Discrete Cosine Transform - Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

DCT (Discrete Cosine Transform) is an N-input sequence x(n) , 0≤n≤N-1 , as a linear transformation or combination of complex exponentials. As a result, the DFT coefficients are in general, complex even if x(n) is real.

Suppose, we try to find out an orthogonal transformation which has N×N structure that expressed a real sequence x(n) as a linear combination of cosine sequence. We already know that −

And 

This is possible if N point sequence x(n) is real and even. Thus,   The resulting DFT itself is real and even. These things make it clear that we could possibly device a discrete cosine transform, for any N point real sequence by taking the 2N point DFT of an “Even extension” of sequence.

DCT is, basically, used in image and speech processing. It is also used in compression of images and speech signals.

DCT is defined by,