IEEE Circuits and Systems Magazine - Q4 2022 - 58

CAS 101
manish okade (national Institute of technology, rourkela)
and Jayanta mukherjee (Indian Institute of technology, Kharagpur)
Discrete Cosine Transform: A Revolutionary Transform
That Transformed Human Lives
t
he widely known Discrete Fourier Transform
(DFT), the frequency-domain representation of a
finite-length time-domain sequence is an orthogonal
transform and has been known for a very long time
and has found many application. The popularity of the
DFT increased tremendously after the publication of the
Fast Fourier Transform (FFT) algorithm by Cooley and
Tukey in 1965 [1]. Orthogonal transforms offer many advantages,
namely fast computational speeds, less storage
space, less rounding off errors, etc. The benefits of
the orthogonal transforms stem from the fact that they
can be factored based on matrix computations. Data
compression, digital
image and video processing are applications where the
orthogonal transforms have made a significant impact.
Unfortunately, the FFT samples are complex-valued
functions even though the original time-domain sequence
has only real-valued samples. In 1974, Nasir
Ahmed with T. Natarajan and K. Rao, proposed the
Discrete Cosine Transform (DCT), also an orthogonal
transform, which is a real-valued frequency-domain
representation of a finite-length time-domain sequence
with real-valued samples [2]. DCT became extremely
popular due to its energy compaction property, data
decorrelation property, and availability of a fast transform
to speed up the computations. Moreover, DCT
comes very close compared to the Karhunen-Loeve
Transform (KLT), which to date is the optimal transform
in the mean square sense yet is limited due to the lack of
a fast transform to compute it as it is signal dependent.
Although it had a humble beginning in the sense that it
was being investigated with the motivation to achieve
an approximation to the KLT along with being computationally
faster, yet the reach and the scale that DCT has
achieved in the 21st century is beyond words. The interesting
story of how DCT was developed can be found in
[3]. Nasir Ahmed's account on the genesis of the DCT
can be seen in the video " The Algorithm That Transform
The World: The Story Of Nasir Ahmed " ' (https://www.
youtube.com/watch?v=I9VXaVVs7WY).
Digital Object Identifier 10.1109/MCAS.2022.3214566
Date of current version: 13 January 2023
58
IEEE cIrcuIts and systEms magazInE
1. Transform Coding
filter design, speech processing,
In transform coding, N discrete-time data samples are firstly
transformed, then the coefficients are individually quantized
and entropy coded into binary bits serving two purposes:
(i) to compact the energy of the original N samples
into coefficients with increasingly smaller variances so
that removing smaller coefficients have negligible reconstruction
errors; and (ii) to decorrelate the original samples
https://www.youtube.com/watch?v=I9VXaVVs7WY https://www.youtube.com/watch?v=I9VXaVVs7WY
IEEE Circuits and Systems Magazine - Q4 2022

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