Signals and Systems - Electrical Engineering

11.4 IIR Filter Design 653

frequency response in 0≤ω≤πis

Hd(ejω)=

{

1 e−jωN 0 ≤ω≤ωc

0 ωc< ω≤π

The desired impulse response for this filter is then found from

hd[n]=

1

2 π

∫ωc

−ωc

1 e−jωNejωndω

The resultinghd[n] will be used as the desired impulse response to approximate.

nExample 11.5

Consider an FIR filter with the following desired magnitude response in 0≤ω≤π:

|Hd(ejω)|=

{

1 0≤ω≤π 4

0 elsewhere in 0≤ω≤π

and zero phase. Find the desired impulse responsehd[n] that we wish to approximate.

Solution

The desired impulse response is computed as follows:

hd[n]=

1

2 π

∫π

−π

Hd(ejω)ejωndω=

1

2 π

π/∫ 4

−π/ 4

ejωndω

=

{

sin(πn/ 4 )/πn n6= 0

0.25 n= 0

which corresponds to the impulse response of a noncausal system. As we will see later, windowing

and shifting ofhd[n] are needed to make it into a causal, finite-length filter. n

11.4 IIR Filter Design

Two possible approaches in the design of IIR filters are:

n Using analog filter design methods and transformations between thes-plane and thez-plane.
n Using optimization techniques.

The first is a frequency transformation approach. Using a mapping between the analog and the
discrete frequencies, we obtain the specifications for an analog filter from the discrete filter speci-
fications. Applying well-known analog filter design methods, we then design the analog filter from
the transformed specifications. The discrete filter is finally obtained by transforming the designed
analog filter.