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Simulation Lab

CONTENTS

S.No Experiments Page. No

1 Basic Operations on Matrices. 3

2 Generation of Various signals and sequences (Periodic

and Aperiodic) such as Unit Impulse, Unit Step,

Square, Saw tooth, Triangular, Sinusoidal, Ramp, Sinc.

6

3 Operations on Signals and Sequences such as Addition,

Multiplication, Scaling, Shifting, Folding,

Computation of Energy and Average Power.

16

4 Finding the Even and Odd parts of Signal/Sequence

and Real and Imaginary Parts of Signal.

23

5 Convolution between Signals and Sequences. 25

6 Auto Correlation and Cross Correlation between

Signals and Sequences

29

7 Computation of unit sample, unit step and sinusoidal

response of the given LTI system and verifying its

physical realiazability and stability properties.

32

8 Gibbs Phenomenon. 38

9 Sampling Theorem Verification. 40

10 Locating the Zeros and Poles and Plotting the Pole-

Zero maps in the S-Plane and Z-Plane for the given

transfer function.

43

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S.No Experiments Page. No

11 Wave form Synthesis using Laplace Transform

12 Verification of Linearity and Time Invariance

Properties of a given Continuous / Discrete System13 Generation of Gaussian noise (Real and Complex),

Computation of its Mean, Mean Square Value and its

Skew, Kurtosis, and PSD , Probability Distribution

Function.

14 Finding the Fourier Transform of a given Signal and

plotting its magnitude and phase spectrum.

15 Removal of Noise by Autocorrelation / Cross

Correlation.

16 Extraction of Periodic Signal Masked by Noise using

Correlation.

17 Verification of Weiner Khinchine Relations.

18 Checking a Random Process for Stationary in Wide

sense.

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MATLAB is an environment for performing calculations and simulations of

a variety of types. MATLAB, short for matrix laboratory, was developed

by Cleve Moler, a professor of mathematics and computer science, in the

late 70s and early 80s. In 1984, Cleve Moler and Jack Little founded The

Math Works which has been developing MATLAB ever since. In the earliest

version of MATLAB, there were about 80 commands, and they allowed for

matrix calculations. Now there are thousands of commands, and there are

many, many types of calculations that one can perform. .Simulink is a

MATLAB add-on that allows one to simulate systems by combining

blocks of various types. the student will learn how to make calculations

using MATLAB and will learn a little about simulating systems using the

simulation tools provided by MATLAB and Simulink .

Some Very Basic Instructions to start MATLAB:

Double click on the MATLAB icon on the computers desktop. This opens

the main MATLAB window. When working with MATLAB there are two

ways of programming. One can either enter commands in the command

window, or one can write functions using the M-file editor. The basic

MATLAB interface is pretty easy to use.

MATLAB does not require that variables be declared. MATLAB allocates

space for the variable the first time the variable is used. To assign a value to

a variable, one uses an equal sign. To set x equal to 2, one writes x = 2.

When one does this, MATLAB replies with: x = 2

MATLAB as a Vector-enabled Calculator:

Two features make MATLAB particularly useful. MATLAB knows how to

work with vectors, and it knows how to produce all sorts of graphs. To

define a vector, an ordered set of numbers that will be represented by the

letter aone types a = [num1 num2 num3 ... numN]. (One can insert

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commas between the different elements, but MATLAB does not require that

one insert them.) MATLAB recognizes this as a command to allocate

memory fora and to assign the set of numbers listed to a.

For example, giving MATLAB the command a = [1 2 3 4 5] causes

MATLAB to respond with a = 1 2 3 4 5

Once a has been defined there are many ways that it can be processed. Most

MATLAB functions are vectorized; they can take vector arguments, and

they will return vectors as answers. When using MATLAB, to calculate the

sine of , one types sin(pi). To calculate the value of sine at the points , 5

one could give MATLAB five separate commands; that is not necessary,

however. Instead, one can type sin (pi*a). (The asterisk

Denotes multiplication. MATLAB does not understand that if one writes

ab one would like MATLAB to multiply to elements. One must always tell

MATLAB to multiply them.) MATLAB will respond to the command with

ans = 1.0e-015 * 0.1225 -0.2449 0.3674 -0.4899 0.6123

In mathematics, one can add vectors and the same is true when using

MATLAB. The following set of commands is legal, and the answer given is

what MATLAB would reply with.

>> a = [1 2 3]

a =

1 2 3

>> b = [4 5 6]

b =

4 5 6

>> a+b

ans = 5 7 9

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MATLAB has many functions that allow one to plot data. We consider some

of the simplest uses of the simplest of the plotting functions: plot. If one

types plot(a), MATLAB plots the values of a against the integers. For

example, giving MATLAB the command plot(1:2:10 ) .By default, plot

interpolates between the points in a plot.) If one gives MATLAB the

command plot(t,a), then MATLAB plots the vector a against the vector t.

Suppose that one gives MATLAB the commands Using the Editor One can

open the MATLAB editor either by typing edit at the MATLAB command

prompt or by clicking on the blank page on the toolbar on the MATLAB

window. (If one would like to open a file in the editor, then one clicks on the

folder on the toolbar, selects the relevant file, and then clicks on it.

MATLAB will open the file in the editor.)

The editor is a damped exponential. sensitive and has many nice features.

When one is finished writing a file, one saves it. MATLAB expects files that

contain MATLAB commands to end with a .m. After saving a file, one can

go to the command window, set

the Current Directory to the directory in which the files was saved, and then

type the files name (without the .m suffix). This causes MATLAB to

perform the commands in the file. One can also cause MATLAB to execute

the commands in the file by hitting F5 while in the editor window. For the

time being we will make use of the editor to save our work and to do the

work in a way that makes it easy to correct mistakes and to print the code

that we have written. Please make sure to save what you are working on at

regular intervals. Comments starting a line with a percent sign tells

MATLAB that that line is a comment. As with all programming languages,

it is very important to comment your code. MATLAB has a help command.

Typing help com_name causes MATLAB to respond with information about

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the command. If one writes a file, places comments at the very beginning of

the file, and types help filename, MATLAB prints the comments that appear

at the beginning of the file. This allows the user to expand the MATLAB

help facility.

Defining a Transfer Function Object:

To define a transfer function object that corresponds to a transfer function

that is rational functiona function that is the quotient of polynomials one

gives MATLAB the command tf (num, den). The vector num contains the

coefficients of the numerator polynomial and the vector den contains the

coefficient of the denominator polynomial. The first element of each vector

is the coefficient of the highest power in the polynomial, and each element

afterwards corresponds to the next lower power of s. Suppose that one has a

system for which G(s) = 1/0.1s + 1

.Giving MATLAB the command G = tf([1],[0.1 1]) cause MATLAB to

respond with Transfer function: 1 / s + 1. Analyzing a Transfer Function

There are many ways that MATLAB can help analyze a transfer function.

Supposing that we have defined the transfer function object G as we did

above. To examine Gs impulse response, one need only give MATLAB the

command impulse (G).To examine the step response all one need do is give

MATLAB the command step(G).By using the magnifying glasses in the

toolbar above the figures, it is possible to zoom in on parts of a figure or to

zoom out from parts of a figure. MATLAB makes it easy to plot Bode plots

as well. The command bode(G) will cause MATLAB to produce the Bode

plots .

Matlab is an interpreted language that means that it translates the command

into machine code when it is executed. If something is changed, it means

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that all consecutive commands must be updated to contain the correct

values. Start by defining a vector s:

>> s= [1,3,5,9,34,32,11] Matlab writes the vector to the screen when return

is hit. To avoid this writing, type a semicolon after the expression, for

example:

>> s2=s.^2;

You can see the contents of s2 by typing

>> s2

Now we want to change the vector s. Set it to

>> s= [3, 23, 66, 43, 11]

Note that the vector s2 has not changed after this command:

>> s2

To update s2 with the new s vector, you must type it again (Use the arrow

keys (up/down) to go backwards and forwards in the command buffer.):

>> s2=s.^2;

And so the correct value is there:

>> s2

Online demo and help: A short introduction to Matlab is retrieved by

typing,

>> intro

and further demonstrations are available through

>> demo

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Try a few of the available demonstrations. All functions in Matlab has a

brief on-line documentation. It is accessed by typing help . For

instance, the functionality of eig is retrieved by typing:

>> help eig

Continue to examine how the following functions work: dir, save, lookfor,

zeros, ones, rand, sin, exp, factor, det, max, min, sum, log, log10, ode23,

datafun, conv.

Variables:

If you do a lot of computing with Matlab, sometimes you would bereminded what variables you actually have used and defined. Then you can

make use of the the two commands below. To see the variables that are

defined you can type:

>> who

To receive more information about the variables type

>> whos

What is the information displayed? Now, if you want to remove a variable

you can use clear:

>> clear s

Check again what variables you have:

>> who

On the other hand, if you need to remove all variables type

>> clear

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Matrices:

Matlab stand for Matrix Laboratory. The whole code of Matlab is made such

that it is very fast if the computing is defined as matrices and vectors. Here

is an examples of a matrix m1 and a vector v1:

Type:

>> m1=[6 8 10 13; 28 0 6 1;1 3 29 31]

>> v1=[3;18;1]

We use these to calculate the product m2=m1T v1:

>> m2=m1'*v1;

Is the result correct? Check! Think how you would write this in a

conventional programming language. Which way do you prefer? Sub-

matrices can be extracted from readily defined matrices. To choose the first

and the last column and the middle row from m1, one writes

>> m1(2,[1,4])

If a whole row or column is needed, for instance

it is possible to use ':'

>> m1(:,[1,4])

Use this technique on m1 and v1 to calculate the product

Operators:

The Matlab language has its own syntax that has to be learnt. We have

already seen some of the operators between matrices and vectors. Matlabcan however act on matrices or on the separate elements in the matrices.

This is decided by which operator is used. See all operators that are available

by typing

>> help ops

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The most important matrix operators are:

multiplication *

division/ and \, (can also be used for solutions of over estimated

systems)addition +

subtraction -

power ^

transpose ' and .', (the former is complex conjugate transpose)

Here is an example how to use some of the operators:

>> m1'*v1+1-[1;2;3;4]

Dotted operators:

Some of the operators on separate elements are preceded by a dot (.), other

are the same as for the matrix operators, i.e. there is no difference between

matrix and element operators such as addition. The element by element

operators are for instance:

multiplication .*

division ./

addition +

subtraction -

power .^

An example how to use them:

>> v1.*[1;2;3]+[4;5;6].^3

Exercise: Calculate the 5th power of the matrix

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Then find the answer to 15, 25, 35, 45, 55, 65, 75, 85, 95 by applying the array

operator (dotted operator) on m3. It is important to know the difference

between array operators and matrix operators!

Matlab functions:

The functions that are available in Matlab are of built-in and m-file types.

Matlab has whole libraries of built in functions, for instance: sum, sin, sign,

abs, gamma, besselj. There are also ready written functions as some

functions provided by Mathworks for an exchange of a lot of money, e.g.

functions for signal processing (signal), partial differential equations (pde),fuzzy logic (fuzzy), system identification (ident), control (control),

optimisation (optim) etc. Try them out by typing help and a word in brackets

above.

>> help signal

Matlab also allows for users to write their own functions. A function works

as a subroutine, where a script is used instead of typing in the command

window. Matlab functions are written in text files with the extension .m,

called m-files. During the following Matlab exercises you will write your

own functions in .m files. An example is the function myfunc given below.

The sign % indicates a comment line. myfunc shows how different signal

processing operations (convolutions, Fourier transforms, etc.) are

implemented in Matlab with one single command.

function r = myfunc(x,y,s)

% MYFUNC is my own-made Matlab function.

% MYFUNC is my first attempt to make a Matlab function.

% It is used by typing:

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% r=myfunc(x,y,s)

% where x, y and s are inputs

% and r is output.

z=conv(x,y);

r=filter (z,y,s);

R=fft(r);

S=fft(s);

l=length(r);

subplot(2 1 1)

plot(abs(S),'b')

hold on

plot(abs(R),'r')

hold off

title('Result of calculations')

ylabel('Absolute values of R, S')

subplot(2 1 2)

plot(s)

hold on

plot(r,'r')

hold off xlabel('time')

title('Input / output signals')

Now type the program in a text file called myfunc.m and save it in your

folder 'CSmatlab'. Use the matlab-editor, evoked by [File-->New-->M-file].

Then test the program by typing:

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>> x=[0.1238 , 0.3715 , 0.3715 , 0.1238];

>> y=[1.0000 , -1.1619 , 0.6959 , -0.1378];

>> s=randn(256,1);

>> r=myfunc(x,y,s);

This is just an example how various complex signal processing operations

such as Fourier Transforms, convolustions etc. can be performed by means

of one simple Matlab command.

Logic:

Normal logic functions included in most programming languages are also

present in Matlab: for end, if else end, relations: > < >= > for k=1:5

y(k)=31^k;

end

>> y

But since Matlab is made for matrix manipulations the above example is

faster to calculate by writing:

>> y=31. ^ [1:5]

Plotting:

Create a vector that contains data of a sinusoid:

>> k=0:39;

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>> s=sin(2*pi*k/40);

Plot this discrete signal with the command

>> stem(k,s)

The signal s could also be thought of as a discrete representation of a

analogue signal. Then we prefer to plot it with another function. First we

create a time vector from 0 to 40 ms.

>> t=(0:39)/1000; % t=time [ms]

>> plot(t,s)

If a logarithmic scale is desired, the following commands can also be used:

loglog, semilogy, semilogx.

DATA PRESENTATION

MATLAB provides a variety of functions for displaying data as 2-D and 3-D

plots of curves and 3-D mesh surface plots, as well as functions for

annotating these graphs. On-line help for each of the commands shown

below is available by typing help command name or doc command

name at the MATLAB prompt.

2-D PLOTS:

The following list summarizes the functions that produce basic line plots of

data. These functions differ only in the way they shape and scale the plot

axes. Each accepts arguments in the form of vectors or matrices and

automatically scales the axis to accommodate the data range.

plot(x,y) Generates a linear plot ofx versus y

errorbar(x,y,e) Generates a linear plot ofx versus y with with error bars

that are symmetric about y and are 2*e(i) long

semilogx(x,y) Generates a plot ofx versus y using a logarithmic scale

for x

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semilogy(x,y) Generates a plot ofx versus y using a logarithmic scale

for y

loglog(x,y) Generates a plot ofx vs. y using a logarithmic scale for x and y

polar(theta,r) Generates a polar plot of angles theta(in rads) versus

magnitudes r

bar(x,y) Generates a bar graph ofy at locations specified by x

hist(y,nb) Generates a histogram of data in vectory in nb number of bins

stairs(x,y) Generates a stair graph of y at locations specified by equally

spaced x

stem(x,y) Generates a discrete impulse plot ofy at locations specified by x

rose(theta,nb) Generates a angle histogram for angles in theta in nb

number of bins

compass(z) Generates a plot that displays the angle and magnitude of the

complex elements ofz as arrows emanating from the origin

quiver(x,y,dx,dy) Generates a plot that displays little arrows at every (x,y)

pair where dx and dy determine the direction and magnitude of the arrows

subplot(n,m,p) Splits the graphics window into a n-by-m matrix of plots

where p will be next window used for a plot command.

PLOTTING OPTIONS

plot_command(x,y,w,z) Generates two plots on the same axes

plot_command(x,y,indicator) Establishes line or marker style and color

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Data Presentation: INDICATORS:

line type indicator point type indicator color indicator

solid - point . red r

dashed -- plus + yellow y

dotted : star * green g

dashdot -. circle O blue b

x-mark x magenta m

cyan c

white w

black k

SCALING

axis(axis) Freezes the current axis scaling for subsequent plots

axis(v) v is a 4-element vector containing [xmin, xmax, ymin, ymax]

axis square Specifies the aspect ratio to be square

axis auto Specifies the aspect ratio to return to the default

hold Freezes the current axis scaling and plot for subsequent plots to share

(another way to make multiple plots on the same graph) The second call to

hold releases the current axis.

ANNOTATION

The following list summarizes the commands for adding annotation to plots.

title(text) Writes the text string as a title at the top of the

current plot

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xlabel(text) Writes the text string beneath the x-axis on the current plot

ylabel(text) Writes the text string beside the y-axis on the current plot

legend (str1,...) Puts a legend on the current plot using the specified

strings as labels

text(text) Writes the text string at the point specified by (x,y) on the

current plot

gtext(text) Writes the text string at the point specified by a mouse click

grid Adds grid lines to the current plot

SCREEN CONTROL

shg Show graph window

clg Clear graph window

figure Create a graph window

delete Delete a graph window

axes Control the graph axis properties

gcf Get current figure handle

gca Get current axis handle

set Change object property values

get Get object property values

Experiment.1

Aim: Write a MATLAB program to perform the Basic Operations onMatrices.

% Basic Matrix Operationsclc;

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close all;

% Addition of 2 matrices

A = [ 1 2 3; 3 4 5; 6 7 8];B = [ 1 2 3; 2 4 7; 3 5 8];

C=A+B

% Subtraction of 2 MatricesD=A-B

% Multiplication of 2 Matrices

E=A*B% Division of 2 Matrices

F=A/B

% Inverse of 2 Matrices

G=inv(A)%Element by Element Multiplication

H=A.*B

% Transpose of matrices

A_transpose=A'B_transpose=B'

% DeterminantK=det(A)

L=det(B)

% Eigen Values and Eigen Vectors

[V,E]=eig(A)[V,E]=eig(B)

Output:

C=

2 4 6

5 8 129 12 16

D =

0 0 01 0 -2

3 2 0

E =

14 25 4126 47 77

44 80 131

F =

1.0000 0 0

1.0000 -2.0000 2.00001.0000 -5.0000 5.0000

G =

1.0e+015 *

-2.7022 4.5036 -1.8014

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5.4043 -9.0072 3.6029

-2.7022 4.5036 -1.8014

H =1 4 9

6 16 35

18 35 64A_transpose =

1 3 6

2 4 73 5 8

B_transpose =

1 2 32 4 5

3 7 8

K =

0L = 1

V =-0.2656 -0.7444 0.4082

-0.4912 -0.1907 -0.8165

-0.8295 0.6399 0.4082

E =14.0664 0 0

0 -1.0664 0

0 0 -0.0000V =

-0.2797 -0.2360 - 0.4332i -0.2360 + 0.4332i

-0.6148 0.7650 0.7650-0.7374 -0.3864 + 0.1486i -0.3864 - 0.1486i

E =

13.3063 0 00 -0.1531 + 0.2274i 0

0 0 -0.1531 - 0.2274i

Experiment.2

Aim: Write a MATLAB program to generate Various Signals and Sequences(Periodic and Aperiodic) such as Unit Impulse, Unit Step, Square, Sawtooth, Triangular, Sinusoidal, Ramp, and Sinc function.

% to plot continuous time signals%

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clear all

t=-1:.01:1;

x1=sin(2*pi*1.5*t);subplot(2,2,1), plot(t,x1);

xlabel('.........> t ');

ylabel('x1(t)');title(' sinusoidal signal');

% to plot an exponential signal%

x2=exp(-2*t);subplot(2,2,2), plot(t,x2);

xlabel('.........> t ');

ylabel('x2(t)');

title(' exponential signal');

% to plot sawtooth signal %

t1=-20:0.01:20;

x3=sawtooth(t1);subplot(2,2,3), plot(t1,x3);

xlabel('.........> t1 ');ylabel('x3(t)');

title(' swatooth signal');

%to plot a square signal5t1=-20:0.01:20;

x4=square(t1);

subplot(2,2,4), plot(t1,x4);xlabel('.........> t1 ');

ylabel('x4(t)');

title(' square signal');clear all

% T0 PLOT A SINC FUNCTION %t1=-20:0.01:20;

x5=sinc(t1/2);

subplot(2,2,1), plot(t1,x5);

xlabel('.........> t1 ');ylabel('x5(t)');

axis([ -20 20 -1.5 1.5 ] )

title(' SINC FUNCTION');

% TO PLOT A RECTANGULAR PULSE%

x6=rectpuls(t1/10);subplot(2,2,2), plot(t1,x6);

xlabel('.........> t1 ');

ylabel('x6(t)');

axis([ -20 20 -1.5 1.5 ] )

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title(' RECTANGULAR PULSE');

% TO PLOT A TRIANGULAR PULSE%

x7=tripuls(t1/10);subplot(2,2,3), plot(t1,x7);

xlabel('.........> t1 ');

ylabel('x7(t)');axis([ -20 20 -1.5 1.5 ] )

title(' TRIANGULAR PULSE');

%TO PLOT A SIGNUM FUNCTION%

x8=sign(t1/3);

subplot(2,2,4), plot(t1,x8);xlabel('.........> t1 ');ylabel('x8(t)');axis([ -20 20 -1.5 1.5 ] )

title(' SIGNUM FUNCTION');

Output Waveforms:

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% TO GENERATE A UNIT STEP SEQUENCE%

N=31;

x1=ones(1,N);n=0:1:N-1;

subplot(2,2,1), stem(n,x1);

xlabel('n ');ylabel('x1(n)');

title('UNIT STEP SEQUENCE');

% TO GENERATE A SINUSOIDAL SEQUENCE %

x2=2*cos(.1*pi*n);



title('SINUSOIDAL SEQUENCE');

% TO GENERATE A EXPONENTIAL SEQUENCE%x3=.6.^(n) ;

subplot(2,2,3), stem(n,x3);xlabel('n ');

ylabel('x3(n)');

title('EXPONENTIAL SEQUENCE');

% TO GENERATE A UNIT IMPULSE SEQUENCE %

n=-4:1:4;

x4=[zeros(1,4),ones(1,1),zeros(1,4);]subplot(2,2,4), stem(n,x4);

xlabel('n ');

ylabel('x4(n)');title('UNIT IMPULSE SEQUENCE');

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Output Waveforms:

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% TO GENERATE A UNIT RAMP SEQUENCE %

n=0:1:10;x1=n ;



title('UNIT RAMP SEQUENCE');

%% TO GENERATE A UNIT STEP SEQUENCE%

n=-4:1:4;

x2=[zeros(1,4),ones(1,5)];


ylabel('x2(n)');

title('UNIT STEP SEQUENCE');

% TO GENERATE A SINUSOIDAL SEQUENCE %

n=-20:.25:20x3=2*sin(.5*pi*n);


xlabel('n ');

ylabel('x3(n)');title('SINUSOIDAL SEQUENCE');

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Output Waveforms:

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Experiment.3.

Aim: Write a MATLAB program to perform Operations on Signals andSequences such as Addition, Multiplication, Scaling, Shifting, Folding,Computation of Energy and Average Power.

% ADDTION OF TWO SEQUENCES %

n=0:1:3;

x=[1,2,2,3];n1=length(x);

subplot(2,2,1),stem(n,x);

xlabel('n'); ylabel('input sequence');

% INPUT SEQUENCE2 %

n=0:1:3;

y=[1,6,4,3];

n2=length(y);subplot(2,2,2),stem(n,y);

xlabel('n'); ylabel('input sequence');% ADDITION OF x and y %

n=0:1:6;

R1=x+y;

subplot(2,2,3), stem(R1);xlabel('n'); ylabel('ADDITION OF X and Y ');

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Output Waveforms:

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% Operations on Signals and Sequences%

% ADDITION OF TWO SINUSOIDAL SIGNALS %

clc;t=0:.01:pi;

y1=sin(2*pi*t);

subplot(3,1,1);plot(t,y1);

title('sinusoidal signal 1');

y2=cos(3*pi*t);subplot(3,1,2);

plot(y2);

title('sinusoidal signal 2');

h=y1+y2;subplot(3,1,3);

plot(h);

title('ADDITION OF TWO SINE WAVES');

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Output Waveforms:

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% OPERATIONS ON SIGNALS %

% TIME SCALING ,TIME SHIFTING %clear all;

tmin=-15; tmax=20;

t=tmin:.1:tmax;y0=y(t);

y1=y(t+4);

y2=2*y(t-3);y3=y(2*t);

y4=y(2*t-3);

y5=y(t/2);

ymax=max([max(y0),max(y1),max(y2),max(y3),max(y4),max(y5)]);ymin=min([min(y0),min(y1),min(y2),min(y3),min(y4),min(y5)]);

subplot(3,2,1),plot(t,y0);

xlabel('t'),ylabel(y0);

axis([tmin tmax ymin ymax]);subplot(3,2,2),plot(t,y1);

xlabel('t'),ylabel(y1);axis([tmin tmax ymin ymax]);

subplot(3,2,3),plot(t,y2);








axis([tmin tmax ymin ymax]);

function x=y(t);

x1=t+5;x2=11+4*t;

x3=24-9*t;

x4=t-6;x=x1.*(-5

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Output Waveforms:

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Experiment 4.

Aim: Write a MATLAB program to Find the Even and Odd parts ofSignal/Sequence and Real and Imaginary Parts of Signal.

% program to plot function y(t),y(-t),odd and even parts of y(t)% consider y.m file which acts as function here

clear all;

close all;clc;

%generate given signal

nmin=-10;

nmax=10;n=nmin:1:nmax;

% given signal

y0=y(n);

%time reversal of signaly1=y0(end:-1:1);

ymax=max([max(y0),max(y1)]);ymin=min([min(y0),min(y1)]);

subplot(2,2,1),plot(n,y0);

xlabel('\itt');ylabel('y({\itt})');

subplot(2,2,2),plot(n,y1);xlabel('\itt');ylabel('y({-\itt})');

% Even part of a signal

ye=(y0+y1)/2;% odd part of a signal

yo=(y0-y1)/2;

subplot(2,2,3),plot(n,ye);xlabel('\itt'),ylabel('y_e({\itt})');

subplot(2,2,4),plot(n,yo);

xlabel('\itt'),ylabel('y_o({\itt})');

function x=y(t)

x1=t+5;x2=11+(4*t);x3=24-(9*t);x4=t-6;

x=x1.*(-5

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Output Waveforms :

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Experiment 5.

Aim: Write a MATLAB program to find the Convolution between Signalsand Sequences.

% Convolution of two given sequences %

clear allx=[1,2,1,2,1,3,2];

N1=length(x);

n=0:1:N1-1;

subplot(2,2,1),stem(n,x);xlabel('n'),ylabel('x(n)');

title('input sequence of x(n)');

h=[1,-1,2,-2,1,1];

N2=length(h);n1=0:1:N2-1;

subplot(2,2,2),stem(n1,h);xlabel('n'),ylabel('h(n)');

title('impulse sequence of x(n)');

y=conv(x,h)

n2=0:1:N1+N2-2;subplot(2,1,2),stem(n2,y);

xlabel('n'),ylabel('y(n)');

title('Convolution of two sequences of x(n) and h(n)');

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Output Waveforms:

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% Convolution of 2 signals %clc;

t=0:.01:pi;

y1=sin(2*pi*t);subplot(3,1,1);

plot(t,y1);

y2=cos(3*pi*t);subplot(3,1,2);

plot(y2);

h=conv(y1,y2);

subplot(3,1,3);plot(h);

Output Waveforms:

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Experiment 6.

Aim: Write a MATLAB program to find the Auto Correlation and CrossCorrelation between Signals and Sequences.

% TO GENERATE THE AUTO CORRELATION and CROSS CORRELATION%% INPUT SEQUENCE1 %

n=0:1:3;

x=[1,2,2,3];n1=length(x);

subplot(2,2,1),stem(n,x);

xlabel('n'); ylabel('input sequence');

% INPUT SEQUENCE2 %

n=0:1:3;y=[1,6,4,3];

n2=length(y);

subplot(2,2,2),stem(n,y);

xlabel('n');ylabel('input sequence');

% AUTO CORRELATION OF x and y %

n=0:1:6;R1=xcorr(x,x);

subplot(2,2,3), stem(R1);

xlabel('n'); ylabel('ACR sequence');

% CROSS CORRELATION OF x and y %

n=0:1:6;

R2=xcorr(x,y);

subplot(2,2,4), stem(R2);xlabel('n');

ylabel('CCR sequence');

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Output Waveforms:

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% ACR and CCR OF TWO SINUSOIDAL SIGNALS%

% TO GENERATE A SINUSOIDAL SEQUENCE %n=-20:.25:20

x1=2*sin(.5*pi*n);


ylabel('x1(n)');

title('SINUSOIDAL SIGNAL1');% TO GENERATE A SINUSOIDAL SEQUENCE %

x2=2*cos(.1*pi*n);



title('SINUSOIDAL SIGNAL2');

% AUTO CORRELATION OF x1 and x2 %

n=0:1:6;R1=xcorr(x1,x2);

subplot(4,1,3), stem(R1);xlabel('n'); ylabel('ACR OF X1 and X2');

% CROSS CORRELATION OF x and y %

n=0:1:6;

R2=xcorr(x1,x2);subplot(4,1,4), stem(R2);

xlabel('n'); ylabel('CCR OF X1 and X2');

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Output Waveforms:

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Experiment 7.

Aim: Write a MATLAB program to compute the unit sample, unit step andsinusoidal response of the given LTI system and verifying its physical

realiazability and stability properties.

% IMPULSE RESPONSE OF A DIGITAL FILTER %

clear all;

b=input('enter the numerator coefficients');

a=input('enter the denominator coefficients');

[H,T]=impz(b,a); stem(T,H);xlabel('n'),ylabel('h(n)');

title('IMPULSE RESPONSE OF DIGITAL FILTER');

Output: H = 1.000 ,.7,.37, .1750, .0781, .0337, .0142, .0059, .0024, .0010 T= 0 1 2 3 4 5

6 7 8 9Output Waveforms:

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%IMPULSE RESPONSE OF A ANALOGFILTER %

clear all;close all;

clc;

% T=0:.01:200;b=input('enter the numerator coefficients');


[H,T]=impulse(b,a); plot(H);

xlabel('t'), ylabel('h(t)');title('IMPULSE RESPONSE OF ANALOG FILTER');

Output Waveforms:

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%STEP RESPONSE OF A ANALOGFILTER %

close all;

clc;b=input('enter the numerator coefficients');


[H]=step(b,a);plot(H);

xlabel('t')

ylabel('s(t)')

title('STEP RESPONSE OF ANALOG FILTER');

Output Waveforms:

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% SINUSOIDAL RESPONSE OF LINEAR SYSTEM %

clear all;

s=tf('s');g=(s+1)/(s^2+5*s+6);

t=0:0.1:15;

u=sin(t);lsim(g,u,t);

Output Waveforms:

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% EXPONENTIAL RESPONSE OF LINEAR SYSTEM %

clear all;

s=tf('s');g=(s+1)/(s^2+5*s+6);

t=0:0.1:15;

u=exp(-3*t);lsim(g,u,t);

Output Waveforms:

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Experiment 8.

Aim: Write a MATLAB program to verify the Gibbs phenomenon.

% plot square wave and verify Gibbs phenomena using first 10 terms of Fourier series

clear all;

T=input('enter the time period of the square wave\n');n1=input('enter the number of cycles to be plotted\n');

n=input('enter number of harmonics to be considered apart from dc\n');

k=n1*T/2;

i=0;for t=-k:k/100:k

x=0;

for j=1:2:(2*n-1)

xnew=x+(4*(cos((t*2*pi*j/T)-(pi*floor(j/2))))/(j*pi));x=xnew;

endi=i+1;

p(:,i)=x;

end

t=-k:k/100:k;plot(t,p);

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Output Waveforms :

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Experiment 9.

Aim: Write a MATLAB program to verify the Sampling Theorem.

% Sampling Theorem %

clc;clear all;

close all;

t=-100:.01:100;fm=.02;

x=cos(2*pi*t*fm);

subplot(2,2,1),plot(t,x);

xlabel('time in sec'),ylabel('x(t)');title('continous time signal');

fs1=.02;

n=-2:2;

x1=cos(2*pi*fm*n/fs1);subplot(2,2,2),stem(n,x1);

hold onsubplot(2,2,2);

plot(n,x1);

title('discrete time signal x(n) with fs2fm');

xlabel('n'),ylabel('x(n)');

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Output Waveforms:

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Experiment 10.

Aim: Write a MATLAB program to locate the Zeros and Poles and plottingthe Pole-Zero maps in S-plane and Z-plane for the given transfer function.

% POLE-ZERO PLOT %

close all; clc;

b=input ('enter the numerator coefficients');a=input ('enter the denominator coefficients');

zplane (b, a); title ('POLE-ZERO PLOT');

Output: enter the numerator coefficients [1 5]enter the denominator coefficients [1 3 2]

Output Waveform:

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Experiment 11.

Aim: Write a MATLAB program to verify the Linearity, Time Invariance and

Stability of a Discrete time System.

% VERIFICATION OF LINEARITY OF DISCRETE SYSTEM %

n=0:40; a=2; b=-3;

x1=cos(2*pi*0.1*n);

x2=cos(2*pi*0.4*n);

x=a*x1+b*x2;

ic=[0 0];

num=[2.2403 2.4908 2.2403];

den=[1 -0.4 0.75];

y1=filter(num,den,x1,ic);

y2=filter(num,den,x2,ic);

y=filter(num,den,x,ic);yt=a*y1+b*y2;

d=y-yt;

subplot(3,1,1), stem(n,y); grid

title('LINEARITY OF DISCRETE TIME SYSTEM ');

subplot(3,1,2), stem(n,yt); grid

subplot(3,1,3), stem(n,d); grid

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Output Waveforms:

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% VERIFICATION OF TIME INVARIANCE OF DTS %

n=0:40;D=10;x=3*cos(2*pi*0.1*n)-2*cos(2*pi*0.4*n);

xd=[zeros(1,D) x];

num=[2.2403 2.4908 2.2403];den=[1 -0.4 0.75];

ic=[0 0];

y=filter(num,den,x,ic)yd=filter(num,den,xd,ic)

d=y-yd(1+D:41+D);

subplot(3,1,1),stem(y),grid;

title('TIME INVARIANCE OF DTS' );subplot(3,1,2),stem(yd),grid;

subplot(3,1,3),stem(d),grid;

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% TO CHECK THE STABILITY OF THE SYSTEM %

num=[1 0.8];

den=[1 1.5 .9];

N=200;h=impz(num,den,N+1);

sum=0;

n=0:N;

for k=1:N+1

if abs(h(k))

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Output Waveforms:

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Experiment 12.

Aim: Write a MATLAB program to generate the Gaussian noise (Real and

Complex), computation of mean, M.S value and its skew, and PSD ,

Probability Distributation function.

% Generation of Gaussian Noise and its mean %

clear;

N=100;

sigw1=sqrt(4);

sigw2=sqrt(9);

rho=-0.4;

x1=rand(1,N);

x2=rand(1,N);

y1=sqrt(-2*log(x1)).*cos(2*pi*x2);

y2=sqrt(-2*log(x1)).*sin(2*pi*x2);

t11=sigw1;

t21=rho*sigw2;

t22=sigw2*sqrt(1-rho^2);w1=t11*y1;

w2=t21*y1+t22*y2;

wmean=[mean(w1) mean(w2)];

wcov=[cov(w1,1) cov(w2,1)];

tmp=corrcoef(w1,w2);

rho_hat=tmp(2,1);

cov_err=100*(sqrt(wcov)-([sigw1 sigw2].^2))./...

([sigw1 sigw2].^2)

rho_err=100*(rho_hat-rho)./rho

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Output:

cov_err =

-50.0337 -68.3450

rho_err =

-10.2492

wmean =

0.4082 -0.3190

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Experiment 13.

Aim: Write a MATLAB program to obtain the Fourier transform of the signal

using Fast Fourier Transform.

% FOURIER TRANSFORM OF A SEQUENCE USING FFT %

N=50;

xn=[1 3 5 8 2 8 9 3 9 7];

subplot(3,1,1); stem(xn);

title('Fourier Transform of the sequence');

Xk=fft(xn,N);

k=0:1:N-1;

subplot(3,1,2); stem(k,abs(Xk));

xlabel('k');

ylabel('Magnitude of Xk');

subplot(3,1,3); stem(k,angle(Xk));

xlabel('k');

ylabel('arg(Xk)');

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Output Waveform:

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1. MATLABBasics

MATLAB is started by clicking the mouse on the appropriate icon and is ended by typing

exit or by using the menu option. After each MATLAB command, the "return" or "enter"key must be depressed.

A. Definition of Variables

Variables are assigned numerical values by typing the expression directly, for example,

typing

a = 1+2

yields: a = 3

The answer will not be displayed when a semicolon is put at the end of an expression, for

example type a = 1+2;.

MATLAB utilizes the following arithmetic operators:

+ addition- subtraction* multiplication/ division^ power operator' transpose

A variable can be assigned using a formula that utilizes these operators and eithernumbers or previously defined variables. For example, since a was defined previously,

the following expression is valid

b = 2*a;

To determine the value of a previously defined quantity, type the quantity by itself:

b

yields: b = 6

If your expression does not fit on one line, use an ellipsis (three or more periods at theend of the line) and continue on the next line.

c = 1+2+3+...5+6+7;

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There are several predefined variables which can be used at any time, in the same manner

as user-defined variables:

i sqrt(-1)j sqrt(-1)pi 3.1416...

For example,

y= 2*(1+4*j)

yields: y = 2.0000 + 8.0000i

There are also a number of predefined functions that can be used when defining a

variable. Some common functions that are used in this text are:

abs magnitude of a number (absolute value for real numbers)

angle angle of a complex number, in radianscos cosine function, assumes argument is in radianssin sine function, assumes argument is in radiansexp exponential function

For example, with y defined as above,

c = abs(y)

yields: c = 8.2462

c = angle(y)

yields: c = 1.3258

With a=3 as defined previously,

c = cos(a)

yields: c = -0.9900

c = exp(a)

yields: c = 20.0855

Note that exp can be used on complex numbers. For example, with y =2+8i as defined above,

c = exp(y)

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yields: c = -1.0751 + 7.3104i

which can be verified by using Euler's formula:

c = exp(2)cos(8) + je(exp)2sin(8)

back to list of contents

B. Definition of Matrices

MATLAB is based on matrix and vector algebra; even scalars are treated as 1x1 matrices.

Therefore, vector and matrix operations are as simple as common calculator operations.

Vectors can be defined in two ways. The first method is used for arbitrary elements:

v = [1 3 5 7];

creates a 1x4 vector with elements 1, 3, 5 and 7. Note that commas could have been used

in place of spaces to separate the elements. Additional elements can be added to the

vector:

v(5) = 8;

yields the vectorv = [1 3 5 7 8]. Previously defined vectors can be used to define a newvector. For example, with v defined above

a = [9 10];

b = [v a];

creates the vectorb = [1 3 5 7 8 9 10].

The second method is used for creating vectors with equally spaced elements:

t = 0:.1:10;

creates a 1x101 vector with the elements 0, .1, .2, .3,...,10. Note that the middle number

defines the increment. If only two numbers are given, then the increment is set to a

default of 1:

k = 0:10;

creates a 1x11 vector with the elements 0, 1, 2, ..., 10.

Matrices are defined by entering the elements row by row:

M = [1 2 4; 3 6 8];

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creates the matrix

There are a number of special matrices that can be defined:

null matrix: M = [ ];nxm matrix of zeros: M = zeros(n,m);nxm matrix of ones: M = ones(n,m);nxn identity matrix: M = eye(n);

A particular element of a matrix can be assigned:

M(1,2) = 5;

places the number 5 in the first row, second column.

In this text, matrices are used only in Chapter 12; however, vectors are used throughout

the text. Operations and functions that were defined for scalars in the previous section

can also be used on vectors and matrices. For example,

a = [1 2 3];b = [4 5 6];c = a + b

yields:

c = 5 7 9

Functions are applied element by element. For example,

t = 0:10;x = cos(2*t);

creates a vector x with elements equal to cos(2t) for t = 0, 1, 2, ..., 10.

Operations that need to be performed element-by-element can be accomplished by

preceding the operation by a ".". For example, to obtain a vectorx that contains theelements of x(t) = tcos(t) at specific points in time, you cannot simply multiply the vectort with the vectorcos(t). Instead you multiply their elements together:

t = 0:10;x = t.*cos(t);


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C. General Information

Matlab is case sensitive so "a" and "A" are two different names.

Comment statements are preceded by a "%".

On-line help for MATLAB can be reached by typinghelp for the full menu or typinghelp followed by a particular function name or M-file name. For example, help cosgives help on the cosine function.

The number of digits displayed is not related to the accuracy. To change the format of the

display, type format short e for scientific notation with 5 decimal places, format longe for scientific notation with 15 significant decimal places and format bank for placingtwo significant digits to the right of the decimal.

The commands who and whos give the names of the variables that have been defined in

the workspace.

The command length(x) returns the length of a vector x and size(x) returns thedimension of the matrix x.


D. M-files

M-files are macros of MATLAB commands that are stored as ordinary text files with theextension "m", that is filename.m. An M-file can be either a function with input and

output variables or a list of commands. All of the MATLAB examples in this textbookare contained in M-files that are available at the MathWorks ftp site, ftp.mathworks.com

in the directory pub/books/heck.

The following describes the use of M-files on a PC version of MATLAB. MATLAB

requires that the M-file must be stored either in the working directory or in a directory

that is specified in the MATLAB path list. For example, consider using MATLAB on aPC with a user-defined M-file stored in a directory called "\MATLAB\MFILES". Then to

access that M-file, either change the working directory by typing cd\matlab\mfiles from

within the MATLAB command window or by adding the directory to the path.

Permanent addition to the path is accomplished by editing the \MATLAB\matlabrc.m

file, while temporary modification to the path is accomplished by typingpath(path,'\matlab\mfiles') from within MATLAB.

The M-files associated with this textbook should be downloaded from the MathWorks ftp

site and copied to a subdirectory named "\MATLAB\KAMEN" and then this directoryshould be added to the path. The M-files that come with MATLAB are already in

appropriate directories and can be used from any working directory.

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As example of an M-file that defines a function, create a file in your working directory

named yplusx.m that contains the following commands:

function z = yplusx(y,x)z = y + x;

The following commands typed from within MATLAB demonstrate how this M-file is

used:

x = 2;y = 3;z = yplusx(y,x)

MATLAB M-files are most efficient when written in a way that utilizes matrix or vector

operations. Loops and if statements are available, but should be used sparingly since they

are computationally inefficient. An example of the use of the command foris

for k=1:10,x(k) = cos(k);end

This creates a 1x10 vector x containing the cosine of the positive integers from 1 to 10.

This operation is performed more efficiently with the commands

k = 1:10;x = cos(k);

which utilizes a function of a vector instead of a for loop. An if statement can be used todefine conditional statements. An example is

if(a =4)b = 2;elseb = 3;

end

The allowable comparisons between expressions are >=,

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Whatever comment is between the quotation marks is displayed to the screen when the

M-file is running, and the user must enter an appropriate value.

2. Fourier Analysis

Commands covered:

dft

idft

fftifft

contfft

Thedft command uses a straightforward method to compute the discrete Fouriertransform. Define a vector x and compute the DFT using the command

X = dft(x)

The first element in X corresponds to the value of X(0).

The command idft uses a straightforward method to compute the inverse discrete Fourier

transform. Define a vector X and compute the IDFT using the command

x = idft(X)

The first element of the resulting vector x is x[0].

For a more efficient but less obvious program, the discrete Fourier transform can be

computed using the command fft which performs a Fast Fourier Transform ofa sequence of numbers. To compute the FFT of a sequence x[n] whichis stored in the vector x, use the command

X = fft(x)

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Used in this way, the command fft is interchangeable with the command dft.For more computational efficiency, the length of the vector x should beequal to an exponent of 2, that is 64, 128, 512, 1024, 2048, etc. Thevector x can be padded with zeros to make it have an appropriatelength. MATLAB does this automatically by using the following

command where N is defined to be an exponent of 2:

X = fft(x,N);

The longer the length ofx, the finer the grid will be for the FFT. Due toa wrap around effect, only the first N/2 points of the FFT have anymeaning.

The ifft command computes the inverse Fourier transform:

x = ifft(X);

The FFT can be used to approximate the Fourier transform of acontinuous-time signal as shown in Section 6.6 of the textbook. Acontinuous-time signal x(t) is sampled with a period of T seconds, thenthe DFT is computed for the sampled signal. The resulting amplitudemust be scaled and the corresponding frequency determined. An M-filethat approximates the Fourier Transform of a sampled continuous-timesignal can be downloaded from contfft.m. Let a vector x be defined asthe sampled continuous-time signal x(t) and let T be the samplingtime.

[X,w] = contfft(x,T);

The outputs are the Fourier transform stored in the vector X and thecorresponding frequency vector w.

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3. Continuous Time System Analysis

Note, the recent versions of Matlab utilize a state space model to represent

a system (where a system sys is defined as sys = ss(A,B,C,D)). Many of the

commands that are listed below have sys as the preferred input argumentrather than num and den. In many cases, the online help for Matlab does

not even indicate the argument list as shown below; however, in most

cases, the argument list as shown below still works. The authors purposely

choose not to present the material in the book or in this tutorial using sys

since it may obscure details for junior and sophomore-level students. For

more details on this notation, see Section 3.F.

A. Transfer Function Representation

Commands covered:

tf2zp

zp2tfcloop

feedback

parallel

series

Transfer functions are defined in MATLAB by storing the coefficients of the numerator

and the denominator in vectors. Given a continuous-time transfer function

where and Store the

coefficients of B(s) and A(s) in the vectors and .In this text, the names of the vectors are generally chosen to be num and den, but any

other name could be used. For example,

is defined by

num = [2 3];

den = [1 4 0 5];

Note that all coefficients must be included in the vector, even zero coefficients.

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A transfer function may also be defined in terms of its zeros, poles and gain:

To find the zeros, poles and gain of a transfer function from the vectors num and denwhich contain the coefficients of the numerator and denominator polynomials, type

[z,p,k] = tf2zp(num,den)

The zeros are stored in z, the poles are stored in p, and the gain is stored in k. To find the

numerator and denominator polynomials from z, p, and k, type

[num,den] = zp2tf(z,p,k)

The overall transfer function of individual systems in parallel, series or feedback can be

found using MATLAB. Store the transfer function G in numG and denG, and the transferfunction H in numH and denH.

To reduce the general feedback system to a single transfer function, T(s) = G(s)/

(1+G(s)H(s)) type

[numT,denT] = feedback(numG,denG,numH,denH);

For a unity feedback system, let numH = 1 and denH = 1 before applying theabove algorithm. Alternately, use the command

[numT,denT] = cloop(numG,denG,-1);

To reduce the series system to a single transfer function, T(s) = G(s)H(s) type

[numT,denT] = series(numG,denG,numH,denH);

To reduce the parallel system to a single transfer function, T(s) = G(s) + H(s) type

[numT,denT] = parallel(numG,denG,numH,denH);

(Parallel is not available in the Student Version.)


B. Time Simulations

Commands covered:

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residue

step

impulselsim

The analytical method to find the time response of a system requirestaking the inverse Laplace Transform of the output Y(s). MATLAB aidesin this process by computing the partial fraction expansion of Y(s)using the command residue. Store the numerator and denominator coefficients ofY(s) in num and den, then type

[r,p,k] = residue(num,den)

The residues are stored in r, the corresponding poles are stored in p, and the gain is stored

in k. Once the partial fraction expansion is known, an analytical expression for y(t) can

be computed by hand.

A numerical method to find the response of a system to a particular input is available inMATLAB. First store the numerator and denominator of the transfer function in numand den, respectively. To plot the step response, type

step(num,den)

To plot the impulse response, type

impulse(num,den)

For the response to an arbitrary input, use the command lsim. Create avector t which contains the time values in seconds at which you want MATLAB tocalculate the response. Typically, this is done by entering

t = a:b:c;

where a is the starting time, b is the time step and c is the end time. Forsmooth plots, chooseb so that there are at least 300 elements in t (increase asnecessary). Define the input x as a function of time, for example, aramp is defined as x = t. Then plot the response by typing

lsim(num,den,x,t);

To customize the commands, the time vector can be defined explicitlyand the step response can be saved to a vector. Simulating theresponse for five to six time constants generally is sufficient to showthe behavior of the system. For a stable system, a time constant iscalculated as 1/Re(-p) where p is the pole that has the largest real part(i.e., is closest to the origin).

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For example, consider a transfer function defined by

The step response y is calculated and plotted from the followingcommands:

num = 2; den = [1 2];t = 0:3/300:3; % for a time constant of 1/2y = step(num,den,t);plot(t,y)

For the impulse response, simply replace the word step with impulse. Forthe response to an arbitrary input stored in x, type

y = lsim(num,den,x,t);plot(t,y)


C. Frequency Response Plots

Commands covered:

freqs

bodelogspace

log10semilogxunwrap

To compute the frequency response H of a transfer function, store the numerator and

denominator of the transfer function in the vectors num and den. Define a vector w that

contains the frequencies for which H) is to be computed, for example w = a:b:c where a isthe lowest frequency, c is the highest frequency and b is the increment in frequency. The

command

H = freqs(num,den,w)

returns a complex vector H that contains the value of the frequency response for eachfrequency in w.

To draw a Bode plot of a transfer function which has been stored in the vectors num and

den, type

bode(num,den)

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To customize the plot, first define the vector w which contains the frequencies at which

the Bode plot will be calculated. Since w should be defined on a log scale, the command

logspace is used. For example, to make a Bode plot ranging in frequencies from 0.1 to100, define w by

w = logspace(-1,2);

The magnitude and phase information for the Bode plot can then be found be executing:

[mag,phase] = bode(num,den,w);

To plot the magnitude in decibels, convert mag using the following command:

magdb = 20*log10(mag);

To plot the results on a semilog scale where the y-axis is linear and the x-axis is

logarithmic, type

semilogx(w,magdb)

for the log-magnitude plot and type

semilogx(w,phase)

for the phase plot. The phase plot may contain jumps of 2 which may not be desired.To remove these jumps, use the command unwrap prior to plotting the phase.

semilogx(w,unwrap(phase))


D. Analog Filter Design

Commands covered:

buttap

cheb1abzp2tf

lp21plp2bp

lp2hplp2bs

MATLAB contains commands for various analog filter designs, including those for

designing a Butterworth filter and a Type I Chebyshev filter. The commands buttap and

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cheb1ab are used to design lowpass Butterworth and Type I Chebyshev filters,

respectively, with cutoff frequencies of 1 rad/sec. For an n-pole Butterworth filter, type

[z,p,k] = buttap(n)

where the zeros of the filter are stored in z, the poles are stored in p and the gain of thefilter is in k. For an n-pole Type I Chebyshev filter with Rp decibels of ripple in the

passband, type

[z,p,k] = cheb1ab(n,Rp)

To find the numerator and denominator polynomials of the resulting filter from z, p andk; type

[b,a] = zp2tf(z,p,k)

where a contains the denominator coefficients and b contains the numerator coefficients.Frequency transformations from one lowpass filter to another with a different cutoff

frequency, or from lowpass to highpass, or lowpass to bandstop or lowpass to bandpasscan be performed in MATLAB. These transformations can be used with either the

Butterworth filters or the Chebyshev filters. Suppose b and a store the numerator and

denominator of a transfer function of a lowpass filter with cutoff frequency 1 rad/sec. Tomap to a lowpass filter with cutoff frequency Wo and numerator and denominator

coefficients stored in b1 and a1, type

[b1,a1] = lp2lp(b,a,Wo)

To map to a highpass filter with cutoff frequency Wo, type

[b1,a1] = lp2hp(b,a,Wo)

To map to a bandpass filter with bandwidth Bw centered at the frequency Wo, type

[b1,a1] = lp2bp(b,a,Wo,Bw)

To map to a bandstop filter with stopband bandwidth Bw centered about the frequency

Wo, type

[b1,a1] = lp2bs(b,a,Wo,Bw)


E. Control Design

Commands covered:

rlocus

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Consider a feedback loop as shown in Figure 1 where G(s)H(s) = KP(s) and K is a gain

and P(s) contains the poles and zeros of the controller and of the plant. The root locus is a

plot of the roots of the closed loop transfer function as the gain is varied. Suppose that thenumerator and denominator coefficients of P(s) are stored in the vectors num and den.

Then the following command computes and plots the root locus:

rlocus(num,den)

To customize the plot for a specific range of K, say for K ranging from 0 to 100, then usethe following commands:

K = 0:100;

r = rlocus(num,den,K);

plot(r,'.')

The graph contains dots at points in the complex plane that are closed loop poles for

integer values of K ranging from 0 to 100. To get a finer grid of points, use a smallerincrement when defining K, for example, K = 0:.5:100. The resulting matrix r contains

the closed poles for all of the gains defined in the vector K. This is particularly useful tocalculate the closed loop poles for one particular value of K. Note that if the root locus

lies entirely on the real axis, then using plot(r,'.') gives inaccurate results.


F. State Space Representation

Commands Covered:

ssstep

lsim

ss2tf

tf2ssss2ss

The standard state space representation is used in MATLAB, i.e.,

where x is nx1 vector, u is mx1, y is px1, A is nxn, B is nxm, and C is pxn. The response

of a system to various inputs can be found using the same commands that are used for

transfer function representations: step, impulse, and lsim. The argument list contains theA, B, C, and D matrices instead of the numerator and denominator vectors. Alternately,

the system can be combined into one model using the command

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sys = ss(A,B,C,D);

Then, sys can be used as an input argument for the other commands. For example, the

step response is obtained by typing either of the following commands:

[y,t,x] = step(A,B,C,D);[y,t,x] = step(sys);

The states are stored in x, the outputs in y and the time vector, which is automatically

generated, is stored in t. The rows of x and y contain the states and outputs for the time

points in t. Each column of x represents a state. For example, to plot the second stateversus time, type

plot(t,x(:,2))

To find the response of an arbitrary input or to find the response to initial conditions, use

lsim. Define a time vector t and an input matrix u with the same number of rows as in tand the number of columns equaling the number of inputs. An optional argument is the

initial condition vector x0. The command is then given as

[y,x] = lsim(A,B,C,D,u,t,x0) or [y,x] = lsim(sys,u,t,x0)

You can find the transfer function for a single-input/single-output (SISO) system using

the command:

[num,den] = ss2tf(A,B,C,D);

The numerator coefficients are stored in num and the denominator coefficients are storedin den.

Given a transformation matrix P, the ss2ss function will perform the similarity transform.

Store the state space model in A, B, C and D and the transformation matrix in P.

[Abar,Bbar,Cbar,Dbar]=ss2ss(A,B,C,D,P) or [Abar,Bbar,Cbar,Dbar] = ss2ss(sys,P)

performs the similarity transform z=Px resulting in a state space system that is defined as:

where .

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4. Discrete-Time System Analysis

A. Convolution

Commands covered:

conv

deconv

To perform discrete time convolution, x[n]*h[n], define the vectors x and h with elements

in the sequences x[n] and h[n]. Then use the command

y = conv(x,h)

This command assumes that the first element in x and the first element in h correspond ton=0, so that the first element in the resulting output vector corresponds to n=0. If this is

not the case, then the output vector will be computed correctly, but the index will have to

be adjusted. For example,

x = [1 1 1 1 1];h = [0 1 2 3];

y = conv(x,h);

yields y = [0 1 3 6 6 6 5 3]. If x is indexed as described above, then y[0] = 0, y[1] = 1, ....

In general, total up the index of the first element in h and the index of the first element in

x, this is the index of the first element in y. For example, if the first element in hcorresponds to n = -2 and the first element in x corresponds to n = -3, then the first

element in y corresponds to n = -5.

Care must be taken when computing the convolution of infinite duration signals. If thevector x has length q and the vector h has length r, then you musttruncate the vector y to have length min(q,r). See the comments inProblem 3.7 of the textbook for additional information.

The command conv can also be used to multiply polynomials: suppose that thecoefficients of a(s) are given in the vector a and the coefficients of b(s) are given in thevector b, then the coefficients of the polynomial a(s)b(s) can be found as the elements of

the vector defined by ab = conv(a,b).

The command deconv is the inverse procedure to the convolution. In this text, it is used

as a means of dividing polynomials. Given a(s) and b(s) with coefficients stored in a and

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b, then the coefficients of c(s) = b(s)/a(s) are found by using the command c =

deconv(b,a).


B. Transfer Function Representation

For a discrete-time transfer function, the coefficients are stored in descending powers of z

or ascending powers of . For example,

then define the vectors as

num = [2 3 4];den = [1 5 6];


C. Time Simulations

Commands Covered:

recur

convdstep

dimpulsefilter

There are three methods to compute the response of a system described by the followingrecursive relationship

The first method uses the command recur and is useful when there are nonzero initial

conditions. This command is available from the MathWorks ftp site and a shortened

version is given in Figure C.5 of the textbook. The inputs to the function are the

coefficients stored in the vectors and , the initial

conditions on x and on y are stored in the vectors x0 = [x[n0-M], x[n0-M+1],...,x[n0-1]]

and y0 = [y[n0-N], y[n0-N+1],...,y[n0-1]]], and the time indices for which the solution

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needs to be calculated are stored in the vector n where n0 represents the first element in

this vector. To use recur, type

y = recur(a,b,n,x,x0,y0);

The output is a vector y with elements y[n]; the first element of y corresponds to thetime index n0. For example, consider the system described by

y[n] - 0.6y[n-1] + 0.08y[n-2] = x[n-1]

where x[n] = u[n] and with initial conditions y[-1] = 2, y[-2] = 1, andx[-1] = x[-2] = 0. To compute the response y[n] for n = 0, 1,...,10, type

a = [-0.6 0.08]; b = [0 1];x0 = 0; y0 = [1 2];n = 0:10;

x = ones(1,11);y = recur(a,b,n,x,x0,y0);

The vector y contains the values of y[n] for n = 0,1,...,10.

The second method to compute the response uses convolution and isuseful when the initial conditions on y are zero. This method involvesfirst finding the impulse response of the system, h[n], and thenconvolving h[n] with x[n] as discussed in Section 4.A. For example,consider the system described above with zero initial conditions, thatis, y[-1]=y[-2]=0. The impulse response for this system is

. The commands to compute y[n] are

n = 0:10;x = ones(1,11);h = 5*(0.4).^n - 5*(.02).^n;y = conv(x,h);y = y(1:length(n));

The vector y contains the values of y[n] for n = 0,1,...,10. Note that thevector was truncated to length(n) because both x[n] and h[n] are infiniteduration signals. See the comments in Section 4.A regarding theconvolution of infinite duration signals.

The third method of solving for the response requires that the transferfunction of the system be known. The commands dstep and dimpulsecompute the unit step response and the unit impulse response, respectively while the

command filter computes the response to initial conditions and to arbitrary inputs. The

denominator coefficients are stored as and the numerator coefficients

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are stored as where there are N-M zeros padded on theend of the coefficients. For example, consider the system given abovewith initial conditions y[-1] = y[-2] = 0. To compute the step responsefor n=0 to n=10, type the commands

n = 0:10;num = [0 1 0]; den = [1 -0.6 0.08];y = dstep(num,den,length(n));

The response can then be plotted using the stem plot. To compute the impulse response,simply replace dstep with dimpulse in the above commands.

To compute the response to an arbitrary input, store the input sequence in the vector x.

The command

y = filter(num,den,x);

is used to compute the system response. If the system has nonzero initial conditions, theinitial conditions can be stored in a vector v0. For a first order system where N=M=1,

define . For a second order system where N=M=2, define

. To compute the response withnonzero initial conditions, type

y = filter(num,den,x,zi);

For example, consider the previous system with the initial conditions y[-1] = 2 and y[-2]

= 1 and input x[n] = u[n]. Type the following commands to compute y[n].

n = 0:10; x = ones(1:11);num = [0 1 0]; den = [1 -0.6 0.08];

zi = [0.6*2-0.08*1, -0.08*2];

y = filter(num,den,x,zi);


D. Frequency Response Plots

Commands covered:

freqz

The DTFT of a system can be calculated from the transfer function using freqz. Definethe numerator and the denominator of the transfer function in num and den. The

command

[H,Omega] = freqz(num,den,n,'whole');

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computes the DTFT for n points equally spaced around the unit circle at the frequencies

contained in the vector Omega. The magnitude of H is found from abs(H) and the phase

of H is found from angle(H). To customize the range for , define a vector Omega of

desired frequencies, for example Omega = -pi:2*pi/300:pi defines a vector of length 301

with values that range from - to . To get the DTFT at these frequencies, type

H = freqz(num,den,Omega);


E. Digital Filter Design

Commands covered:

bilinearbutter

cheby1hamminghanning

The analog prototype method of designing IIR filters can be done by first designing an

analog filter with the desired characteristics as shown in Section 3.D, then mapping the

filter to the discrete-time domain. Store the numerator and denominator of the analogfilter, H(s), in the vectors num and den, and let T be the sampling period. Then the

numerator and denominator of the digital filter H(z) is found from the following

command

[numd,dend] = bilinear(num,den,1/T)

Alternately, the commands butter and cheby1 automatically design the analog filter and

then use the bilinear transformation to map the filter to the discrete-time domain.

Lowpass, highpass, bandstop, and bandpass filters can be designed using this method.

The digital cutoff frequencies must be specified; these should be normalized by . Todesign a digital lowpass filter based on the analog Butterworth filter, use the commands:

[num,den] = butter(n,Omegac)

where n is the number of poles and Omegac is the normalized digital cutoff frequency,

. To design a highpass filter with cutoff frequency Omegac, use thecommands

[num,den] = butter(n,Omegac,'high')

To design a bandpass filter with passband from Omega1 to Omega2, define Omega =

[Omega1,Omega2] and use the command

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[num,den] = butter(n,Omega)

To design a bandstop filter with stopband from Omega1 to Omega2, define Omega =

[Omega1, Omega2] and use the command

[num,den] = butter(n,Omega,'stop')

The design for an nth order Type I Chebyshev filter is accomplished using the samemethods as for butter except that "butter" is replaced by "cheby1":

[num,den] = cheby1(n,Omegac); % for a lowpass filter

[num,den] = cheby1(n,Omegac,'high'); % for a highpass filter

If Omega has two elements,

[num,den] = cheby1(n,Omega); % for a bandpass

[num,den] = cheby1(n,Omega,'stop'); % for a bandstop

The windows used in FIR filter design are given by

w = boxcar(N) % rectangular window

w = hamming(N)w = hanning(N)

These commands are used to truncate the infinite impulse response of an ideal digital

filter with the result being an FIR filter with length N.

The Signal Processing Toolbox also provides commands for computing the FIR filterdirectly. To obtain an FIR filter with length N and cutoff frequency Omegac (normalized

by ) use the command

h = fir1(N-1,Omegac)

The vector h contains the impulse response of the FIR where h(1) is the value of h[0].

A length N highpass filter with normalized cutoff frequency Omegac is designed byusing the command

h = fir1(N-1,Omegac,'high')

A bandpass with passband from Omega1 to Omega2 is obtained by typing

h = fir1(N-1,Omega)

where Omega = [Omega1,Omega2]. A bandstop filter with stopband from Omega1 toOmega2 is obtained by typing

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h = fir1(N-1,Omega,'stop')

where Omega = [Omega1,Omega2]. The fir1 command uses the Hamming window by

default. Other windows are obtained by adding an option of 'hanning' or 'boxcar' to thearguments; for example,

h = fir1(N-1,Omegac,'high',boxcar(N))

creates a highpass FIR filter with cutoff frequency Omegac using a rectangular window.


F. Digital Control Design

Commands covered:

bilinearc2dmhybrid

An analog controller G(s) can be mapped to a digital controller D(z) using the bilinear

transformation or the step response matching method. Store the numerator anddenominator of G(s) in num and den. Then the numerator and denominator of D(z) is

found from the bilinear transformation using the commands

[numd,dend] = bilinear(num,den,1/T)

where T is the sampling frequency. To use the step invariant method, use the commands

[numd,dend] = c2dm(num,den,T,'zoh')

To simulate the response of a continuous-time plant G(s) with a digital controller D(z),use the command hybrid. Consider the block diagram in Figure 11.25 of the text. The

numerator and denominator coefficients of the plant are stored in numG and denG; the

numerator and denominator coefficients of the controller are stored in numD and denD;the reference input signal is stored in r; and the sampling time is stored in T. The

increments in the time vector should selected to be the sampling time divided by aninteger, for example, t = 0:b:Tend where there is some integer m such that bm=T. Thecommand is used as

[y,ud] = hybrid(numG,denG,numD,denD,T,t,r);

The outputs of the command are the system response, y, and the control signal that is

input to the plant, ud. The M-file contains a loop which computes the discrete-time

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control and then simulates the continuous-time plant for T seconds with the constant

control. The process repeats for the next T second interval.

G. State Space Representation

Commands Covered:

dlsimdstep

dimpulse

Most of the commands for the continuous time state space representation also work for

the discrete time state space. For example, ss2tf, tf2ss, and ss2ss for discrete time areused exactly the same way as for the continuous time case discussed inSection 3.F.

There is a discrete time version of the command lsim, which is used as follows:

[y,x] = dlsim(A,B,C,D,u,n);

where the output is stored in y, the states are stored in x, the input is stored in u and the

time index is stored in n.

5. Plotting

Commands covered:

plot

xlabel

ylabeltitle

gridaxis

stem

subplot

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The command most often used for plotting is plot, which creates linear plots of vectors

and matrices; plot(t,y) plots the vector t on the x-axis versus vector y on the y-axis. There

are options on the line type and the color of the plot which are obtained usingplot(t,y,'option'). The linetype options are '-' solid line (default), '--' dashed line, '-.' dot

dash line, ':' dotted line. The points in y can be left unconnected and delineated by a

variety of symbols: + . * o x. The following colors are available options:

r red

b blue

g green

w white

k black

For example, plot(t,y,'--') uses a dashed line, plot(t,y,'*') uses * at all the points defined in

t and y without connecting the points, and plot(t,y,'g') uses a solid green line. The options

can also be used together, for example, plot(t,y,'g:') plots a dotted green line.

To plot two or more graphs on the same set of axes, use the command plot(t1,y1,t2,y2),

which plots y1 versus t1 and y2 versus t2.

To label your axes and give the plot a title, type

xlabel('time (sec)')ylabel('step response')

title('My Plot')

Finally, add a grid to your plot to make it easier to read. Type

grid

The problem that you will encounter most often when plotting functions is thatMATLAB will scale the axes in a way that is different than you want them to appear.

You can easily override the autoscaling of the axes by using the axis command after the

plotting command:

axis([xmin xmax ymin ymax]);

where xmin, xmax, ymin, and ymax are numbers corresponding to the limits you desirefor the axes. To return to the automatic scaling, simply type axis.

For discrete-time signals, use the command stem which plots each point with a small

open circle and a straight line. To plot y[k] versus k, type stem(k,y). To obtain filledcircles, as the plots appear in the textbook, use the command stem(k,y,'filled').

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To plot more than one graph on the screen, use the command subplot(mnp) which

partitions the screen into an mxn grid where p determines the position of the particular

graph counting the upper left corner as p=1. For example,

subplot(211),semilogx(w,magdb);

subplot(212),semilogx(w,phase);

plots the bode plot with the log-magnitude plot on top and the phase plot below. Titles

and labels can be inserted immediately after the appropriate semilogx command or plotcommand. To return to a full screen plot, type subplot(111).

6. Loading and Saving Data

When using MATLAB, you may wish to leave the program but save the vectors andmatrices you have defined. To save the file to the working directory, type

save filename

where "filename" is a name of your choice. To retrieve the data later, type

load filename

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