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Lab 3 - Pole-Zero Plots and Frequency Responses

DUE DATE: OCT 16, 2000

Objectives

This exercise analyzes how the magnitude and phase of a pole or zero influences the sytem's magnitude response. It will first explore the relationship between the pole-zero plot and the magnitude response of a system. Once this relationship is understood, a filter with a desired magnitude response can be designed by strategically placing its poles and zeros. Finally, a system in which the effects of the poles and zeros cancel each other out will be examined.

Introduction

Use the J-DSP program to do this exercise. Before starting, become familiar with the following three J-DSP blocks as they will be very helpful in doing the problems below. Each of the three can be found under the filter blocks menu.

The Pole-Zero Block. This block can be used to create a pole-zero plot. The Pole-Zero block can be connected to either the bottom of a filter block or to a Freq-Resp block. Connecting the pole-zero block to the bottom of a filter block will automatically set the filter coefficients of that filter block so that its poles and zeros are at the locations specified in the Pole-Zero block. Connecting the Pole-Zero block to a Freq-Resp block will display, in the Freq-Resp window, the frequency response of the filter having poles and zeros specified in the Pole-Zero block.

Poles and zeros can be placed on the plot by using either the mouse or the keyboard. To place them using the mouse, select "graphical" from the pop-down menu, press the "Place Poles" or "Place Zeros" button and then click on the plot in the desired location of the zero or pole. To place poles and zeros using the keyboard, select one of the manual options from the pop-down menu, choose either the pole or zero radio button, enter the location using the keyboard and press the enter button on the right edge of the window. To delete a pole or zero, select the pole or zero to be deleted from the list of poles and zeros to the right of the plot by clicking on it and then press the delete button at the bottom of the window. To move a pole or zero, press the move PZ button and then click on the pole or zero in the plot and while holding the mouse button down drag it to a new location. The reset button will erase all the poles and zeros from the plot.

The Freq-Resp block. The Freq-Resp block can be connected to the top of a filter block, to a Pole-Zero block as explained above, or to a filter design block such as IIR Fltr, FIR Fltr or Kaiser. The Freq-Resp block will show the frequency response of the filter to which it is connected or if connected to the Pole-Zero block, the frequency response of a filter having poles and zeros specified in the Pole-Zero block.

The PZ-Plot block. Connect the PZ-Plot block to the top of a filter block to see a plot of the poles and zeros of that filter or to any of the filter design blocks to see a plot of their poles and zeros. The filter design blocks include IIR Fltr, FIR Fltr and Kaiser. For this lab, use the J-DSP program which requires Netscape 4.6 or higher to run. Push the ``Start'' button below to begin.

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Problems

In all problems, with the exception of number 3, use linear scaling for the magnitude.

Problem 1: Pole-Zero Plots

Find the poles and zeros of the following transfer functions. Use the Pole-Zero block in J-DSP to plot the poles and zeros and the Freq-Resp block to view the system's frequency response. Plot the frequency response of each one using linear scaling.

• (a) Plot the frequency response of this system and name the plot as graph1
  

  

• (b) Plot the frequency response of this system and name the plot as graph2
  

  

• (c) Plot the frequency response of this system and name the plot as graph3
  

  

QUIZ QUESTIONS

1.1 What effect does a pole in a system's transfer function tend to have on the magnitude of the frequency response?
a) It creates a valley in the magnitude response.
b) It creates a peak in the magnitude response.
c) It has no effect.

1.2 What effect does a zero in a system's transfer function tend to have on the magnitude of the frequency response?
a) It creates a valley in the magnitude response.
b) It creates a peak in the magnitude response.
c) It has no effect.

1.3 Where are the poles and zeros located in part (a)? H(z)=(1-1.2z^-1)/( 1 - 0.5z^-1)
a) One zero at (1/1.2) and one pole at (1/0.5).
b) One zero at 1.2 and one pole at 0.5.
c) One pole at 1.2 and one zero at 0.5.

1.4 Where are the zeros located in the transfer function in part (b), H(z)=1-z^-3.

1.5 How many poles does the transfer function H(z)=1/( 1 - 0.85z^-5) have?
a) 1 pole at z=0.85^(1/5) and 4 complex poles which form 2 complex conjugate pairs.
b) 5 repeated poles at z=0.85^(1/5).
c) 1 pole at z=0.85^(1/5).

1.6 What effect do poles at the origin(0 + 0i) have on the magnitude of the frequency response of a system?
a) They cause a peak at frequency of 0 radians.
b) They have no effect.
c) They cause a valley at frequency of 0 radians.

1.7 The transfer function in part a has a pole and a zero both located at 0 radians. Why is there a valley in the magnitude response at 0 radians instead of a peak?
a) Zeros always have a stronger influence that poles when located at the same frequcncy.
b) The magnitude of the zero is greater that the magnitude of the pole.
c) The zero is closer to the unit circle than the pole.

Problem 2: Varying the magnitude of poles and zeros.

Consider a system which has poles at

and a zero at

where

• (i) r = 0.96
• (ii) r = 0.71
• (iii) r = 0.14

• (a) Derive analytically the impulse response of the system and show its dependence on r. Sketch the impulse response for r=0.96 using J-DSP. Mark the plot of the impulse response as graph4
• (b) Plot the frequency and phase response for case (i) Use linear scaling. Name the plot as graph5.
• (c) Plot the frequency and phase response for case (ii) Use linear scaling.Mark the plot as graph6.
• (d) Plot the frequency and phase response for case (iii) Use linear scaling. Save the plot as graph7.
• (e) Observe the differences in the the three plots.

QUIZ QUESTIONS

2.1 If h₁[n]=aⁿu[n] and h₂[n]=a^(n-1)u[n-1], what is true about the magnitude of the frequency response of h₁[n] and h₂[n]?
a) They will have different shapes.
b) They will be exactly the same.
c) The will differ only by a gain factor.

2.2 What form does the analytical impulse response in problem 2 have?
a) rⁿcos(a*n)u[n] with |r|<1 A sinusoidally decaying exponential.
b) aⁿu[n] with |a|<1. A decaying exponential. .
c) cos(a*n)u[n] An oscillator.

2.3 In problem 2, as the poles move away from the unit circle
a) the peaks in the frequency response become sharper.
b) the peaks become smaller.
c) the bandwidth of the sytem decreases.

Problem 3: Lowpass Filter/ Highpass Filter Design by Pole-Zero Placement

For this problem, you will design filters by pole and zero placement using J-DSP's Pole-Zero block. You may want to use the following set-up to do the design.

Double click on the Pole-Zero block and then Freq-Resp block so you can see each block's respective window at the same time. Place the poles and zeros on the pole-zero plot at the desired locations. When all poles and zeros have been placed, press the move PZ button in the Pole-Zero window and move the poles and zeros around by clicking them and dragging them to new locations. As you move the poles and zeros, the frequency response will be immediately updated. Adjust the location of the poles and zeros until the desired response is obtained.

For this problem, plot the magnitude responses in decibels.

• (a) Design a lowpass filter using pole-zero placement with approximate cutoff frequency . Use three sets of zeros and two sets of poles. Plot of the frequency response using a decibel scale and name the plot as graph8.

• (b) Design a highpass filter using pole-zero placement with an approximate cutoff frequency of . Use two sets of zeros and five sets of poles. Plot the frequency response using a decibel scale and name the plot as graph9.

QUIZ QUESTIONS

3.1 What kind of filter does the following pole-zero diagram respresent where x's are poles and o's are the zeros?

a) Lowpass filter
b) Highpass filter
c) Bandpass filter
d) Bandstop filter
e) Allpass filter

Problem 4: An interesting frequency response

Consider the following system:

• (a) Find the poles and zeros of the transfer function.
• (b) Plot the frequency and phase response of the system on a linear scale.Name the plot as graph10.
• (c) Examine the frequency and phase response.

QUIZ QUESTIONS

4.1 What is the magnitude of all the poles of the system in problem 4?
a) 1.1
b) 0.9
c) 0.8
d) 1.2

4.2 What is the magnitude of all the zeros of the system in problem 4?
a) 1.1
b) 0.9
c) 0.8
d) 1.2

4.3 What can be said about the distance of the poles and zeros from the unit circle?
a) The poles and zeros are about the same distance from the unit circle.
b) The zeros are much farther from the unit circle than the poles.
c) The poles are much farther from the unit circle than the zeros.

4.4 The system in this problem is
a) a lowpass filter.
b) an allpass filter.
c) a highpass filter.
d) a bandpass filter.

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Copyright 2000 Andreas Spanias, MIDL, Arizona State University JDSP and Report Submission Software Developed by ASU-MIDL For questions contact Prof. Spanias spanias@asu.edu.