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## 9.2    PARALLEL FIR FILTERS

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

### 9.2.1    Formulation of Parallel FIR Filtering Using Polyphase Decomposition

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 X0(z2) and X1 (z2) are the z-transforms of x(2k) and x(2k + 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 H0(z2) and H1(z2) are of length N/2 and are referred to as even subfilter and odd subfilter, ...

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