# 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?

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.