IEEE Circuits and Systems Magazine - Q4 2022 - 59

be symmetrically extended at its endpoint in two ways.
First, the extended sequence is symmetric about the end
point itself, referred to as the whole-sample symmetric
(WS) extension [5]. Second, the symmetric extension is
centered about the midpoint between the end sample,
referred to as the half-sample symmetric (HS) extension.
Likewise, two types of antisymmetric extensions about
a sample point of a sequence are possible, referred to as
the whole-sample antisymmetric (WA) extension and the
half-sample antisymmetric extension. This results in 16
distinct types of periodic extensions, of which 8 are symmetric-periodic,
leading to 8 different types of DCTs, and
8 are antisymmetric-periodic, leading to 8 different types
of DSTs. In this article, the focus is on DCTs, especially
the Type-I and Type-II DCTs, due to the advantages their
properties offer and their applicability in a wide range of
applications. The governing equations for the Type-I DCT
XI and Type-II DCT XII are given below [6]:
Xk
I
22¦D2 kx n
n 0
N
N
Xk
II
where
D k
-
®
°
°¯
1,
1
2
,
ifkN
otherwise.
0 or
(3)
If the sequence x(n)(n = 0, 1, ...N − 1) is represented by Ndimensional
column vector x. The Type-II DCT of x can
be expressed in matrix form as
X = CN · x
(4)
where the (k, n)th element of CN is given by
CN kn
,,
,.
01
D
N
k cos
kn N
dd
22 1
kn
N
S
2
(5)
As in the conventional literature, by the word DCT,
it implies Type-II DCT, we also follow the same convention.
A Fast DCT (FDCT) algorithm [7] following an approach
similar to Fast Fourier Transform (FFT) exists,
i.e., by splitting the DCT sum into odd and even terms,
making DCT popular in the research community.
3. DCT Properties
The popularity of the Type-II DCT is due to the number
of interesting properties [5] as summarized below:
1) Linearity and Orthogonality: Since DE
xn yno
hi
fourth QuartEr 2022
Xk YkDCTDCT holds, DCT is said to be
22¦D kx n
N
n
N
1
cos, dd0 kN
2
S
nk
N
(1)
cos, dd
2
S
kn
N
1
kN
01
(2)
satisfy linearity. Linear DCT transform matrix C is
real and orthogonal i.e. C−1 = CT where
ck n
,
-
®
°
°¯
D
D
,
k
cosotherwise .
00
21
2
for k
S
kn
N
,
(6)
DCT is related to the DFT of a symmetrically extended
signal, which gives less discontinuity at
the boundaries and better energy compaction,
unlike the DFT, which introduces discontinuities.
2) Energy Conservation and Decorrelation: Consider
the transform as y = Cx in vector form. Energy
conservation property is derived as follows.
 
yCxCxCxC Cx
22
22 2 x
()TT T
T
22xx x 2
(7)
A large part of signal energy is packed in a few
transform coefficients, typically in the low-frequency
range, commonly known as the energy
compaction property. The decorrelation property
removes the high amount of correlation that exists
in the spatial domain signal when DCT is applied
to it. This can be validated by estimating the
covariance matrix of the transformed signal, i.e.,
E[(y − E(y))(y − E(y))T]where the off-diagonal elements
in the covariance matrix tend to be small.
Due to this property, the angles between vectors
are preserved.
3) Matrix Factorization: CN, the N-point DCT matrix can
be factored into a product of a few sparse matrices.
CPN
N
1
2
ª
¬
«
«
«
CN
2
2
JC JNN N
22 2
º
¼
»
»
»
ª
¬
«
«
«
IJ
JI
NN
NN
22
22
º
¼
»
»
»
(8)
where, PN, IN, JN, and 0N are the N × N permutation
matrix, identity, reverse identity, and zero
matrices, respectively. This property is useful in
designing fast matrix multiplication operations for
various algorithms.
4) Subband Relationship and Approximate DCT Computation:
Let the sequence x(n), 0 ≤ n ≤ (N − 1) be an
N-point sequence with even N. Let x(n) be decomposed
into two subbands xL(n) and xH(n) of length
N
2
each as follows:
xn
L
xn
H
xn xn
xn xn
22 1
2
22 1
2
,, 1 } .
2
n 01,
N
(9)
IEEE cIrcuIts and systEms magazInE
59
IEEE Circuits and Systems Magazine - Q4 2022

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