Recall the standard (non-polyphase) resampler in Figure 1.
For simplicity, assume that L>MLM
. Since the length of an FIR filter is inversely proportional
to the transition bandwidth (recalling Kaiser's formula), and the transition
bandwidth is directionally proportional to the cutoff frequency, we model the
lowpass filter length as N=αLN
αL, where
αα is a constant that determines the filter's (and
thus the resampler's) performance (independent of LL
and MM). To compute one output point, we require
MM filter outputs, each requiring N=αL
NαL
multiplies, giving a total of αLM
αLM
multiplies per output.
In the polyphase implementation, calculation of one output point requires the
computation of only one polyphase filter output. With N=αL
NαL
master filter taps and LL
branches, the polyphase filter length is αα,
so that only αα multiplies are required per
output. Thus, the polyphase implementation saves a factor of LM
LM multiplies over the
standard implementation!
