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Decimation reduces sampling rate by integer factor M after anti-aliasing filtering; interpolation increases rate by factor L by inserting zeros and low-pass filtering. Polyphase filter structures decompose filters into parallel paths operating at reduced rates, enabling efficient implementation. Multirate systems are fundamental in audio codecs, communication systems, and signal processing applications.
From sampling theory, you know that a signal with bandwidth B Hz requires a sampling rate of at least 2B Hz to be reconstructed perfectly. From aliasing, you know what goes wrong when that rule is violated: high-frequency components fold back into the baseband and are indistinguishable from genuine low-frequency content. Multirate processing asks a more practical question: what if you have already sampled a signal and need to work with it at a *different* rate — either lower (to save computation or storage) or higher (to interface with another system)? The answers are decimation and interpolation, and both require careful anti-aliasing logic you already know how to reason about.
Decimation by M means keeping only every M-th sample and discarding the rest. Naively doing this would compress the time axis by M, which in the frequency domain expands the spectrum by M — any content between f_s/(2M) and f_s/2 would alias into the lower-frequency band. The solution is to low-pass filter the signal to bandwidth f_s/(2M) *before* discarding samples. This anti-aliasing filter ensures the content that would alias has been removed. The combined operation — filter then downsample — is decimation. The result is a signal at rate f_s/M that faithfully represents the original signal's content up to f_s/(2M). The computational payoff is that all subsequent processing happens at the lower rate, with fewer multiplications per second.
Interpolation by L is the reverse: you want more samples per second from a signal that currently has fewer. The procedure is to insert L−1 zeros between every existing sample (upsampling), then apply a low-pass filter at cutoff f_s/2 (in the new higher rate). The zero-insertion step creates a signal at rate L·f_s, but examining its DTFT reveals that the original spectrum now repeats at multiples of the original f_s — in other words, imaging artifacts (the interpolation analog of aliasing). The low-pass filter removes these images, leaving a smoothly interpolated signal. Conceptually, the filter "fills in" the inserted zeros with values that make the waveform continuous and bandlimited. Rational rate conversions by a factor L/M are achieved by cascading interpolation by L with decimation by M.
The efficiency bottleneck in both operations is the anti-aliasing or anti-imaging filter. A direct implementation of an N-tap FIR filter applied before decimation by M would compute N multiplications per input sample, even though M−1 out of every M outputs are immediately discarded — clearly wasteful. Polyphase decomposition resolves this by rearranging the filter computation. The filter's coefficients are split into M sub-filters (polyphase components), each of which operates at the lower rate. The computation now costs N/M multiplications per output sample — exactly what you'd expect after moving filtering to the lower rate. This factor-of-M efficiency gain is what makes multirate processing practical in real-time systems. In audio codecs (MP3, AAC), video processing, and software-defined radio, signals are routinely processed at different rates in different stages, and polyphase filter banks are the core computational structure enabling all of it.