Coders 129
excitation [6]. In this case, the prediction coefficients can be coded in
one of the many ways that we have already discussed. Such analy-
sis/synthesis coders are called LPC vocoders. A vocoder of this type
was standardized by the Department of Defense as the Federal Stan-
dard FS1015. This system is also called LPC-10 (or LPC-10e) because
a 10th-order covariance linear predictive analysis is used to estimate
the vocal tract system. The LPC-10 system has a bit rate of 2400 bps
using a frame size of 22.5 ms with 12 bits/frame allocated to pitch,
voicing and gain and the remaining bits allocated to the vocal tract
filter coded as PARCOR coefficients.
7.4.2 Residual-Excited Linear Predictive Coding
In Section 7.2, we presented Figure 7.4 as motivation for the use of
the source/system model in digital speech coding. This figure shows
an example of inverse filtering of the speech signal using a predic-
tion error filter, where the prediction error (or residual) is significantly
smaller and less lowpass in nature. The speech signal can be recon-
structed from the residual by passing it through the vocal tract sys-
tem H(z) = 1/A(z). None of the methods discussed so far attempt
to directly code the prediction error signal in an open-loop manner.
ADPCM and analysis-by-synthesis systems derive the excitation to the
synthesis filter by a feedback process. The two-state model attempts
to construct the excitation signal by direct analysis and measurement
of the input speech signal. Systems that attempt to code the resid-
ual signal directly are called residual-excited linear predictive (RELP)
Direct coding of the residual faces the same problem faced in
ADPCM or CELP: the sampling rate is the same as that of the input,
and accurate coding could require several bits per sample. Figure 7.11
shows a block diagram of a RELP coder [128]. In this system, which is
quite similar to voice-excited vocoders (VEV) [113, 130], the problem
of reducing the bit rate of the residual signal is attacked by reducing
its bandwidth to about 800 Hz, lowering the sampling rate, and coding
the samples with adaptive quantization. Adaptive delta modulation
was used in [128], but APCM could be used if the sampling rate is
130 Digital Speech Coding
] [n x ] [n r
& Spectrum
] [n x
] [n d
} {α
] [ ˆ n
} {α
] [n d
Fig. 7.11 Residual-excited linear predictive (RELP) coder and decoder.
lowered to 1600 Hz. The 800 Hz band is wide enough to contain several
harmonics of the highest pitched voices. The reduced bandwidth resid-
ual is restored to full bandwidth prior to its use as an excitation signal
by a nonlinear spectrum flattening operation, which restores higher
harmonics of voiced speech. White noise is also added according to an
empirically derived recipe. In the implementation of [128], the sam-
pling rate of the input was 6.8 kHz and the total bit rate was 9600 kbps
with 6800 bps devoted to the residual signal. The quality achieved at
this rate was not significantly better than the LPC vocoder with a
two-state excitation model. The principle advantage of this system is
that no hard V/UV decision must be made, and no pitch detection is
required. While this system did not become widely used, its basic prin-
ciples can be found in subsequent open-loop coders that have produced
much better speech quality at bit rates around 2400 bps.
7.4.3 Mixed Excitation Systems
While two-state excitation allows the bit rate to be quite low, the qual-
ity of the synthetic speech output leaves much to be desired. The output
of such systems is often described as "buzzy," and in many cases, errors
in estimating pitch period or voicing decision cause the speech to sound
unnatural if not unintelligible. The weaknesses of the two-state model
7.4 Open-Loop Coders 131
for excitation spurred interest in a mixed excitation model where a
hard decision between V and UV is not required. Such a model was
first proposed by Makhoul et al. [75] and greatly refined by McCree
and Barnwell [80].
Figure 7.12 depicts the essential features of the mixed-excitation
linear predictive coder (MELP) proposed by McCree and Barn-
well [80]. This configuration was developed as the result of careful
experimentation, which focused one-by-one on the sources of distor-
tions manifest in the two-state excitation coder such as buzziness and
tonal distortions. The main feature is that impulse train excitation
and noise excitation are added instead of switched. Prior to their addi-
tion, they each pass through a multi-band spectral shaping filter. The
gains in each of five bands are coordinated between the two filters so
that the spectrum of e[n] is flat. This mixed excitation helps to model
short-time spectral effects such as "devoicing" of certain bands during
voiced speech.
In some situations, a "jitter" parameter ∆P is invoked
to better model voicing transitions. Other important features of the
MELP system are lumped into the block labeled "Enhancing Filters."
This represents adaptive spectrum enhancement filters used to enhance
formant regions,
and a spectrally flat "pulse dispersion filter" whose
Fig. 7.12 Mixed-excitation linear predictive (MELP) decoder.
Devoicing is evident in frame 9 in Figure 5.6.
Such filters are also used routinely in CELP coders and are often referred to as post
132 Digital Speech Coding
purpose is to reduce "peakiness" due to the minimum-phase nature of
the linear predictive vocal tract system.
These modifications to the basic two-state excitation LPC vocoder
produce marked improvements in the quality of reproduction of the
speech signal. Several new parameters of the excitation must be esti-
mated at analysis time and coded for transmission, but these add only
slightly to either the analysis computation or the bit rate [80]. The
MELP coder is said to produce speech quality at 2400 bps that is com-
parable to CELP coding at 4800 bps. In fact, its superior performance
led to a new Department of Defense standard in 1996 and subsequently
to MIL-STD-3005 and NATO STANAG 4591, which operates at 2400,
1200, and 600 bps.
7.5 Frequency-Domain Coders
Although we have discussed a wide variety of digital speech coding
systems, we have really only focused on a few of the most important
examples that have current relevance. There is much room for varia-
tion within the general frameworks that we have identified. There is
one more class of coders that should be discussed. These are called
frequency-domain coders because they are based on the principle
of decomposing the speech signal into individual frequency bands.
Figure 7.13 depicts the general nature of this large class of coding
systems. On the far left of the diagram is a set (bank) of analysis band-
pass filters. Each of these is followed by a down-sampler by a factor
appropriate for the bandwidth of its bandpass input. On the far right
in the diagram is a set of "bandpass interpolators" where each interpo-
lator is composed of an up-sampler (i.e., raising the sampling rate by
the same factor as the corresponding down-sampling box) followed by
a bandpass filter similar to the analysis filter. If the filters are carefully
designed, and the outputs of the down-samplers are connected to the
corresponding inputs of the bandpass interpolators, then it is possi-
ble for the output ˆ x[n] to be virtually identical to the input x[n] [24].
Such perfect reconstruction filter bank systems are the basis for a wide-
ranging class of speech and audio coders called subband coders [25].
When the outputs of the filter bank are quantized by some sort of
quantizer, the output ˆ x[n] is not equal to the input, but by careful
7.5 Frequency-Domain Coders 133
Fig. 7.13 Subband coder and decoder for speech audio.
design of the quantizer the output can be indistinguishable from the
input perceptually. The goal with such coders is, of course, for the total
composite bit rate
to be as low as possible while maintaining high
In contrast to the coders that we have already discussed, which
incorporated the speech production model into the quantization pro-
cess, the filter bank structure incorporates important features of the
speech perception model. As discussed in Chapter 3, the basilar mem-
brane effectively performs a frequency analysis of the sound impinging
on the eardrum. The coupling between points on the basilar membrane
results in the masking effects that we mentioned in Chapter 3. This
masking was incorporated in some of the coding systems discussed so
far, but only in a rudimentary manner.
To see how subband coders work, suppose first that the individ-
ual channels are quantized independently. The quantizers can be any
quantization operator that preserves the waveform of the signal. Typ-
ically, adaptive PCM is used, but ADPCM could be used in princi-
ple. Because the down-sampled channel signals are full band at the
lower sampling rate, the quantization error affects all frequencies in
The total bit rate will be the sum of the products of the sampling rates of the down-
sampled channel signals times the number of bits allocated to each channel.
134 Digital Speech Coding
that band, but the important point is that no other bands are affected
by the errors in a given band. Just as important, a particular band
can be quantized according to its perceptual importance. Further-
more, bands with low energy can be encoded with correspondingly low
absolute error.
The simplest approach is to pre-assign a fixed number of bits to
each channel. In an early non-uniformly spaced five channel system,
Crochiere et al. [25] found that subband coders operating at 16 kbps
and 9.6 kbps were preferred by listeners to ADPCM coders operating
at 22 and 19 kbps, respectively.
Much more sophisticated quantization schemes are possible in the
configuration shown in Figure 7.13 if, as suggested by the dotted boxes,
the quantization of the channels is done jointly. For example, if the
channel bandwidths are all the same so that the output samples of the
down-samplers can be treated as a vector, vector quantization can be
used effectively [23].
Another approach is to allocate bits among the channels dynami-
cally according to a perceptual criterion. This is the basis for modern
audio coding standards such as the various MPEG standards. Such sys-
tems are based on the principles depicted in Figure 7.13. What is not
shown in that figure is additional analysis processing that is done to
determine how to allocate the bits among the channels. This typically
involves the computation of a fine-grained spectrum analysis using the
DFT. From this spectrum it is possible to determine which frequen-
cies will mask other frequencies, and on the basis of this a threshold of
audibility is determined. Using this audibility threshold, bits are allo-
cated among the channels in an iterative process so that as much of the
quantization error is inaudible as possible for the total bit budget. The
details of perceptual audio coding are presented in the recent textbook
by Spanias [120].
7.6 Evaluation of Coders
In this chapter, we have discussed a wide range of options for digital
coding of speech signals. These systems rely heavily on the techniques of
linear prediction, filter banks, and cepstrum analysis and also on models
7.6 Evaluation of Coders 135
for speech production and speech perception. All this knowledge and
technique is brought to bear on the problem of reducing the bit rate of
the speech representation while maintaining high quality reproduction
of the speech signal.
Throughout this chapter we have mentioned bit rates, complexity
of computation, processing delay, and quality as important practical
dimensions of the speech coding problem. Most coding schemes have
many operations and parameters that can be chosen to tradeoff among
the important factors. In general, increasing the bit rate will improve
quality of reproduction; however, it is important to note that for many
systems, increasing the bit rate indefinitely does not necessarily con-
tinue to improve quality. For example, increasing the bit rate of an
open-loop two-state-excitation LPC vocoder above about 2400 bps does
not improve quality very much, but lowering the bit rate causes notice-
able degradation. On the other hand, improved pitch detection and
source modeling can improve quality in an LPC vocoder as witnessed
by the success of the MELP system, but this generally would come
with an increase in computation and processing delay.
In the final analysis, the choice of a speech coder will depend on
constraints imposed by the application such as cost of coder/decoder,
available transmission capacity, robustness to transmission errors,
and quality requirements. Fortunately, there are many possibilities to
choose from.
Text-to-Speech Synthesis Methods
In this chapter, we discuss systems whose goal is to convert ordinary
text messages into intelligible and natural sounding synthetic speech
so as to transmit information from a machine to a human user. In
other words, we will be concerned with computer simulation of the
upper part of the speech chain in Figure 1.2. Such systems are often
referred to as text-to-speech synthesis (or TTS) systems, and their gen-
eral structure is illustrated in Figure 8.1. The input to the TTS sys-
tem is text and the output is synthetic speech. The two fundamental
processes performed by all TTS systems are text analysis (to deter-
mine the abstract underlying linguistic description of the speech) and
speech synthesis (to produce the speech sounds corresponding to the
text input).
Fig. 8.1 Block diagram of general TTS system.
8.1 Text Analysis 137
8.1 Text Analysis
The text analysis module of Figure 8.1 must determine three things
from the input text string, namely:
(1) pronunciation of the text string: the text analysis pro-
cess must decide on the set of phonemes that is to be spoken,
the degree of stress at various points in speaking, the into-
nation of the speech, and the duration of each of the sounds
in the utterance
(2) syntactic structure of the sentence to be spoken: the
text analysis process must determine where to place pauses,
what rate of speaking is most appropriate for the material
being spoken, and how much emphasis should be given to
individual words and phrases within the final spoken output
(3) semantic focus and ambiguity resolution: the text anal-
ysis process must resolve homographs (words that are spelled
alike but can be pronounced differently, depending on con-
text), and also must use rules to determine word etymology
to decide on how best to pronounce names and foreign words
and phrases.
Figure 8.2 shows more detail on how text analysis is performed.
The input to the analysis is plain English text. The first stage of pro-
cessing does some basic text processing operations, including detecting
the structure of the document containing the text (e.g., email message
versus paragraph of text from an encyclopedia article), normalizing the
text (so as to determine how to pronounce words like proper names or
homographs with multiple pronunciations), and finally performing a
linguistic analysis to determine grammatical information about words
and phrases within the text. The basic text processing benefits from
an online dictionary of word pronunciations along with rules for deter-
mining word etymology. The output of the basic text processing step
is tagged text, where the tags denote the linguistic properties of the
words of the input text string.
138 Text-to-Speech Synthesis Methods
Fig. 8.2 Components of the text analysis process.
8.1.1 Document Structure Detection
The document structure detection module seeks to determine the loca-
tion of all punctuation marks in the text, and to decide their significance
with regard to the sentence and paragraph structure of the input text.
For example, an end of sentence marker is usually a period, ., a ques-
tion mark, ?, or an exclamation point, !. However this is not always the
case as in the sentence, "This car is 72.5 in. long" where there are two
periods, neither of which denote the end of the sentence.
8.1.2 Text Normalization
Text normalization methods handle a range of text problems that occur
in real applications of TTS systems, including how to handle abbrevi-
ations and acronyms as in the following sentences:
Example 1: "I live on Bourbon St. in St. Louis"
Example 2: "She worked for DEC in Maynard MA"
8.1 Text Analysis 139
where, in Example 1, the text "St." is pronounced as street or saint,
depending on the context, and in Example 2 the acronym DEC can
be pronounced as either the word "deck" (the spoken acronym) or
the name of the company, i.e., Digital Equipment Corporation, but is
virtually never pronounced as the letter sequence "D E C."
Other examples of text normalization include number strings like
"1920" which can be pronounced as the year "nineteen twenty" or the
number "one thousand, nine hundred, and twenty," and dates, times
currency, account numbers, etc. Thus the string "$10.50" should be
pronounced as "ten dollars and fifty cents" rather than as a sequence
of characters.
One other important text normalization problem concerns the pro-
nunciation of proper names, especially those from languages other than
8.1.3 Linguistic Analysis
The third step in the basic text processing block of Figure 8.2 is a
linguistic analysis of the input text, with the goal of determining, for
each word in the printed string, the following linguistic properties:
• the part of speech (POS) of the word
• the sense in which each word is used in the current context
• the location of phrases (or phrase groups) with a sentence (or
paragraph), i.e., where a pause in speaking might be appro-
• the presence of anaphora (e.g., the use of a pronoun to refer
back to another word unit)
• the word (or words) on which emphasis are to be placed, for
prominence in the sentence
• the style of speaking, e.g., irate, emotional, relaxed, etc.
A conventional parser could be used as the basis of the linguistic anal-
ysis of the printed text, but typically a simple, shallow analysis is per-
formed since most linguistic parsers are very slow.
140 Text-to-Speech Synthesis Methods
8.1.4 Phonetic Analysis
Ultimately, the tagged text obtained from the basic text processing
block of a TTS system has to be converted to a sequence of tagged
phones which describe both the sounds to be produced as well as the
manner of speaking, both locally (emphasis) and globally (speaking
style). The phonetic analysis block of Figure 8.2 provides the process-
ing that enables the TTS system to perform this conversion, with the
help of a pronunciation dictionary. The way in which these steps are
performed is as follows.
8.1.5 Homograph Disambiguation
The homograph disambiguation operation must resolve the correct pro-
nunciation of each word in the input string that has more than one pro-
nunciation. The basis for this is the context in which the word occurs.
One simple example of homograph disambiguation is seen in the phrases
"an absent boy" versus the sentence "do you choose to absent yourself."
In the first phrase the word "absent" is an adjective and the accent is
on the first syllable; in the second phrase the word "absent" is a verb
and the accent is on the second syllable.
8.1.6 Letter-to-Sound (LTS) Conversion
The second step of phonetic analysis is the process of grapheme-to-
phoneme conversion, namely conversion from the text to (marked)
speech sounds. Although there are a variety of ways of performing this
analysis, perhaps the most straightforward method is to rely on a stan-
dard pronunciation dictionary, along with a set of letter-to-sound rules
for words outside the dictionary.
Figure 8.3 shows the processing for a simple dictionary search for
word pronunciation. Each individual word in the text string is searched
independently. First a "whole word" search is initiated to see if the
printed word exists, in its entirety, in the word dictionary. If so, the
conversion to sounds is straightforward and the dictionary search begins
on the next word. If not, as is the case most often, the dictionary search
attempts to find affixes (both prefixes and suffixes) and strips them
8.1 Text Analysis 141
Fig. 8.3 Block diagram of dictionary search for proper word pronunciation.
from the word attempting to find the "root form" of the word, and
then does another "whole word" search. If the root form is not present
in the dictionary, a set of letter-to-sound rules is used to determine the
best pronunciation (usually based on etymology of the word) of the
root form of the word, again followed by the reattachment of stripped
out affixes (including the case of no stripped out affixes).
8.1.7 Prosodic Analysis
The last step in the text analysis system of Figure 8.2 is prosodic anal-
ysis which provides the speech synthesizer with the complete set of
synthesis controls, namely the sequence of speech sounds, their dura-
tions, and an associated pitch contour (variation of fundamental fre-
quency with time). The determination of the sequence of speech sounds
is mainly performed by the phonetic analysis step as outlined above.
The assignment of duration and pitch contours is done by a set of pitch
142 Text-to-Speech Synthesis Methods
and duration rules, along with a set of rules for assigning stress and
determining where appropriate pauses should be inserted so that the
local and global speaking rates appear to be natural.
8.2 Evolution of Speech Synthesis Systems
A summary of the progress in speech synthesis, over the period 1962–
1997, is given in Figure 8.4. This figure shows that there have been
3 generations of speech synthesis systems. During the first genera-
tion (between 1962 and 1977) formant synthesis of phonemes using
a terminal analog synthesizer was the dominant technology using
rules which related the phonetic decomposition of the sentence to
formant frequency contours. The synthesis suffered from poor intel-
ligibility and poor naturalness. The second generation of speech syn-
thesis methods (from 1977 to 1992) was based primarily on an LPC
representation of sub-word units such as diphones (half phones). By
carefully modeling and representing diphone units via LPC param-
eters, it was shown that good intelligibility synthetic speech could
be reliably obtained from text input by concatenating the appropri-
ate diphone units. Although the intelligibility improved dramatically
Fig. 8.4 Time line of progress in speech synthesis and TTS systems.
8.2 Evolution of Speech Synthesis Systems 143
over first generation formant synthesis, the naturalness of the syn-
thetic speech remained low due to the inability of single diphone units
to represent all possible combinations of sound using that diphone
unit. A detailed survey of progress in text-to-speech conversion up
to 1987 is given in the review paper by Klatt [65]. The synthesis
examples that accompanied that paper are available for listening at
The third generation of speech synthesis technology was the period
from 1992 to the present, in which the method of "unit selection syn-
thesis" was introduced and perfected, primarily by Sagisaka at ATR
Labs in Kyoto [108]. The resulting synthetic speech from this third
generation technology had good intelligibility and naturalness that
approached that of human-generated speech. We begin our discussion
of speech synthesis approaches with a review of early systems, and then
discuss unit selection methods of speech synthesis later in this chapter.
8.2.1 Early Speech Synthesis Approaches
Once the abstract underlying linguistic description of the text input has
been determined via the steps of Figure 8.2, the remaining (major) task
of TTS systems is to synthesize a speech waveform whose intelligibility
is very high (to make the speech useful as a means of communication
between a machine and a human), and whose naturalness is as close to
real speech as possible. Both tasks, namely attaining high intelligibility
along with reasonable naturalness, are difficult to achieve and depend
critically on three issues in the processing of the speech synthesizer
"backend" of Figure 8.1, namely:
(1) choice of synthesis units: including whole words, phones,
diphones, dyads, or syllables
(2) choice of synthesis parameters: including LPC features,
formants, waveform templates, articulatory parameters, sinu-
soidal parameters, etc.
(3) method of computation: including rule-based systems or
systems which rely on the concatenation of stored speech
144 Text-to-Speech Synthesis Methods
8.2.2 Word Concatenation Synthesis
Perhaps the simplest approach to creating a speech utterance corre-
sponding to a given text string is to literally splice together prerecorded
words corresponding to the desired utterance. For greatest simplicity,
the words can be stored as sampled waveforms and simply concatenated
in the correct sequence. This approach generally produces intelligible,
but unnatural sounding speech, since it does not take into account the
"co-articulation" effects of producing phonemes in continuous speech,
and it does not provide either for the adjustment of phoneme dura-
tions or the imposition of a desired pitch variation across the utter-
ance. Words spoken in continuous speech sentences are generally much
shorter in duration than when spoken in isolation (often up to 50%
shorter) as illustrated in Figure 8.5. This figure shows wideband spec-
trograms for the sentence "This shirt is red" spoken as a sequence of
isolated words (with short, distinct pauses between words) as shown
at the top of Figure 8.5, and as a continuous utterance, as shown at
the bottom of Figure 8.5. It can be seen that, even for this trivial
Fig. 8.5 Wideband spectrograms of a sentence spoken as a sequence of isolated words (top
panel) and as a continuous speech utterance (bottom panel).
8.2 Evolution of Speech Synthesis Systems 145
example, the duration of the continuous sentence is on the order of half
that of the isolated word version, and further the formant tracks of the
continuous sentence do not look like a set of uniformly compressed for-
mant tracks from the individual words. Furthermore, the boundaries
between words in the upper plot are sharply defined, while they are
merged in the lower plot.
The word concatenation approach can be made more sophisticated
by storing the vocabulary words in a parametric form (formants, LPC
parameters, etc.) such as employed in the speech coders discussed in
Chapter 7 [102]. The rational for this is that the parametric represen-
tations, being more closely related to the model for speech produc-
tion, can be manipulated so as to blend the words together, shorten
them, and impose a desired pitch variation. This requires that the con-
trol parameters for all the words in the task vocabulary (as obtained
from a training set of words) be stored as representations of the words.
A special set of word concatenation rules is then used to create the
control signals for the synthesizer. Although such a synthesis system
would appear to be an attractive alternative for general purpose syn-
thesis of speech, in reality this type of synthesis is not a practical
There are many reasons for this, but a major problem is that there
are far too many words to store in a word catalog for word concatena-
tion synthesis to be practical except in highly restricted situations. For
example, there are about 1.7 million distinct surnames in the United
States and each of them would have to be spoken and stored for a gen-
eral word concatenation synthesis method. A second, equally important
limitation is that word-length segments of speech are simply the wrong
sub-units. As we will see, shorter units such as phonemes or diphones
are more suitable for synthesis.
Efforts to overcome the limitations of word concatenation followed
two separate paths. One approach was based on controlling the motions
of a physical model of the speech articulators based on the sequence of
phonemes from the text analysis. This requires sophisticated control
It should be obvious that whole word concatenation synthesis (from stored waveforms) is
also impractical for general purpose synthesis of speech.
146 Text-to-Speech Synthesis Methods
rules that are mostly derived empirically. From the vocal tract shapes
and sources produced by the articulatory model, control parameters
(e.g., formants and pitch) can be derived by applying the acoustic theory
of speech production and then used to control a synthesizer such those
used for speech coding. An alternative approach eschews the articulatory
model and proceeds directly to the computation of the control signals
for a source/system model (e.g., formant parameters, LPC parameters,
pitch period, etc.). Again, the rules (algorithm) for computing the control
parameters are mostly derived by empirical means.
8.2.3 Articulatory Methods of Synthesis
Articulatory models of human speech are based upon the application of
acoustic theory to physical models such as depicted in highly stylized
form in Figure 2.1 [22]. Such models were thought to be inherently
more natural than vocal tract analog models since:
• we could impose known and fairly well understood phys-
ical constraints on articular movements to create realistic
motions of the tongue, jaw, teeth, velum, etc.
• we could use X-ray data (MRI data today) to study the
motion of the articulators in the production of individual
speech sounds, thereby increasing our understanding of the
dynamics of speech production
• we could model smooth articulatory parameter motions
between sounds, either via direct methods (namely solving
the wave equation), or indirectly by converting articulatory
shapes to formants or LPC parameters
• we could highly constrain the motions of the articu-
latory parameters so that only natural motions would
occur, thereby potentially making the speech more natural
What we have learned about articulatory modeling of speech is
that it requires a highly accurate model of the vocal cords and of the
vocal tract for the resulting synthetic speech quality to be considered
acceptable. It further requires rules for handling the dynamics of the
8.2 Evolution of Speech Synthesis Systems 147
articulator motion in the context of the sounds being produced. So
far we have been unable to learn all such rules, and thus, articula-
tory speech synthesis methods have not been found to be practical for
synthesizing speech of acceptable quality.
8.2.4 Terminal Analog Synthesis of Speech
The alternative to articulatory synthesis is called terminal analog speech
synthesis. In this approach, each sound of the language (phoneme) is
characterized by a source excitation function and an ideal vocal tract
model. Speech is produced by varying (in time) the excitation and the
vocal tract model control parameters at a rate commensurate with the
sounds being produced. This synthesis process has been called terminal
analog synthesis because it is based on a model (analog) of the human
vocal tract production of speech that seeks to produce a signal at its
output terminals that is equivalent to the signal produced by a human
The basis for terminal analog synthesis of speech is the
source/system model of speech production that we have used many
times in this text; namely an excitation source, e[n], and a transfer
function of the human vocal tract in the form of a rational system
function, i.e.,
H(z) =
1 −
, (8.1)
where S(z) is the z-transform of the output speech signal, s[n],
E(z) is the z-transform of the vocal tract excitation signal, e[n],
and ¦b
¦ = ¦b
, b
, b
, . . . , b
¦ and ¦a
¦ = ¦a
, a
, . . . , a
¦ are the (time-
varying) coefficients of the vocal tract filter. (See Figures 2.2 and 4.1.)
In the designation "terminal analog synthesis", the terms analog and terminal result from
the historical context of early speech synthesis studies. This can be confusing since today,
"analog" implies "not digital" as well as an analogous thing. "Terminal" originally implied
the "output terminals" of an electronic analog (not digital) circuit or system that was an
analog of the human speech production system.
148 Text-to-Speech Synthesis Methods
For most practical speech synthesis systems, both the excitation signal
properties (P
and V/UV ) and the filter coefficients of (8.1) change
periodically so as to synthesize different phonemes.
The vocal tract representation of (8.1) can be implemented as a
speech synthesis system using a direct form implementation. However,
it has been shown that it is preferable to factor the numerator and
denominator polynomials into either a series of cascade (serial) reso-
nances, or into a parallel combination of resonances. It has also been
shown that an all-pole model (B(z) =constant) is most appropriate for
voiced (non-nasal) speech. For unvoiced speech a simpler model (with
one complex pole and one complex zero), implemented via a paral-
lel branch is adequate. Finally a fixed spectral compensation network
can be used to model the combined effects of glottal pulse shape and
radiation characteristics from the lips and mouth, based on two real
poles in the z-plane. A complete serial terminal analog speech syn-
thesis model, based on the above discussion, is shown in Figure 8.6
[95, 111].
The voiced speech (upper) branch includes an impulse generator
(controlled by a time-varying pitch period, P
), a time-varying voiced
signal gain, A
, and an all-pole discrete-time system that consists
of a cascade of 3 time-varying resonances (the first three formants,
), and one fixed resonance F
Fig. 8.6 Speech synthesizer based on a cascade/serial (formant) synthesis model.
8.3 Unit Selection Methods 149
The unvoiced speech (lower) branch includes a white noise
generator, a time-varying unvoiced signal gain, A
, and a reso-
nance/antiresonance system consisting of a time-varying pole (F
) and
a time-varying zero (F
The voiced and unvoiced components are added and processed by
the fixed spectral compensation network to provide the final synthetic
speech output.
The resulting quality of the synthetic speech produced using a ter-
minal analog synthesizer of the type shown in Figure 8.6 is highly vari-
able with explicit model shortcomings due to the following:
• voiced fricatives are not handled properly since their mixed
excitation is not part of the model of Figure 8.6
• nasal sounds are not handled properly since nasal zeros are
not included in the model
• stop consonants are not handled properly since there is no
precise timing and control of the complex excitation signal
• use of a fixed pitch pulse shape, independent of the pitch
period, is inadequate and produces buzzy sounding voiced
• the spectral compensation model is inaccurate and does not
work well for unvoiced sounds.
Many of the short comings of the model of Figure 8.6 are alleviated by
a more complex model proposed by Klatt [64]. However, even with a
more sophisticated synthesis model, it remains a very challenging task
to compute the synthesis parameters. Nevertheless Klatt's Klattalk
system achieved adequate quality by 1983 to justify commercializa-
tion by the Digital Equipment Corporation as the DECtalk system.
Some DECtalk systems are still in operation today as legacy systems,
although the unit selection methods to be discussed in Section 8.3 now
provide superior quality in current applications.
8.3 Unit Selection Methods
The key idea of a concatenative TTS system, using unit selection
methods, is to use synthesis segments that are sections of prerecorded
150 Text-to-Speech Synthesis Methods
natural speech [31, 50, 108]. The word concatenation method discussed
in Section 8.2.2 is perhaps the simplest embodiment of this idea; how-
ever, as we discussed, shorter segments are required to achieve better
quality synthesis. The basic idea is that the more segments recorded,
annotated, and saved in the database, the better the potential quality
of the resulting speech synthesis. Ultimately, if an infinite number of
segments were recorded and saved, the resulting synthetic speech would
sound natural for virtually all possible synthesis tasks. Concatenative
speech synthesis systems, based on unit selection methods, are what is
conventionally known as "data driven" approaches since their perfor-
mance tends to get better the more data that is used for training the
system and selecting appropriate units.
In order to design and build a unit selection system based on using
recorded speech segments, several issues have to be resolved, including
the following:
(1) What speech units should be used as the basic synthesis
building blocks?
(2) How the synthesis units are selected (extracted) from natural
speech utterances?
(3) How the units are labeled for retrieval from a large database
of units?
(4) What signal representation should be used to represent the
units for storage and reproduction purposes?
(5) What signal processing methods can be used to spectrally
smooth the units (at unit junctures) and for prosody modi-
fication (pitch, duration, amplitude)?
We now attempt to answer each of these questions.
8.3.1 Choice of Concatenation Units
The units for unit selection can (in theory) be as large as words and
as small as phoneme units. Words are prohibitive since there are essen-
tially an infinite number of words in English. Subword units include
syllables (about 10,000 in English), phonemes (about 45 in English,
but they are highly context dependent), demi-syllables (about 2500
8.3 Unit Selection Methods 151
in English), and diphones (about 1500–2500 in English). The ideal
synthesis unit is context independent and easily concatenates with
other (appropriate) subword units [15]. Based on this criterion, the
most reasonable choice for unit selection synthesis is the set of diphone
Before any synthesis can be done, it is necessary to prepare an inven-
tory of units (diphones). This requires significant effort if done manu-
ally. In any case, a large corpus of speech must be obtained from which
to extract the diphone units as waveform snippets. These units are then
represented in some compressed form for efficient storage. High quality
coding such as MPLP or CELP is used to limit compression artifacts.
At the final synthesis stage, the diphones are decoded into waveforms
for final merging, duration adjustment and pitch modification, all of
which take place on the time waveform.
8.3.2 From Text to Diphones
The synthesis procedure from subword units is straightforward, but far
from trivial [14]. Following the process of text analysis into phonemes
and prosody, the phoneme sequence is first converted to the appropriate
sequence of units from the inventory. For example, the phrase "I want"
would be converted to diphone units as follows:
Text Input: I want.
Phonemes: /#/ AY/ /W/ /AA/ /N/ /T/ /#/
Diphones: /# AY/ /AY-W/ /W-AA/ /AA-N/ /N-T/ /T- #/,
where the symbol # is used to represent silence (at the beginning and
end of each sentence or phrase).
The second step in online synthesis is to select the most appropri-
ate sequence of diphone units from the stored inventory. Since each
diphone unit occurs many times in the stored inventory, the selection
A diphone is simply the concatenation of two phonemes that are allowed by the language
constraints to be sequentially contiguous in natural speech. For example, a diphone based
upon the phonemes AY and W would be denoted AY−W where the − denotes the joining
of the two phonemes.
152 Text-to-Speech Synthesis Methods
of the best sequence of diphone units involves solving a dynamic pro-
gramming search for the sequence of units that minimizes a specified
cost function. The cost function generally is based on diphone matches
at each of the boundaries between diphones, where the diphone match
is defined in terms of spectral matching characteristics, pitch matching
characteristics, and possibly phase matching characteristics.
8.3.3 Unit Selection Synthesis
The "Unit Selection" problem is basically one of having a given set of
target features corresponding to the spoken text, and then automat-
ically finding the sequence of units in the database that most closely
match these features. This problem is illustrated in Figure 8.7 which
shows a target feature set corresponding to the sequence of sounds
(phonemes for this trivial example) /HH/ /EH/ /L/ /OW/ from the
word /hello/.
As shown, each of the phonemes in this word have mul-
tiple representations (units) in the inventory of sounds, having been
extracted from different phonemic environments. Hence there are many
versions of /HH/ and many versions of /EH/ etc., as illustrated in
Figure 8.7. The task of the unit selection module is to choose one of
each of the multiple representations of the sounds, with the goal being
to minimize the total perceptual distance between segments of the cho-
sen sequence, based on spectral, pitch and phase differences throughout
the sounds and especially at the boundary between sounds. By spec-
ifying costs associated with each unit, both globally across the unit,
"Trained" Perceptual
Fig. 8.7 Illustration of basic process of unit selection. (After Dutoit [31].)
If diphones were used, the sequence would be /HH-EH/ /EH-L/ /L-OW/.
8.3 Unit Selection Methods 153
Fig. 8.8 Online unit selection based on a Viterbi search through a lattice of alternatives.
and locally at the unit boundaries with adjacent units, we can find
the sequence of units that best "join each other" in the sense of min-
imizing the accumulated distance across the sequence of units. This
optimal sequence (or equivalently the optimal path through the combi-
nation of all possible versions of each unit in the string) can be found
using a Viterbi search (dynamic programming) [38, 99]. This dynamic
programming search process is illustrated in Figure 8.8 for a 3 unit
search. The Viterbi search effectively computes the cost of every pos-
sible path through the lattice and determines the path with the low-
est total cost, where the transitional costs (the arcs) reflect the cost
of concatenation of a pair of units based on acoustic distances, and
the nodes represent the target costs based on the linguistic identity of
the unit.
Thus there are two costs associated with the Viterbi search, namely
a nodal cost based on the unit segmental distortion (USD) which is
defined as the difference between the desired spectral pattern of the
target (suitably defined) and that of the candidate unit, throughout
the unit, and the unit concatenative distortion (UCD) which is defined
as the spectral (and/or pitch and/or phase) discontinuity across the
boundaries of the concatenated units. By way of example, consider
a target context of the word "cart" with target phonemes /K/ /AH/
/R/ /T/, and a source context of the phoneme /AH/ obtained from the
154 Text-to-Speech Synthesis Methods
Fig. 8.9 Illustration of unit selection costs associated with a string of target units and a
presumptive string of selected units.
source word "want" with source phonemes /W/ /AH/ /N/ /T/. The
USD distance would be the cost (specified analytically) between the
sound /AH/ in the context /W/ /AH/ /N/ versus the desired sound
in the context /K/ /AH/ /R/.
Figure 8.9 illustrates concatenative synthesis for a given string of
target units (at the bottom of the figure) and a string of selected units
from the unit inventory (shown at the top of the figure). We focus
our attention on Target Unit t
and Selected Units θ
and θ
. Asso-
ciated with the match between units t
and θ
is a USD Unit Cost
and associated with the sequence of selected units, θ
and θ
, is a
UCD concatenation cost. The total cost of an arbitrary string of N
selected units, Θ = ¦θ
, θ
, . . . , θ
¦, and the string of N target units,
T = ¦t
, t
, . . . , t
¦ is defined as:
d(Θ, T) =
, t
) +
, θ
), (8.2)
where d
, t
) is the USD cost associated with matching target unit t
with selected unit θ
and d
, θ
) is the UCD cost associated with
concatenating the units θ
and θ
. It should be noted that when
selected units θ
and θ
come from the same source sentence and
are adjacent units, the UCD cost goes to zero, as this is as natural a
concatenation as can exist in the database. Further, it is noted that
generally there is a small overlap region between concatenated units.
The optimal path (corresponding to the optimal sequence of units) can
be efficiently computed using a standard Viterbi search ([38, 99]) in
8.3 Unit Selection Methods 155
which the computational demand scales linearly with both the number
of target and the number of concatenation units.
The concatenation cost between two units is essentially the spectral
(and/or phase and/or pitch) discontinuity across the boundary between
the units, and is defined as:
, θ
) =
, θ
), (8.3)
where p is the size of the spectral feature vector (typically p = 12 mfcc
coefficients (mel-frequency cepstral coefficients as explained in Sec-
tion 5.6.3), often represented as a VQ codebook vector), and the extra
two features are log power and pitch. The weights, w
are chosen dur-
ing the unit selection inventory creation phase and are optimized using
a trial-and-error procedure. The concatenation cost essentially mea-
sures a spectral plus log energy plus pitch difference between the two
concatenated units at the boundary frames. Clearly the definition of
concatenation cost could be extended to more than a single boundary
frame. Also, as stated earlier, the concatenation cost is defined to be
zero (d
, θ
) = 0) whenever units θ
and θ
are consecutive (in
the database) since, by definition, there is no discontinuity in either
spectrum or pitch in this case. Although there are a variety of choices
for measuring the spectral/log energy/pitch discontinuity at the bound-
ary, a common cost function is the normalized mean-squared error in
the feature parameters, namely:
, θ
) =
(m) − f
, (8.4)
where f
(l) is the kth feature parameter of the lth frame of segment θ
m is the (normalized) duration of each segment, and σ
is the variance
of the kth feature vector component.
The USD or target costs are conceptually more difficult to under-
stand, and, in practice, more difficult to instantiate. The USD cost
associated with units θ
and t
is of the form:
, t
) =
, (8.5)
156 Text-to-Speech Synthesis Methods
where q is the number of features that specify the unit θ
or t
, i = 1, 2, . . . , q is a trained set of target weights, and T
() can be
either a continuous function (for a set of features, f
, such as segmental
pitch, power or duration), or a set of integers (in the case of cate-
gorical features, f
, such as unit identity, phonetic class, position in
the syllable from which the unit was extracted, etc.). In the latter
case, φ
can be looked up in a table of distances. Otherwise, the local
distance function can be expressed as a simple quadratic distance of
the form
. (8.6)
The training of the weights, w
, is done off-line. For each phoneme
in each phonetic class in the training speech database (which might be
the entire recorded inventory), each exemplar of each unit is treated
as a target and all others are treated as candidate units. Using this
training set, a least-squares system of linear equations can be derived
from which the weight vector can be solved. The details of the weight
training methods are described by Schroeter [114].
The final step in the unit selection synthesis, having chosen the opti-
mal sequence of units to match the target sentence, is to smooth/modify
the selected units at each of the boundary frames to better match
the spectra, pitch and phase at each unit boundary. Various smooth-
ing methods based on the concepts of time domain harmonic scaling
(TDHS) [76] and pitch synchronous overlap add (PSOLA) [20, 31, 82]
have been proposed and optimized for such smoothing/modification at
the boundaries between adjacent diphone units.
8.4 TTS Applications
Speech technology serves as a way of intelligently and efficiently
enabling humans to interact with machines with the goal of bring-
ing down cost of service for an existing service capability, or providing
new products and services that would be prohibitively expensive with-
out the automation provided by a viable speech processing interactive
8.5 TTS Future Needs 157
system. Examples of existing services where TTS enables significant
cost reduction are as follows:
• a dialog component for customer care applications
• a means for delivering text messages over an audio connection
(e.g., cellphone)
• a means of replacing expensive recorded Interactive Voice
Response system prompts (a service which is particularly
valuable when the prompts change often during the course
of the day, e.g., stock price quotations).
Similarly some examples of new products and services which are
enabled by a viable TTS technology are the following:
• location-based services, e.g., alerting you to locations of
restaurants, gas stations, stores in the vicinity of your current
• providing information in cars (e.g., driving directions, traffic
• unified messaging (e.g., reading e-mail and fax messages)
• voice portal providing voice access to web-based services
• e-commerce agents
• customized News, Stock Reports, Sports scores, etc.
• giving voice to small and embedded devices for reporting
information and alerts.
8.5 TTS Future Needs
Modern TTS systems have the capability of producing highly intelligi-
ble, and surprisingly natural speech utterances, so long as the utterance
is not too long, or too syntactically complicated, or too technical. The
biggest problem with most TTS systems is that they have no idea as
to how things should be said, but instead rely on the text analysis for
emphasis, prosody, phrasing and all so-called suprasegmental features
of the spoken utterance. The more TTS systems learn how to produce
context-sensitive pronunciations of words (and phrases), the more nat-
ural sounding these systems will become. Hence, by way of example,
158 Text-to-Speech Synthesis Methods
the utterance "I gave the book to John" has at least three different
semantic interpretations, each with different emphasis on words in the
utterance, i.e.,
I gave the book to John, i.e., not to Mary or Bob.
I gave the book to John, i.e., not the photos or the
I gave the book to John, i.e., I did it, not someone else.
The second future need of TTS systems is improvements in the unit
selection process so as to better capture the target cost for mismatch
between predicted unit specification (i.e., phoneme name, duration,
pitch, spectral properties) and actual features of a candidate recorded
unit. Also needed in the unit selection process is a better spectral dis-
tance measure that incorporates human perception measures so as to
find the best sequence of units for a given utterance.
Finally, better signal processing would enable improved compression
of the units database, thereby making the footprint of TTS systems
small enough to be usable in handheld and mobile devices.
Automatic Speech Recognition (ASR)
In this chapter, we examine the process of speech recognition by
machine, which is in essence the inverse of the text-to-speech problem.
The driving factor behind research in machine recognition of speech
has been the potentially huge payoff of providing services where humans
interact solely with machines, thereby eliminating the cost of live agents
and significantly reducing the cost of providing services. Interestingly,
as a side benefit, this process often provides users with a natural and
convenient way of accessing information and services.
9.1 The Problem of Automatic Speech Recognition
The goal of an ASR system is to accurately and efficiently convert
a speech signal into a text message transcription of the spoken words,
independent of the device used to record the speech (i.e., the transducer
or microphone), the speaker's accent, or the acoustic environment in
which the speaker is located (e.g., quiet office, noisy room, outdoors).
That is, the ultimate goal, which has not yet been achieved, is to per-
form as well as a human listener.
160 Automatic Speech Recognition (ASR)
Fig. 9.1 Conceptual model of speech production and speech recognition processes.
A simple conceptual model of the speech generation and speech
recognition processes is given in Figure 9.1, which is a simplified version
of the speech chain shown in Figure 1.2. It is assumed that the speaker
intends to express some thought as part of a process of conversing
with another human or with a machine. To express that thought, the
speaker must compose a linguistically meaningful sentence, W, in the
form of a sequence of words (possibly with pauses and other acoustic
events such as uh's, um's, er's etc.). Once the words are chosen, the
speaker sends appropriate control signals to the articulatory speech
organs which form a speech utterance whose sounds are those required
to speak the desired sentence, resulting in the speech waveform s[n]. We
refer to the process of creating the speech waveform from the speaker's
intention as the Speaker Model since it reflects the speaker's accent and
choice of words to express a given thought or request. The processing
steps of the Speech Recognizer are shown at the right side of Figure 9.1
and consist of an acoustic processor which analyzes the speech signal
and converts it into a set of acoustic (spectral, temporal) features, X,
which efficiently characterize the speech sounds, followed by a linguistic
decoding process which makes a best (maximum likelihood) estimate of
the words of the spoken sentence, resulting in the recognized sentence
W. This is in essence a digital simulation of the lower part of the speech
chain diagram in Figure 1.2.
Figure 9.2 shows a more detailed block diagram of the overall speech
recognition system. The input speech signal, s[n], is converted to the
sequence of feature vectors, X = ¦x
, x
, . . . , x
¦, by the feature analy-
sis block (also denoted spectral analysis). The feature vectors are com-
puted on a frame-by-frame basis using the techniques discussed in the
earlier chapters. In particular, the mel frequency cepstrum coefficients
9.2 Building a Speech Recognition System 161
Fig. 9.2 Block diagram of an overall speech recognition system.
are widely used to represent the short-time spectral characteristics. The
pattern classification block (also denoted as the decoding and search
block) decodes the sequence of feature vectors into a symbolic repre-
sentation that is the maximum likelihood string,
W that could have
produced the input sequence of feature vectors. The pattern recogni-
tion system uses a set of acoustic models (represented as hidden Markov
models) and a word lexicon to provide the acoustic match score for each
proposed string. Also, an N-gram language model is used to compute
a language model score for each proposed word string. The final block
in the process is a confidence scoring process (also denoted as an utter-
ance verification block), which is used to provide a confidence score for
each individual word in the recognized string. Each of the operations
in Figure 9.2 involves many details and, in some cases, extensive digital
computation. The remainder of this chapter is an attempt to give the
flavor of what is involved in each part of Figure 9.2.
9.2 Building a Speech Recognition System
The steps in building and evaluating a speech recognition system are
the following:
(1) choose the feature set and the associated signal processing
for representing the properties of the speech signal over time
162 Automatic Speech Recognition (ASR)
(2) choose the recognition task, including the recognition word
vocabulary (the lexicon), the basic speech sounds to repre-
sent the vocabulary (the speech units), the task syntax or
language model, and the task semantics (if any)
(3) train the set of speech acoustic and language models
(4) evaluate performance of the resulting speech recognition
Each of these steps may involve many choices and can involve signifi-
cant research and development effort. Some of the important issues are
summarized in this section.
9.2.1 Recognition Feature Set
There is no "standard" set of features for speech recognition. Instead,
various combinations of acoustic, articulatory, and auditory features
have been utilized in a range of speech recognition systems. The most
popular acoustic features have been the (LPC-derived) mel-frequency
cepstrum coefficients and their derivatives.
A block diagram of the signal processing used in most modern
large vocabulary speech recognition systems is shown in Figure 9.3.
The analog speech signal is sampled and quantized at rates between
8000 up to 20,000 samples/s. A first order (highpass) pre-emphasis net-
work (1 − αz
) is used to compensate for the speech spectral falloff
at higher frequencies and approximates the inverse to the mouth trans-
mission frequency response. The pre-emphasized signal is next blocked
into frames of N samples, with adjacent frames spaced M samples
apart. Typical values for N and M correspond to frames of duration
15–40 ms, with frame shifts of 10 ms being most common; hence adja-
cent frames overlap by 5–30 ms depending on the chosen values of N
and M. A Hamming window is applied to each frame prior to spec-
tral analysis using either standard spectral analysis or LPC methods.
Following (optionally used) simple noise removal methods, the spec-
tral coefficients are normalized and converted to mel-frequency cep-
stral coefficients via standard analysis methods of the type discussed
in Chapter 5 [27]. Some type of cepstral bias removal is often used
prior to calculation of the first and second order cepstral derivatives.
9.2 Building a Speech Recognition System 163
Fig. 9.3 Block diagram of feature extraction process for feature vector consisting of mfcc
coefficients and their first and second derivatives.
Typically the resulting feature vector is the set of cepstral coefficients,
and their first- and second-order derivatives. It is typical to use about
13 mfcc coefficients, 13 first-order cepstral derivative coefficients, and
13 second-order derivative cepstral coefficients, making a D = 39 size
feature vector.
9.2.2 The Recognition Task
Recognition tasks vary from simple word and phrase recognition sys-
tems to large vocabulary conversational interfaces to machines. For
example, using a digits vocabulary, the task could be recognition of a
string of digits that forms a telephone number, or identification code,
or a highly constrained sequence of digits that form a password.
9.2.3 Recognition Training
There are two aspects to training models for speech recognition, namely
acoustic model training and language model training. Acoustic model
164 Automatic Speech Recognition (ASR)
training requires recording each of the model units (whole words,
phonemes) in as many contexts as possible so that the statistical
learning method can create accurate distributions for each of the
model states. Acoustic training relies on accurately labeled sequences of
speech utterances which are segmented according to the transcription,
so that training of the acoustic models first involves segmenting the
spoken strings into recognition model units (via either a Baum–Welch
[10, 11] or Viterbi alignment method), and then using the segmented
utterances to simultaneously build model distributions for each state of
the vocabulary unit models. The resulting statistical models form the
basis for the pattern recognition operations at the heart of the ASR
system. As discussed in Section 9.3, concepts such as Viterbi search are
employed in the pattern recognition process as well as in training.
Language model training requires a sequence of text strings that
reflect the syntax of spoken utterances for the task at hand. Generally
such text training sets are created automatically (based on a model of
grammar for the recognition task) or by using existing text sources,
such as magazine and newspaper articles, or closed caption transcripts
of television news broadcasts, etc. Other times, training sets for lan-
guage models can be created from databases, e.g., valid strings of tele-
phone numbers can be created from existing telephone directories.
9.2.4 Testing and Performance Evaluation
In order to improve the performance of any speech recognition system,
there must be a reliable and statistically significant way of evaluating
recognition system performance based on an independent test set of
labeled utterances. Typically we measure word error rate and sentence
(or task) error rate as a measure of recognizer performance. A brief
summary of performance evaluations across a range of ASR applica-
tions is given in Section 9.4.
9.3 The Decision Processes in ASR
The heart of any automatic speech recognition system is the pattern
classification and decision operations. In this section, we shall give a
brief introduction to these important topics.
9.3 The Decision Processes in ASR 165
9.3.1 Mathematical Formulation of the ASR Problem
The problem of automatic speech recognition is represented as a statis-
tical decision problem. Specifically it is formulated as a Bayes maximum
a posteriori probability (MAP) decision process where we seek to find
the word string
W (in the task language) that maximizes the a pos-
teriori probability P(W[X) of that string, given the measured feature
vector, X, i.e.,
W = argmax
Using Bayes' rule we can rewrite (9.1) in the form:
W = argmax
. (9.2)
Equation (9.2) shows that the calculation of the a posteriori probability
is decomposed into two terms, one that defines the a priori probability
of the word sequence, W, namely P(W), and the other that defines the
likelihood that the word string, W, produced the feature vector, X,
namely P(X[W). For all future calculations we disregard the denom-
inator term, P(X), since it is independent of the word sequence W
which is being optimized. The term P(X[W) is known as the "acoustic
model" and is generally denoted as P
(X[W) to emphasize the acous-
tic nature of this term. The term P(W) is known as the "language
model" and is generally denoted as P
(W) to emphasize the linguistic
nature of this term. The probabilities associated with P
(X[W) and
(W) are estimated or learned from a set of training data that have
been labeled by a knowledge source, usually a human expert, where the
training set is as large as reasonably possible. The recognition decoding
process of (9.2) is often written in the form of a 3-step process, i.e.,
W = argmax
. .. .
Step 3
. .. .
Step 1
. .. .
Step 2
, (9.3)
where Step 1 is the computation of the probability associated with the
acoustic model of the speech sounds in the sentence W, Step 2 is the
computation of the probability associated with the linguistic model of
the words in the utterance, and Step 3 is the computation associated
166 Automatic Speech Recognition (ASR)
with the search through all valid sentences in the task language for the
maximum likelihood sentence.
In order to be more explicit about the signal processing and com-
putations associated with each of the three steps of (9.3) we need to
more explicit about the relationship between the feature vector, X, and
the word sequence W. As discussed above, the feature vector, X, is a
sequence of acoustic observations corresponding to each of T frames of
the speech, of the form:
X = ¦x
, x
, . . . , x
¦, (9.4)
where the speech signal duration is T frames (i.e., T times the frame
shift in ms) and each frame, x
, t = 1, 2, . . . , T is an acoustic feature
vector of the form:
= (x
, x
, . . . , x
) (9.5)
that characterizes the spectral/temporal properties of the speech signal
at time t and D is the number of acoustic features in each frame.
Similarly we can express the optimally decoded word sequence, W, as:
W = w
, w
, . . . , w
, (9.6)
where there are assumed to be exactly M words in the decoded string.
9.3.2 The Hidden Markov Model
The most widely used method of building acoustic models (for both
phonemes and words) is the use of a statistical characterization known
as Hidden Markov Models (HMMs) [33, 69, 97, 98]. Figure 9.4 shows
a simple Q = 5-state HMM for modeling a whole word. Each HMM
state is characterized by a mixture density Gaussian distribution that
characterizes the statistical behavior of the feature vectors within the
states of the model [61, 62]. In addition to the statistical feature densi-
ties within states, the HMM is also characterized by an explicit set of
state transitions, a
, which specify the probability of making a tran-
sition from state i to state j at each frame, thereby defining the time
sequence of the feature vectors over the duration of the word. Usually
the self-transitions, a
are large (close to 1.0), and the jump transitions,
, a
, a
, a
, in the model, are small (close to 0).
9.3 The Decision Processes in ASR 167
Fig. 9.4 Word-based, left-to-right, HMM with 5 states.
The complete HMM characterization of a Q-state word model (or a
sub-word unit like a phoneme model) is generally written as λ(A, B, π)
with state transition matrix A = ¦a
, 1 ≤ i, j ≤ Q¦, state observation
probability density, B = ¦b
), 1 ≤ j ≤ Q¦, and initial state distribu-
tion, π = ¦π
, 1 ≤ i ≤ Q¦ with π
set to 1 for the "left-to-right" models
of the type shown in Figure 9.4.
In order to train the HMM (i.e., learn the optimal model param-
eters) for each word (or sub-word) unit, a labeled training set of sen-
tences (transcribed into words and sub-word units) is used to guide
an efficient training procedure known as the Baum–Welch algorithm
[10, 11].
This algorithm aligns each of the various words (or sub-
word units) with the spoken inputs and then estimates the appro-
priate means, covariances and mixture gains for the distributions in
each model state. The Baum–Welch method is a hill-climbing algo-
rithm and is iterated until a stable alignment of models and speech
is obtained. The details of the Baum–Welch procedure are beyond
the scope of this chapter but can be found in several references on
Speech Recognition methods [49, 99]. The heart of the training pro-
cedure for re-estimating HMM model parameters using the Baum–
Welch procedure is shown in Figure 9.5. An initial HMM model is
used to begin the training process. The initial model can be ran-
domly chosen or selected based on a priori knowledge of the model
The Baum–Welch algorithm is also widely referred to as the forward–backward method.
168 Automatic Speech Recognition (ASR)
Fig. 9.5 The Baum–Welch training procedure based on a given training set of utterances.
parameters. The iteration loop is a simple updating procedure for com-
puting the forward and backward model probabilities based on an input
speech database (the training set of utterances) and then optimizing
the model parameters to give an updated HMM. This process is iter-
ated until no further improvement in probabilities occurs with each new
It is a simple matter to go from the HMM for a whole word,
as shown in Figure 9.4, to an HMM for a sub-word unit (such as a
phoneme) as shown in Figure 9.6. This simple 3-state HMM is a basic
sub-word unit model with an initial state representing the statistical
characteristics at the beginning of a sound, a middle state representing
the heart of the sound, and an ending state representing the spec-
tral characteristics at the end of the sound. A word model is made
by concatenating the appropriate sub-word HMMs, as illustrated in
Fig. 9.6 Sub-word-based HMM with 3 states.
9.3 The Decision Processes in ASR 169
Fig. 9.7 Word-based HMM for the word /is/ created by concatenating 3-state subword
models for the sub-word units /ih/ and /z/.
Figure 9.7, which concatenates the 3-state HMM for the sound /IH/
with the 3-state HMM for the sound /Z/, giving the word model for
the word "is" (pronounced as /IH Z/). In general the composition of
a word (from sub-word units) is specified in a word lexicon or dictio-
nary; however once the word model has been built it can be used much
the same as whole word models for training and for evaluating word
strings for maximizing the likelihood as part of the speech recognition
We are now ready to define the procedure for aligning a sequence
of M word models, w
, w
, . . . , w
with a sequence of feature vectors,
X = ¦x
, x
, . . . , x
¦. The resulting alignment procedure is illustrated
in Figure 9.8. We see the sequence of feature vectors along the horizon-
tal axis and the concatenated sequence of word states along the vertical
axis. An optimal alignment procedure determines the exact best match-
ing sequence between word model states and feature vectors such that
the first feature vector, x
, aligns with the first state in the first word
model, and the last feature vector, x
, aligns with the last state in the
Mth word model. (For simplicity we show each word model as a 5-state
HMM in Figure 9.8, but clearly the alignment procedure works for any
size model for any word, subject to the constraint that the total num-
ber of feature vectors, T, exceeds the total number of model states, so
that every state has at least a single feature vector associated with that
state.) The procedure for obtaining the best alignment between feature
vectors and model states is based on either using the Baum–Welch sta-
tistical alignment procedure (in which we evaluate the probability of
every alignment path and add them up to determine the probability of
the word string), or a Viterbi alignment procedure [38, 132] for which
170 Automatic Speech Recognition (ASR)
Fig. 9.8 Alignment of concatenated HMM word models with acoustic feature vectors based
on either a Baum–Welch or Viterbi alignment procedure.
we determine the single best alignment path and use the probability
score along that path as the probability measure for the current word
string. The utility of the alignment procedure of Figure 9.8 is based on
the ease of evaluating the probability of any alignment path using the
Baum–Welch or Viterbi procedures.
We now return to the mathematical formulation of the ASR prob-
lem and examine in more detail the three steps in the decoding
Equation (9.3).
9.3.3 Step 1 — Acoustic Modeling
The function of the acoustic modeling step (Step 1) is to assign prob-
abilities to the acoustic realizations of a sequence of words, given the
observed acoustic vectors, i.e., we need to compute the probability that
the acoustic vector sequence X = ¦x
, x
, . . . , x
¦ came from the word
sequence W = w
, w
, . . . , w
(assuming each word is represented as an
HMM) and perform this computation for all possible word sequences.
9.3 The Decision Processes in ASR 171
This calculation can be expressed as:
, x
, . . . , x
, w
, . . . , w
). (9.7)
If we make the assumption that each frame, x
, is aligned with
word i(t) and HMM model state j(t) via the function w
and if we
assume that each frame is independent of every other frame, we can
express (9.7) as the product
, (9.8)
where we associate each frame of X with a unique word and state,
, in the word sequence. Further, we calculate the local probability
given that we know the word from which frame t came.
The process of assigning individual speech frames to the appropriate
word model in an utterance is based on an optimal alignment process
between the concatenated sequence of word models and the sequence
of feature vectors of the spoken input utterance being recognized. This
alignment process is illustrated in Figure 9.9 which shows the set of
T feature vectors (frames) along the horizontal axis, and the set of M
words (and word model states) along the vertical axis. The optimal
segmentation of these feature vectors (frames) into the M words is
shown by the sequence of boxes, each of which corresponds to one of
the words in the utterance and its set of optimally matching feature
We assume that each word model is further decomposed into a set
of states which reflect the changing statistical properties of the fea-
ture vectors over time for the duration of the word. We assume that
each word is represented by an N-state HMM model, and we denote
the states as S
, j = 1, 2, . . . , N. Within each state of each word there
is a probability density that characterizes the statistical properties of
the feature vectors in that state. We have seen in the previous section
that the probability density of each state, and for each word, is learned
during a training phase of the recognizer. Using a mixture of Gaussian
densities to characterize the statistical distribution of the feature vec-
tors in each state, j, of the word model, which we denote as b
172 Automatic Speech Recognition (ASR)
Fig. 9.9 Illustration of time alignment process between unknown utterance feature vectors
and set of M concatenated word models.
we get a state-based probability density of the form:
) =
, µ
], (9.9)
where K is the number of mixture components in the density function,
is the weight of the kth mixture component in state j, with the
constraint c
≥ 0, and N is a Gaussian density function with mean
vector, µ
, for mixture k for state j, and covariance matrix, U
, for
mixture k in state j. The density constraints are:
= 1, 1 ≤ j ≤ N (9.10)
= 1, 1 ≤ j ≤ N. (9.11)
We now return to the issue of the calculation of the probability of
frame x
being associated with the j(t)th state of the i(t)th word in
9.3 The Decision Processes in ASR 173
the utterance, P
), which is calculated as
= b
), (9.12)
The computation of (9.12) is incomplete since we have ignored the
computation of the probability associated with the links between word
states, and we have also not specified how to determine the within-word
state, j, in the alignment between a given word and a set of feature
vectors corresponding to that word. We come back to these issues later
in this section.
The key point is that we assign probabilities to acoustic realizations
of a sequence of words by using hidden Markov models of the acoustic
feature vectors within words. Using an independent (and orthograph-
ically labeled) set of training data, we "train the system" and learn
the parameters of the best acoustic models for each word (or more
specifically for each sound that comprises each word). The parameters,
according to the mixture model of (9.9) are, for each state of the model,
the mixture weights, the mean vectors, and the covariance matrix.
Although we have been discussing acoustic models for whole words,
it should be clear that for any reasonable size speech recognition task,
it is impractical to create a separate acoustic model for every possible
word in the vocabulary since each word would have to be spoken in
every possible context in order to build a statistically reliable model
of the density functions of (9.9). Even for modest size vocabularies
of about 1000 words, the amount of training data required for word
models is excessive.
The alternative to word models is to build acoustic-phonetic models
for the 40 or so phonemes in the English language and construct the
model for a word by concatenating (stringing together sequentially)
the models for the constituent phones in the word (as represented in a
word dictionary or lexicon). The use of such sub-word acoustic-phonetic
models poses no real difficulties in either training or when used to
build up word models and hence is the most widely used representation
for building word models in a speech recognition system. State-of-the-
art systems use context-dependent phone models as the basic units of
recognition [99].
174 Automatic Speech Recognition (ASR)
9.3.4 Step 2 — The Language Model
The language model assigns probabilities to sequences of words, based
on the likelihood of that sequence of words occurring in the context
of the task being performed by the speech recognition system. Hence
the probability of the text string W ="Call home" for a telephone
number identification task is zero since that string makes no sense for
the specified task. There are many ways of building Language Models
for specific tasks, including:
(1) statistical training from text databases transcribed from
task-specific dialogs (a learning procedure)
(2) rule-based learning of the formal grammar associated with
the task
(3) enumerating, by hand, all valid text strings in the language
and assigning appropriate probability scores to each string.
The purpose of the language model, or grammar, is to enable the
computation of the a priori probability, P
(W), of a word string, W,
consistent with the recognition task [59, 60, 106]. Perhaps the most
popular way of constructing the language model is through the use
of a statistical N-gram word grammar that is estimated from a large
training set of text utterances, either from the task at hand or from a
generic database with applicability to a wide range of tasks. We now
describe the way in which such a language model is built.
Assume we have a large text training set of word-labeled utterances.
(Such databases could include millions or even tens of millions of text
sentences.) For every sentence in the training set, we have a text file
that identifies the words in that sentence. If we make the assumption
that the probability of a word in a sentence is conditioned on only
the previous N − 1 words, we have the basis for an N-gram language
model. Thus we assume we can write the probability of the sentence
W, according to an N-gram language model, as
, w
, . . . , w
) (9.13)
, w
, . . . , w
), (9.14)
9.3 The Decision Processes in ASR 175
where the probability of a word occurring in the sentence only depends
on the previous N − 1 words and we estimate this probability by
counting the relative frequencies of N-tuples of words in the train-
ing set. Thus, for example, to estimate word "trigram" probabilities
(i.e., the probability that a word w
was preceded by the pair of words
, w
)), we compute this quantity as
, w
) =
, w
, w
, w
, (9.15)
where C(w
, w
, w
) is the frequency count of the word triplet
(i.e., the trigram of words) consisting of (w
, w
, w
) as it occurs
in the text training set, and C(w
, w
) is the frequency count of
the word doublet (i.e., bigram of words) (w
, w
) as it occurs in
the text training set.
9.3.5 Step 3 — The Search Problem
The third step in the Bayesian approach to automatic speech recogni-
tion is to search the space of all valid word sequences from the language
model, to find the one with the maximum likelihood of having been spo-
ken. The key problem is that the potential size of the search space can
be astronomically large (for large vocabularies and high average word
branching factor language models), thereby taking inordinate amounts
of computing power to solve by heuristic methods. Fortunately, through
the use of methods from the field of Finite State Automata Theory,
Finite State Network (FSN) methods have evolved that reduce the
computational burden by orders of magnitude, thereby enabling exact
maximum likelihood solutions in computationally feasible times, even
for very large speech recognition problems [81].
The basic concept of a finite state network transducer is illustrated
in Figure 9.10 which shows a word pronunciation network for the word
/data/. Each arc in the state diagram corresponds to a phoneme in the
word pronunciation network, and the weight is an estimate of the proba-
bility that the arc is utilized in the pronunciation of the word in context.
We see that for the word /data/ there are four total pronunciations,
176 Automatic Speech Recognition (ASR)
Fig. 9.10 Word pronunciation transducer for four pronunciations of the word /data/.
(After Mohri [81].)
namely (along with their (estimated) pronunciation probabilities):
(1) /D/ /EY/ /D/ /AX/ — probability of 0.32
(2) /D/ /EY/ /T/ /AX/ — probability of 0.08
(3) /D/ /AE/ /D/ /AX/ — probability of 0.48
(4) /D/ /AE/ /T/ /AX/ — probability of 0.12.
The combined FSN of the 4 pronunciations is a lot more efficient than
using 4 separate enumerations of the word since all the arcs are shared
among the 4 pronunciations and the total computation for the full FSN
for the word /data/ is close to 1/4 the computation of the 4 variants
of the same word.
We can continue the process of creating efficient FSNs for each
word in the task vocabulary (the speech dictionary or lexicon), and
then combine word FSNs into sentence FSNs using the appropriate
language model. Further, we can carry the process down to the level of
HMM phones and HMM states, making the process even more efficient.
Ultimately we can compile a very large network of model states, model
phones, model words, and even model phrases, into a much smaller
network via the method of weighted finite state transducers (WFST),
which combine the various representations of speech and language and
optimize the resulting network to minimize the number of search states
(and, equivalently, thereby minimize the amount of duplicate computa-
tion). A simple example of such a WFST network optimization is given
in Figure 9.11 [81].
Using the techniques of network combination (which include net-
work composition, determinization, minimization, and weight push-
ing) and network optimization, the WFST uses a unified mathematical
9.4 Representative Recognition Performance 177
Fig. 9.11 Use of WFSTs to compile a set of FSNs into a single optimized network to
minimize redundancy in the network. (After Mohri [81].)
framework to efficiently compile a large network into a minimal rep-
resentation that is readily searched using standard Viterbi decoding
methods [38]. Using these methods, an unoptimized network with 10
states (the result of the cross product of model states, model phones,
model words, and model phrases) was able to be compiled down to
a mathematically equivalent model with 10
states that was readily
searched for the optimum word string with no loss of performance or
word accuracy.
9.4 Representative Recognition Performance
The block diagram of Figure 9.1 represents a wide range of possibil-
ities for the implementation of automatic speech recognition. In Sec-
tion 9.2.1, we suggest how the techniques of digital speech analysis
discussed in Chapters 4–6 can be applied to extract a sequence of fea-
ture vectors from the speech signal, and in Section 9.3 we describe the
most widely used statistical pattern recognition techniques that are
employed for mapping the sequence of feature vectors into a sequence
of symbols or words.
Many variations on this general theme have been investigated over
the past 30 years or more, and as we mentioned before, testing and
evaluation is a major part of speech recognition research and devel-
opment. Table 9.1 gives a summary of a range of the performance of
some speech recognition and natural language understanding systems
178 Automatic Speech Recognition (ASR)
Table 9.1 Word error rates for a range of speech recognition systems.
Corpus Type of Vocabulary Word
speech size error rate (%)
Connected digit strings
(TI Database)
Spontaneous 11 (0–9, oh) 0.3
Connected digit strings
(AT&T Mall
Spontaneous 11 (0–9, oh) 2.0
Connected digit strings
Conversational 11 (0–9, oh) 5.0
Resource management
Read speech 1000 2.0
Airline travel information
system (ATIS)
Spontaneous 2500 2.5
North American business
Read text 64,000 6.6
Broadcast News Narrated News 210,000 ≈15
Switchboard Telephone 45,000 ≈27
Call-home Telephone 28,000 ≈35
that have been developed so far. This table covers a range of vocabulary
sizes, speaking styles, and application contexts [92].
It can be seen that for a vocabulary of 11 digits, the word error
rates are very low (0.3% for a very clean recording environment for
the TI (Texas Instruments) connected digits database [68]), but when
the digit strings are spoken in a noisy shopping mall environment the
word error rate rises to 2.0% and when embedded within conversational
speech (the AT&T HMIHY (How May I Help You) c (system) the word
error rate increases significantly to 5.0%, showing the lack of robustness
of the recognition system to noise and other background disturbances
[44]. Table 9.1 also shows the word error rates for a range of DARPA
tasks ranging from
• read speech of commands and informational requests about
a naval ships database (the resource management system or
RM) with a 1000 word vocabulary and a word error rate of
• spontaneous speech input for booking airlines travel (the air-
line travel information system, or ATIS [133]) with a 2500
word vocabulary and a word error rate of 2.5%
9.5 Challenges in ASR Technology 179
• read text from a range of business magazines and newspa-
pers (the North American business task, or NAB) with a
vocabulary of 64,000 words and a word error rate of 6.6%
• narrated news broadcasts from a range of TV news providers
like CNBC, (the broadcast news task) with a 210,000 word
vocabulary and a word error rate of about 15%
• recorded live telephone conversations between two unrelated
individuals (the switchboard task [42]) with a vocabulary of
45,000 words and a word error rate of about 27%, and a
separate task for live telephone conversations between two
family members (the call home task) with a vocabulary of
28,000 words and a word error rate of about 35%.
9.5 Challenges in ASR Technology
So far, ASR systems fall far short of human speech perception in all
but the simplest, most constrained tasks. Before ASR systems become
ubiquitous in society, many improvements will be required in both
system performance and operational performance. In the system area
we need large improvements in accuracy, efficiency, and robustness in
order to utilize the technology for a wide range of tasks, on a wide
range of processors, and under a wide range of operating conditions.
In the operational area we need better methods of detecting when a
person is speaking to a machine and isolating the spoken input from
the background, we need to be able to handle users talking over the
voice prompts (so-called barge-in conditions), we need more reliable
and accurate utterance rejection methods so we can be sure that a word
needs to be repeated when poorly recognized the first time, and finally
we need better methods of confidence scoring of words, phrases, and
even sentences so as to maintain an intelligent dialog with a customer.
We have attempted to provide the reader with a broad overview of
the field of digital speech processing and to give some idea as to the
remarkable progress that has been achieved over the past 4–5 decades.
Digital speech processing systems have permeated society in the form of
cellular speech coders, synthesized speech response systems, and speech
recognition and understanding systems that handle a wide range of
requests about airline flights, stock price quotations, specialized help
desks etc.
There remain many difficult problems yet to be solved before digital
speech processing will be considered a mature science and technology.
Our basic understanding of the human articulatory system and how the
various muscular controls come together in the production of speech is
rudimentary at best, and our understanding of the processing of speech
in the human brain is at an even lower level.
In spite of our shortfalls in understanding, we have been able to cre-
ate remarkable speech processing systems whose performance increases
at a steady pace. A firm understanding of the fundamentals of acous-
tics, linguistics, signal processing, and perception provide the tools for
Conclusion 181
building systems that work and can be used by the general public. As
we increase our basic understanding of speech, the application systems
will only improve and the pervasiveness of speech processing in our
daily lives will increase dramatically, with the end result of improving
the productivity in our work and home environments.
We wish to thank Professor Robert Gray, Editor-in-Chief of Now Pub-
lisher's Foundations and Trends in Signal Processing, for inviting us
to prepare this text. His patience, technical advice, and editorial skill
were crucial at every stage of the writing. We also wish to thank
Abeer Alwan, Yariv Ephraim, Luciana Ferrer, Sadaoki Furui, and Tom
Quatieri for their detailed and perceptive comments, which greatly
improved the final result. Of course, we are responsible for any weak-
nesses or inaccuracies that remain in this text.
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Supplemental References
The specific references that comprise the Bibliography of this text
are representative of the literature of the field of digital speech pro-
cessing. In addition, we provide the following list of journals and
books as a guide for further study. Listing the books in chronologi-
cal order of publication provides some perspective on the evolution of
the field.
Speech Processing Journals
• IEEE Transactions on Signal Processing. Main publication of IEEE Sig-
nal Processing Society.
• IEEE Transactions on Speech and Audio. Publication of IEEE Signal
Processing Society that is focused on speech and audio processing.
• Journal of the Acoustical Society of America. General publication of the
American Institute of Physics. Papers on speech and hearing as well as
other areas of acoustics.
• Speech Communication. Published by Elsevier. A publication of the
European Association for Signal Processing (EURASIP) and of the
International Speech Communication Association (ISCA).
192 Supplemental References
General Speech Processing References
• Speech Analysis, Synthesis and Perception, J. L. Flanagan, Springer-
Verlag, Second Edition, Berlin, 1972.
• Linear Prediction of Speech, J. D. Markel and A. H. Gray, Jr., Springer
Verlag, Berlin, 1976.
• Digital Processing of Speech Signals, L. R. Rabiner and R. W. Schafer,
Prentice-Hall Inc., 1978.
• Speech Analysis, R. W. Schafer and J. D. Markel (eds.), IEEE Press
Selected Reprint Series, 1979.
• Speech Communication, Human and Machine, D. O'Shaughnessy,
Addison-Wesley, 1987.
• Advances in Speech Signal Processing, S. Furui and M. M. Sondhi, Mar-
cel Dekker Inc., New York, 1991.
• Discrete-Time Processing of Speech Signals, J. Deller, Jr., J. H. L.
Hansen, and J. G. Proakis, Wiley-IEEE Press, Classic Reissue, 1999.
• Acoustic Phonetics, K. N. Stevens, MIT Press, 1998.
• Speech and Audio Signal Processing, B. Gold and N. Morgan, John Wiley
and Sons, 2000.
• Digital Speech Processing, Synthesis and Recognition, S. Furui, Second
Edition, Marcel Dekker Inc., New York, 2001.
• Discrete-Time Speech Signal Processing, T. F. Quatieri, Prentice Hall
Inc., 2002.
• Speech Processing, A Dynamic and Optimization-Oriented Approach,
L. Deng and D. O'Shaughnessy, Marcel Dekker, 2003.
• Springer Handbook of Speech Processing and Speech Communication,
J. Benesty, M. M. Sondhi and Y Huang (eds.), Springer, 2008.
• Theory and Application of Digital Speech Processing, L. R. Rabiner and
R. W. Schafer, Prentice Hall Inc., 2009.
Speech Coding References
• Digital Coding of Waveforms, N. S. Jayant and P. Noll, Prentice Hall
Inc., 1984.
• Practical Approaches to Speech Coding, P. E. Papamichalis, Prentice
Hall Inc., 1987.
• Vector Quantization and Signal Compression, A. Gersho and R. M.
Gray, Kluwer Academic Publishers, 1992.
• Speech Coding and Synthesis, W. B. Kleijn and K. K. Paliwal, Elsevier,
• Speech Coding, A Computer Laboratory Textbook, T. P. Barnwell and
K. Nayebi, John Wiley and Sons, 1996.
Supplemental References 193
• A Practical Handbook of Speech Coders, R. Goldberg and L. Riek, CRC
Press, 2000.
• Speech Coding Algorithms, W. C. Chu, John Wiley and Sons, 2003.
• Digital Speech: Coding for Low Bit Rate Communication Systems, Sec-
ond Edition, A. M. Kondoz, John Wiley and Sons, 2004.
Speech Synthesis
• From Text to Speech, J. Allen, S. Hunnicutt and D. Klatt, Cambridge
University Press, 1987.
• Acoustics of American English, J. P. Olive, A. Greenwood and J. Cole-
man, Springer-Verlag, 1993.
• Computing Prosody, Y. Sagisaka, N. Campbell and N. Higuchi, Springer-
Verlag, 1996.
• Progress in Speech Synthesis, J. VanSanten, R. W. Sproat, J. P. Olive
and J. Hirschberg (eds.), Springer-Verlag, 1996.
• An Introduction to Text-to-Speech Synthesis, T. Dutoit, Kluwer Aca-
demic Publishers, 1997.
• Speech Processing and Synthesis Toolboxes, D. Childers, John Wiley and
Sons, 1999.
• Text To Speech Synthesis: New Paradigms and Advances, S. Narayanan
and A. Alwan (eds.), Prentice Hall Inc., 2004.
• Text-to-Speech Synthesis, P. Taylor, Cambridge University Press, 2008.
Speech Recognition and Natural Language Processing
• Fundamentals of Speech Recognition, L. R. Rabiner and B. H. Juang,
Prentice Hall Inc., 1993.
• Connectionist Speech Recognition-A Hybrid Approach, H. A. Bourlard
and N. Morgan, Kluwer Academic Publishers, 1994.
• Automatic Speech and Speaker Recognition, C. H. Lee, F. K. Soong and
K. K. Paliwal (eds.), Kluwer Academic Publisher, 1996.
• Statistical Methods for Speech Recognition, F. Jelinek, MIT Press, 1998.
• Foundations of Statistical Natural Language Processing, C. D. Manning
and H. Schutze, MIT Press, 1999.
• Spoken Language Processing, X. Huang, A. Acero and H.-W. Hon, Pren-
tice Hall Inc., 2000.
• Speech and Language Processing, D. Jurafsky and J. H. Martin, Prentice
Hall Inc., 2000.
• Mathematical Models for Speech Technology, S. E. Levinson, John Wiley
and Sons, 2005.
194 Supplemental References
Speech Enhancement
• Digital Speech Transmission, Enhancement, Coding and Error Conceal-
ment, P. Vary and R. Martin, John Wiley and Sons, Ltd., 2006.
• Speech Enhancement, Theory and Practice, P. C. Loizou, CRC Press,
Audio Processing
• Applications of Digital Signal Processing to Audio and Acoustics,
H. Kahrs and K. Brandenburg (eds.), Kluwer Academic Publishers,
• Audio Signal Processing and Coding, A. Spanias, T. Painter and V. Atti,
John Wiley and Sons, 2007.