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Lab 2 - The Z-Transform and Frequency Responses

DUE DATE: OCT 16,2000

Objectives

The objective of this lab is to be able to represent a LTI system by its system function in the z-domain. This z-domain system function will then be used to determine the system's frequency response and its impulse response. Furthermore, the system function will be used to filter given input signals.

For each problem format the required graphs using paint brush option. You are also required to submit the block diagram of your system in problem 1a, 6a (i) and 6b (ii) Make a note of the filter coefficients that you use in each problem so you can answer the questions at the end of this lab. It may be a good idea to look at the quiz for this lab before getting started to know what is required.

Introduction

For this lab, you are required to use a new Java progam called J-DSP. To become familiar with it, we recommend that you first work through the Introduction to J-Dsp.

The z-transform of the impulse response of a LTI system can be written in the following form:

The a_i's and the b_i's are called the filer coefficients of the system with a₀ always being equal to one. To create an LTI system with J-DSP, first, compute the z-transform of the system's impulse response. Determine the filter coefficients from this z-domain function and enter them into a filter block in the J-DSP program. If the input to a filter block is x[n], then the output will be

where * denotes convolution. If the input to the filter block is an impulse, the output will be the inverse z-transform of the system function. Besides the filter block, other J-DSP blocks that will be useful to you in this lab include the Signal Generator Block, the plot block and the FFT block. For all frequency responses in this lab, use a Fast Fourier Transform of size 256 and plot on a linear scale.

J-DSP

For this lab, use the J-DSP program which requires Netscape 4.6 or higher to run.or useIE to load the webpage. Push the ``Start'' button below to begin.

Your browser does not recognize the applet tag.

REPORT

All reports must be typed and not exceed 10 pages including figures.The labs 2 & 3 must be submitted as one report.The report must include the following:
Introduction:
(around ½ page) State objective of the lab.
Give an introduction to Z- Transform and Pole-Zero plots and frequency responses.
Analysis:
For all problems give theoretical justification and give equations as needed.Briefly work out the steps in obtaining the Z-transform.
Experiment:
Give experimental results.
Cut and Paste graphs wherever applicable in your document. You can use Alt+ Print Scrn to port to clipboard. For all figures use figure captions e.g.,
Fig 1: Graph 1:Impulse Response
Quiz Questions:
A set of quiz questions are presented at the end of each problem.The answers for these questions are to be included in the same problem.
Discussion:
Summarize all results and their significance (around ½ page).Here some explanation for the graphs is expected.

Problems

Problem 1: Exponential Sequences

• (a) Design and simulate a digital filter that has the impulse response

  

  Mark the block diagram of your system as graph1
  Obtain graphs of the following using JDSP:
  - The impulse response (Mark the impulse response as graph2)
  - The magnitude of the frequency response on a linear scale. Use the Fast Fourier Transform (FFT) of size 256. (mark the magnitude response as graph3)

• (b) Redo (a) for

  

  (Mark impulse response as graph4)
  (Mark magnitude response plotted on a linear scale as graph5)

• (c) Redo (a) for

  

  (Mark impulse response as graph6)
  (Mark magnitude response plotted on a linear scale as graph7)

QUIZ QUESTIONS

1.1 The filter in part a is a
a) highpass filter.
b) lowpass filter.
c) bandpass filter.

1.2 To calculate the z-transform in part b, the following z-transformproperty is useful
a) multiplication by an exponentialsequence.
b) differentiation and time shiftingof X[z].
c) Linearity Property .

1.3 The filter in part c is
a) an IIR filter.
b) a FIR filter.

1.4 The region of convergence of the z-transform of the filter in part bis:
a) |z| < 0.9.
b) |z| > 0.9.
c) |z| >= 0.9.

1.5 The filter in part c is
a) stable.
b) unstable.

1.6 The region of convergence of the z-transform of the filter in part c
a) contains the unit circle.
b) does not contain the unit circle.

1.7 If the region of convergence of the z-transform of a system containsthe unit circle, the system is
a) stable.
b) unstable.

1.8 What are the values of the non-zero filter block coefficients in parta? (Note: a₀ is always 1)
a) a₁=1 b₀=0.9
b) a₁=-0.9 b₀=1.0
c) a₁=0.9 b₀=1.0a₁=0.81

1.9 What are the values of the non-zero filter block coefficients in partb? (Note: a₀ is always 1)
a) b₀=0 b₁=1.0 a₁=-1.8a₂=0.81
b) b₀=1.0 b₁=1.0a₁=1.8 a₂=1.0
c) b₀=0 b₁=1.0 a₁=2.0a₂=2.4

1.10 List the non-zero b filter coefficients for part c.

 

1.11 List the non-zero a filter coefficients for part c starting with a₁.

 

Problem 2: Digital Oscillator

Design and simulate a digital oscillator where the impulse response has the form

with

Obtain a graph of the frequency response using a linear scale. (Mark the graph as graph8)

QUIZ QUESTIONS

2.1 The period of the impulse response in problem 2 is
a) 4 samples.
b) 8 samples.
c) 16 samples.

2.2 A maximum in the magnitude of the frequency response occurs at whatfrequency?
a) pi/2.
b) pi/3.
c) pi/4.
d) 2*pi/3.

2.3 List the non-zero b filter coefficients.

2.4 List the non-zero a filter coefficients starting with a₁.

Problem 3:

Consider a system whose impulse response is:

The input signal to the system is

Do the following which will include 3 graphs.

• (i) Simulate y[n] = x[n]*h[n] when . To generate x[n], open the signal generator block. Set the signal to a sinusoid with frequency 0.25, pulsewidth 256 and gain 1. Mark the plot of y[n] as graph9
  
• (ii) Simulate y[n] = x[n]*h[n] when . To generate x[n], set the frequency to 0.5 and the pulsewidth to 256. Mark the plot of y[n] as graph10
  
• (iii) Plot the frequency response using a linear scale and Mark the plot of the frequency response as graph11
  
• (iv) State the reasons for the behavior of this system.

QUIZ QUESTIONS

3.1 If the input to this system is a sinusoid with frequency pi/2, theoutput at steady state will be a sinusoid scaled by a factor of
a) 1.4
b) 3.6
c) 2.2

3.2 The output of the system in part a is
a) 1 for all values of n>1
b) 0 for all values of n>1
c) 0 for all values of n>0
d) 1 for all values of n>0

3.3 The frequency of y[n], the output of the system, in part b is
a) pi/2
b) pi/3
c) pi/4
d) none of the above

3.4 List the non-zero b filter coefficients

3.5 Give some reasons for the behavior of the system in this problem.

Problem 4: Symmetric Impulse Response

Consider the following system:

• (a) Simulate the system and plot the phase response. Mark the phase response as graph12
  
• (b) What can be said about the phase response?

QUIZ QUESTIONS

4.1 Which statement is true about the impulse response of the system inproblem 4?
a) It is symmetric about n=0
c) It is symmetric about n=2.5
c) It is antisymmetric about n=0

4.2 Which of the following is true about the phase response?
a) It has a constant value at allfrequencies.
b) It has a constant slope at allfrequencies.
c) It has neither constant value norconstant slope.

4.3 What is the effect of the phase response in this sytem on a signalthat is applied at its input?
a) It changes the signal's frequency.
b) It has no effect on the signal atall.
c) It causes a constant delay in thesignal from input to output for signals of all frequencies.

4.4 List the non-zero b filter coefficients.

Problem 5: Pole-Zero Plots

The figure below shows the pole-zero configuration of two causal linear time-invariant systems.



Do the following:

• (i) Determine H_a(z), the system function for the pole-zero plot in figure a.
• (ii) Obtain the impulse response of system (a) by simulation. (Mark impulse resp. as graph13)
• (iii) Determine the response of system (a) to a triangular input signal with amplitude 1 and length 12 samples. (Mark the output as graph14)
• (iv) Determine H_b(z), the system function for the pole-zero plot in figure b.
• (v) Obtain the impulse response of system (b) by simulation. (Mark the impulse resp. as graph15)
• (vi) Determine the response of system (b) to a triangular input signal with amplitude 1 and length 12 samples. (Mark the output as graph16)

QUIZ QUESTIONS

5.1 The pole-zero plot in figure a represents
a) an IIR filter.
b) a FIR filter.

5.2 The pole-zero plot in figure b represents
a) a FIR filter.
b) an IIR filter.

5.3 The output of system a to the triangle input is
a) positive for all values of n.
b) negative for all values of n.
c) alternating positive and negative.

5.4 What are the values of the non-zero filter block coefficients in parta? (Note: a₀ is always 1)
a) b₀=1 a₁=-0.9a₂=0.325 a₃=-0.05
b) b₀=1.0 a₁=1.0a₂=1.8 a₃=0.9
c) b₀=1 a₁=1.0 a₂=0.9
d)None of these

5.5 What are the values of the non-zero filter block coefficients in partb? (Note: a₀ is always 1)
a) b₀=1 b₁=-0.9b₂=0.9
b) b₀=1 b₁=0.6 b₂=1.8
c) b₀=1 b₁=1.0 b₂=0.5
d)None of these

Problem 6: Cascaded and Parallel Configurations of Systems

(a) Consider the following sequence, y[n], which is the convolution of two casual sequences.

• (i) Create a system using two filter blocks such that the impulse response of the system is y[n] given above. Use a=0.5 and b=0.25. Plot the impulse response of the system.(Mark impulse response as graph17) Mark the block diagram of the system as graph18.
• (ii) Now create a system using only one filter block such that the impulse response is also y[n]. Again use a=0.5 and b=0.25. Verify that the impulse response of this system is the same as the impulse response of the system in the previous part.

(b) Consider a system that has the following impulse response:

• (i) Implement the system using two filter blocks and one add block. Plot the impulse response(Mark the impulse response as graph19) and the block diagram of the system as graph20.
• (ii) Implement the same system using only one filter block. Verify that the impulse response is the same as that of the system in the previous part.

  QUIZ QUESTIONS

  6.1 A cascade connection of 2 systems is equivalent to
  a) 1 system, whose impulse response isthe convolution of the impulse responses of the 2 cascaded systems
  b) 1 system, whose impulse response isthe sum of the impulse responses of the 2 cascaded systems
  c) 1 system, whose impulse response isthe product of the impulse responses of the 2 cascaded systems

  6.2 A parallel connection of 2 systems is equivalent to
  a) 1 system, whose impulse response isthe convolution of the impulse responses of the 2 cascaded systems
  b) 1 system, whose impulse response isthe sum of the impulse responses of the 2 cascaded systems
  c) 1 system, whose impulse response isthe product of the impulse responses of the 2 cascaded systems

  6.3 The poles of the system function is part a, ii are located at
  a) 0.5 and 0.25
  b) -0.5 and 0.25
  c) 1 and 0.5

  6.4 What are the values of the filter block coefficients of the twofilters in part a (i)? (Note: a₀ is always 1)
  a) b₀=1 a₁=-0.5and b₀=-0.25 a₁=1
  b) b₀=-0.8 a₁=1and b₀=-0.25 a₁=1
  c) b₀=1 a₁=-0.5and b₀=1 a₁=-0.25

  6.5 What are the values of the filter block coefficients of the singlefilter in part a (ii)? (Note: a₀ is always 1)
  a) b₀=1 a₁=-0.75a₂=0.125 a₃=0.25
  b) b₀=1 a₁=-0.75a₂=0.125
  c) b₀=1 a₁=-0.5a₂=1 a₃=-0.25

  6.6 What are the values of the filter block coefficients of the twofilters in part b (i)? (Note: a₀ is always 1)
  a) b₀=1 a₁=-0.5and b₀=2 a₁=0.9
  b) b₀=1 a₁=0.5and b₀=2 a₁=-0.9
  c) b₀=1 a₁=0.75and b₀=1 a₁=0.9

  6.7 What are the values of the filter blockcoefficients of the single filter in part b (ii)? (Note: a₀ isalways 1)
  a) b₀=3 b₁=0.75a₁=1.8 a₂=0.25
  b) b₀=1 a₁=0.75a₂=1.8
  c) b₀=3 b₁=-0.1a₁=0.4 a₂=-0.45

  Did you find any bugs in the J-DSP program?If so, please describe them. How difficult was it to learn to use J-DSP? Wasanything not intuitive? Don't be afraid to be critical. Sincere comments willbe appreciated.
  
  
  
  
  
  
  
  
  
  
  

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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.