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102 Chapter 3 Digital Signal Processing
where
Note that Equations(3.27) and (3.28), which describe the symmetric cosine
transform, have identical forms. Hence, only one algorithm needs to be imple-
mented. Further, the rows and columns are either symmetric or antisymmetric.
This fact can be exploited to reduce the number of arithmetic operations as well as
the hardware cost.
Unfortunately, the DC component will appear in some of the higher-frequency
components, thus rendering the SDCT unsuitable for image coding applications.
3.19.4 MSDCT (Modified Symmetric Discrete Cosine
Transform)
The MSDCT (modified discrete symmetric DCT) was developed by Sikstrom et al.
[27, 30, 37, 38] for transform coding. Its distinguishing feature is that the basis
vectors are symmetric or antisymmetric and that the DC component that may oth-
erwise appear as an error in the other frequency components is suppressed at the
expense of a slightly nonorthogonal transform. The fact that the transform is not
strictly orthogonal causes an error, but it is much smaller than the error caused by
quantization of the data word length by one bit. Hence, this effect is in practice
negligible. Further the forward and inverse transforms are identical.
The MSDCT (modified symmetric discrete cosine transform) is defined as
The IMSDCT — Inverse MSDCT is
where
Equations (3.29) and (3.30), which describe the modified symmetric cosine
transform and its inverse, also have identical forms [30, 34, 37, 45]. Hence, it is
necessary to implement only one algorithm.
EXAMPLE 3.8
Show that the MSDCT is not orthogonal. Use the MSDCT with N = 8 for the sake
of simplicity.
