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 ?j?) in the transfer function. Its magnitude contribution to the frequency response at z = e j ? is ( e j ? ? e j?)( e j ? ? e j?). When ? = ?, the magnitude will reach zero, since the first factor ( e j? ? e j?) = 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 ( z ? re j?)( z ? re ?j?), where 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?) =

Copyright Elsevier Inc. 2008 under license agreement with Books24x7

Products & Services
Liquid Level Switches
Liquid level switches are used to detect specific liquid levels or interfaces between liquids. These level measurements are indicated via an electrical switching action.
Electrical Receptacles
Electrical receptacles, outlets, and wall sockets accept plugs and provide current to run electrical devices.
Signal Filters
Signal filters block or decrease (attenuate) unwanted frequencies or signal wave characteristics.
Color Meters and Appearance Instruments
Color meters and appearance instruments are used to measure the properties of paints and coatings including color, gloss, haze and transparency.
Circuit Breakers
Circuit breakers are mechanical switching devices capable of breaking currents under specified abnormal circuit conditions.

Topics of Interest

8.8 Realization Structures of Infinite Impulse Response Filters In this section, we will realize the designed IIR filter using direct form I and direct form II. We will then realize a higher-order...

LECTURE 53: RELATIONSHIP BETWEEN THE DTFT AND THE Z-TRANSFORM We have seen that the discrete-time Fourier transform (DTFT) of a system's impulse response (the frequency response of the system) exists...

D.1 Sinusoidal Steady-State Response Analysis of the sinusoidal steady-state response of digital filters will lead to the development of the magnitude and phase responses of digital filters. Let us...

LECTURE 26: QUALITATIVE EVALUATION OF THE FREQUENCY RESPONSE FROM THE POLE-ZERO PLOT In this chapter, we will analyze the behavior of stable continuous-time linear time-invariant (LTI) systems by...

9.2.2   PI Control Design by Pole Placement Consider the closed-loop system with PI control in Figure 9.9. We have four design goals for the PI controller: (1) the closed-loop system is stable;...