# How does this Odd stacking DFT work?

I have just read on page 3 of this paper:

https://www.onsemi.com/site/pdf/Filtering_Hearing_Ai.pdf

That an "odd stacking" DFT can be created, essentially sacrificing the DC and Nyquist bands to offset the band centre frequencies by $Fs/(2N)$.

For the synthesis it describes the math as:

$Y[k] = \sum^{n=0}_ {N-1} y(n) W^{-(k+v)n}, k=0,1,...,N-1$

where v is 0 for an even-stacking DFT and 0.5 for an odd-stacking DFT.

So it appears to just be a $\pi$ phase offset on the complex exponential used on the DFT.

Have I interpreted this correctly? Can anyone explain further how this causes the offsetting of the frequency bands?

• Actually, I see that I did not interpret is correctly! The the complex exponential $W$ is being raised to the power of $-(k+v)n$, meaning that the offset of 0.5 will increase the frequency by 0.5 of the bandwidth. – kippertoffee Jun 6 '18 at 12:48
• So I see how this works in a DFT. For bonus points, could anyone comment on how this would be implemented in an FFT, where there is no explicit DFT matrix involved? – kippertoffee Jun 6 '18 at 12:50