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TIIR Construction: A One-Pole Example

Consider an FIR filter having a truncated geometric sequence $ \{h_0,h_0 p, \ldots,h_0 p^N\}$ as an impulse response. This filter has the same impulse response for the first $ N+1$ terms as the one-pole IIR filter with transfer function

$\displaystyle H_{\rm IIR}(z)=\frac{h_0}{1-p z^{-1}}.

Subtracting off the tail of the impulse response gives

H_{\rm FIR}(z) &=& h_0 +h_0 p z^{-1} + \cdots+h_0 p^{N}z^{-N}\\
&=&\left\{h_0 +h_0 p z^{-1} +\cdots\;\right\}\\
&& \quad{}-\left\{h_0 p^{N+1}z^{-(N+1)} +h_0
p^{(N+2)} z^{-(N+2)} +\cdots\;\right\}\\
&=&\frac{h_0}{1-p z^{-1}}
- p^{N+1} z^{-(N+1)}\frac{h_0}{1-p z^{-1}}\\
&=& h_0 \frac{1- p^{N+1} z^{-(N+1)}}{1- p z^{-1}}

The time-domain recursion for this filter is

y[n]&=& \sum_{k=0}^N h_0p^k x[n-k]\\
&=& p y[n-1] +h_0\left(x[n]-p^{N+1} x[n-(N+1)]\right)

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``Horn Modeling'', by Julius O. Smith III, (From Lecture Overheads, Music 420).
Copyright © 2019-02-05 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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