Laboratory Report
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Laboratory Report
ELEC 342 Laboratory experiment #2
Additional MATLAB features, Properties of Signals and Systems, Convolution and System
Response
By
Anas Senouci (40132281)
Laboratory made on February 1
st
, 2022 Lab section: MJ-X
Lab instructor: Mohebbi, Ali
Due date: February 14, 2022
OBJECTIVES
Part I of this lab will go over some more MATLAB programming language capabilities,
such as loops, conditional selection and array processing.
Part II will demonstrate how to utilise the MATLAB convolution function show the
relation between input, output and transfer function. The goal will be to understand how
MATLAB can verify various properties of signals and systems such as: linearity, time
variation, evenness, oddness will be verified using simple MATLAB scripts.
THEORY
The theory behind this lab was mainly to learn new concepts like loops, the logic behind
functions properties like linearity and time variation. TASKS, RESULTS AND DISCUSSION
Part I
Questions 1-a:
The first question aims to find the energy of a signal and use the disp
function of MATLAB. get
familiar with MATLAB's subplot and hold commands. The total energy of x[n] found was 285
and the total energy of y[n] was 15333. The result can be found in appendix I with the code used.
The graphs created (Figure 1) show that y[n] grow exponentially since y[n]=x[n]^2.
Questions 1-b:
The part of the question is the same as the first, but the input was changed to be a sinusoidal
function. The energy of x[n] was found to be 5 and the energy of y[n] was found to be 3.75. The
Figure 2 in the Appendix II show the signals. Questions 2-a:
Question 2 is all about understanding the properties of signal, mainly linearity. In the first part, a
function sin and a function cos were given as being dual using a system. To find out if the signals
were linear, the law of superposition and homogeneity was applied to the systems (scaling and
additivity). When the scaled outputs y1 and y2 are added together, they produce the same result
as when the scaled inputs x1 and x2 are summed. This is what Figure 3 shows, the results are
similar whose systems are linear. Questions 2-b:
This question refers to the concept of linearity but also of temporal variance. An input is given,
and the output is squared. Is this system linear and time invariant? This is what we will discover
in two ways, with an output equal to 1 or 0 and an output of our much wider choice. The trap is
that 0 squared gives back 0 so we could believe that some results are the same while for the rest
of the figures, the law of additivity and scaling are not respected. We find using the formulas in
Appendix IV and Figure 4 that the system y[n] = x2[n] is not linear and is invariant over time.
Also, the system y[n] = 2x[n] + 5δ[n] is not linear and is time variant. Figures 6 and 7 show that
the delay of n (the time value), the systems results are not the same for outputs and inputs.
Question 3-a:
For the last task of the first part, the concept of even, odd and mirror components are introduced
with
the
MATLAB
functions
abs
and
exp.
The
function
p has been used. The mirror of this function can be found in Figure 12, the only one has chosen
to do was to reverse the sign of n. Subsequently, the odd and even components were found
thanks to the functions: even= (x[n]+x[-n])/2 and odd=(x[n]-x[-n])/2. It is important to say that
this signal has no even part so the signal itself is odd over the interval n: 0 to 10.
Question 3-b:
The same process was done as the first part of the question but with a new system: This system has no odd component, so it is therefore, even. Question 3-c:
The first method to generate the MATLAB arrays x1 and x2 uses the initialisation of an array n
of size 20. Then, the array is used as the variable n of the x1 equation. This method is easy to
understand and uses only two lines of code. The other method used a for loop to fill elements in
the array. It is a bit more complex to understand and needs more time from the computer. The
first method is ideal.
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Part II
Question 1:
This question introduces the concept of output response of signals and difference equation. A for
loop was used to the array y that represents . The Figure 15 shows the output
response.
Question 2:
This question introduces the function conv
of MATLAB which is used to make convolutions.
The question asks to compute the system response of question 1 by convolving H[n] and x[n].
The answer is shows by Figure 16. The main difference between the output in the first and
second question is the number of values from the conv function is bigger than the values from
the for loop (method one). The graphs however (Figure 15 and 16) are the same.
Question 3-a:
In this question, it is asked to find the linearity of an unknown signal. The linearity was tested
with the additivity and scaling methods with random inputs. In this case, the constants of scaling
are 5 and 3 because in the case of exponentials, the output will differ. Putting something different
and not equal to 1 or 0 gives very few possibilities to make mistakes. The output will show if the
system in linear or not, if the two outputs are similar, then the system is linear. Observations
were made, when different inputs were fed to the system, different outputs showed as arrays. The
system is linear, but more proof is needed like feeding a ramp function to the system. If the
output of the system is a line, then the system is linear.
Question 3-b:
The time variance of the system can be checked by shifting the original input by a constant and
see if the output is the same as the non-shifted one. In the experiment, the constant 1 has been
shifting the input and output. The system is time variant.
CONCLUSION
The lab explores the use of the logic and concept of the MATLAB program that are used to find
the properties of signals. Several basic functions and commands needed to be used to answer
questions about linearity, convolution, time variance of different systems. Simple proofs were
required to promote learning of computer behavior and operations.
APPENDIX I
%Question 1 a):
clear all
; clc;
n=0:9; x=n; y=x.^2; subplot(2,1,1);
stem(n,x);
xlabel(
'n'
);
ylabel(
'Amplitude'
);
title(
'x[n]'
)
subplot(2,1,2);
stem(n,y);
xlabel(
'n'
);
ylabel(
'Amplitude'
);
title(
'y[n]'
);
Ex=0;
Ey=0;
for i = 1:10
Ex=Ex+x(i)^2;
Ey=Ey+y(i)^2;
end
disp(
'Energy of signal x[n] is '
);
disp(Ex);
disp(
'Energy of signal y[n] is '
);
disp(Ey);
Energy of signal x[n] is 285
Energy of signal y[n] is 15333
Figure 1. Part I Q1(a)
APPENDIX II
%Question 1 b):
clear all
; clc;
n=0:9; x=sin((2*pi)/10*n);
y=x.^2; subplot(2,1,1);
stem(n,x);
xlabel(
'n'
);
ylabel(
'Amplitude'
);
title(
'x[n]'
);
subplot(2,1,2);
stem(n,y);
xlabel(
'n'
);
ylabel(
'Amplitude'
);
title(
'y[n]'
);
Ex=0;
Ey=0;
for i = 1:10
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Ex=Ex+x(i)^2;
Ey=Ey+y(i)^2;
end
disp(
'Energy of signal x[n] is '
);
disp(Ex)
disp(
'Energy of signal y[n] is '
);
disp(Ey)
Energy of signal x[n] is 5.0000
Energy of signal y[n] is 3.7500
Figure 2. Part I Q1(b)
APPENDIX III
%Question 2 a):
clear all
; clc;
n=[0:10];
x1=sin((2*pi/10)*n);
x2=cos((2*pi/10)*n);
x3=x1+x2;
y1=2.*x1;
y2=2.*x2;
y3=2.*x3;
if (y3 == (y1+y2))
disp(
"Output is consistent with a linear system"
);
else
disp(
"System is not linear"
);
end
subplot(2,3,1);
stem(n,x1);
xlabel(
'n'
);ylabel(
'x1(n)'
);title(
'x1(n)=sin(2*pi*n/10)'
)
subplot(2,3,2);
stem(n,x2);
xlabel(
'n'
);ylabel(
'x2(n)'
);title(
'x2(n)=cos(2*pi*n/10)'
)
subplot(2,3,3);
stem(n,x3);
xlabel(
'n'
);ylabel(
'x2(n)'
);title(
'x3(n)=x1(n)+x2(n)'
)
subplot(2,3,4);
stem(n,y1);
xlabel(
'n'
);ylabel(
'y1(n)'
);title(
'y1(n)=2*x1(n)'
)
subplot(2,3,5);
stem(n,y2);
xlabel(
'n'
);ylabel(
'y2(n)'
);title(
'y2(n)=2*x2(n)'
)
subplot(2,3,6);
stem(n,y3);
xlabel(
'n'
);ylabel(
'y3(n)'
);title(
'y3=y1(n)+y2(n)'
)
Output is consistent with a linear system
Figure 3. Part I Q2(a)
APPENDIX IV
%Question 2 b):
clear all
; clc;
%i)
%Linearity n[0,1]:
n=0:20;
for i=1:length(n)
x1(i)=1;
x2(i)=1;
end
x3=x1+x2;
y3=x3.^2;
y1=x1.^2;
y2=x2.^2;
y4=y1+y2;
subplot(2,1,1);
stem(n,y3);
xlabel(
'n'
);
ylabel(
'Amplitude'
);
title(
'y3[n]'
);
subplot(2,1,2);
stem(n,y4);
xlabel(
'n'
);
ylabel(
'Amplitude'
);
title(
'y4[n]'
);
if y3==y4
disp(
'Output consistent with a linear system'
)
else
disp(
'Non-linear system'
)
end
%Time invariance n[0,1]:
figure
subplot(2,2,1);
stem(n,y3);
xlabel(
'n'
);
ylabel(
'Amplitude'
);
title(
'y3[n]'
);
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subplot(2,2,2);
stem(n-1,y3);
xlabel(
'n'
);
ylabel(
'Amplitude'
);
title(
'y3[n-1]'
);
subplot(2,2,3);
stem(n,x3);
xlabel(
'n'
);
ylabel(
'Amplitude'
);
title(
'x3[n]'
);
subplot(2,2,4);
stem(n-1,x3);
xlabel(
'n'
);
ylabel(
'Amplitude'
);
title(
'x3[n-1]'
);
if y3(1)== x3(1).^2;
disp(
'Time invariant'
)
else
disp(
'Time variant'
)
end
%Linearity bigger n:
for i=1:length(n)
x1(i)=i;
x2(i)=i+13;
end
x3=x1+x2;
y1=x1.^2;
y2=x2.^2;
y3=x3.^2;
y4=y1+y2;
figure
subplot(2,1,1);
stem(n,y3);
xlabel(
'n'
);
ylabel(
'Amplitude'
);
title(
'y3[n]'
);
subplot(2,1,2);
stem(n,y4);
xlabel(
'n'
);
ylabel(
'Amplitude'
);
title(
'y4[n]'
);
if y3==y4
disp(
'Output consistent with a linear system'
)
else
disp(
'Non-linear system'
)
end
%Time invariance bigger n:
figure
subplot(2,2,1);
stem(n,y3);
xlabel(
'n'
);
ylabel(
'Amplitude'
);
title(
'y3[n]'
);
subplot(2,2,2);
stem(n-4,y3);
xlabel(
'n'
);
ylabel(
'Amplitude'
);
title(
'y3[n-4]'
);
subplot(2,2,3);
stem(n,x3);
xlabel(
'n'
);
ylabel(
'Amplitude'
);
title(
'x3[n]'
);
subplot(2,2,4);
stem(n-4,x3);
xlabel(
'n'
);
ylabel(
'Amplitude'
);
title(
'x3[n-4]'
);
r=6;
if y3(r+2)== x3(r+2).^2;
disp(
'Time invariant'
)
else
disp(
'Time variant'
)
end
n[0,1]:
Non-linear system
Time invariant
Bigger n:
Non-linear system
Time invariant
Figure 4.
Part I Q2 (b) i)
Figure 5.
Part I Q2 (b) i)
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Figure 6.
Part I Q2 (b) i)
Figure 7.
Part I Q2 (b) i)
APPENDIX V
%Question 2 b):
clear all
; clc;
%ii)
%Linearity n[0,1]:
n=0:20;
for i=1:length(n)
x1(i)=0;
x2(i)=1;
d1(i)=dirac(0);
d2(i)=dirac(1);
end
x3=x1+x2;
y3=x3.^2;
y1=2*x1+5*d1;
y2=2*x2+5*d2;
y4=y1+y2;
subplot(2,1,1);
stem(n,y3);
xlabel(
'n'
);
ylabel(
'Amplitude'
);
title(
'y3[n]'
);
subplot(2,1,2);
stem(n,y4);
xlabel(
'n'
);
ylabel(
'Amplitude'
);
title(
'y4[n]'
);
if y3==y4
disp(
'Output consistent with a linear system'
)
else
disp(
'Non-linear system'
)
end
%Time invariance n[0,1]:
figure
subplot(2,2,1);
stem(n,y3);
xlabel(
'n'
);
ylabel(
'Amplitude'
);
title(
'y3[n]'
);
subplot(2,2,2);
stem(n-1,y3);
xlabel(
'n'
);
ylabel(
'Amplitude'
);
title(
'y3[n-1]'
);
subplot(2,2,3);
stem(n,x3);
xlabel(
'n'
);
ylabel(
'Amplitude'
);
title(
'x3[n]'
);
subplot(2,2,4);
stem(n-1,x3);
xlabel(
'n'
);
ylabel(
'Amplitude'
);
title(
'x3[n-1]'
);
if y3(1)== 2*x3(1)+5*dirac(1)
disp(
'Time invariant'
)
else
disp(
'Time variant'
)
end
%Linearity bigger n:
for i=1:length(n)
x1(i)=i;
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x2(i)=i+12;
d1(i)=dirac(i);
d2(i)=dirac(i+12);
end
x3=x1+x2;
y3=x3.^2;
y1=2*x1+5*d1;
y2=2*x2+5*d2;
y4=y1+y2;
figure
subplot(2,1,1);
stem(n,y3);
xlabel(
'n'
);
ylabel(
'Amplitude'
);
title(
'y3[n]'
);
subplot(2,1,2);
stem(n,y4);
xlabel(
'n'
);
ylabel(
'Amplitude'
);
title(
'y4[n]'
);
if y3==y4
disp(
'Output consistent with a linear system'
)
else
disp(
'Non-linear system'
)
end
%Time invariance bigger n:
figure
subplot(2,2,1);
stem(n,y3);
xlabel(
'n'
);
ylabel(
'Amplitude'
);
title(
'y3[n]'
);
subplot(2,2,2);
stem(n-4,y3);
xlabel(
'n'
);
ylabel(
'Amplitude'
);
title(
'y3[n-4]'
);
subplot(2,2,3);
stem(n,x3);
xlabel(
'n'
);
ylabel(
'Amplitude'
);
title(
'x3[n]'
);
subplot(2,2,4);
stem(n-4,x3);
xlabel(
'n'
);
ylabel(
'Amplitude'
);
title(
'x3[n-4]'
);
if y3(6)== 2*x3(6)+5*dirac(6);
disp(
'Time invariant'
)
else
disp(
'Time variant'
)
end
n[0,1]:
Non-linear system
Time variant
Bigger n:
Non-linear system
Time variant
Figure 8.
Part I Q2 (b) ii)
Figure 9.
Part I Q2 (b) ii)
Figure 10.
Part I Q2 (b) ii)
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Figure 11.
Part I Q2 (b) ii)
APPENDIX VI
%Question 3 a):
clear all
; clc;
n=0:10;
x1=exp(-2.*abs(n)).*sin( (2*pi/36)*n);
subplot(2,2,1);
stem(n,x1);
ylabel x[n]
;
xlabel n
;
title 'x[n]'
;
% x1 mirror:
x2 = exp(-2.*abs(n)).*sin( (2*pi/36)*-n);
subplot(2,2,2)
stem(-n,x2);
ylabel x[n]
;
xlabel n
;
title 'Mirror of x[n]'
;
% x1 even:
e=(1/2)*(x1+x2);
subplot(2,2,3);
stem(n,e);
ylabel x[n]
;
xlabel n
;
title 'Even component of x[n]'
;
% x1 odd
o = (1/2) * ( x1 - x2);
subplot(2,2,4);
stem(n,o);
ylabel x[n]
;
xlabel n
;
title 'Odd component of x[n]'
;
Figure 12.
Part 1 Q3 (a)
APPENDIX VII
%Question 3 b):
clear all
; clc;
n=-5:5;
x1 = (-1).^(n-1);
subplot(2,2,1);
stem(n,x1);
ylabel x[n]
;
xlabel n
;
title 'x[n]'
;
% x1 mirror:
x2 = (-1).^(-n-1);
subplot(2,2,2)
stem(-n,x2);
ylabel x[n]
;
xlabel n
;
title 'Mirror of x[n]'
;
% x1 even:
e=(1/2)*(x1+x2);
subplot(2,2,3);
stem(n,e);
ylabel x[n]
;
xlabel n
;
title 'Even component of x[n]'
;
% x1 odd
o = (1/2) * ( x1 - x2);
subplot(2,2,4);
stem(n,o);
ylabel x[n]
;
xlabel n
;
title 'Odd component of x[n]'
;
Figure 13.
Part 1 Q3 (b)
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APPENDIX IIX
%Question 3 c):
clear all
; clc;
n = [1 : 20 ];
x1 = sin((2*pi/40) * n) .* cos((2*pi/40) * n);
for index = 1 : 20
x2(index) = sin((2*pi/40) * index) * cos((2*pi/40) * index);
end
subplot(2,1,1)
stem(n, x1)
title 'Elegant method making full use of MATLABs array capabilities'
xlabel n
ylabel x[n]
subplot(2,1,2)
stem(n, x2)
title 'Gets the job done but it is a lot of work and we are not in the MATLAB mindset'
xlabel n
ylabel x[n] Figure 14.
Part 1 Q3 (c)
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APPENDIX IX
%Part II:
%Question 1:
clear all
; clc;
n=0:9;
x=zeros(1:10);
x(2:4)=1;
y(1)=0;
for (i=1:10)
if (i==1)
y(i)=x(i);
else
y(i)=x(i)+((1/4).*y(i-1));
end
end
stem (n,y)
xlabel n
ylabel y[n]
title 'Output Response y[n]'
Figure 15.
Part 2 Q1
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APPENDIX X
%Part II:
%Question 2:
clear all
; clc;
n=0:9;
x=zeros(1:10);
x(2:4)=1;
y(1)=0;
h=(1/4).^n;
y=conv(x(1:9),h(1:9));
stem(y)
title 'Output response y[n]'
xlabel n
ylabel y[n]
Figure 16.
Part 2 Q2
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APPENDIX XI
%Part II:
%Question 3:
clear all
; clc;
%a)
x=[1 2 3];
y=Sys1(x);
n=0:10;
y1=5*y;
y2=3*y;
y3=y1+y2;
x1=Sys1(5*x);
x2=Sys1(3*x);
x3=x1+x2;
if y3==x3
disp(
'Output consistent with a linear system'
)
else
disp(
'Non-linear system'
)
end
disp(x3);
disp(y3);
%b)
if y3(1)== x3(1);
disp(
'Time invariant'
)
else
disp(
'Time variant'
)
end
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Excess-3 code is significant for arithmetic operations as it overcomes shortcoming encountered while using 8421 BCD code to add two decimal digits whose sum exceeds 9. Excess-3 arithmetic uses different algorithm than normal non-biased BCD or binary positional number system. An electronics company has hired your services to design a code converter that converts Binary Coded Decimal (BCD) code for it. Design the converter.
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Recommended textbooks for you
- Delmar's Standard Textbook Of ElectricityElectrical EngineeringISBN:9781337900348Author:Stephen L. HermanPublisher:Cengage Learning
Delmar's Standard Textbook Of Electricity
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ISBN:9781337900348
Author:Stephen L. Herman
Publisher:Cengage Learning