1.10 Computational savings of polyphase interpolation/decimation

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 $NM$ multiplies per output.

For polyphase decimation, we have $\frac{N}{M}$ multiplies per branch and $M$ branches, giving a total of $N$ multiplies per output. The assumption of $\frac{N}{M}$ multiplies per branch follows from the fact that $h(n)$ is downsampled by $M$ to create each polyphase filter. Thus, we conclude that thestandard implementation requires $M$ times as many operations as its polyphase counterpart. (For decimation, we count multiplesper output, rather than per input, to avoid confusion, since only every $M^{th}$ input produces an output.)

From this result, it appears that the number of multiplications required by polyphase decimation isindependent of the decimation rate $M$ . However, it should be remembered that the length $N$ of the $\frac{\pi }{M}$ -lowpass FIR filter $H(z)$ will typically be proportional to $M$ . This is suggested, e.g. , by the Kaiser FIR-length approximation formula $$N\approx \frac{-10\lg ({\delta }_p{\delta }_s)-13}{2.324\Delta (\omega )}$$ where $\Delta (\omega )$ in the transition bandwidth in radians, and ${\delta }_p$ and ${\delta }_s$ are the passband and stopband ripple levels. Recall that, to preserve a fixed signal bandwidth, the transitionbandwidth $\Delta (\omega )$ will be linearly proportional to the cutoff $\frac{\pi }{M}$ , 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 byfactor $L$ 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 thanper output, to avoid confusion, since $M$ outputs are generated in parallel.)