Signals And Systems
Lab Manual
Name: _________________________________________
Reg. No:___________ Section: _________ Group: ______
University of Engineering & Technology, Taxila
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Checked by:
Date:
Software Requirements
S.No.
1
Software Title
Matlab
Description
Computing mathematics laboratory
Table of Contents
Lab No.
Lab # 1
Lab Title
Page
Familiarize with the Matlab environment and running
some basic Matlab commnds in Matlab
Writing mathematical expressions and familiarizing
with some more advanced matlab commands
Understanding loops implementation in matlab
4
Lab # 4
Implementing complex algorithm in matlab using
loops
11
Lab # 5
12
Lab # 6
Lab # 7
Implementing (1) Decibel (2) Taking input from the
user (3) Root mean square (4) Harmonic mean (5)
Selection sort
Implementing Fourier transform in matlab
Verifying Fourier transform properties in matlab
Lab # 8
Lab # 9
Convolution of two rectangular pulses
Convolution of any two arbitrary inputs
15
16
Lab # 10
Lab # 11
Implementing Z-tranform in Matlab
Implementing Sampling in matlab
17
18
Lab # 12
Lab # 13
Lab # 14
Lab # 15
Lab # 16
Verfiying Nyguist theorem
Modulating a signal based on carrier frequency
Upsampling and downsampling of signal
Implementing a moving average filter
Implementing a moving average filter and plotting
the magnitude and phase of its fourier transform.
19
20
21
22
23
Lab # 2
Lab # 3
8
10
13
14
Experiment # 1
Lab Title:
Basic Matlab Tutorial
Learning
Objectives:
Familiarize with the Matlab environment and running some
basic Matlab commnds in Matlab
Equipment
Required:
Matlab
Variables and Arrays
The functional unit of data in any Matlab program is array. An Array is
a collection of data values organized into rows and columns and
known by a single name.
Arrays can be classified as either vectors or matrices. The term VECTOR
is usually used to describe an array with only one dimension. While the
term MATRIX is usually used to describe an array with two or more
dimensions.
Size of array is specified by the number of rows and the number of
columns in the array, with the number of rows mentioned first. The total
number of elements in the array will be the product of the number of
rows and the number of columns.
Try using these values:
[3.4]
[1.0 2.0 3.0]
[1.0; 2.0; 3.0]
[1, 2, 3; 4, 5, 6]
[1, 2, 3, 4, 5, 6]
[]
Not all the elements of an array need to be defined when it is
created. If a specific array element is defined and onwe or more of the
elements before it are not, then the earlier elements are automatically
created and initialized to zero. For example, if c is not previously
defined, the statement
C(2,3)=5
Will produce matrix 0 0 0
005
Similarly the array d=[1 2]. Then the statement
d(4)=4
will produce d=[1 2 0 4].
Initializing with shortcut expressions
Matlab provides a special shortcut notation for these circumstances
using the COLON operator. The colon operator specifies a whole series
of values by specifying the first vales in the series, the stepping
increment and then the last value in the series. The general form of a
colon operator is
First: increment: last
For example
x=1:2:10
will generate x= 1 3 5 7 9
Try using angles=(0.01:0.01:1.00)*pi;
Transpose Operator(‘):
This operator swaps the rows and columns of any array that it is applied
to.
For example try using:
F=[1:4]
And F=[1:4]’
Initializing with Built-in functions:
Try using these:
A=zeros(2)
B=zeros(2,3)
C=[1 2; 3 4]
D=zeros(size(C))
E=ones(3)
Ones(3,1)
Eye(3)
Eye(3,2)
Length(C) //Generate the longest dimension of the
array.
Zeros can be used to create an all zero array of any desired size. If
function has a single square array; it will generate a square array using
the single argument as both the number of rows and columns. size
function returns two values containing the number of rows and
columns in an array.
Initializing variables with keyboard input:
Input function displays a prompt string in the command window and
then waits for the user to type in a response. For example , consider the
following statement:
My_val=input(‘enter an input value’);
Multidimensioanl Array
Matlab allows us to create arrays as many dimensions as necessary for
any given problem. These arrays have one subscript for each
dimension and an individual element in the array will be the product of
the maximum value of each subscript. For example the following two
statements create a 2X2X3 array C:
C(:,:,1)=[1 2 3; 4 5 6];
C(:,:,2)=[7 8 9; 10 11 12];
Whos C
SubArrays
It is possible to select and use subsets of arrays as though they were
separate arrays. To select a portion of array, just include a list of all of
the elements to be selected in the parenthesis after the rray name. For
example:
Arr1=[1.1 2.2 3.3 4.4 5.5]
Try using Arr1(3)
And Arr1([1 4])
And also Arr1(1:2:5)
End Function:
It is very useful for creating array subscripts. When used in an array
subscript; end returns the highest value taken on by that subscript. For
example. Suppose that array arr3 is defined as follows:
Arr3=[ 1 2 3 4 5 6 7 8];
Then Arr1(5:end) would be the array [5 6 7 8] and array(end) would
generate 8.
The value returned by end is always the highest value of a given
subscript. Is end appears in different subscripts, it can return different
values within the same expression. For example, suppose that the 3X4
array Arr4 is defined as follows:
Arr4=[1 2 3 4; 5 6 7 8; 9 10 11 12];
Then the expression
Arr4(2:end, 2:end)
would return the array.(Try this one).
TASKS
Question # 01:
This M-file calculates and plot the function sin(x) for 0<=x<=6
X=0:0.1:6;
Y=sin(X);
plot(X,Y);
plot(X,Y)
Question # 02:
The following Matlab statement plot the function y(x)=2e-0.2x for the range
0<=x<=10
X=0:0.1:10;
Y=2*exp(0.2*X);
Plot(X,Y);
Question # 03:
suppose that u=1 and v=3. Evaluate the following expressions using
Matlab.
(a) 4u/3v
(b) 2v-2/(u+v)2
(c) v3/v3-u3
(d) 4/3 pi.v2
Question # 04:
(a) Generate a 6X6 Matrix.
(b) Generate a 6X1 Matrix
(c) Generate a 3X4 Matrix
Question # 05
Answer the following questions for the array shown below.
1.1 0.0 2.1 -3.5 6.0
0.0 1.1 -6.6 2.8 3.4
1.1 0.1 0.3 -0.4 1.3
-1.4 5.1 0.0 1.1 0.0
(a) what is the size of the array?
(b) What is the value of the array(4,1)?
(c) What is the size and value of array( : , 1 : 2 )?
(d) What is the size and Value of the (array[1 3],end)?
Question # 06
Are the following Matlab variable names legal or illegal? Why?
(a) dog1
(b) 1dog
(c) Do_you_know_the_way__to_lahore
(d) _help
Question # 07
Determine the size and contents of the follwings arrays. Note that the
later arrays may depend on the definition of arrays defined earlier in
this exercise
(a) a=1:2:5
(b) b=[a’ a’ a’]
(c) c=b(1:2:3,1:2:3);
(d) d=a+b(2,:);
(e) w=[zeros(1,3) ones(3,1)’ 3:5’]
(f) b([1 3],2 )=b([3 1],2);
Question # 08
Assume that the array is defined as follows; determine the contents of
the following subarrays
1.1
0.0
2.1
1.4
0.0
1.1
0.1
5.1
(a) array(3,:)
(b) array(:,3)
(c) array(1:2:3,[3 3 4])
(d) array([1 1],:)
2.1
6.6
0.3
0.0
3.5 6.0
2.8 3.4
0.4 1.3
1.1 0.0
Experiment # 2
Lab Title:
Basic Matlab Tutorial
Learning
Objectives:
Writing mathematical expressions and familiarizing with
some more advanced matlab commands
Equipment
Required:
Matlab
Question # 01:
Assume that value has been initialized to 10pi and determine what is
printed out by eah of the following statements:
(a) disp([‘value=’num2str(value)]);
(b) disp([‘value=’int2str(value)]);
(c) fprintf(‘value=%e\n’,value);
(d) fprintf(‘value=%f\n’,value);
(e) fprintf(‘value=%12.4f\n’,value);
Question # 02:
Evaluate the following expressions
(a) 11/5+6
(b) (11/5)+6
(c) 11/(5+6)
(d) 3^2^3
(e) (3^2)^3
(f) 3^(2^3)
(g) round(-11/5)+6
(h) ceil(-11/5)+6
(i) floor(-11/5)+6
Question # 03:
Evaluate the following Matlab expressions
(a) 5.5>=5
(b) 20>20
(c) xor(17-pi<15,pi<3)
(d) true>false
(e) ~~(35/17)==(35/17)
(f) (17<=8)==(3/2==1)
(g) 17.5&&(3.3>2.)
Question # 04:
The following statements are intended to alert a user to dangerously high
oral thermometer readings(values are in degree Farenheit ). Are they
correct or incorrect? If they are incorrect, why and correct them
if temp<97.5
disp(‘Temperature below normal’)
elseif temp>97.5
disp(‘temperature normal’)
elseif temp>99.5
disp(‘temperature slightly high)
elseif temp>103.0
disp(‘temperature dangerously high’)
end
Question # 05:
The cost of sending a packet by an express delivery service is $12.00 for the
first two pounds and $4.50 for each pound or fraction thereof over two
pounds. If the package weighs more than 70 pounds, a $ 15.00 excess
weight surcharge is added to cost. No package over 100 pounds will be
accepted. Write a program that accepts the weight of a package in
pounds and computes the cost of mailing the package. Be sure to handle
the case of overweight packages.
Question # 06:
Implement the following function f(x,y) as follows:
X+Y
for
X>=0 and Y>=0
2
X+Y
for
X>=0 and Y<0
2
X + Y for
X<0 and Y>=0
X2+Y2 for
X<0 and Y<0
Question # 07:
Try these built-in functions in Matlab
Abs(x)
Acos(x)
Asin(x)
Atan(x)
Atan2(y,x)
Ceil(x)
Fix(x)
Floor(x)
Round(x)
Char(x)
Experiment # 3
Lab Title:
Basic Matlab Tutorial
Learning
Objectives:
Understanding loops implementation in matlab
Equipment
Required:
Matlab
Question # 01:
Implement unit impulse function in Matlab
Question # 02:
Implement Unit step function in matlab
Question # 03:
Classify signal as even or odd in Matlab
Experiment # 4
Lab Title:
Basic Matlab Tutorial
Learning
Objectives:
Implementing complex algorithms in matlab using loops
Equipment
Required:
Matlab
Question # 01:
Calculate factorial of a number
Question # 02:
Implement the equation y(x)=x2-3x+2 for all the values of x between -1
and 3 with the increment of 0.1.
Question # 03:
Take a signal and scale it.
Experiment # 5
Lab Title:
Basic Matlab Tutorial
Learning
Objectives:
Implementing (1) Decibel (2) Taking input from the user (3)
Root mean square (4) Harmonic mean (5) Selection sort
Equipment
Required:
Matlab
Question # 01:
Engineers often measure the ratio of two power measurements in decibels
or dB. The equation for the ratio of two power measurements in decibel is
dB=10log10p2/p1
where p2 is the power level being measured and P1 is some reference
power level. Assume that the reference power level P1 is 1 Watt and write
a program that calculates the decibel level corresponding to power
levels between 1 and 20 watts in increment of 02 Watt steps. Plot the dBversus-power level on a log-linear scale.
Question # 02:
Write a matlab program that will accept an arbitrary number of possible
input values and calculate both the arithmetic mean and geometric
mean of the numbers. Use a While loop to get the input values and
terminate the inputs when the user inputs a negative value.
Question # 03:
Write a matlab program that will accept an arbitrary number of inputs
and caluculate the rms average of the numbers. Propmpt the user for the
number of values to be entered and use a FOR loop to read in the
numbers.
Question # 04:
Write a matlab program that will read an arbitrary number of possible
input values and caluculate the harmonic mean of the numbers. Use any
method you desire to read in the input values.
Question # 05:
Write a matlab program for the selection sort. The program should accept
5 input values and sort them according to the criteria
Question # 06:
Modify the selection sort so that it accepts a second parameter which
may be “UP” or “DOWN” . if the input argement is UP, sort the values in
ascending order. If the argument is “DOWN” , sort the values in
descending order. If the argument is missing, the default case is to sort the
values in ascending order. (Be sure to handle the case of invalid
arguments, and be sure to include the proper help information in your
functon)
Experiment # 6
Lab Title:
Basic Matlab Tutorial
Learning
Objectives:
Implementing Fourier transform in matlab
Equipment
Required:
Matlab
Question # 01:
Look for the Matlab commands
fft
Ifft
Conv
Question # 02:
Generate a signal as sum of two sinusoidal of frequency f1=400Hz and
f2=800Hz.Use fft to find the Fourier transform.
Question # 03:
Calculate Fourier transform of e^-at u(t).Plot original signal, magnitude
and phase of its Fourier transform and verify it from text book example#4.1
Question # 04:
Convolve the signals x1(t)=e^at and x2(t)=sint for 100 samples. Verify that
“convolution in time domain equals multiplication in frequency domain.”
Question # 05:
Prove that FT of sinc is gate & ifft of gate is sinc.
Experiment # 7
Lab Title:
Basic Matlab Tutorial
Learning
Objectives:
Verifying fourier transform properties.
Equipment
Required:
Matlab
Question # 01:
Verify following fourier transform properties in matlab
Linearity: a . x[n] + b . y[n]  a. X + b.Y
Shifting: x[n-nd]
 exp(-jwnd)X
Inverting: x[-n]
 X(exp(-jw))
Compute the LHS and take its Fourier Transform.
Compute the RHS.
Compare the two: both results should be equal.
Experiment # 8
Lab Title:
Basic Matlab Tutorial
Learning
Objectives:
Convolution
Equipment
Required:
Matlab
Question # 01:
Convolution of any two rectangular pulses
Experiment # 9
Lab Title:
Basic Matlab Tutorial
Learning
Objectives:
Generalized Convolution
Equipment
Required:
Matlab
Question # 01:
Make your own convolution code; perform convolution of two
rectangular pulses; show the step by step procedure (via flip and drag) of
convolution; compare your results with the built in Convolution command.
Experiment # 10
Lab Title:
Basic Matlab Tutorial
Learning
Objectives:
Implementing Z-Tranform in matlab
Equipment
Required:
Matlab
Question # 01:
Find the Z-transform of
(1) Impulse
(2) Unit step
(3) Delayed impulse by one unit to the right
(4) Delayed Unit step by one unit to the right
(5) anU(n)
(6) Sin wn
(7) Coswn
Question # 02:
Find the inverse Z-transform of all the answers collected from question # 01
and verify the answer that what you got in question 1 was right
Question # 03:
Take any two vectors and find their roots
Question # 04:
Display the pole-zero plot of
A=[0 1 1]
B=[1 -2 3]
Question # 05:
Try to make your own matlab code for the Z-transform; (Bonus marks)
Experiment # 11
Lab Title:
Basic Matlab Tutorial
Learning
Objectives:
Implementing sampling in matlab
Equipment
Required:
Matlab
Question # 01:
Create a signal x(t)=coswo(t). Generate the impulse train p(t) and sample
the signal using x(t)=x(t)p(t). Plot observed samples in Time Domain.
Question # 02:
Now Plot spectra of x(t) & p(t) and convolve both spectras. Also obtain
spectra of sampled signal from question 1.
Question # 03:
Study Aliasing from question 2. Consider the following frequencies for
Sampling.
a)f0=fs/2
b)f0=3/2fs
Experiment # 12
Lab Title:
Basic Matlab Tutorial
Learning
Objectives:
Verifying nyguist theorm
Equipment
Required:
Matlab
Question # 01:
With reference to the previous lab; Take any arbitrary signal; sample it;
Verify nyguist theorem.
Experiment # 13
Lab Title:
Basic Matlab Tutorial
Learning
Objectives:
Modulating a signal based on carrier frequency
Equipment
Required:
Matlab
Question # 01:
The experiment implements a gui-based tutorial of the effect of
multiplying a cosine with a carrier frequency ‘fc’ with a given signal. And
the effect are monitored both in time and frequency domain.
The parameters that the gui would take:
‘fc’: the carrier frequency
‘fs’: the sampling frequency
The gui would show the input signal as sampled
The resulted signal after multiplication with the carrier frequency should be
displayed.
Display frequency spectrum of both the original signal and the modulated
signal.
Experiment # 14
Lab Title:
Basic Matlab Tutorial
Learning
Objectives:
Upsampling and downsampling of signal
Equipment
Required:
Multimedia
Question # 01:
Upsampling is the process of increasing the sampling rate of a signal. The
upsampled signal satisfies the Nyquist-Shannon sampling theorem if the
original signal does. Consider a discrete signal f(k) on a radian frequency
digital frequency range. Let L denote the upsampling factor.
1. Add L-1 zeros between each sample in f(k). Or, equivalently define
2. Filter with a low-pass filter which, theoretically, should be the sinc
filter with frequency cut off at pi/L.
Downsampling (or "subsampling") is the process of reducing the sampling
rate of a signal. This is usually done to reduce the data rate or the size of
the data.
If the sampling theorem is not satisfied then the resulting digital signal will
have aliasing. To ensure that the sampling theorem is satisfied, a low-pass
filter is used as an anti-aliasing filter to reduce the bandwidth of the signal
before the signal is downsampled.
Let M denote the downsampling factor.
1. Filter the signal to ensure that the sampling theorem is satisfied. This
filter should, theoretically, be the sinc filter with frequency cutoff at
pi/M. Let the filtered signal be denoted g(k).
2. Reduce the data by picking out every Mth sample: h(k) = g(Mk).
Data rate reduction occurs in this step.
Experiment # 15
Lab Title:
Basic Matlab Tutorial
Learning
Objectives:
Implementing a moving average filter
Equipment
Required:
Multimedia
Question # 01:
Implement Moving average filter in matlab
Experiment # 16
Lab Title:
Basic Matlab Tutorial
Learning
Objectives:
Implementing a moving average filter and plotting the
magnitude and phase of its fourier transform.
Equipment
Required:
Multimedia
Question # 01:
Implementing a moving average filter and plotting the magnitude and
phase of its Fourier transform.