Computational Savings of Polyphase Interpolation/Decimation Assume that we design FIR LPF H z with N taps, requiring N multiplies per output. For standard decimation by factor M, we have N multiplies per intermediate sample and M intermediate samples per output, giving N M multiplies per output. For polyphase decimation, we have N M multiplies per branch and M branches, giving a total of N multiplies per output. The assumption of N M multiplies per branch follows from the fact that h n is downsampled by M to create each polyphase filter. Thus, we conclude that the standard implementation requires M times as many operations as its polyphase counterpart. (For decimation, we count multiples per output, rather than per input, to avoid confusion, since only every Mth input produces an output.) From this result, it appears that the number of multiplications required by polyphase decimation is independent of the decimation rate M. However, it should be remembered that the length N of the M -lowpass FIR filter H z will typically be proportional to M. This is suggested, e.g., by the Kaiser FIR-length approximation formula N -10 10 δ p δ s 13 2.324 Δ ω where Δ ω in the transition bandwidth in radians, and δ p and δ s are the passband and stopband ripple levels. Recall that, to preserve a fixed signal bandwidth, the transition bandwidth Δ ω will be linearly proportional to the cutoff M, so that N will be linearly proportional to M. In summary, polyphase decimation by factor M requires N multiplies per output, where N is the filter length, and where N is linearly proportional to M. Using similar arguments for polyphase interpolation, we could find essentially the same result. Polyphase interpolation by factor L requires N multiplies per input, where N is the filter length, and where N is linearly proportional to the interpolation factor L. (For interpolation we count multiplies per input, rather than per output, to avoid confusion, since M outputs are generated in parallel.)