Is there mathematical relationship between the FFT and DCT transforms

1. What is FFT and DCT?
2. How to calculate DCT?
3. Is DCT The real part of DFT?
4. What is the main difference between DCT and DFT?

What is FFT and DCT?

FFT is the algorithm which is used for the fast computation of DFT. DCT is similar to the discrete Fourier transform (DFT), but using only real numbers (cosine as twiddle factor).

How to calculate DCT?

The DCT Transform Matrix

The two-dimensional DCT of A can be computed as B=T*A*T' . Since T is a real orthonormal matrix, its inverse is the same as its transpose. Therefore, the inverse two-dimensional DCT of B is given by T'*B*T .

Is DCT The real part of DFT?

DCT is NOT the real part of the DFT rather it is related to the DFT of a symmetrically extended signal/image. The energy of signal/image is packed mostly in only a few DCT coefficients (i.e. only a few significant X(k)'s), hence making DCT very useful for data compression applications.

What is the main difference between DCT and DFT?

DCTs are equivalent to DFTs of roughly twice the length, operating on real data with even symmetry (since the Fourier transform of a real and even function is real and even), whereas in some variants the input or output data are shifted by half a sample. There are eight standard DCT variants, of which four are common.