Adaptive Modulation over Nakagami Fading Channels.
We first study the capacity of Nakagami multipath fading (NMF) channels  with an average power constraint for three power and rate adaptation  policies. We obtain closed-form solutions for NMF  channel capacity for each power and rate adaptation strategy.  Results show that rate adaptation is the key to increasing  link spectral efficiency.  We then analyze  the performance of practical constant-power variable-rate M-QAM schemes  over NMF channels. We obtain closed-form  expressions for the outage probability, spectral efficiency and  average bit-error-rate (BER) assuming perfect channel  estimation and negligible time delay between channel estimation  and signal set adaptation. We also  analyze the impact of time delay on the BER of adaptive M-QAM.
Introduction
The radio spectrum available for wireless services is extremely scarce, while demand for
these services is growing at a rapid pace [1]. Hence spectral efficiency is of primary concern
in the design of future wireless data communications systems. In this paper we first
investigate the theoretical spectral efficiency limits of adaptive transmission in Nakagami
multipath fading (NMF) channels [2]. We then propose and study adaptive multi-level
quadrature amplitude modulation (M-QAM) schemes which improve link spectral efficiency
(R=W [Bits/Sec/Hz]), defined as the average transmitted data rate per unit band-width
for a specified average transmit power and bit-error-rate (BER). We also evaluate
the performance of these schemes relative to the theoretical spectral efficiency limit.
Mobile radio links can exhibit severe multipath fading which leads to serious degradation
in the link carrier-to-noise ratio (CNR) and consequently a higher BER. Fading
compensation such as an increased link budget margin or interleaving with channel coding
are typically required to improve link performance. However, these techniques are
designed relative to the worst-case channel conditions, resulting in poor utilization of the
full channel capacity a good percentage of the time (i.e., under negligible or shallow fading
conditions). Adapting certain parameters of the transmitted signal to the channel
fading leads to better utilization of the channel capacity. The basic concept of adaptive
transmission is real-time balancing of the link budget through adaptive variation of the
March 13, 1998
transmitted power level, symbol transmission rate, constellation size, coding rate/scheme,
or any combination of these parameters [3], [4], [5], [6], [7]. Thus, without wasting power or
sacrificing BER, these schemes provide a higher average link spectral efficiency by taking
advantage of the time-varying nature of wireless channels: transmitting at high speeds under
favorable channel conditions and responding to channel degradation through a smooth
reduction of their data throughput. Good performance of these schemes requires accurate
channel estimation at the receiver and a reliable feedback path between that estimator
and the transmitter. Furthermore since outage probability of such schemes can be quite
high, especially for channels with low average CNR, buffering of the input data may be
required, and adaptive systems are therefore best suited to applications without stringent
delay constraints.
The Shannon capacity of a channel defines its maximum possible rate of data transmission
for an arbitrarily small BER, without any delay or complexity constraints. Therefore
the Shannon capacity represents an optimistic bound for practical communication schemes,
and also serves as a bench-mark against which to compare the spectral efficiency of adaptive
transmission schemes [8]. In [9] the capacity of a single-user flat-fading channel with
perfect channel measurement information at the transmitter and receiver was derived for
various adaptive transmission policies. In this paper we apply the general theory developed
in [9] to obtain closed-form expressions for the capacity of NMF channels under different
adaptive transmission schemes. In particular, we consider three adaptive policies: optimal
simultaneous power and rate adaptation, constant power with optimal rate adaptation,
and channel inversion with fixed rate. We then present numerical results showing that
rate adaptation is the key to achieving high link spectral efficiency. Rate adaptation can
be achieved through a variation of the symbol time duration [3] or constellation size [5].
The former method requires complicated hardware and results in a variable-bandwidth
system, whereas the latter technique is better suited for hardware implementation, since
it results in a variable-throughput system with a fixed bandwidth. Based on these advantages
we analyze the performance of constant-power variable-rate M-QAM schemes for
spectrally efficient data transmission over NMF channels. Similar analysis has been presented
in [6] for a variable-power variable-rate M-QAM in Rayleigh fading and log-normal
March 13, 1998
Feedback Path
Channel
Nakagami Channel
Slowly Varying
Modulator
Input
Transmitter
Channel
Estimator
Constellation Size
Selector
Pilot
Demodulator
Receiver
Data
Output
Data
AGC Carrier
Recovery
Data
Fig. 1. Adaptive communication system model.
shadowing, and in [10] for constant-power variable-rate M-QAM in Rayleigh fading. We
extend the results of [6], [10] to constant-power variable-rate M-QAM by analyzing the
resulting spectral efficiency and BER for the more general NMF distribution. We also
analyze the impact of time delay on the performance of adaptive M-QAM.
The remainder of this paper is organized as follows. In Section II we outline the channel
and communication system models. In Section III we derive the capacity of NMF channels
for the optimal adaptive policy, constant power policy, and channel inversion policy, and
we present some numerical examples comparing (i) the NMF channel capacity with the
capacity of an additive white Gaussian noise (AWGN) channel, and (ii) the NMF channel
capacity for the various adaptive policies. In Section IV we propose and evaluate the
performance of an adaptive constant-power variable-rate M-QAM system assuming perfect
channel estimation and negligible time delay. The BER degradation due to time delay is
analyzed in Section V. A summary of our results is presented in Section VI.
II. System and Channel Models
A. Adaptive Communication System Model
A block diagram of the adaptive communication system is shown in Fig. 1. A pilot
tone continually sends a known "channel sounding" sequence so that the channel-induced
envelope fluctuation ff and phase shift OE can be extracted at the channel estimation stage.
Based on this channel gain estimate -
ff, a decision device selects the rate and power to be
transmitted, configures the demodulator accordingly, and informs the transmitter about
that decision via the feed back path. The constellation size assignment for the proposed
March 13, 1998
constant-power variable-rate M-QAM scheme will be discussed in more detail in Section
IV-A. The transmission system keeps its configuration unchanged (i.e, no re-adaptation)
for a duration - t [s]. Meanwhile the phase estimate -
OE is used at the receiver for full
compensation of the phase variation (i.e., ideal coherent phase detection), whereas the
channel gain estimate -
ff is used on a continuous basis by the automatic gain controller
(AGC)/demodulator for symbol-by-symbol maximum-likelihood detection.
For satisfactory operation the modulator and demodulator must be configurated at any
instant for the same constellation size. Efficient error control schemes are therefore required
to insure an error-free feedback path. However such schemes inevitably introduce a certain
time delay - fb [s], which may include decoding/ARQ delay, and propagation time via the
feedback path. Hence, even if perfect channel estimates are available at the receiver, the
system will not be able to adapt to the actual channel fading but rather to at best a - fb
delayed version of it. In practice, the choice of the power and/or constellation is based on
a channel estimate at time t, but the data are sent over the channel at time t +- such that
is the rate at which we change the constellation size and power.
The goal is to operate with the smallest possible - fb to minimize the impact of feedback
delay, and with the largest possible - t to minimize the rate of system reconfiguration. This
issue will be further discussed in Section V.
B. Channel Model and Fading Statistics
We consider a slowly-varying flat-fading channel changing at a rate much slower than
the symbol data rate, so the channel remains roughly constant over hundreds of symbols.
The multipath fading environment can be characterized by different statistical models.
For NMF channels the probability distribution function (PDF) of the channel gain ff is
given by [2, (11)]
\Omega
\Gamma(m) exp
\Gammam ff 2
\Omega
is the average received power, m is the Nakagami fading parameter
(m - 1=2), and \Gamma(:) is the gamma function [11]. The received CNR, fl, is then gamma
March 13, 1998
distributed according to the PDF, p fl (fl), given by
\Gamma(m) exp
\Gammam
where fl is the average received CNR. The phase OE of the Nakagami fading is uniformly
distributed over [0,2-].
The Nakagami fading represents a wide range of multipath channels via the m fading
parameter [2]. For instance, the Nakagami-m distribution includes the one-sided Gaussian
distribution which corresponds to worst-case fading) and the Rayleigh distribution
special cases. In addition, when m ? 1, a one-to-one mapping between the
Rician factor and the Nakagami fading parameter allows the Nakagami-m distribution to
closely approximate the Rice distribution [2]. Finally, and perhaps most importantly, the
Nakagami-m distribution often gives the best fit to urban [12] and indoor [13] multipath
propagation.
III. Capacity of Nakagami Fading Channels
A. Optimal Adaptation
Given an average transmit power constraint, the channel capacity of a fading channel
with received CNR distribution p fl (fl) and optimal power and rate adaptation (!C ? opra
[Bit/Sec]) is given in [9] as
log 2
where W [Hz] is the channel bandwidth and fl o is the optimal cutoff CNR level below
which data transmission is suspended. This optimal cutoff must satisfy the equation
To achieve the capacity (3), the channel fade level must be tracked at both the receiver and
transmitter, and the transmitter has to adapt its power and rate accordingly, allocating
high power levels and rates for good channel conditions (fl large), and lower power levels
and rates for unfavorable channel conditions (fl small). Since no data is sent when
the optimal policy suffers a probability of outage P out , equal to the probability of no
March 13, 1998
transmission, given by
Substituting (2) in (4) we find that fl must satisfy
where \Gamma(:; :) is the complementary incomplete gamma function [11]. For the special case
of the Rayleigh fading channel reduces to
e \Gammafl
is the exponential integral of first order [11]. Let
Note that df(x)
Moreover, from (8), lim x!0
there is a unique positive x o for which f(x
equivalently, there is a unique fl which satisfies (6). An asymptotic expansion of (6) shows
that as Our numerical results show that fl o increases as fl increases, so
lies in the interval [0,1].
Substituting (2) in (3), and defining the integral J n (-) as
we can rewrite the channel capacity !C? opra as
Jm
The evaluation of J n (-) for n a positive integer is derived in [14, Appendix A]. Using
that result we obtain the NMF channel capacity per unit bandwidth
[Bits/Sec/Hz] under the optimal power and rate adaptation policy as
which can also be written as
denotes the Poisson distribution defined by
For the special case of the Rayleigh fading channel, using (7) in (12) for m=1, the optimal
capacity per unit bandwidth reduces to the simple expression
e \Gammafl
Using (2) in the probability of outage equation (5) yields
B. Constant Transmit Power
With optimal rate adaptation to channel fading with a constant transmit power, the
channel capacity !C? ora [Bits/Sec] becomes [9]
!C? ora was previously introduced by Lee [15], [16] as the average channel capacity of a
flat-fading channel, since it is obtained by averaging the capacity of an AWGN channel
over the distribution of the received CNR. In fact, (16) represents the capacity of the
fading channel without transmitter feedback (i.e. with the channel fade level known at
the receiver only) [17], [18], [19].
Substituting (2) into (16) and defining the integral I n (-) as
I
Z +1t
the channel capacity !C? ora of a NMF channel can be written as
Im
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The evaluation of I n (-) for n a positive integer is derived in [14, Appendix B]. Using that
result, we can rewrite !C? ora =W [Bits/Sec/Hz] as
One may also express (20) in terms of the Poisson distribution as [16]
Note that Yao and Sheikh [20] provided a closed-form expression for the capacity of NMF
channels in terms of the complementary incomplete gamma function. However their derivation
is different then ours and their resulting expression [20, (7)] contains m order deriva-
tives. For the special case of the Rayleigh fading channel reduces to
C. Channel Inversion with Fixed Rate
The channel capacity when the transmitter adapts its power to maintain a constant CNR
at the receiver (i.e., inverts the channel fading) was also investigated in [9]. This technique
uses fixed-rate modulation and a fixed code design, since the channel after channel
inversion appears as a time-invariant AWGN channel. As a result, channel inversion with
fixed rate is the least complex technique to implement, assuming good channel estimates
are available at the transmitter and receiver. The channel capacity with this technique
(!C? cifr [Bits/Sec]) is derived from the capacity of an AWGN channel and is given in [9]
as
Channel inversion with fixed rate suffers a large capacity penalty relative to the other
techniques, since a large amount of the transmitted power is required to compensate for
the deep channel fades. Another approach is to use a modified inversion policy which
inverts the channel fading only above a fixed cutoff fade depth fl . The capacity with this
truncated channel inversion and fixed rate policy (!C ? tifr [Bits/Sec]) was derived in [9]
to be
R +1
where P out is given by (5). The cutoff level fl o can be selected to achieve a specified outage
probability or, alternatively (as shown in Figures 2, 3, and 4), to maximize (24).
By substituting the CNR distribution (2) in (23) we find that the capacity per unit
bandwidth of a NMF channel with total channel inversion, !C ? cifr =W , is given for all
Thus the capacity of a Rayleigh fading channel zero in this case. Note that
the capacity of this policy for a NMF channel is the same as the capacity of an AWGN
channels with equivalent CNR=
With truncated channel inversion the capacity per unit bandwidth !C? tifr =W [Bits/Sec/Hz]
can be expressed in terms of fl and fl o by substituting (2) into (24), which yields
\Gamma(m; mfl
For the special case of the Rayleigh fading channel 1), the capacity per unit band-width
with truncated channel inversion reduces to
e \Gammafl
D. Numerical Results

Figures

2, 3, and 4 show the capacity per unit bandwidth as a function of fl for a NMF
channel under the three different adaptive policies for
respectively. We see from these figures that the capacity of NMF channels
is always smaller than the capacity of an AWGN channel for fl - 0 dB but converges
to it as the m parameter increases or, equivalently, as the amount of fading decreases.
We also see that optimal power and rate adaptation yields a small increase in capacity
over just optimal rate adaptation, and this small increase in capacity diminishes as the
average received CNR and/or fading parameter m increase. Note finally that fixed rate
transmission with channel inversion suffers the largest capacity penalty. However, this
penalty diminishes as the amount of fading decreases.
Average Received CNR [dB]
Capacity
per
Unit
Bandwidth
AWGN Channel Capacity
Optimal Power and Rate
Optimal Rate and Constant Power
Truncated Channel Inversion
Total Channel Inversion
Fig. 2. Capacity per unit bandwidth for a Rayleigh fading channel (m=1) under different adaption
policies.
IV. Adaptive M-QAM Modulation
A. Proposed Adaptive Schemes
The BER of coherent M-QAM with two-dimensional Gray coding over an additive white
Gaussian noise (AWGN) channel assuming perfect clock and carrier recovery can be well
approximated by [6]
Exact expressions for the BER of "square" M-QAM (when the number of bits per symbol
n is even) are known [21, Chapter 5], and are plotted by the solid lines in Fig. 5. On the
other hand, tight upper-bounds on the BER of "non-square" M-QAM (when the number of
bits per symbol n is odd) are also available [22, p. 283], and are plotted by the cross/solid
lines in Fig. 5. For comparison, the dash lines in this figure show the BER approximation
for different values of M . Note that the approximate BER expression upper bounds
the exact BER for M - 4 and for which is the BER range of interest. We
will use this approximation when needed in our analysis since it is "invertible" in the sense
that it provides a simple closed-form expression for the link spectral efficiency of M-QAM
as a function of the CNR and the BER. In addition, (28) and its inverse are very simple
functions which lead, as shown below, to closed-form analytical expressions and insights
March 13, 1998
ALOUINI AND GOLDSMITH: ADAPTIVE MODULATION OVER NAKAGAMI FADING CHANNELS 11
Average Received CNR [dB]
Capacity
per
Unit
Bandwidth
AWGN Channel Capacity
Optimal Power and Rate
Optimal Rate and Constant Power
Truncated Channel Inversion
Total Channel Inversion
Fig. 3. Capacity per unit bandwidth for a Nakagami fading channel with m=2, and for different adaption
policies.
that are unattainable with more complicated BER expressions.
Assuming ideal Nyquist pulses and given a fixed CNR (fl) and BER (BER 0 ) the spectral
efficiency of continuous-rate M-QAM can be approximated by inverting (28), giving
R
where adaptive continuous rate (ACR) M-QAM scheme responds
to the instantaneous channel CNR fluctuation by varying the number of bits per symbol
according to (29). In the context of this paper, continuous-rate means that the number
of bits per symbol is not restricted to integer values. While continuous-rate M-QAM is
possible [23] it is more practical to study the performance of adaptive discrete rate (ADR)
M-QAM, where the constellation size M n is restricted to 2 n for n a positive integer. In
this case the scheme responds to the instantaneous channel CNR fluctuation by varying its
constellation size as follows. The CNR range is divided into N fading regions, and the
constellation size M n is assigned to the nth region When the received
CNR is estimated to be in the nth region, the constellation size M n is transmitted.
Suppose we set a target BER, BER 0 . The region boundaries (or switching thresholds)
are then set to the CNR required to achieve the target BER 0 using M n -QAM over
March 13, 1998
Average Received CNR [dB]
Capacity
per
Unit
Bandwidth
AWGN Channel Capacity
Optimal Power and Rate
Optimal Rate and Constant Power
Truncated Channel Inversion
Total Channel Inversion
Fig. 4. Capacity per unit bandwidth for a Nakagami fading channel with m=4, and for different adaption
policies.
Carrier-to-Noise-Ratio CNR g [dB]
Bit
Rate
BER
Approximation (3)
Exact
Upper Bound
Fig. 5. BER for M-QAM versus CNR.
Assignment of Constellation Size Relative to Received CNR for a Target BER
Received Carrier-to-Noise-Ratio CNR g [dB]
Constellation
Size
logM
Continuous Rate Adaptive M-QAM (4)
Discrete Rate Adaptive M-QAM
Fig. 6. Number of bits per symbol versus CNR.
an AWGN channel. Specifically
denotes the inverse complementary error function. When the switching
thresholds are chosen according to (30), the system will operate with a BER below the
target BER, as will be confirmed in Section IV-D. Note in particular that all the
are chosen according to (28). Since (28) is an upper-bound of the BER only
for M - 4, fl 1 is chosen according to the exact BER performance of 2-QAM (BPSK). The
thick line in Fig. 6 shows the number of bits per symbol as a function of the received CNR
for ADR M-QAM with 8-regions, along with the corresponding switching thresholds. For
comparison the thin line in this figure shows the bits per symbol of ACR M-QAM.
B. Outage Probability
Since no data is sent when the received CNR falls below fl 1 , the ADR M-QAM scheme
suffers an outage probability, P out , of
Figs. 7 shows the outage probability for various values of the Nakagami fading parameter
and for target BERs of 10 \Gamma3 and 10 \Gamma6 , respectively.
Probability of Outage
Average Received Carrier-to-Noise-Ratio g- [dB]
Probability
of
Outage
out
Target BER=10 -3
Target BER=10 -6
Fig. 7. Outage probability in Nakagami fading.
C. Achievable Spectral Efficiency
Integrating (29) over (2) and following the same steps of Section III-B which obtained
(20), we find the average link spectral efficiency, !R? acr =W , of the ACR M-QAM over
NMF channels as
!R? acr
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Achievable Rates in Rayleigh Fading (m=1) for a Target BER
Average Received Carrier-to-Noise-Ratio g- [dB]
Spectral
Efficiency
Regions
Regions
Regions
Capacity (Optimal Rate and Constant Power)
Continuous Rate Adaptive M-QAM
Discrete Rate Adaptive M-QAM
Non Adaptive 2-QAM (B-PSK)
Fig. 8. Achievable spectral efficiency for a target BER of 10 \Gamma3 and
The average link spectral efficiency, of the ADR M-QAM over NMF
channels is just the sum of the data rates (log 2 [M n associated with the individual
weighted by the probability a
that the CNR fl falls in
the nth region:
where the a n s can be expressed as
a
Figs. 8, show the average link spectral efficiency of ACR M-QAM (32) and
ADR M-QAM (33) for a target BER
tively. The Shannon capacity using constant-power and variable-rate (20) is also shown
for comparison, along with the spectral efficiency of nonadaptive 2-QAM (BPSK). This
latter efficiency is found by determining the value of the average received CNR for which
the average BER of nonadaptive BPSK over Nakagami fading channel, as given by (38),
equals the target BER. Note that the achievable spectral efficiency of ACR M-QAM comes
within 5 dB of the Shannon capacity limit. ADR M-QAM suffers a minimum additional
1.2 dB penalty, whereas nonadaptive BPSK suffers a large spectral efficiency penalty.
Achievable Rates in Nakagami Fading (m=2) for a Target BER
Average Received Carrier-to-Noise-Ratio g- [dB]
Average
Spectral
Efficiency
Regions
Regions
Regions
Capacity (Optimal Rate and Constant Power)
Continuous Rate Adaptive M-QAM
Discrete Rate Adaptive M-QAM
Non Adaptive 2-QAM (B-PSK)
Fig. 9. Achievable spectral efficiency for a target BER of 10 \Gamma3 and 2.
3013579Achievable Rates in Nakagami Fading (m=4) for a Target BER
Average Received Carrier-to-Noise-Ratio g- [dB]
Average
Spectral
Efficiency
Regions
Regions
Regions
Capacity (Optimal Rate and Constant Power)
Continuous Rate Adaptive M-QAM
Discrete Rate Adaptive M-QAM
Non Adaptive 2-QAM (B-PSK)
Fig. 10. Achievable spectral efficiency for a target BER of 10 \Gamma3 and
D. Average Bit Error Rate
ACR M-QAM always operates at the target BER. However, since the choice of M n in
ADR M-QAM is done in a conservative fashion, this discrete technique operates at an
average smaller than the target BER. This BER can be computed
exactly as the ratio of the average number of bits in error over the total average number
March 13, 1998
of transmitted bits
where
Using (2) and the approximation (28) in (36) BER n can be expressed in closed-form as
(b
where
BER n can also be computed exactly by using the exact expressions for the BER(M n ; fl)
as given in [21, Chapter 5] and [10].
Figs. 11, 12, and 13 show the average BER for ADR M-QAM for a target BER of 10 \Gamma3
and for respectively. The BER calculations based on the
approximation (37) are plotted in solid lines whereas the exact average BERs are plotted
by the star/solid lines. The average BER of nonadaptive BPSK over Nakagami fading
channel is given by
s
denotes the Gauss' hypergeometric function [11]. We plot (38) in Figs.
11, 12, and 13 in dash lines for comparison with (35).
In these figures we observe similar trends in the average BER for various m parameters.
For instance we see that the average BER of ADR M-QAM is always below the 10 \Gamma3 target
BER. Recall that the approximation (28) lower bounds the exact BER for M=2 and that
ADR M-QAM often uses the 2-QAM constellation (B-PSK) at low average CNRs. This
explains why the average BER based on the approximation (37) lower bounds the exact
average BER for Conversely because of the fact that the approximation (28)
upper bounds the exact BER for M ? 2 and because ADR M-QAM often uses the high
constellation sizes at high average CNRs the closed-form approximate average BER for
ADR M-QAM tightly upper-bounds the exact average BER for
March 13, 1998
Average BER in Rayleigh Fading (m=1) for a Target BER
Average Received Carrier-to-Noise-Ratio g- [dB]
Average
Bit
Rate

Regions
Regions
Regions
ACR M-QAM
ADR M-QAM (Approx.)
ADR M-QAM (Exact)
Non-Adaptive 2-QAM
Fig. 11. Average BER for a target BER of 10 \Gamma3 and
M-QAM uses the largest available constellation often when the average CNR is large, the
average BER prediction as fl increases becomes dominated by the BER performance of
that constellation.
V. Impact of Time Delay
Recall from Section II-A that the choice of the constellation size is based on a channel
estimate at time t, whereas the data are sent over the channel at time t - such that
. If a delay of - fb degrades BER significantly, then this adaptive technique
will not work, since - fb is an inherent and unavoidable parameter of the system. However,
if a delay of - fb has a small impact on the BER then we should choose - t as large
as possible so that we meet the BER requirement while minimizing the rate of system
reconfiguration. In this section we analyze the impact of time delay on the performance
of adaptive M-QAM over NMF channels, assuming perfect channel estimates.
Average BER in Nakagami Fading (m=2) for a Target BER
Average Received Carrier-to-Noise-Ratio g- [dB]
Average
Bit
Rate

Regions
Regions
Regions
ACR M-QAM
ADR M-QAM (Approx.)
ADR M-QAM (Exact)
Non-Adaptive 2-QAM
Fig. 12. Average BER for a target BER of 10 \Gamma3 and 2.
A. Fading Correlation
Investigating the impact of time delay requires the second-order statistics for the channel
variation, which are known for Nakagami fading. Let ff and ff - denote the channel gains
at a time t and t respectively. For a slowly-varying channel we can assume that the
average received power remains constant over the time delay -
Under these conditions the joint these two correlated Nakagami-m
distributed channel gains is given by [2, (126)]
\Omega
I
ae)\Omega
exp
ae)\Omega
where I m\Gamma1 (.) is the (m \Gamma 1)th-order modified Bessel function of the first kind [11], and
ae is the correlation factor between ff and ff - . Since Nakagami fading assumes isotropic
scattering of the multipath components, ae can be expressed in terms of the time delay - , the
mobile speed, v [m/s], and the wavelength of the carrier frequency - c [m] as
where J 0 (:) is the zero-order Bessel function of the first kind [11], and f
the maximum Doppler frequency shift [24, p. 31].
The PDF of ff - conditioned on ff, p ff - =ff (ff - =ff), is given by
Average BER in Nakagami Fading (m=4) for a Target BER
Average Received Carrier-to-Noise-Ratio g- [dB]
Average
Bit
Rate

Regions
Regions
Regions
ACR M-QAM
ADR M-QAM (Approx.)
ADR M-QAM (Exact)
Non-Adaptive 2-QAM
Fig. 13. Average BER for a target BER of 10 \Gamma3 and
Inserting (1) and (39) in (40) and expressing the result in terms of the CNRs fl and fl -
yields
ae
I
exp
B. Analysis
B.1 Adaptive Continuous Rate M-QAM
For all delays - let the communication system be configured according to fl (CNR at
such that M(fl) is given by
The constellation size M(fl) is based on the value fl at time t, but that constellation is
transmitted over the channel at time t when fl has changed to fl - . Since M does
not depend on fl - (CNR at time - delay does not affect the link spectral efficiency
as calculated in Section IV-C. However, delay affects the instantaneous BER,
which becomes a function of the "mismatch" between fl - and fl:
Integrating (43) over the conditional PDF (41) yields the average BER conditioned on
fl, BER(fl), as
Inserting (41) and (43) in (44), BER(fl) can be written in a closed-form with the help of
the generalized Marcum Q-function of order m, Qm (:; :) [25, p. 299, (11.63)]
exp
Qm
Using the recurrence relation [25, p. 299, (11.64)]
x
I m (2 p
we get that for all x, Qm (x; which can be shown to equal 1. Therefore
reduces to:
exp
Although this formula was derived for integer m it is also valid for all non-integer values
of m - 1=2. Averaging (47) over the PDF of fl (2) yields the average BER, !BER? acr ,
as
Finally, using (47) in (48) and making the substitution
yields
exp
where
Since this analysis assumes continuous rate adaptation and since M n (fl) - M(fl) for all
fl, (49) represents an upper-bound on the average BER degradation for ADR M-QAM, as
will be confirmed in the following sections.
B.2 Adaptive Discrete Rate M-QAM
The constellation size M n is chosen based on the value of fl according to the ADR M-QAM
scheme described in Section IV-A. However the constellation is transmitted over the
channel when fl has changed to fl - . As in Section V-B-1, we can easily see that the link
spectral efficiency of ADR M-QAM is unaffected by time delay. However, delay affects
!BER? adr , which can be computed as in (35) with BER n replaced by BER 0
Using again the generalized Marcum Q-functions it can be shown that
(b
where
Note that as ae
reduces to BER n (37), as
expected.
C. Numerical Results
Figs. 14 and 15 show adr as a function of the normalized
time delay f D - for different values of the Nakagami m parameter, for a target BER of
respectively. It can be seen from Figs. 14 and 15 that a normalized time
delay up to about 10 \Gamma2 can be tolerated without a noticeable degradation in the average
BER. For example, for a 900 MHz carrier frequency and a target BER of 10 \Gamma3 , a time
delay up to 3.33 ms can be tolerated for pedestrians with a speed of 1 m/s (3.6 km/hr),
and a time delay up to 0.133 ms can be tolerated for mobile vehicles with a speed of 25
m/s (90 km/hr). Comparing Figs. 14 and 15 we see that systems with the lower BER
requirements of 10 \Gamma6 are more sensitive to time delay, as they will suffer a higher "rate of
increase" in BER. For example, in Rayleigh fading, systems with a 10 \Gamma3 BER requirement
suffer about one order of magnitude degardation for f D
systems with a 10 \Gamma6 BER requirement suffer about four order of magnitude degardation
for the same range of f D - . However, in both cases these systems will be able to operate
satisfactorily if the normalized delay is below the critical value of 10 \Gamma2 .
Average BER Degradation due to Time Delay for a Target BER 0 =10 -3
Normalized Time Delay f D t
Average
Bit
Rate

Adaptive Continuous Rate M-QAM
Adaptive Discrete Rate M-QAM
Fig. 14. Average BER vs. normalized time delay for a BER 0 of 10 \Gamma3 , fl=20 dB, and 5 fading regions.
VI. Conclusion
We have studied the capacity of NMF channels with an average power constraint for three
power and rate adaptation policies. We obtain closed-form solutions for NMF channel
capacity for each power and rate adaptation strategy. Our results show that optimal power
and rate adaptation yields a small increase in capacity over just optimal rate adaptation
with constant power, and this small increase in capacity diminishes as the average received
carrier-to-noise ratio, and/or the m parameter increases. Fixed rate transmission with
channel inversion suffers the largest capacity penalty. However, this penalty diminishes as
the amount of fading decreases. Based on these results we conclude that rate rather than
power adaptation is the key to increasing link spectral efficiency. We therefore proposed
and studied the performance of constant-power variable-rate M-QAM schemes over NMF
channels assuming perfect channel estimation and negligible time delay. We determined
their spectral efficiency performance and compared this to the theoretical maximum. Our
March 13, 1998
Average BER Degradation due to Time Delay for a Target BER 0 =10 -6
Normalized Time Delay f D t
Average
Bit
Rate

Adaptive Continuous Rate M-QAM
Adaptive Discrete Rate M-QAM
Fig. 15. Average BER vs. normalized time delay for a BER 0 of 10 \Gamma6 , fl=20 dB, and 5 fading regions.
results show that for a target BER of 10 \Gamma3 , the spectral efficiency of adaptive continuous
rate M-QAM comes within 5 dB of the Shannon capacity limit and adaptive discrete rate
M-QAM comes within 6.2 dB of this limit. We also analyzed the impact of time delay
on the BER of adaptive M-QAM. Results show that systems with low BER requirements
will be more sensitive to time delay but will still be able to operate satisfactorily if the
normalized time delay is below the critical value of 10 \Gamma2 .



--R

"Wireless data communications,"
"The m-distribution- A general formula of intensity distribution of rapid fading,"
"Variable-rate transmission for Rayleigh fading channels,"
"Symbol rate and modulation level controlled adaptive modula- tion/TDMA/TDD for personal communication systems,"
"Variable rate QAM for mobile radio,"
"Variable-rate variable-power M-QAM for fading channels,"
"Adaptive modulation system with variable coding rate concatenated code for high quality multi-media communication systems,"
"Variable-rate coded M-QAM for fading channels,"
"Capacity of fading channels with channel side information,"
"Upper bound performance of adaptive modulation in a slow Rayleigh fading channel,"
Table of Integrals
"A statistical model for urban multipath propagation,"
"Indoor mobile radio channel at 946 MHz: measurements and modeling,"
"Capacity of Rayleigh fading channels under different adaptive transmission and diversity techniques."
"Estimate of channel capacity in Rayleigh fading environment,"
"Comment on "
"Channels with block interference,"
"Information theoretic considerations for cellular mobile radio,"
"A Gaussian channel with slow fading,"
"Evaluation of channel capacity in a generalized fading channel,"
Modern Quadrature Amplitude Modulation.
New York
"Efficient modulation for band-limited channels,"

Special Functions- An Introduction to the Classical Functions of Mathematical Physics
--TR

--CTR
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and Nakagami fading;adaptive modulation techniques;link spectral efficiency