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?) =


Products & Services
IC Electronic Filters
IC electronic filters are frequency-selective circuits that consist of devices such resistors, capacitors, inductors, transistors, or operational amplifiers coupled with reactive components.
Passive Filters
Passive filters are implemented using only passive components such as resistors, capacitors and inductors. These filters do not produce any amplification of the input signal.
Active Band Pass Filters
Active band pass filters are used to attenuate frequencies below and above a range of frequencies (i.e., the bandwidth or passband of the filter).
Active Low Pass Filters
Active low pass filters pass signals from low frequencies and reject signals from high frequencies.
Active Band Reject Filters
Active band reject filters are tuned circuits that prevent the passage of signals within a specified band of frequencies. These devices are also known as bandstop or notch filters.

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;...