# Overview of iir filter design

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## Iir filter

$y(n)=-\sum_{k=1}^{M-1} {a}_{k}y(n-k)+\sum_{k=0}^{M-1} {b}_{k}x(n-k)$ $H(z)=\frac{{b}_{0}+{b}_{1}z^{-1}+{b}_{2}z^{-2}+\mathrm{...}+{b}_{M}z^{-M}}{1+{a}_{1}z^{-1}+{a}_{2}z^{-2}+\mathrm{...}+{a}_{M}z^{-M}}$

## Iir filter design problem

Choose $\{{a}_{i}\}$ , $\{{b}_{i}\}$ to best approximate some desired $\left|{H}_{d}(w)\right|$ or, (occasionally), ${H}_{d}(w)$ .

As before, different design techniques will be developed for different approximation criteria.

• ## Bilinear transform

Maps $()$ L optimal (and other) analog filter designs to $()$ L optimal digital IIR filter designs.
• ## Prony's method

Quasi- $(, L)$ optimal method for time-domain fitting of a desired impulse response ( ad hoc ).
• ## Lp optimal design

$(, L)$ optimal filter design ( $1< p$ ) using non-linear optimization techniques.

## Comments on iir filter design methods

The bilinear transform method is used to design "typical" $()$ L magnitude optimal filters. The $(, L)$ optimization procedures are used to design filters for which classical analogprototype solutions don't exist. The program by Deczky ( DSP Programs Book , IEEE Press) is widely used. Prony/Linear Prediction techniquesare used often to obtain initial guesses, and are almost exclusively used in data modeling, system identification, andmost applications involving the fitting of real data (for example, the impulse response of an unknown filter).

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