# Conversion between DCT to DFT frequency domains

It is well known that the DFT can be interpreted as a filter bank. The same can be done for the DCT, for instance Fig. 3.29, pag. 129 in [1] shows a plot of the frequency response of a DCT filter bank. I am wondering how this plot (and similar plots for other transforms like DST, etc.) was obtained. For the DFT, we have the prototype filter $H_0(z)=\sum_{n=0}^{N-1}z^{-n}$. The frequency response of the $k$-th DFT bin is $H_k(z)=H_0(z\, e^{-j\frac{2\pi}{N}k})$. Using the transformation $z=e^{j\omega}$ we can readily find the frequency response for all bins (see the plot below).

As far as I know there is no z-transform equivalent for the DCT, so it's not clear to me how that plot was obtained. Any ideas?

[1] Malvar, H.S., Signal processing with lapped transforms, 1992, Artech House

• Not sure if I understand your confusion, but as far as I can see, it is simply the prototype filter and all its cosine-modulated versions. The prototype is rectangular in the time domain, i.e. a sinc function in the frequency domain. – Matt L. May 10 '13 at 9:32