Purdue University: ECE438 - Digital Signal Processing with Applications 1

ECE438 - Laboratory 3:
Frequency Analysis
By Prof. Charles Bouman and Prof. Mireille Boutin
Fall 2015

1 Introduction
In this experiment, we will use Fourier series and Fourier transforms to analyze continuous-
time and discrete-time signals and systems. The Fourier representations of signals involve the
decomposition of the signal in terms of complex exponential functions. These decompositions
are very important in the analysis of linear time-invariant (LTI) systems, due to the property
that the response of an LTI system to a complex exponential input is a complex exponential
of the same frequency! Only the amplitude and phase of the input signal are changed.
Therefore, studying the frequency response of an LTI system gives complete insight into its
behavior.
In this experiment and others to follow, we will use the Simulink extension to Matlab.
Simulink is an icon-driven dynamic simulation package that allows the user to represent a
system or a process by a block diagram. Once the representation is completed, Simulink
may be used to digitally simulate the behavior of the continuous or discrete-time system.
Simulink inputs can be Matlab variables from the workspace, or waveforms or sequences
generated by Simulink itself. These Simulink-generated inputs can represent continuous-
time or discrete-time sources. The behavior of the simulated system can be monitored
using Simulink’s version of common lab instruments, such as scopes, spectrum analyzers and
network analyzers.

Questions or comments concerning this laboratory should be directed to Prof. Mireille Boutin, School
of Electrical and Computer Engineering, Purdue University, West Lafayette IN 47907
Purdue University: ECE438 - Digital Signal Processing with Applications 2

2 Background Exercises

INLAB REPORT:
Submit these background exercises with the lab report.

2.1 Synthesis of Periodic Signals

Each signal given below represents one period of a periodic signal with period T0 .
1. Period T0 = 2. For t ∈ [0, 2]:
1
s(t) = rect(t − )
2
2. Period T0 = 1. For t ∈ [− 12 , 21 ]:
1
s(t) = rect(2t) −
2
For each of these two signals, do the following:
i. Compute the Fourier series expansion in the form
X
∞
s(t) = a0 + Ak sin(2πkf0 t + θk )
k=1

where f0 = 1/T0 .
Hint: You may want to use one of the following references:
Sec. 4.1 of “Digital Signal Processing”, by Proakis and Manolakis, 1996;
Sec. 4.2 of “Signals and Systems”, by A. Oppenheim and A. Willsky, 1983;
Sec. 3.3 of “Signals and Systems”, A. Oppenheim and A. Willsky, 1997.
Note that in the expression above, the function in the summation is sin(2πkf0 t + θk ),
rather than a complex sinusoid. The formulas in the above references must be modified
to accommodate this. You can compute the cos/sin version of the Fourier series, then
convert the coefficients.
ii. Sketch the signal on the interval [0, T0 ].

3 Getting Started with Simulink

In this section, we will learn the basics of Simulink and build a simple system.

Help on Simulink
https://engineering.purdue.edu/VISE/ee438L/matlab/help/pdf/simulink.pdf
Down load Lab3Utilities.zip
https://engineering.purdue.edu/VISE/ee438L/lab3/data/Lab3Utilities.zip
Purdue University: ECE438 - Digital Signal Processing with Applications 3

Full Block Library

Sine Wave Scope Spectrum Network
Simulink Blocks
Analyzer Analyzer

CT System Analysis using

Synthesizer Modulator
a Network Analyzer

Experiment 1 Experiment 2 Experiment 3

CT System Analysis using

DT System Analysis
a Unit Step

Experiment 4 Experiment 5

Figure 1: Simulink utilities for lab 3.

To get the library of Simulink functions for this laboratory, download the file Lab3Utilities.zip.
Once Matlab is started, type “Lab3” to bring up the library of Simulink components shown
in Fig. 1. This library contains a full library of Simulink blocks, a spectrum analyzer and
network analyzer designed for this laboratory, a sine wave generator, a scope, and pre-design
systems for each of the experiments that you will be running.

Sine Wave Scope

Spectrum
Analyzer

Figure 2: Simulink model for the introductory example.

In order to familiarize yourself with Simulink, you will first build the system shown in
Fig. 2. This system consists of a sine wave generator that feeds a scope and a spectrum
analyzer.

1. Open a window for a new system by using the New option from the File pull-down
menu, and select Model.
2. Drag the Sine Wave, Scope, and Spectrum Analyzer blocks from the Lab3 window into
the new window you created.
3. Now you need to connect these three blocks. With the left mouse button, click on the
output of the Sine Wave and drag it to the input of the Scope. Now use the right
button to click on the line you just created, and drag to the input of the Spectrum
Analyzer block. Your system should now look like Fig. 2.
Purdue University: ECE438 - Digital Signal Processing with Applications 4

4. Double click on the Scope block to make the plotting window for the scope appear.
5. Set the simulation parameters by selecting Configuration Parameters from the Simu-
lation pull-down menu. Under the Solver tab, set the Stop time to 50, and the Max
step size to 0.02. Then select OK. This will allow the Spectrum Analyzer to make a
more accurate calculation.
6. Start the simulation by using the Start option from the Simulation pull-down menu.
A standard Matlab figure window will pop up showing the output of the Spectrum
Analyzer.
7. Change the frequency of the sine wave to 7π rad/sec by double clicking on the Sine
Wave icon and changing the number in the Frequency field. Restart the simulation.
Observe the change in the waveform and its spectral density. If you want to change the
time scaling in the plot generated by the spectrum analyzer, from the Matlab prompt
use the subplot(2,1,1) and axis() commands.
8. When you are done, close the system window you created by using the Close option
from the File pull-down menu.

4 Continuous-Time Frequency Analysis

Help on printing figures in Simulink

https://engineering.purdue.edu/VISE/ee438L/matlab/help/pdf/print.pdf

In this section, we will study the use and properties of the continuous-time Fourier
transform with Simulink. The Simulink package is especially useful for continuous-time
systems because it allows the simulation of their behavior on a digital computer.

4.1 Synthesis of Periodic Signals

Double click the icon labeled Synthesizer to bring up a model as shown in Fig. 3. This system
may be used to synthesize periodic signals by adding together the harmonic components of a
Fourier series expansion. Each Sin Wave block can be set to a specific frequency, amplitude
and phase. The initial settings of the Sin Wave blocks are set to generate the Fourier series
expansion
X13
4
x(t) = 0 + sin(2πkt) .
k=1 kπ
k odd

These are the first 8 terms in the Fourier series of the periodic square wave shown in Fig. 4.
Run the model by selecting Start under the Simulation menu. A graph will pop up
that shows the synthesized square wave signal and its spectrum. This is the output of the
Purdue University: ECE438 - Digital Signal Processing with Applications 5

Constant

Sine Wave1

Sine Wave2

Sine Wave3

Spectrum
Analyzer
Sine Wave4 Sum

Sine Wave5 Scope1

Sine Wave6

Sine Wave7

Mux

Scope2

Mux

Figure 3: Simulink model for the synthesizer experiment.

Spectrum Analyzer. After the simulation runs for a while, the Spectrum Analyzer element
will update the plot of the spectral energy and the incoming waveform. Notice that the
energy is concentrated in peaks corresponding to the individual sine waves. Print the output
of the Spectrum Analyzer.
You may have a closer look at the synthesized signal by double clicking on the Scope1
icon. You can also see a plot of all the individual sine waves by double clicking on the Scope2
icon.
Synthesize the two periodic waveforms listed in section 2.1 of the background exercises.
Do this by setting the frequency, amplitude, and phase of each sinewave generator to the
proper values. For each case, print the output of the Spectrum Analyzer.

INLAB REPORT:
Hand in plots of the Spectrum Analyzer output for each of the three synthesized waveforms.
For each case, comment on how the synthesized waveform differs from the desired signal,
and on the structure of the spectral density.
Purdue University: ECE438 - Digital Signal Processing with Applications 6

0.8

0.6

0.4

0.2

Amplitude
0

−0.2

−0.4

−0.6

−0.8

−1

0 0.5 1 1.5 2 2.5 3

Time in seconds

Figure 4: The desired waveform for the synthesizer experiment.

4.2 Modulation Property

Scope
Repeating
Sequence

Product

Sine Wave
Spectrum
Analyzer

Figure 5: Simulink model for the modulation experiment.

Double click the icon labeled Modulator to bring up a system as shown in Fig. 5. This
system modulates a triangular pulse signal with a sine wave. You can control the duration
and duty cycle of the triangular envelope and the frequency of the modulating sine wave.
The system also contains a spectrum analyzer which plots the modulated signal and its
spectrum.
Generate the following signals by adjusting the Time values and Output values of the
Repeating Sequence block and the Frequency of the Sine Wave. The Time values vector
contains entries spanning one period of the repeating signal. The Output values vector
contains the values of the repeating signal at the times specified in the Time values vector.
Note that the Repeating Sequence block does NOT create a discrete time signal. It creates a
continuous time signal by connecting the output values with line segments. Print the output
of the Spectrum Analyzer for each signal.

1. Triangular pulse duration of 1 sec; period of 2 sec; modulating frequency of 10 Hz

(initial settings of the experiment).

2. Triangular pulse duration of 1 sec; period of 2 sec; modulating frequency of 15 Hz.

3. Triangular pulse duration of 1 sec; period of 3 sec; modulating frequency of 10 Hz.

Purdue University: ECE438 - Digital Signal Processing with Applications 7

4. Triangular pulse duration of 1 sec; period of 6 sec; modulating frequency of 10 Hz.

Notice that the spectrum of the modulated signal consists of a comb of impulses in the
frequency domain, arranged around a center frequency.

INLAB REPORT:
Hand in plots of the output of the Spectrum Analyzer for each signal. Answer following
questions:

• What effect does changing the modulating frequency have on the spectral density?

• Why does the spectrum have a comb structure and what is the spectral distance
between impulses? Why?

• What would happen to the spectral density if the period of the triangle pulse were to
increase toward infinity? (in the limit)

4.3 System Analysis

Scope Network
Analyzer

Analog Butterworth
LP Filter

Figure 6: Simulink model for the continuous-time system analysis experiment using a network
analyzer.

Double click the icon labeled CT System Analysis using a Network Analyzer to bring up a
system as shown in Fig. 6. This system includes a Network Analyzer model for measuring the
frequency response of a system. The Network Analyzer works by generating a weighted chirp
signal (shown on the Scope) as an input to the system-under-test. The analyzer measures
the frequency response of the input and output of the system and computes the transfer
function. By computing the inverse Fourier transform, it then computes the impulse response
of the system. Use this setup to compute the frequency and impulse response of the given
fourth-order Butterworth filter with a cut-off frequency of 1Hz. Print the figure showing the
magnitude response, the phase response and the impulse response of the system. To use the
tall mode to obtain a larger printout, type
orient(’tall’);
directly before you print.
Purdue University: ECE438 - Digital Signal Processing with Applications 8

du/dt

Step Derivative Scope

Analog Butterworth
LP Filter

Figure 7: Simulink model for the continuous-time system analysis experiment using a unit
step.

An alternative method for computing the impulse response is to input a step function
into the system and then to compute the derivative of the output. The model for doing
this is given in the CT System Analysis using a Unit Step block. Double click on this icon
and compute the impulse response of the filter using this setup (Fig. 7). Make sure that
the characteristics of the filter are the same as in the previous setup. After running the
simulation, print the graph of the impulse response from the scope.

INLAB REPORT:
Hand in the printout of the output of the Network Analyzer (magnitude and phase of the
frequency response, and the impulse response) and the plot of the impulse response obtained
using a unit step. What are the advantages and disadvantages of each method?

5 Discrete-Time Frequency Analysis

In this section of the laboratory, we will study the use of the discrete-time Fourier transform.

5.1 Discrete-Time Fourier Transform

The DTFT (Discrete-Time Fourier Transform) is the Fourier representation used for finite
energy discrete-time signals. For a discrete-time signal, x(n), we denote the DTFT as the
function X(ω) given by the expression
X
∞
X(ω) = x[n]e−jωn .
n=−∞

Since X(ω) is a periodic function of ω with a period of 2π, we need only to compute X(ω)
for −π < ω < π.
Write a Matlab function X = DTFT(x,n0,dw) that computes the DTFT of the discrete-
time signal x. Here n0 is the time index corresponding to the 1st element of the x vector,
and dw is the spacing between the samples of the Matlab vector X. For example, if x is a
vector of length N, then its DTFT is computed by

X
N
X(w) = x[n]e−jw(n+n0−1)
n=1
Purdue University: ECE438 - Digital Signal Processing with Applications 9

where w is a vector of values formed by w=(-pi:dw:pi).

√
Hint: In Matlab, j or i is defined as −1. However, you may also compute this value using
the Matlab expression i = sqrt(-1).
For the following signals use your DTFT function to

i. Compute X(ω)
ii. Plot the magnitude and the phase of X(ω) in a single plot using the subplot command.
Hint: Use the abs() and angle() commands.

1. x[n] = δ[n]
2. x[n] = δ[n − 5]
3. x[n] = (0.5)n u[n]

INLAB REPORT:
Hand in a printout of your Matlab function. Also hand in plots of the DTFT’s magnitude
and phase for each of the three signals.

5.2 Magnitude and Phase of the Frequency Response of a Discrete-

Time Systems

INLAB REPORT:
Submit these exercises with the lab report.
Consider the discrete-time system described by the following difference equation:

y(n) = 0.9y[n − 1] + 0.3x[n] + 0.24x[n − 1]

Assume that the system is causal.

i. Draw a system diagram.

ii. Obtain the impulse response of the system by replacing x[n] with δ[n] in the above
equation. (Use causality to set up the initial conditions.)
iii. Use your answer in (ii) to obtain the frequency response of the system.
iV. Find the frequency response of the system using another method. Specifically, take
the DTFT of the left-hand-side and right-hand-side of the difference equation, and
then use linearity and the time-shifting property of the DTFT along with the fact that
Y (ω)
H(ω) = X(ω) .

v. Use Matlab to compute and plot the magnitude and phase responses, |H(ω)| and
6 H(ω), for −π < ω < π. You may use Matlab commands phase and abs.
Purdue University: ECE438 - Digital Signal Processing with Applications 10

Mux

Sine Wave Mux Scope

1/z
1

Unit Delay1 Sum1

Gain1

1/z 1

Unit Delay 2 Gain2 Sum 2

Gain3

Figure 8: Incomplete Simulink setup for the discrete-time system analysis experiment.

5.3 System Analysis

Help on printing Simulink system windows

matlab/help/pdf/print.pdf/matlab/help/pdf/print.pdf

Double click the icon labeled DT System Analysis to bring up an incomplete block
diagram as shown in Fig. 8. It is for a model that takes a discrete-time sine signal, processes
it according to a difference equation and plots the multiplexed input and output signals in a
graph window. Complete this block diagram such that it implements the following difference
equation given in section 5.2.

y(n) = 0.9y[n − 1] + 0.3x[n] + 0.24x[n − 1]

You are provided with the framework of the setup and the building blocks that you will
need. You can change the values of the Gain blocks by double clicking on them. After
you complete the setup, adjust the frequency of Sine Wave to the following frequencies:
ω = π/16, ω = π/8, and ω = π/4. For each frequency, make magnitude response measure-
ments using the input and output sequences shown in the graph window. Compare your
measurements with the values of the magnitude response |H(ejω )| which you computed in
the background exercises at these frequencies.
An alternative way of finding the frequency response is taking the DTFT of the impulse
response. Use your DTFT function to find the frequency response of this system from
its impulse response. The impulse response was calculated in section 5.2 of the background
exercises. Plot the impulse response, and the magnitude and phase of the frequency response
in the same figure using the subplot command.
Purdue University: ECE438 - Digital Signal Processing with Applications 11

INLAB REPORT:
Hand in the following:

• Printout of your completed block diagram

• Table of both the amplitude measurements you made and their theoretical values.

• Printout of the figure with the impulse response, and the magnitude and phase of the
frequency response.