IEEE Circuits and Systems Magazine - Q4 2022 - 60

The relationship between the DCT of subbands (of
N
2
Xk
LH
2
2
cossin
SS
k
N
where,
Xk
L 0,
-
®
°
°°
¯
°
°
°
and
-Sk
Sk 21 xn k
n 0
H ¦()n
®
°
°
°
°
¯
°
°
°
°
SN k ,
H
,
2
H
N
2
H,
N
1
N
2
d k N 1.dd
1
(12)
Since DCT packs most of the energy in the low-frequency
coefficients, Eq. (10) can be approximated
to contain only the cosine term referred as subband
approximation of DCT given below.
Xk
|
-
®
°
°¯
2cos Xk k 01
2
S
0,
k
N
L }1
2
N
,, ,
otherwise .
(13)
5) Decimation and Interpolation Property: These properties
are useful for decimating signals directly in
the transform domain as well as carrying out the
interpolation operations. Let us denote Type-I DCT
and Type-II DCT of x(n) as X1(k) = C1(x(n)), k.n = 0,
1, 2, ..., N and Xk Cx nk n
II
2 } ,,
012
N
,, ,, ,
2
respectively, where C1(.) and C2(.), refers to the
corresponding transformation functions as given
in Eqs. (1) and (2). Then, the decimation is
expressed as
xn xn C
21
d 2
kn 012
,, ,, ,
}
N
2
1 Xk XN k
II
2
1
(14)
Similarly, the interpolation is expressed as
xn C
u 1
1 Xk XN k
2
II
kn 012
60
IEEE cIrcuIts and systEms magazInE
,, ,, , }N
^`
II
,
(15)
^`
,
Cu nNCx lC ym nl mN11 120
Cu nNCx lC ym
nl Nm N
22 1
010
2
ddd d
,,
,
(18)
7) Integer Cosine Transforms (ICT): ICTs are derived
from a DCT matrix as its integer approximations
and preserves its properties of orthogonality,
symmetry, relative order, and sign of the elements
of the matrix. The motivation for utilization of
ICTs is to achieve fast computations. For example,
H.264 video codec utilizes a 4-point integer transform
which is useful during matrix multiplication
operations.
4. Applications
The DCT is a core technology behind almost all international
standards for image and video compression,
including the ISO JPEG image compression standard
(1992) and a series of video coding standards from ITU
and ISO: H.261 (1988), MPEG-1 (1993), MPEG-2 (1995),
H.263 (1988), MPEG-4 (1998), H.264/AVC (2003), H.265/
HEVC (2013). The impact of these standards in todays
life is immense. Every digital camera and mobile phone
camera supports the JPEG format, which is also used
in essentially all websites. Other image formats using
wavelets exist (JPEG 2000), but the low computation
cost of JPEG, thanks to efficient DCT algorithms
and its good compression performance, made JPEG
much more popular. The impact of the ISO/ITU-T video
formats based on the DCT is immense: early digital
television and DVDs used the MPEG-2 format, and early
videoconferencing systems used the H.263 format. Today,
almost all digital video systems use either H.264
fourth QuartEr 2022
,, ,
dd
(17)
0dd1
2
k
N
,
Xk ,
L
k
dd
k
N
2
XN k 11
2
Ld d
N
kN
(11)
N
2
1
Xk
2
2
k
N
where,
point), XL(k) and SH(k), with the original DCT
(of N point) of the sequence, X(k), is given by
Sk kN,01 dd
(10)
xn
u
-
®
°
°¯
0,
§
©
x¨
n 1
2
·
¹
¸,
n
n
even
odd
(16)
6) Convolution-Multiplication Property: Convolution
between two sequences x(n) and y(n) is defined
as the convolution between their symmetrically
extended sequences of the same general period.
When both the extended sequences are periodic,
the convolution operation is defined as the
circular convolution.
If both are antiperiodic,
the operation is defined as the skew-circular
convolution. Let un xn yn xn
2
 ,
and
y(n) are such that their symmetric extensions
produce a general period of 2 N. The convolution
multiplication property for DCT-I and DCTII
are given as
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