Lab 2: Sinusoidal Signals, Complex Exponentials,
and Manipulation of Audio Signals
ELG3125 A – Signal and System Analysis
Fall 2017
School of Engineering and Computer Science
University of Ottawa
Professor: Martin Bouchard
Group Members:
Duc Duy Pham, 7812628
Vinh-Khiem Justin Tran, 8199665
Experiment Date: September 26th, 2017
Report Due Date: October 10th, 2017
Table of Contents
1.0 Objectives.................................................................................................................................1
2.0 Reading and Writing Audio Files...........................................................................................1
2.1 Results of Reading and Writing Audio Files..........................................................................1
3.0 Sinusoidal Signals and Periodicity.........................................................................................2
3.1 Results of Sinusoidal Signals and Periodicity........................................................................2
4.0 Complex Exponential Signals.................................................................................................8
4.1 Results of Complex Exponential Signals...............................................................................8
Appendix A: Reading and Writing Audio Files Matlab Source Code....................................10
Appendix B: Sinusoidal Signals and Periodicity Matlab Source Code...................................11
Appendix C: Complex Exponential Signals Matlab Source Code..........................................17
ii
1.0 Objectives
The objectives as stated in the lab manual is, “… to learn to read, listen to, and write
audio files from Matlab, and to experiment with the properties of sinusoidal signals and complex
exponentials in continuous time and discrete time.”
2.0 Reading and Writing Audio Files
In this part of the lab, one will learn how to listen to a generated wave. They will then
learn how to write the wave to a sound file and then read from the sound file. The read sound
wave will then be verified against the original generated wave.
2.1 Results of Reading and Writing Audio Files
The verification of the generated wave and the wave read from the audio file are shown
in Figure 2.1, below. As can be seen, the two signals overlap each other on the plot. This proves
that the wave that was read is a match of the generated wave.
Figure 2.1. Comparison of generated wave and read wave.
1
3.0 Sinusoidal Signals and Periodicity
In this part of the lab, one will learn about the different properties of continuous time and
discrete time sinusoidal signals.
3.1 Results of Sinusoidal Signals and Periodicity
Question 1
ω1 =
π
2
ω2 =3
ω3 =√ 2
Figure 3.1. Comparison between three sinusoidal signals with different angular frequencies in
continuous time.
 Regardless of the angular frequency, the sinusoidal signal in continuous time are always
periodic.
Question 2
π
ω1 = ( Periodic )
2
ω2 =3 ( Aperiodic )
ω3 =√ 2 ( Aperiodic )
2
Figure 3.2. Comparison between three sinusoidal signals with different angular frequencies in
discrete time.
 Sinusoidal signals in discrete time are not always periodic.
Question 3
2
ω1 = 2 π=π
4
3
3
ω2 = 2 π= π
4
2
4
ω3 = 2 π=2 π
4
3
Figure 3.3. Comparison between three sinusoidal signals, in discrete time, which have the same
period but are distinct (different values of m, constant values of N).
 While sinusoidal signals, in discrete time, can have the same period, they are distinct.
Question 4
1
ω1 = ×2 π
5
ω2 =ω1 +2 π =
12
π
5
ω3 =ω1 +2 ×2 π=
22
π
5
Figure 3.4. Comparison between three sinusoidal signals, in discrete time, with a difference of
multiple of 2 π in the angular frequencies
 One can add a multiple of 2 π in the angular frequency of a sinusoidal signal in
discrete time and still get the same resulting signal.
Question 5
low frequency , closer
ω1=0.00123 ( ¿0 )
ω2 =1.55567
4
high frequency , closer
ω 3=2.99978 ( ¿π )
Figure 3.5. Comparison between three sinusoidal signals in discrete time with near 0/near π
angular frequency.
 In discrete time, the sinusoidal signals with frequencies near 0 generate the slowest
change and those with frequencies near π generate the fastest change.
Question 6
ω1 =π → T 1=2
ω2 =2 π →T 2=1
T 3 =2 ( which isthe LCM of 2∧1 )
5
Figure 3.6.1. Addition of two sinusoidal signals in continuous time (with LCM).
π
ω1 = → T 1 =6
3
π
ω2 = → T 2=4
2
T 3 =12 ( which isthe LCM of 6∧4 )
Figure 3.6.2. Addition of two sinusoidal signals in discrete time (with LCM).
 In either continuous time or discrete time, adding two sinusoidal signals will result in a
sinusoidal signal whose period will be the LCM of the two original periods.
6
Question 7
ω1 =10→ T 1=
π
5
π
ω2 = → T 2=4
2
T 3 is indeterminate
Figure 3.7. Addition of two sinusoidal signals in discrete time (no LCM).
 When the two original periods have no LCM, their resulting signal addition will not be
periodic. In other words, it will have a infinite period.
Question 8
ω1 =3 ( non− periodic )
π
ω2 = ( periodic )
2
7
Figure 3.8. Addition of non-periodic sinusoidal signal and periodic sinusoidal signal.
 In discrete time, adding a non-periodic sinusoidal signal with a periodic sinusoidal signal
will result in a non-periodic one.
4.0 Complex Exponential Signals
In this part of the lab, one will learn the effects of a positive frequency and a negative
frequency on the result of a complex exponential signal. They will then verify that the real part
of a complex exponential is a cosine wave and that the imaginary part of a complex exponential
is a sine wave.
4.1 Results of Complex Exponential Signals
The effects of positive frequency and negative frequency on complex exponentials are
shown in Figure 4.1, below. As can be seen, a positive frequency on a complex exponential
causes the resulting three-dimensional plot to form a corkscrew shape that is rotating clockwise.
A negative frequency results in a corkscrew shape that is rotating counter clockwise.
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Figure 4.1. Comparison of positive frequency and negative frequency on complex exponentials.
The verification of the real part of a complex exponential representing a cosine wave and
the imaginary part of a complex exponential representing a sine wave are shown in Figure 4.2,
below. As can be seen, the expected results are indeed confirmed.
9
Figure 4.2. Verification of real and imaginary parts of complex exponential.
10
Appendix A: Reading and Writing Audio Files Matlab Source Code
clear all
clc
% Create wave and play wave
n=0:1:16000;
y=sin(2*pi*0.1*n)+sin(2*pi*0.2*n);
adjust=0.64983591643110114696039250089352;
y1=adjust*y;
Fs=8000;
sound(y1,Fs)
% Write wave to file
filename='Lab2_Part1.wav';
audiowrite(filename,y1,Fs);
% Read wave file
clear Fs
[z,Fs]=audioread(filename);
% Display and playing audioread
pause(1)
figure
subplot(2,1,1)
stem(n(1:100),y1(1:100),'black')
grid()
title('Generated Wave')
xlabel('n (samples)')
ylabel('y[n]')
subplot(2,1,2)
stem(n(1:100),z(1:100),'yellow')
grid()
title('Read Wave')
xlabel('n (samples)')
ylabel('z[n]')
sound(z,Fs)
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Appendix B: Sinusoidal Signals and Periodicity Matlab Source Code
Question 1
w1 = pi/2;
w2 = 3;
w3 = sqrt(2);
t = 0:0.0001:8;
y1 = sin(w1*t);
y2 = sin(w2*t);
y3 = sin(w3*t);
%plot the 3 graphs
figure
subplot(3,1,1)
plot(t,y1)
grid on
title('Sinusoidal Signal sin(pi/2t)')
xlabel('t (s)')
legend('sin(pi/2t)','Location','northeast')
subplot(3,1,2)
plot(t,y2)
grid on
title('Sinusoidal Signal sin(3t)')
xlabel('t (s)')
legend('sin(3t)','Location','northeast')
subplot(3,1,3)
plot(t,y3)
grid on
title('Sinusoidal Signal sin(sqrt(2)t)')
xlabel('t (s)')
legend('sin(sqrt(2)t)','Location','northeast')
Question 2
w1 = pi/2;
w2 = 3;
w3 = sqrt(2);
n = -8:1:8;
y1 = sin(w1*n);
y2 = sin(w2*n);
y3 = sin(w3*n);
%plot the 3 graphs
figure
subplot(3,1,1)
stem(n,y1)
grid on
title('Sinusoidal Signal sin(pi/2n)')
xlabel('n (samples)')
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legend('sin(pi/2n)','Location','northeast')
subplot(3,1,2)
stem(n,y2)
grid on
title('Sinusoidal Signal sin(3n)')
xlabel('n (samples)')
legend('sin(3n)','Location','northeast')
subplot(3,1,3)
stem(n,y3)
grid on
title('Sinusoidal Signal sin(sqrt(2)n)')
xlabel('n (samples)')
legend('sin(sqrt(2)n)','Location','northeast')
Question 3
w1 = pi;
w2 = 3/2*pi;
w3 = 2*pi;
n = -8:1:8;
y1 = sin(w1*n);
y2 = sin(w2*n);
y3 = sin(w3*n);
%plot the 3 graphs
figure
subplot(3,1,1)
stem(n,y1)
grid on
title('Sinusoidal Signal sin(pi*n)')
xlabel('n (samples)')
legend('sin(pi*n)','Location','northeast')
subplot(3,1,2)
stem(n,y2)
grid on
title('Sinusoidal Signal sin(3/2*pi*n)')
xlabel('n (samples)')
legend('sin(3/2*pi*n)','Location','northeast')
subplot(3,1,3)
stem(n,y3)
grid on
title('Sinusoidal Signal sin(2*pi*n)')
xlabel('n (samples)')
legend('sin(2*pi*n)','Location','northeast')
Question 4
w1 =(1/5)*2*pi;
w2 = w1+2*pi;
w3 = w2+2*pi;
13
n = -8:1:8;
y1 = sin(w1*n);
y2 = sin(w2*n);
y3 = sin(w3*n);
%plot the 3 graphs
figure
subplot(3,1,1)
stem(n,y1)
grid on
title('Sinusoidal Signal sin((1/5)*2*pi*n)')
xlabel('n (samples)')
legend('sin((1/5)*2*pi*n)','Location','northeast')
subplot(3,1,2)
stem(n,y2)
grid on
title('Sinusoidal Signal sin((w1+2*pi)*n)')
xlabel('n (samples)')
legend('sin((w1+2*pi)*n)','Location','northeast')
subplot(3,1,3)
stem(n,y3)
grid on
title('Sinusoidal Signal sin((w2+2*pi)*n)')
xlabel('n (samples)')
legend('sin((w2+2*pi)*n)','Location','northeast')
Question 5
w1 = 0.00123;
w2 = 1.55567;
w3 = 2.99978;
n = -8:1:8;
y1 = sin(w1*n);
y2 = sin(w2*n);
y3 = sin(w3*n);
%plot the 3 graphs
figure
subplot(3,1,1)
stem(n,y1)
grid on
title('Sinusoidal Signal sin(0.00123n)')
xlabel('n (samples)')
legend('sin(0.00123n)','Location','northeast')
subplot(3,1,2)
stem(n,y2)
grid on
title('Sinusoidal Signal sin(1.55567n)')
xlabel('n (samples)')
legend('sin(1.55567n)','Location','northeast')
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subplot(3,1,3)
stem(n,y3)
grid on
title('Sinusoidal Signal sin(2.99978n)')
xlabel('n (samples)')
legend('sin(2.99978n)','Location','northeast')
Question 6
Continuous
w1 = pi;
w2 = 2*pi;
t = -8:0.00001:8;
y1 = sin(w1*t);
y2 = sin(w2*t);
y3=y1+y2;
%plot the 3 graphs
figure
subplot(3,1,1)
plot(t,y1)
grid on
title('Sinusoidal Signal sin(pi*t)')
xlabel('t (s)')
legend('sin(pi*t)','Location','northeast')
subplot(3,1,2)
plot(t,y2)
grid on
title('Sinusoidal Signal sin(2*pi*t)')
xlabel('t (s)')
legend('sin(2*pi*t)','Location','northeast')
subplot(3,1,3)
plot(t,y3)
grid on
title('Sinusoidal Signal y3=sin(pi*t)+sin(2*pi*t)')
xlabel('t (s)')
legend('y3=sin(pi*t)+sin(2*pi*t)','Location','northeast')
Discrete
w1 = pi/3;
w2 = pi/2;
n = -20:1:20;
y1 = sin(w1*n);
y2 = sin(w2*n);
y3=y1+y2;
%plot the 3 graphs
figure
15
subplot(3,1,1)
stem(n,y1)
grid on
title('Sinusoidal Signal sin(pi/3*n)')
xlabel('n (samples)')
legend('sin(pi/3*n)','Location','northeast')
subplot(3,1,2)
stem(n,y2)
grid on
title('Sinusoidal Signal sin(pi/2*n)')
xlabel('n (samples)')
legend('sin(pi/2*n)','Location','northeast')
subplot(3,1,3)
stem(n,y3)
grid on
title('Sinusoidal Signal y3=sin(pi/3*n)+sin(pi/2*n)')
xlabel('n (samples)')
legend('y3=sin(pi/3*n)+sin(pi/2*n)','Location','northeast')
Question 7
w1 = 10;
w2 = pi/2;
n = -50:1:50;
y1 = sin(w1*n);
y2 = sin(w2*n);
y3=y1+y2;
%plot the 3 graphs
figure
subplot(3,1,1)
stem(n,y1)
grid on
title('Sinusoidal Signal sin(10n)')
xlabel('n (samples)')
legend('sin(10n)','Location','northeast')
subplot(3,1,2)
stem(n,y2)
grid on
title('Sinusoidal Signal sin(pi/2*n)')
xlabel('n (samples)')
legend('sin(pi/2*n)','Location','northeast')
subplot(3,1,3)
stem(n,y3)
grid on
title('Sinusoidal Signal y3=sin(10n)+sin(pi/2*n)')
xlabel('n (samples)')
legend('y3=sin(10n)+sin(pi/2*n)','Location','northeast')
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Question 8
w1 = 3;
w2 = pi/2;
n = -20:1:20;
y1 = sin(w1*n);
y2 = sin(w2*n);
y3=y1+y2;
%plot the 3 graphs
figure
subplot(3,1,1)
stem(n,y1)
grid on
title('Sinusoidal Signal sin(3n)')
xlabel('n (samples)')
legend('sin(3n)','Location','northeast')
subplot(3,1,2)
stem(n,y2)
grid on
title('Sinusoidal Signal sin(pi/2*n)')
xlabel('n (samples)')
legend('sin(pi/2*n)','Location','northeast')
subplot(3,1,3)
stem(n,y3)
grid on
title('Sinusoidal Signal y3=sin(3n)+sin(pi/2n)')
xlabel('n (samples)')
legend('y3=sin(3n)+sin(pi/2*n)','Location','northeast')
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Appendix C: Complex Exponential Signals Matlab Source Code
clear all
clc
% Base variables
dt=0.01;
t = 0:dt:3;
fp=1;
fn=-1;
% Positive frequency exponential
zp=exp(j*2*pi*fp*t);
% Negative frequency exponential
zn=exp(j*2*pi*fn*t);
% Compare positive and negative frequnecy
figure
plot3(t,real(zp),imag(zp),'black');
hold on
plot3(t,real(zn),imag(zn),'yellow');
hold off
grid()
title('Effects of Positive Frequency and Negative Frequency on Complex
Exponentials')
xlabel('t (s)')
legend('Positive Frequency','Negative Frequency')
% Check real = cos imag = sin
figure
subplot(2,1,1)
plot(t,real(zp),'black')
grid()
title('Re(exp)=cos')
xlabel('t (s)')
ylabel('Re(exp)')
subplot(2,1,2)
plot(t,imag(zp),'yellow')
grid()
title('Im(exp)=sin')
xlabel('t (s)')
ylabel('Im(exp)')
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