# DCT step in MFCC process

During MFCC process FFT is done to get the power spectrum but last step is DCT in order to get back to time domain. Cannot DCT be used in the first case to convert the signal to frequency domain?

DCT is not used only for going back to time domain, it is also used due to it's "de-correlating properties". By saying that I mean compression of most of the information in smaller number of coefficients. Although in theory we could use the DCT to calculate the spectrum, but I never saw anyone doing that. Usually thing is simple:

• want to analyse the spectrum? Use DFT
• want to compress and de-correlate data? Use some flavour of DCT

Also one very important thing. In case of DCT, the equivalence property of convolution and multiplication in frequency domain doesn't hold: $$x[n]\star y[n] \nLeftrightarrow X_{DCT}[k]\cdot Y_{DCT}[k]$$ which is the fundamental property (homomorphic) used when calculating the MFCC's. It allows to perform the Cepstral Mean Normalization and remove the "convolutional distortions", such as vocal tract. For more info please refer to my previous answer.

• Thanks @jojek for your reply.But I want to know if there are problems in using DCT for power spectrum.Why people have not use DCT for MFCC as DCT is computationally faster than FFT?
– mun
Dec 9 '14 at 15:15