Subfilters

Subfilters are the constituent components of a larger filtering operation, commonly arising in polyphase decomposition and filter-bank designs. In multirate digital signal processing, a single filter can be represented as multiple subfilters that operate on different phase or sample streams. This modular view supports efficient computation, especially when the system decimates or interpolates the signal.

In polyphase form, an FIR filter H(z) of length L with decimation by M can be decomposed into M subfilters E_0(z), E_1(z), ..., E_{M-1}(z). The overall transfer is H(z) = sum_{q=0}^{M-1} z^{-q} E_q(z^M). Each subfilter collects every Mth coefficient of H, making it responsible for a residue class modulo M. The input sequence is partitioned into M streams; each subfilter processes one stream, and the outputs are combined (interleaved) to yield the same result as the original filter, often with reduced multiplications.

Subfilters appear in filter banks, wavelet analysis, and communication systems, where they enable efficient implementation and flexible scaling. They allow designers to tailor localized frequency responses and to optimize hardware or software resources, such as memory bandwidth and parallel execution.

Design considerations include phase alignment, latency, numerical precision, and reconstruction properties in perfect-reconstruction filter banks. While subfilters are not unique to a specific algorithm, their correct combination maintains the intended overall response and stability of the system.