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# The Discrete Cosine Transform

Gilbert Strang
Massachusetts Institute of Technology
http://www-math.mit.edu/gs
gs@math.mit.edu
to appear in SIAM Review

Abstract
Each Discrete Cosine Transform uses N real basis vectors whose components are
cosines. In the DCT-4, for example, the th component of vk is cos( + 12 )( + 12 ) N .
j j k

These basis vectors are orthogonal and the transform is extremely useful in image
processing. If the vector x gives the intensities along a row of pixels, its cosine series
P k vk has the coecients
c k
c = (x ; vk )
=N . They are quickly computed from an FFT.
But a direct proof of orthogonality, by calculating inner products, does not reveal how
natural these cosine vectors are.
We prove orthogonality in a di erent way. Each DCT basis contains the eigenvectors
of a symmetric \second di erence" matrix. By varying the boundary conditions we get
the established transforms DCT-1 through DCT-4. Other combinations lead to four
additional cosine transforms. The type of boundary condition (Dirichlet or Neumann,
centered at a meshpoint or a midpoint) determines the applications that are appropriate
for each transform. The centering also determines the period: N 1 or N in the
established transforms, N
1
2 or N + 12 in the other four. The key point is that all
these \eigenvectors of cosines" come from simple and familiar matrices.

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0. Introduction
Just as the Fourier series is the starting point in transforming and analyzing periodic func-
tions, the basic step for vectors is the Discrete Fourier Transform (DFT). It maps the \time
domain" to the \frequency domain." A vector with N components is written as a combi-
nation of N special basis vectors vk . Those are constructed from powers of the complex
number w = e2i=N :
 
vk = 1; wk ; w2k ; : : : ; w(N 1)k ; k = 0; 1; : : : ; N 1 :

The vectors vk are the columns of the Fourier matrix F = FN . Those columns are orthog-
onal. So the inverse of F is its conjugate transpose, divided by k vk k2 = N . The discrete
Fourier series x = ck vk is x = F c. The inverse c = F 1x uses ck = (x; vk )=N for the
P

## (complex) Fourier coecients.

Two points to mention, about orthogonality and speed, before we come to the purpose
of this note. For these DFT basis vectors, a direct proof of orthogonality is very ecient:
X1
N
(wk )j (w` )j = (wwkww`) 1 1 :
k ` N
(vk ; v` ) =
j =0

## The numerator is zero because wN = 1. The denominator is nonzero because k 6= `. This

proof of (vk ; v` ) = 0 is short but not very revealing. I want to recommend a di erent proof,
which recognizes the vk as eigenvectors. We could work with any circulant matrix, and we
will choose below a symmetric A0 . Then linear algebra guarantees that its eigenvectors vk
are orthogonal.
Actually this second proof, verifying that A0 vk = k vk , brings out a central point of
Fourier analysis. The Fourier basis diagonalizes every periodic constant coecient operator.

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Each frequency k (or 2k=N ) has its own frequency response k . The complex exponential
vectors vk are important in applied mathematics because they are eigenvectors!
The second key point is speed of calculation. The matrices F and F 1 are full, which
normally means N 2 multiplications for the transform and the inverse transform: y = Fx
and x = F 1 y. But the special form Fjk = wjk of the Fourier matrix allows a factorization
into very sparse and simple matrices. This is the Fast Fourier Transform (FFT). It is easiest
when N is a power 2L . The operation count drops from N 2 to 21 NL, which is an enormous
saving. But the matrix entries (powers of w) are complex.
The purpose of this note is to consider real transforms that involve cosines. Each matrix
of cosines yields a Discrete Cosine Transform (DCT). There are four established types, DCT-
1 through DCT-4, which di er in the boundary conditions at the ends of the interval. (This
di erence is crucial. The DCT-2 and DCT-4 are constantly applied in image processing;
they have an FFT implementation and they are truly useful.) All four types of DCT are
orthogonal transforms. The usual proof is a direct calculation of inner products of the N
basis vectors, using trigonometric identities.
We want to prove this orthogonality in the second (indirect) way. The basis vectors of
cosines are actually eigenvectors of symmetric second-di erence matrices. This proof seems
more attractive, and ultimately more useful. It also leads us, by selecting di erent boundary
conditions, to four less familiar cosine transforms. The complete set of eight DCT's was
found in 1985 by Wang , and we want to derive them in a simple way. We begin now
with the DFT.

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1. The Periodic Case and the DFT
The Fourier transform works perfectly for periodic boundary conditions (and constant coef-
cients). For a second di erence matrix, the constant diagonals contain 1 and 2 and 1.
The diagonals with 1 loop around to the upper right and lower left corners, by periodicity,
to produce a circulant matrix:
2 3
2 1 1
6
6 1 2 1 7
7
A 6
0=6 ... 7
7 :
6 7
4 1 2 1 5
1 1 2
For this matrix A0 , and every matrix throughout the paper, we look at three things:

## 1. The interior rows

2. The boundary rows (rows 0 and N 1)
3. The eigenvectors
The interior rows will be the same in every matrix! The j th entry of A0 u is uj 1 +
2uj uj+1, which corresponds to u00. This choice of sign makes each matrix positive
de nite (or at least semide nite). No eigenvalues are negative.
At the rst and last rows (j = 0 and j = N 1), this second di erence involves u 1

## produces the 1 entries that appear in the corners of A0 .

Note: The numbering throughout this paper goes from 0 to N 1, since SIAM is glad to
be on very friendly terms with the IEEE. But we still use i for
p
1! No problem anyway,
since the DCT is real.

4
We now verify that vk = (1; wk ; w2k ; : : : ; w(N 1)k ) is an eigenvector of A0 . It is periodic
because wN = 1. The j th component of A0 vk = k vk is the second di erence
 
w(j 1)k + 2wjk w(j+1)k = w k + 2 wk wjk
 
= e 2ik=N + 2 e2ik=N wjk
 
= 2 2 cos 2Nk wjk :

A0 is symmetric and those eigenvalues k = 2 2 cos 2Nk are real. The smallest is 0 = 0,
corresponding to the eigenvector v0 = (1; 1; : : : ; 1). In applications it is very useful to have
this at DC vector (direct current in circuit theory, constant gray level in image processing)
as one of the basis vectors.
Since A0 is a real symmetric matrix, its orthogonal eigenvectors can also be chosen real.
In fact the real and imaginary parts of the vk must be eigenvectors:
 
ck = Re vk = 1; cos 2Nk ; cos 4Nk ; : : : ; cos 2(N N 1)k
 
sk = Im vk = 0; sin 2Nk ; sin 4Nk ; : : : ; sin 2(N N 1)k :

The equal pair of eigenvalues k = N k gives the two eigenvectors ck and sk . The excep-
tions are 0 = 0 with one eigenvector c0 = (1; 1; : : : ; 1), and for even N also N=2 = 4 with
cN=2 = (1; 1; : : : ; 1; 1). Those two eigenvectors have length pN , while the other ck and
sk have length pN=2. It is these exceptions that make the real DFT (sines together with
p
cosines) less attractive than the complex form. That factor 2 is familiar from ordinary
Fourier series. It will appear in the k = 0 term for the DCT-1 and DCT-2, always with the
at basis vector (1; 1; : : : ; 1).

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We expect the cosines alone, without sines, to be complete over a half-period. In Fourier
series this changes the interval from [ ; ] to [0; ]. Periodicity is gone because cos 0 6=
cos . The di erential equation is still u00 = u. The boundary condition that produces
cosines is u0 (0) = 0. Then there are two possibilities, Neumann and Dirichlet, at the other
boundary:

## Zero slope: u0 () = 0 gives eigenfunctions uk (x) = cos kx 

Zero value: u() = 0 gives eigenfunctions uk (x) = cos k + 12 x :
The two sets of cosines are orthogonal bases for L2 [0; ]. The eigenvalues from u00k = uk

are  = k2 and  = k + 12 2 .
All our attention now goes to the discrete case. The key point is that every boundary
condition has two fundamental approximations. At each boundary, the condition on u
can be imposed at a meshpoint or at a midpoint. So each problem has four basic discrete
approximations. (More than four, if we open up to further re nements in the boundary
conditions | but four are basic.) Often the best choices use the same centering at the two
ends | both meshpoint centered or both midpoint centered.
In our problem, u0 (0) = 0 at one end and u0 () = 0 or u() = 0 at the other end yield
eight possibilities. Those eight combinations produce eight cosine transforms. Starting
from u(0) = 0 instead of u0 (0) = 0, there are also eight sine transforms. Our purpose is to
organize this approach to the DCT (and DST), by describing the second di erence matrices
and identifying their eigenvectors.

6
Each of the eight (or sixteen) matrices has the tridiagonal form
2 3

6
6 1 2 1 7
7
A
6
= 66 1 2 1 7
7 : (1)
6    7
7
4 1 2 1 5
 
The boundary conditions decide the eigenvectors, with four possibilities at each end: Dirich-
let or Neumann, centered at a meshpoint or a midpoint. The reader may object that
symmetry requires o -diagonal 1's in the rst and last rows. The meshpoint Neumann
condition produces 2. So we admit that the eigenvectors in that case need a rescaling at
p
the end (only involving 2) to be orthogonal. The result is a beautifully simple set of basis
vectors. We will describe their applications in signal processing.

## 2. The Discrete Cosine Transform

The discrete problem is so natural, and almost inevitable, that it is really astonishing
that the DCT was not discovered until 1974 . Perhaps this time delay illustrates an
underlying principle. Each continuous problem (di erential equation) has many discrete
approximations (di erence equations). The discrete case has a new level of variety and
complexity, often appearing in the boundary conditions.
In fact the original paper by Ahmed, Natarajan, and Rao derived the DCT-2 basis
as approximations to the eigenvectors of an important matrix, with entries jj kj. This
is the covariance matrix for a useful class of signals. The number  (near 1) measures
the correlation between nearest neighbors. The true eigenvectors would give an optimal
\Karhunen-Loeve basis" for compressing those signals. The simpler DCT vectors are close
to optimal (and independent of ).
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The four standard types of DCT are now studied directly from their basis vectors (recall
that j and k go from 0 to N 1). The j th component of the kth basis vector is
p
DCT-1: cos jk N 1 (divide by 2 when j or k is 0 or N 1)
1   p
DCT-2: cos j + 2 k N (divide by 2 when k = 0)
 p
DCT-3: cos j k + 12 N (divide by 2 when j = 0)
 
DCT-4: cos j + 21 k + 12 N

Those are the orthogonal columns of the four DCT matrices C1 , C2 , C3 , C4 . The matrix C3
with top row p12 (1; 1; : : : ; 1) is the transpose of C2 . All columns of C2 , C3 , C4 have length
p
N=2. The immediate goal is to prove orthogonality.
p
Proof! These four bases (including the rescaling by 2) are eigenvectors of symmetric
second di erence matrices. Thus each basis is orthogonal. We start with matrices A1 , A2 ,
A3 , A4 in the form (1), whose eigenvectors are pure (unscaled) cosines. Then symmetrizing
p
these matrices introduces the 2 scaling; the eigenvectors become orthogonal. Three of the
matrices were studied in an unpublished manuscript  by David Zachmann, who wrote
down the explicit eigenvectors. His paper is very useful. He noted earlier references for the
eigenvalues; a complete history would be virtually impossible.
We have seen that A0 , the periodic matrix with 1, 2, 1 in every row, shares the
same cosine and sine eigenvectors as the second derivative. The cosines are picked out by a
zero-slope boundary condition in the rst row.

## 3. Boundary Conditions at Meshpoints and Midpoints

There are two natural choices for the discrete analogue of u0 (0) = 0:

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Symmetry around the meshpoint j = 0: u 1 = u1
Symmetry around the midpoint j = 21 : u 1 = u0
The rst is called whole -sample symmetry in signal processing, the second is half -sample.
Symmetry around 0 extends (u0 ; u1 ; : : :) evenly across the left boundary to (: : : ; u1 ; u0 ;
u1; : : :). Midpoint symmetry extends the signal to (: : : ; u1 ; u0 ; u0 ; u1; : : :) with u0 repeated.
Those are the simplest re ections of a discrete vector. We substitute the two options for
u 1 in the second di erence u1 + 2u0 u 1 that straddles the boundary:
Symmetry at meshpoint: u 1 = u1 yields 2u0 2u1
Symmetry at midpoint: u 1 = u0 yields u0 u1 .
Those are the two possible top rows for the matrix A:

meshpoint:

= 2 2 and midpoint:

= 1 1:

At the other boundary there are the same choices in replacing u0 () = 0. Substituting
uN = uN 2 or uN = uN 1 in the second di erence uN 2 + 2uN 1 uN gives the two
forms for the Neumann condition in the last row of A:

## meshpoint:   = 2 2 and midpoint:   = 1 1 :

The alternative at the right boundary is the Dirichlet condition u() = 0. The meshpoint
condition uN = 0 removes the last term of uN 2 + 2uN 1 uN . The midpoint condition
uN + uN 1 = 0 is simple too, but the resulting matrix will be a little surprising. The 2
turns into 3:

## meshpoint:   = 1 2 and midpoint:   = 1 3 :

Now we have 2  4 = 8 combinations. Four of them give the standard basis functions of
cosines, listed above. Those are the DCT-1 to DCT-4, and they come when the centering
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is the same at the two boundaries: both meshpoint centered or both midpoint centered.
Zachmann  makes the important observation that all those boundary conditions give
second-order accuracy around their center points. Finite di erences are one-sided and less
accurate only with respect to the wrong center! We can quickly write down the matrices
A1 to A4 that have these cosines as eigenvectors.

## 4. The Standard Cosine Transforms

Notice especially that the denominator in the cosines (which is N 1 or N ) agrees with
the distance between \centers." This distance is an integer, measuring from meshpoint to
meshpoint or from midpoint to midpoint. We also give the diagonal matrix D that makes
D 1AD symmetric and makes the eigenvectors orthogonal:
2 3
DCT-1 2 2
6 1 2 1 7
Centers j = 0 and N 1 A
6
1=6   
7
7
Componentsp cos jk N 1 p  6
4  1 2 1
7
5
D1 = diag 2; 1; : : : ; 1; 2 2 2
2 3
DCT-2 1 1
6 1 2 1 7
Centers j = 12 and N  12 A
6
2=6   
7
7
Components cos j + 12 k N 6
4 1 2 1
7
5
D2 = I 1 1
2 3
DCT-3 2 2
6 1 2 1 7
Centers j = 0 and N  A
6
3=6   
7
7
Componentsp cos j k + 12 N 6
4 1 2 1
7
5
D3 = diag( 2; 1; : : : ; 1) 1 2

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2 3
DCT-4 1 1
6 1 2 1 7
Centers j = 12 and N  12  A
6
4=6   
7
7
Components cos j + 12 k + 21 N 6
4 1 2 1
7
5
D4 = I 1 3

Recently Sanchez et al  provided parametric forms for all matrices that have the DCT
bases as their eigenvectors. These are generally full matrices of the form \Toeplitz plus
near-Hankel." Particular tridiagonal matrices (not centered di erences) were noticed by
Kitajima, Rao, Hou, and Jain. We hope that the pattern of second di erences with di erent
centerings will bring all eight matrices into a common structure. Perhaps each matrix
deserves a quick comment.

## DCT-1: The similarity transformation D1 1 A1 D1 yields a symmetric matrix. This mul-

tiplies the eigenvector matrix for A1 by D1 1 . (Notice that Ax = x leads to
(D 1 AD)D 1 x = D 1 x.) The eigenvectors become orthogonal for both odd N and even
p
N , when D1 1 divides the rst and last components by 2:
     
N =3 p12 ; 1; p12 p12 ; 0; p12 p12 ; 1; p12 for k = 0; 1; 2
   
N =4 p12 ; 1; 1; p12 : : : p12 ; 1; 1; p12 for k = 0; 1; 2; 3 :
p p
The rst and last eigenvectors have length N 1, the others have length (N 1)=2.

DCT-2: These basis vectors cos j + 21  k N are the most popular of all, because k = 0
gives the at vector (1; 1; : : : ; 1). Their rst and last components are not exceptional. The
boundary condition u 1 = u0 is a zero derivative centered on a midpoint. Similarly the
right end has uN = uN 1 . When those outside values are eliminated, the boundary rows of
A2 have the neat 1 and 1.
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c0 DC Att. >= 310.6215 dB Mirr Att. >= 320.1639 dB Stopband Att. >= 9.9559 dB Cod. Gain = 8.8259 dB

c1
0

c2
−5

−10

c3

## Magnitude Response (dB)

−15

c4 −20

c5 −25

c6
−30

c7
−35

−40
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
Normalized Frequency

Figure 1: The eight DCT-2 vectors and their Fourier transforms (absolute values).

I believe that this DCT-2 (often just called the DCT) should be in applied mathematics
courses along with the DFT. Figure 1 shows the eight basis vectors (when N = 8). On
the right are the Fourier transforms of those vectors. Maybe you can see the rst curve
je2ij=8 j and especially its second lobe, rising to 13 decibels (which is 20 log10 13) below
the top. This is not a big dropo ! Like the closely connected Gibbs phenomenon, it does
not improve as N increases. A good lowpass lter can drop by 40 or 50 db. The other seven
transforms vanish at zero frequency (no leakage of the direct current DC term). Those
seven vectors are orthogonal to (1; 1; : : : ; 1).

This basis was chosen for the JPEG algorithm in image compression. Each 8  8 block in
the image is transformed by a two-dimensional DCT. We comment below on the undesirable
blocking artifacts that appear when the transform coecients are compressed.

DCT-3: The vectors cos j k + 12  N are the discrete analogues of cos(k + 21 )x. The Neu-
mann condition at the left and Dirichlet condition at the right are centered at meshpoints.
p
For orthogonality we need the factor D3 1 that divides the rst components by 2. This
basis loses to the DCT-4.
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DCT-4: We had never seen the nal entry \3" in the matrix A4 but MATLAB insisted it
was right. Now we realize that a zero boundary condition at a midpoint gives uN  uN 1

## (the extension is antisymmetric ). Then 1, 2, 1 becomes 1, 3. The eigenvectors are

even at the left end and odd at the right end. This attractive property leads to j + 21 and
k + 21 and a symmetric eigenvector matrix C4. Its applications to \lapped transforms" are
described below.

Remember our proof of orthogonality! It is a veri cation that the cosine vectors are
eigenvectors of A1 , A2 , A3 , A4 . For all the 1, 2, 1 rows this needs to be done only
once (and it reveals the eigenvalues  = 2 2 cos ). There is an irreducible minimum of
trigonometry when the j th component of the kth vector ck is cos j in Types 1 and 3, and
cos(j + 12 ) in Types 2 and 4:

## cos(j 1) + 2 cos j cos(j + 1) = (2 2 cos ) cos j

       
cos j 1  + 2 cos j + 1  cos j + 3  = (2 2 cos ) cos j + 1  :
2 2 2 2
This is Ack = k ck on all interior rows. The angle is  = k N 1 for Type 1 and  = k N for
 
Type 2. It is  = k + 12 N for A3 and A4 . This leaves only the rst and last components
of Ack = k ck to be veri ed in each case.
Let us do only the fourth case, for the last row 1, 3 of the symmetric matrix A4 . A
last row of 1, 1 would subtract the j = N 2 component from the j = N 1 component.
Trigonometry gives those components as
    
j=N 1: cos N 1
2 k + 21 N = sin 12 k + 21 N
    
j=N 2: cos N 3
2 k + 21 N = sin 32 k + 21 N :

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We subtract using sin a sin b = 2 cos b+a  sin b a . The di erence is
2 2
   
2 cos k + 12 N sin 12 k + 21 N : (2)

The last row of A4 actually ends with 3, so we still have 2 times the last component
(j = N 1) to include with (2):
     
2 2 cos k + 21 N sin 12 k + 21 N : (3)

This is just k times the last component of ck . The nal row of A4 ck = k ck is veri ed.
There are also Discrete Sine Transforms DST-1 through DST-4. The entries of the basis
vectors sk are sines instead of cosines. These sk are orthogonal because they are eigenvectors
of symmetric second di erence matrices, with a Dirichlet (instead of Neumann) condition at
the left boundary. In writing about the applications to signal processing , we presented
a third proof of orthogonality|which simultaneously covers the DCT and the DST, and
shows their fast connection to the DFT matrix of order 2N . This is achieved by a neat
matrix factorization given by Wickerhauser :
 
e i=4N T R F2N R = C04 0
iS4 :
The entries of S4 are sin(j + 21 )(k + 12 ) N . The connection matrix R is very sparse, with
w = ei=2N :
 
D = diag(1; w;
 : : : ; wN 1 )
R = p1 D D with
2 E E E = antidiag(w; w2 ; : : : ; wN ) :
Since RT and F2N and R have orthogonal columns, so do C4 and S4 .

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5. Cosine Transforms with N 1
2 and N + 21
There are four more combinations of the discrete boundary conditions. Every combination
that produces a symmetric matrix will also produce (from the eigenvectors of that matrix)
an orthogonal transform. But you will see N 1 and N + 21 in the denominators of the
2
cosines, because the distance between centers is no longer an integer. One center is a
midpoint and the other is a meshpoint.
The transforms DCT-5 to DCT-8, when they are spoken of at all, are called \odd."
They are denoted by DCT-I O to DCT-IV O in  and . Three of the tridiagonal matrices
(A5 ; A6 ; A8 ) are quite familiar:
2 3
DCT-5 2 2
Centers j = 0 and N 12
6
6 1 2 1 7
7
Components cos jk N  1 A 5=6
6    7
7
p 2
4 1 2 1 5
D5 = diag( 2; 1; : : : ; 1) 1 1
2 3
DCT-6 1 1
Centers j = 21 and N  1
6
6 1 2 1 7
7
Components cos j + 21 k N  1 A
6=6
6    7
7
p 2
4 1 2 1 5
D6 = diag(1; : : : ; 1; 2) 2 2
2 3
DCT-7 2 2
Centers j = 0 and N 21 
6
6 1 2 1 7
7
Components cos j k + 12 N  1 A 7=6
6    7
7
p 2
4 1 2 1 5
D7 = diag( 2; 1; : : : ; 1) 1 3
2 3
DCT-8 1 1
Centers j = 21 and N 
6
6 1 2 1 7
7

Components cos j + 21 k + 12 N + 1 A
8=6
6    7
7 :
2
4 1 2 1 5
D8 = I 1 2

15
We could study A8 by re ection across the left boundary, to produce the pure Toeplitz 1,
2, 1 matrix (which is my favorite example in teaching). The eigenvectors become discrete
sines on a double interval|almost. The length of the double interval is not 2N , because
the matrix from re ection has odd order. This leads to the new \period length" N + 21 in
the cosines.
Notice that A5 has the boundary conditions (and eigenvector components) in reverse
order from A6 . The rst eigenvectors of A5 and A6 are (1; 1; : : : ; 1), corresponding to k = 0
and  = 0. This \ at vector" can represent a solid color or a xed intensity by itself (this
is terri c compression). The DCT-5 and DCT-6 have a coding gain that is completely
comparable to the DCT-2.
p
So we think through the factors that come from D6 = diag(1; : : : ; 1; 2). The sym-
p
metrized D6 1 A6 D6 has 2 in the two lower right entries, where A6 has 1 and 2. The
p
last components of the eigenvectors are divided by 2; they are orthogonal but less beau-
tiful. We implement the DCT-6 by keeping the matrix C6 with pure cosine entries, and
accounting for the correction factors by diagonal matrices:

4  
2N 1 C6 diag 12 ; 1; : : : ; 1 C6T diag 1; : : : ; 1; 12 = I: (4)

The cosine vectors have squared length 2N 1 , except the all-ones vector that is adjusted
4
by the rst diagonal matrix. The last diagonal matrix corrects the N th components as D6
requires. The inverse of C6 is not quite C6T (analysis is not quite the transpose of synthesis,
as in an orthogonal transform) but the corrections have trivial cost. For N = 2 and k = 1,
the matrix identity (4) involves cos 21 3=2 = 12 and cos 23 3=2 = 1:
 1  1     
4 1 2 2 1 1 1 = 1 1 :
3 1 1 1 1 1 1
2 2

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Malvar has added a further good suggestion: Orthogonalize the last N 1 basis vectors
against the all-ones vector. Otherwise the DC component (which is usually largest) leaks
into the other components. Thus we subtract from each c6k (with k > 0) its projection onto
the at c60 :
c~6k = c6k ( 2N1) (1; 1; : : : ; 1) :
k
(5)

The adjusted basis vectors are now the columns of Ce6 , and (5) becomes
2 1 +1 3
1 2N 2N :::
6 1 7
C6 = Ce6 664 ...
7
7
5
:
1
This replacement in equation (4) also has trivial cost, and that identity becomes Ce6 Ce6 1 = I .
The coecients in the cosine series for x are y = Ce6 1x. Then x is reconstructed from Ce6 y
(possibly after compressing y). You see how we search for a good basis. . .
Transforms 5 to 8 are not used in signal processing. The half-integer periods are a
disadvantage, but re ection o ers a possible way out. The re ected vectors have an integer
\double period" and they overlap.

6. Convolution
The most important algebraic identity in signal processing is the convolution rule. A slightly
awkward operation in the time domain (convolution, from a Toeplitz matrix or a circulant
matrix) becomes beautifully simple in the frequency domain (just multiplication). This
accounts for the absence of matrices in the leading textbooks on signal processing. The
property of time invariance (delay of input simply delays the output) is always the starting
point.

17
We can quickly describe the rules for doubly in nite convolution and cyclic convolution.
A vector h of lter coecients is convolved with a vector x of inputs. The output is y = h  x
with no boundary and y = h c x in the cyclic (periodic) case:
1
X X
yn = hk xn k or yn = hk x` : (6)
1 k+`n(mod N )

## Those are matrix-vector multiplications y = H x. On the whole line (n 2 Z) the doubly

in nite matrix H is Toeplitz; the number hk goes down its kth diagonal. In the periodic
case (n 2 ZN ) the matrix is a circulant; the kth diagonal continues with the same hk onto
the (k N )th diagonal. The eigenvectors of these matrices are pure complex exponentials.
So when we switch to the frequency domain, the matrices are diagonalized. The eigenvectors
are the columns of a Fourier matrix, and F 1 HF is diagonal. Convolution with h becomes
multiplication by the eigenvalues H (!) in the diagonal matrix:
P1  P1  P1
1 hk e 1 x` e = 1 yn e is H (!)X (!) = Y (!) (7)1
ik! i`! in!
P  P  PN 1
N 1 1x
0 hk wk N
0 ` w` = 0 ynwn is H (w)X (w) = Y (w) (7)N

The in nite case (Discrete Time Fourier Transform) allows all frequencies j!j  . The
cyclic case (DFT) allows the N roots of wN = 1. The multiplications in (7) agree with
the convolutions in (6) because e ikx e i`x =e i(k+`)x and wk w` = wk+` . The question is:
What convolution rule goes with the DCT?
A complete answer was found by Martucci . The nite vectors h and x are symmet-
rically extended to length 2N or 2N 1, by re ection. Those are convolved in the ordinary
cyclic way (so the double length DFT appears). Then the output is restricted to the original
N components. This symmetric convolution h s x corresponds in the transform domain to
multiplication of the cosine series.
18
The awkward point, as the reader already knows, is that a symmetric re ection can
match u 1 with u0 or u1 . The centering can be whole sample or half sample at each
boundary. The extension of h can be di erent from the extension of x! This con rms again
that discrete problems have an extra degree of complexity beyond continuous problems.
(And we resist the temptation to compare combinatorics and linear algebra with calculus.)
In the continuous case, we are multiplying two cosine expansions. This corresponds to
symmetric convolution of the coecients in the expansions.

## 7. The DCT in Image Processing

Images are not in nite, and they are not periodic. The image has boundaries, and the
left boundary seldom has anything to do with the right boundary. A periodic extension
can be expected to have a discontinuity. That means a slow decay of Fourier coecients
and a Gibbs oscillation at the jump | the one place where Fourier has serious trouble!
In the image domain this oscillation is seen as \ringing." The natural way to avoid this
discontinuity is to re ect the image across the boundary. With cosine transforms, a double
length periodic extension becomes continuous.
A two-dimensional image may have (512)2 pixels. The gray level of the pixel at position
(i; j ) is given by an integer x(i; j ) (between 0 and 255, thus 8 bits per pixel). That long
vector x can be ltered by x  h, rst a row at a time (j xed) and then by columns (using
the 1D transforms of the rows). This is computationally and algebraically simplest: the 2D
Toeplitz and circulant matrices are formed from 1D blocks.

19
Similarly the DCT-2 is applied to rows and then to columns; 2D is the tensor product
of 1D with 1D. The JPEG compression algorithm (established by the Joint Photographic
Experts Group) divides the image into 8  8 blocks of pixels. Each block produces 64 DCT-2
coecients. Those 64-component vectors from the separate blocks are compressed by the
quantization step that puts coecients into a discrete set of bins. Only the bin numbers
are transmitted. The receiver approximates the true cosine coecient by the value at the
middle of the bin (most numbers go into the zero bin). Figures 2{5 show the images that
the receiver reconstructs at increasing compression ratios and decreasing bit rates:

## 1. The original image (1:1 compression, all 8 bits per pixel)

2. Medium compression (8:1, average 1 bit per pixel)
3. High compression (32:1, average 41 bit per pixel)
4. Very high compression (128:1, average 161 bit per pixel)
You see severe blocking of the image as the compression rate increases. In teleconferencing at
a very low bit rate, you can scarcely recognize your friends. This JPEG standard for image
processing is quick but certainly not great. The newer standards allow for other transforms,
with overlapping between blocks. The improvement is greatest for high compression.
One personal comment on quantization: This more subtle and statistical form of round-
o should have applications elsewhere in numerical analysis. Numbers are not simply
rounded to fewer bits, regardless of size. Nor do we sort by size and keep only the largest
(this is thresholding, when we want to lose part of the signal | it is the basic idea in
denoising). The bit rate is controlled by the choice of bin sizes, and quantization is surpris-
ingly cheap. Vector quantization, which puts vectors into multidimensional bins, is more
20
expensive but in principle more ecient. This technology of coding is highly developed 
and it must have more applications waiting to be discovered.
A major improvement for compression and image coding was Malvar's extension of the
ordinary DCT to a lapped transform. Instead of dividing the image into completely separate
blocks for compression, his basis vectors overlap two or more blocks. The overlapping has
been easiest to develop for the DCT-4, using its even-odd boundary conditions|which the
DCT-7 and DCT-8 share. Those conditions help to maintain orthogonality between the
tail of one vector and the head of another. The basic construction starts with a symmetric
lowpass lter of length 2N . Its coecients p(0); : : : p(2N 1) are modulated (shifted in
frequency) by the DCT-4:
 
The kth basis vector has j th component p(j ) cos (k + 12 )(j + N2+1 ) N .

There are N basis vectors of length 2N , overlapping each block with the next block. The
1D transform matrix becomes block bidiagonal instead of block diagonal. It is still an
orthogonal matrix [4, 9] provided p2 (j ) + p2 (j + N ) = 1 for each j . This is Malvar's
Modulated Lapped Transform, which is heavily used by the Sony mini disc and Dolby AC-3.
(It is included in the MPEG-4 standard for video.) We naturally wonder if this MLT basis
is also the set of eigenvectors for an interesting symmetric matrix. Coifman and Meyer
found the analogous construction  for continuous wavelets.
The success of any transform in image coding depends on a combination of properties|
mathematical, computational, and visual. The relation to the human visual system is
decided above all by experience. This article was devoted to the mathematical property of
orthogonality (which helps the computations). There is no absolute restriction to second

21
di erence matrices, or to these very simple boundary conditions. We hope that the eigen-
vector approach will suggest more new transforms, and that one of them will be fast and
visually attractive.

DCT http://www.cis.ohio-state.edu/hypertext/faq/usenet/

## Author http://www-math.mit.edu/ gs/ 

References
 N. Ahmed, T. Natarajan, and K. R. Rao, Discrete cosine transform, IEEE Trans.
on Computers, C-23 (1974) 90{93.

 R. Coifman and Y. Meyer, Remarques sur l'analyse de Fourier a fen^etre, C. R. Acad.
Sci. Paris 312 (1991) 259{261.

##  N. J. Jayant and P. Noll, Digital Coding of Waveforms, Prentice-Hall (1984).

 H. S. Malvar, Signal Processing with Lapped Transforms, Artech House (1992).

 S. Martucci, Symmetric convolution and the discrete sine and cosine transforms,
IEEE Trans. Signal Processing 42 (1994) 1038{1051.

 K. R. Rao and P. Yip, Discrete Cosine Transforms, Academic Press (1990).

22
 V. Sanchez, P. Garcia, A. Peinado, J. Segura, and A. Rubio, Diagonalizing prop-
erties of the discrete cosine transforms, IEEE Trans. Signal Processing 43 (1995)
2631{2641.

 G. Strang, The search for a good basis, in Numerical Analysis 1997, D. Griths, D.
Higham, and A. Watson, eds., Pitman Lecture Notes, Addison Wesley Longman.

 G. Strang and T. Nguyen, Wavelets and Filter Banks, Wellesley-Cambridge Press
(1996).

 Z. Wang and B. Hunt, The discrete W-transform, Appl. Math. Comput. 16 (1985)
19{48.

(1994).

##  D. Zachmann, Eigenvalues and eigenvectors of nite di erence matrices, unpub-

lished manuscript (1987), http:// .

23
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