This section considers the implementation of parallel FIR filters using algorithmic strength reduction transformation.

This section addresses the formulation of parallel FIR filters using polyphase decomposition, a technique used in multirate signal processing [1].

An *N*-tap FIR filter can be expressed in time domain as

where {*x*(*n*)} is an infinite length input sequence and the sequence {*h*(*n*)} contains FIR filter coefficients of length *N*, or in *z*-domain as

The input sequence {*x*(0), *x*(1), *x*(2), *x*(3), ···} can be decomposed into even- numbered part and odd-number part as follows:

where *X*_{0}(*z*^{2}) and *X*_{1} (*z*^{2}) are the *z*-transforms of *x*(2*k*) and *x*(2*k* + 1) (for 0 ≤ *k* < ∞), respectively. In (9.3), *X*(*z*) is decomposed into two polyphases. Similarly, the length-*N* filter coefficients *H*(*Z*) can be decomposed as

where *H*_{0}(*z*^{2}) and *H*_{1}(*z*^{2}) are of length *N*/2 and are referred to as even subfilter and odd subfilter, ...

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