##### From Digital Signal Processing: Fundamentals and Applications

## 8.7 Pole-Zero Placement Method for Simple Infinite Impulse Response Filters

This section introduces a pole-zero placement method for a simple IIR filter design. Let us first examine effects of the pole-zero placement on the magnitude response in the z-plane shown in Figure 8.31.

Figure 8.31: Effects of pole-zero placement on the magnitude response.

In the z-plane, when we place a pair of complex conjugate zeros at a given point on the unit circle with an angle * ?* (usually we do), we will have a numerator factor of ( *z* ? *e ^{j?}*)(

*z*?

*e*) in the transfer function. Its magnitude contribution to the frequency response at

^{?j?}*z*=

*e*

^{j}^{?}is (

*e*

^{j}^{?}?

*e*)(

^{j?}*e*

^{j}^{?}?

*e*). When ? =

^{j?}*?*, the magnitude will reach zero, since the first factor (

*e*?

^{j?}*e*) = 0 gives a zero magnitude. When a pair of complex conjugate poles are placed at a given point within the unit circle, we have a denominator factor of (

^{j?}*z*?

*re*)(

^{j?}*z*?

*re*), where

^{?j?}*r*is the radius chosen to be less than and close to 1 to place the poles inside the unit circle. The magnitude contribution to the frequency response at ? =

*?*will rise to a large magnitude, since the first factor (

*e*?

^{j?}*re*)

^{j?}*=*

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