Iir filter
$$y(n)=\sum_{k=1}^{M1} {a}_{k}y(nk)+\sum_{k=0}^{M1} {b}_{k}x(nk)$$
$$H(z)=\frac{{b}_{0}+{b}_{1}z^{1}+{b}_{2}z^{2}+\mathrm{...}+{b}_{M}z^{M}}{1+{a}_{1}z^{1}+{a}_{2}z^{2}+\mathrm{...}+{a}_{M}z^{M}}$$
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.
Outline of iir filter design material

Maps
$()$∞
L optimal (and other)
analog filter designs to
$()$∞
L optimal digital IIR filter designs.

Prony's method
Quasi
$(, L)$ optimal method for timedomain fitting of a
desired impulse response (
ad
hoc ).

Lp optimal design
$(, L)$ optimal filter design (
$1< p$∞ ) using nonlinear optimization techniques.
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).