Workshop Number 1- Matlab Tutorial

8/10/2019 Workshop Number 1- Matlab Tutorial

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Signals & System - A Practical Approach Using Matlab 1999 -S.R. Taghizadeh

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Workshop Number 1

Matlab Tutorial

Introduction

The aim of this workshop is to introduce the concept of Matlab. The following topics

are included:

Simple arithmetic operations

Generation of signals, Manipulation and visualisation

Defining Systems and plotting frequency and phase responses

System responses

Spectral Analysis

Filter designing

Tutorial 1: Simple Arithmetic Operations

Once in Matlab environment, you can perform simple arithmetical operations similar to a

hand-held calculator. Some examples are shown below:


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2

4+5

ans =

9

123-76

ans =

47

4^2

ans =

16

7*(5+2)/5

ans =

9.8000

Consider the following example:

You type this.

This is Matlab Prompt.

Matlab generates this

Matlab prompt. Ready for more Command

To raise a number toa power: 4 2

Evaluation of anexpression


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x=156-87

x =

69

Once a variable is defined (like x above), Matlab stores this in the memory. The variable

x may be used to evaluate other expressions. For example:

y=x+1

y =

70

Now there are two variables and their values stored in the memory. To find out how

many variables are currently in the memory, type the command and press enter on

the prompt:

who

Your variables are:

ans x y

Here, a variable x has been defined asx=156-87. Matlab generates the answer as x=69.

Here we create another variable called, 'y' andassign the value of x in the memory plus 1.


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A similar command called performs the same operation, but also gives the size

of each variable:

whos

Name Size Elements Bytes Density Complex

ans 1 by 1 1 8 Full No

x 1 by 1 1 8 Full No

y 1 by 1 1 8 Full No

Grand total is 3 elements using 24 bytes

Note that x and y have size of 1 by 1. The reason for this is that Matlab treats all

variables as a Matrix (Hence the name Matlab which stands for MATrix LABoratory).

This means that all the arithmetical operations are performed as Matrix arithmetic.

Consider the following examples:

x=[1,2,3,4] % This creates a matrix x having one row and 4 columns (i.e. a vector )

x = % and initialises it to 1,2,3 and 4.

1 2 3 4

y=[5,6,7,8]

y =

5 6 7 8

Now lets execute the command :


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whos

Name Size Elements Bytes Density Complex

ans 1 by 1 1 8 Full No

x 1 by 4 4 32 Full No

y 1 by 4 4 32 Full No

Grand total is 9 elements using 72 bytes

As it can be seen, the variables x and y have dimensions of 1 by 4. This means that x

and y are matrices of 1 row and 4 columns each. When multiplying such variables,

matrix rules for multiplication is applied. For example:

z=x*y

??? Error using ==> *

Inner matrix dimensions must agree.

To multiply each member of x by a corresponding member of y, we perform a sample-

by-sample multiplication using the array operator ".*" as follows:

z=x.*y

z =

5 12 21 32


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Size & Length of a variable

Two commands, 'size' and 'length' may be used to determine its size as follows:

size(x)

ans =

1 4

length(x)

ans =

4

Solving Simultaneous Equations

Consider solving the following simultaneous equation:

2x1+6x 2=3

5x1-2x 2=12

Lets put this in matrix form:

123

2562

y x

12

3,

2

1 and

25

62 Q

x

x X P Let

Hence

PX=Q , Therefore X=P -1Q

In Matlab, This is defined and solved as followed:

'size' returns the number of rowsand columns of a variable.

'length' returns the number of elements of a variable.


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clear

P=[2,6;5,-2];

Q=[3;12];

X=inv(P)*Q

X =

2.2941

-0.2647

Question 1

Use Matlab to solve the following linear system:

0325

2

532

321

321

321

x x x

x x x

x x x

Here are the sequence of commands:

P=[2 3 -1;1 -1 1;5 2 -3];

Q=[5;2;0];

X=P\Q

X =

1.0000

2.0000

3.0000

This clears all previously declared variables

';' separates rows from columns

Matlab doesnot echo when a ';' at the end of adeclaration

'inv' inverts a matrix

This creates thematrix:

325

111132

P

This creates the columnvector:

02

5

QThis solves thesimultaneous equationusing: X=P -1Q


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Complex Numbers

In Matlab complex numbers/variables such as:

Z=a+jb may be defined as follows:

a=2;

b=3;

z=a+b*j

z =

2.0000 + 3.0000i

Examples of complex number arithmetic are given below:

z1=2+3*j;

z2=4-5*j;

za=z1+z1za =

4.0000 + 6.0000i

zd=z1-z2

zd =

-2.0000 + 8.0000i

zm=z1*z2

zm =

23.0000 + 2.0000i

zdiv=z1/z2

zdiv =

-0.1707 + 0.5366i

'j' or 'i' may be used to indicate 1 andwhen defining a complex number.

Matlab always uses i to representacomplex numbers.

Defining complex numbers z1 and z2

Addition

Subtraction

Multiplication

Division


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Separating real and imaginary part of a complex number

z=2+3*j;

R=real(z)

R =

2

I=imag(z)

I =

3

Magnitude and argument(phase) of a complex number

z=6+3*j;

magz=abs(z)

magz =

6.7082

phasez=angle(z)

phasez =

0.4636

Note: Given z=a+jb , then

)(tan 1

22

ab

phasez

bamagz

real(z) extracts the real part of z

imag(z) extracts the imaginary part of z

abs(z) returns the magnitude of thecomplex number z.

angle(z) returns the argument (phase) of the complex number z.


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Vector representation of complex numbers

z=2+3*j;

compass(z);

grid;

title('Vector representation of z=2+3j');

xlabel('Real Part');

ylabel('Imaginary Part');

The result of the command compass(z) is shown below:

-2 -1 0 1 2 3-1

-0.5

0

0.5

1

1.5

2

2.5

3

Real Part

I

agi

nar yPar t

Vector representation of z=2+3j

title('Vector representation of z=2+3j'); grid; ylabel('Imaginary Part');

xlabel('Real Part');

angle

magZ


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Trigonometric Functions

Matlab includes the following trigonometric functions

sin/asin: sine/arcsine

cos/acos: cosine/arccosine

tan/atan: tangent/arctan

sinh / asinh: hyperbolic sine/arcsine

cosh/acosh: hyperbolic cosine/arccosine

tanh/atanh:hyperbolic tangent/arcttang

Vector and matrix generation

Vectors may be generated in Matlab using the colon notation (:). For example to

generate a vector x having values from 1 to 8 may be performed as follows:

x=1:8

x =

1 2 3 4 5 6 7 8

Notice the following examples:


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t=-0.2:0.1:0.2

t =

-0.2000 -0.1000 0 0.1000 0.2000

x=10:-1:0

x =

10 9 8 7 6 5 4 3 2 1 0

Subscripting vectors and matrices

Consider vector x:

x=[1.2, 3.4, 5.6, 6.7,3.7,8.7]

Here x has 6 elements. Each element of x may be referenced using the followingnotation:

y=x(1) y has the same value as the first element of vector x.

z=x(4) z has the same value as the 4 th elements of vector x.

Sometimes it may necessary to assign a group of elements of a vector to another variable.

For example:

w=x(1:3) w becomes a three column vector having values [1.2,3.4,5.6].

q=x(5:6) q=[3.7,8.7]

Matrices may also be referenced in the same way. Consider the following matrix:

Here, t takes on values from -0.2 to 0.2 in stepsof 0.1. the general format is as follows:Variable=Starting value:Steps:Final value

Here, values of x start form 10 down to 0 insteps of -1:

Starting ValueSteps

Final Value


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1211109

8765

4321

A

A has 3 rows and 4 coulmns. For example the element in row 2 and column3 is 7, we

reference this as A(2,3).

Consider the following special references which makes Matlab a powerful way of

manipulating matrices.

Suppose A is a 10x10 matrix.

Then

B=A(1:5,3)

B becomes a 5x1 matrix that consists of the first five elements in the third column of A.

This is shown below:

19876543210

2198765439

3219876548

4321987657

5432198766

6543219875

7654321984

8765432193

9876543212

10987654321

A

B=A(1:5,3)


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B=A(:,5)

B becomes the entire 5 th column of A as shown below:

19876543210

2198765439

3219876548

4321987657

5432198766

65432198757654321984

8765432193

9876543212

10987654321

A

B=A(1:5,:)B becomes the first five rows of A as shown below.

19876543210

2198765439

3219876548

4321987657

54321987666543219875

7654321984

8765432193

9876543212

10987654321

A

B=A(1:5,3)

B=A(1:5,:)


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Assume B is a 5x5 matrix and A is a 10x10 matrix as before. Then complex indexing

may be achieved. For example:

A(:,[3 5 9])=B(:,2:4)

This replaces the third, fifth and ninth columns of A with the second third and fourth

columns of B as shown below:

19876543210

2198765439

3219876548

4321987657

5432198766

65432198757654321984

8765432193

9876543212

10987654321

A

6.07.08.09.00.1

5.06.07.08.09.0

4.05.06.07.08.0

3.04.05.06.07.0

2.03.04.05.06.0

1.02.03.04.05.02.01.02.03.04.0

3.02.01.02.03.0

4.03.02.01.02.0

5.04.03.02.01.0

B

Rational Operations

Six relational operators are available for comparing two variables of the same

dimensions.

< Less than

Greater than

>= Greater than or equal

== Equal

~= Not equal


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Some Useful Functions

A group of functions provides basic data analysis capabilities:

max: Maximum value

min: Minimum value

std: standard deviation (square root of variance)

sum: Sum of elements

sort: Sorting

prod: Product of elements

var: Variance

sqrt: Square root

Consider the following examples.

x=[1 2 4 6 9 10 12 3 7]; % placing a ; at the end of a line, causes matlab no to echo the answer.

max(x)

ans =

12

min(x)

ans = 1

sum(x)

ans =

54

sort(x)

ans =

1 2 3 4 6 7 9 10 12

std(x)

ans =

3.8079


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var(x)

ans =

14.5000

sqrt(var(x)) % this evaluates: x

ans =

3.8079


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Generation and Visualisation of Signals

Consider generating and plotting a sinusoidal signal from -2 to +2 in steps of /10.

The function is x(t)=sin(t). Here is the sequences of commands to generate and plot x(t):

t=-2*pi:pi/10:+2*pi;

x=sin(t);

plot(t,x);

grid;

title('x(t)=sin(t)');

xlabel('Time');

ylabel('Amplitude');

Here is the output:

Create a vector of values of t from -2 to +2 insteps of /10. Note that pi is a predefinedconstant in Matlab.

Now generate the samples of x(t).

Now plot samples of x against t

-8 -6 -4 -2 0 2 4 6 8-1

-0.8

-0.6

-0.4

-0.2

0

0 .2

0 .4

0 .6

0 .8

1

Time

AmplItude

x(t)=sin(t)


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Plotting more than one signal on the same set of axis

Consider plotting x1(t) and x2(t) defined as:

x1(t)=10sin(2 f 1t)

x2(t)=8cos(2

f 2t)Assume a sampling frequency of f s=4000Hz with f 1=100Hz and f 2=250Hz.

The discrete version of these signals are:

x1(n)=10sin(2 (f 1/ f s )n)

x2(t)=8cos(2 (f 2/ f s )n)

We will generate 100 samples of each as shown below:

fs=4000; % Defining Sampling Frequency

A1=10; % Amplitude of the first sinewave

A2=8; % Amplitude of the second sinewave

f1=100; % Frequency of the first sinewave

f2=250; % Frequency of the second sinewave

N=100; % Number of samples to generate

n=0:N-1; % time index

x1=A1*sin(2*pi*(f1/fs)*n); % Generating x1(t)

x2=A2*sin(2*pi*(f2/fs)*n); % Genertaing x2(t)

plot(n,x1,'y',n,x2,'c'); % Plot the two signals

grid; % Add a grid to the figure

title('Two Sinewaves'); % Add title

xlabel('Sample Number'); % Label x-axis

ylabel('Amplitude'); % Label y-axis

Here is the output.


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Signals & System - A Practical Approach Using Matlab 20

Two Useful Functions

Two functions which are often used in creating vecotrs and matrices are:'ones' and 'zeros'. Their usage is illustrated below:

x=ones(n,m)

This creates a matrix of size 'n' rows and 'm' columns with their initial values set

to 1.

Y=zeros(n,m)

This creates a matrix of size 'n' rows and 'm' columns with their initial values set

to 1.

Generating 32 samples of an impulse signal: x=[1, zeros(1,31)];

Generating 32 samples of a step signal: x=ones(1,32);

Generating a single pulse:

x1=[zeros(1,8),ones(1,16),zeros(1,8)];

0 10 20 30 40 50 60 70 80 90 100-10

-8

-6

-4

-2

0

2

4

6

8

10Two Sinewaves

Sample Number

AmpLITUDe

Documents

Workshop Number 1- Matlab Tutorial