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Computational Savings of Polyphase/DFT Filterbanks

Module by: Phil Schniter

Summary: This module looks at the computational savings of the polyphase/DFT modulated filterbank implementation by comparing the number of computations performed for various methods.

This module will briefly take a look a the computational savings of the polyphase/DFT modulated filterbank implementation. We will start by looking at our standard analysis bank and then move on to compare this with our other implementation approaches.
Assume that the lowpass filter in the standard analysis bank, Hz H z , has impulse response length NN. To calculate the sub-band output vector { y k m|k=0…M-1} y k m k 0 … M 1 y k m using the standard structure, we have
M decimator outputs vector × M filter outputs decimator outputs × N + 1 multiplications filter output = M 2 N + 1 multiplications vector M decimator outputs vector × M filter outputs decimator outputs × N + 1 multiplications filter output = M 2 N + 1 multiplications vector (1)
where we have included one multiply for the modulator. The calculations above pertain to standard (i.e., not polyphase) decimation. If we implement the lowpass/downsampler in each filterbank branch with a polyphase decimator,
M branch outputs vector × N + 1 multiplications branch output = M N + 1 multiplications vector M branch outputs vector × N + 1 multiplications branch output = M N + 1 multiplications vector (2)
To calculate the same output vector for the polyphase/DFT structure, we have approximately
1 DFT vector × M 2 log 2 M multiplications DFT × M polyphase outputs DFT × N M multiplications polyphase output 1 DFT vector × M 2 log 2 M multiplications DFT × M polyphase outputs DFT × N M multiplications polyphase output (3)
= N + M 2 log 2 M multiplications vector = N + M 2 log 2 M multiplications vector The table below gives some typical numbers. Recall that the filter length NN will be linearly proportional to the decimation factor MM, so that the ratio NM N M determines the passband and stopband performance.
  M=32 M 32 , NM=10 N M 10 M=128 M 128 , NM=10 N M 10
standard 328,704 20,987,904
standard with polyphase 10,272 163,968
polyphase/DFT 400 1,728

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