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General Causal FIR Filters

The most general case--a TDL having a tap after every delay element--is the general causal Finite Impulse Response (FIR) filter, shown in Fig.2.22. It is restricted to be causal because the output $ y(n)$ may not depend on ``future'' inputs $ x(n+1)$ , $ x(n+2)$ , etc. The FIR filter is also called a transversal filter. FIR filters are described in greater detail in [452].

Figure 2.22: The general, causal, finite-impulse-response (FIR) digital filter.
\includegraphics{eps/fir}

The difference equation for the $ M$ th-order FIR filter in Fig.2.22 is, by inspection,

$\displaystyle y(n) = b_0 x(n) + b_1 x(n-1) + b_2 x(n-2) + b_3 x(n-3) + \cdots + b_M x(n-M)
$

and the transfer function is

$\displaystyle H(z) = b_0 + b_1 z^{-1} + b_2 z^{-2} + b_3 z^{-3} + \cdots + b_M z^{-M}
= \sum_{m=0}^M b_m z^{-m} \isdef B(z).
$

The STK class for implementing arbitrary direct-form FIR filters is called Fir. (There is also a class for IIR filters named Iir.) In Matlab and Octave, the built-in function filter is normally used.


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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2014-06-11 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA