# “Undersampled” Polyphase Filter Bank

I'm working on an application where I want to grab high-resolution pieces of the spectrum, but I don't need the whole spectrum-- just a few bands. I've looked into using the Goertzel algorithm (too slow; with the number of points we want the regular FFT is faster), pruned FFTs (doesn't look like much of an improvement in performance), and polyphase filter banks (PFBs) (the problem being that it requires an FFT of the same size that I would have otherwise already computed, defeating the point).

The books that I've looked at so far ("Digital Signal Processing, 4th ed" by Proakis/Manolakis and "Signal Analysis: Wavelets, Filter Banks, Time-Frequency Transforms and Applications" by Mertins) both focus on either "critically sampled", where the decimation rate $D$ equals the number of subbands $N$ or "oversampled" where $N > D$.

Is there a way to produce a computationally efficient PFB such that $D > N$? In my application the loss of information is unimportant because I only need a few subbands of small bandwidth.

• You might be interested in the zoom FFT. – Jason R Jun 12 '15 at 18:40

1. Use an IQ mixer to translate the centre of the slice, $F_c$, to 0 Hz. This is done by multiplying the signal with $e^{-j2\pi F_c/F_s}$, where $F_s$ is the sample frequency.