signals system lab manual

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Table of ContentsLab 1: Getting Started With Matlab .............................................................................................. 2 Lab 2: Introduction to Matrix ......................................................................................................... 9 Lab 3: Programing in Matlab ....................................................................................................... 18 Lab 4: Introduction to Graphics ................................................................................................... 35 Lab 5: Linear Algebra using MATLAB ....................................................................................... 47 Lab 6: Introduction to Signals ...................................................................................................... 64 Lab 7: Signals Transformation ..................................................................................................... 77 Lab 8: Sampling, periodicity, Harmonics and Sound Manipulation ...................................... 84 Lab 9: Convolution and LTI Systems .......................................................................................... 95 Lab 10: Laplace-Transform ....................................................................................................... 104 Lab 11: Z-transform and Inverse Z-transform Analysis...................................................... 108 Lab 12: Fourier series and the Gibbs Phenomenon ............................................................... 114 Lab 13: Discrete-Time Signals in the Frequency Domain .................................................... 119 Lab 14: Discrete Fourier Transform ........................................................................................ 129 Lab 15: Introduction to SIMULINK ........................................................................................ 136



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Lab 1: Getting Started With MatlabThe purpose of this Lab is to present basics of MATLAB. We do not assume any prior knowledge of

this package. This Lab is intended for users running a professional version of MATLAB 7, Release 14

under Windows XP or Latest. Topics discussed in this Lab include the Command Window, numbers

and arithmetic operations, saving and reloading a work, using help, MATLAB demos, interrupting a

running program, long command lines, and MATLAB resources on the Internet.

Main MATLAB Interface

Some MATLAB Development Windows

•Command Window: where you enter commands

•Command History: running history of commands which is preserved across MATLAB sessions

•Current directory: Default is $matlabroot/work

•Workspace: GUI for viewing, loading and saving MATLAB variables

•Array Editor: GUI for viewing and/or modifying contents of MATLAB variables (double-click the

array’s name in the Workspace)

•Editor/Debugger: text editor, debugger; editor works with file types in addition to .m (MATLAB

“m-files”)

MATLAB Editor window



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The MATLAB Editor Window is the window where MATLAB programs are written and edited.

MATLAB offers users to either write commands in the Command Window where they are executed

one by one, or to write the whole program in the Editor Window where the whole program can be

written, saved as *.m file and can be executed at once anytime.

MATLAB Help Window

MATLAB offers a very powerful Help Window where the user can search for anything related to the

MATLAB version being used.

1.1 The Command WindowYou can start MATLAB by double clicking on the MATLAB icon that should be on the desktop of

your computer. This brings up the window called the Command Window. This window allows a user

to enter simple commands. To clear the Command Window type clc and next press the Enter or

Return key. To perform a simple computations type a command and next press the Enter or Return

key. For instance,

s = 1 + 2

s =3

fun = sin(pi/4)

fun =

0.7071

In the second example the trigonometric function sine and the constant π are used. In MATLAB they

are named sin and pi, respectively.

Note that the results of these computations are saved in variables whose names are chosen by the

user. If they will be needed during your current MATLAB session, then you can obtain their values

typing their names and pressing the Enter or Return key. For instance,

s



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s =

3

Variable name begins with a letter, followed by letters, numbers or underscores. MATLAB

recognizes only the first 31 characters of a variable name.

To change a format of numbers displayed in the Command Window you can use one of the several

formats that are available in MATLAB. The default format is called short (four digits after thedecimal point.) In order to display more digits click on File, select Preferences…, and next select a

format you wish to use. They are listed below the Numeric Format. Next click on Apply and OK

and close the current window. You can also select a new format from within the Command Window.

For instance, the following command

format long

changes a current format to the format long. To display more digits of the variable fun type

fun

fun =

0.70710678118655

To change a current format to the default one type

format short

fun

fun =

0.7071

To close MATLAB type exit in the Command Window and next press Enter or Return key. A

second way to close your current MATLAB session is to select File in the MATLAB's toolbar and

next click on Exit MATLAB option. All unsaved information residing in the MATLAB Workspace

will be lost.

1.1.1 Numbers and Arithmetic operations

There are three kinds of numbers used in MATLAB:

integers

real numbers

complex numbers

Integers are entered without the decimal point

xi = 10

xi =

10However, the following number

xr = 10.01

xr =

10.0100

is saved as the real number. It is not our intention to discuss here machine representation of numbers.

This topic is usually included in the numerical analysis courses. Variables realmin and realmax

denote the smallest and the largest positive real numbers in MATLAB. For instance,

realmin

ans =

2.2251e-308



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Complex numbers in MATLAB are represented in rectangular form. The imaginary unit √ 1is

denoted either by i or j

i

ans =

0 + 1.0000i

In addition to classes of numbers mentioned above, MATLAB has three variables representing the

non numbers:

-Inf

Inf

NaN

The – Inf and Inf are the IEEE representations for the negative and positive infinity, respectively.

Infinity is generated by overflow or by the operation of dividing by zero. The NaN stands for the not-

a-number and is obtained as a result of the mathematically undefined operations such as

0.0/0.0 or .

List of basic arithmetic operations in MATLAB include six operations

MATLAB has two division operators / - the right division and \ - the left division. They do not

produce the same resultsrd = 47/3

rd =

15.6667

ld = 47\3

ld =

0.0638

1.1.2 Saving and Reloading your Work

All variables used in the current MATLAB session are saved in the Workspace. You can view the

content of the Workspace by clicking on File in the toolbar and next selecting Show Workspace from

the pull-down menu. You can also check contents of the Workspace typing whos in the Command

Window. For instance,

whos

Name Size Bytes Class

ans 1x1 16 double array (complex)

fun 1x1 8 double array

ld 1x1 8 double array

rd 1x1 8 double array

s 1x1 8 double arrayxi 1x1 8 double array



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xr 1x1 8 double array

Grand total is 7 elements using 64 bytes

shows all variables used in current session. You can also use command who to generate a list of

variables used in current session

who

Your variables are:ans ld s xr

fun rd xi

To save your current workspace select Save Workspace as… from the File menu. Chose a name for

your file, e.g. filename.mat and next click on Save. Remember that the file you just created must be

located in MATLAB's search path. Another way of saving your workspace is to type save filename in

the Command Window. The following command save filename s saves only the variable s.

Another way to save your workspace is to type the command diary filename in the Command

Window. All commands and variables created from now will be saved in your file. The following

command: diary off will close the file and save it as the text file. You can open this file in a text

editor, by double clicking on the name of your file, and modify its contents if you wish to do so.

To load contents of the file named filename into MATLAB's workspace type load filename in

the Command Window.

More advanced computations often require execution of several lines of computer code. Rather than

typing those commands in the Command Window you should create a file. Each time you will need

to repeat computations just invoke your file. Another advantage of using files is the ease to modify its

contents. To learn more about files, see [1], pp. 67- 75 and also Section 2.2 of Lab 2.

1.2 HelpOne of the nice features of MATLAB is its help system. To learn more about a function you are

to use, say rref , type in the Command Window

help svd

SVD Singular value decomposition.

[U,S,V] = SVD(X) produces a diagonal matrix S, of the samedimension as X

and with nonnegative diagonal elements indecreasing order, and unitary

matrices U and V so that

X = U*S*V'.S = SVD(X) returns a vector containing the singular values.

[U,S,V] = SVD(X,0) produces the "economy size"decomposition. If X is m-by-n

with m > n, then only thefirst n columns of U are computed and S is n-by-n.

See also SVDS, GSVD.

Overloaded methods

help sym/svd.m



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If you do not remember the exact name of a function you want to learn more about use command

lookfor followed by the incomplete name of a function in the Command Window. In the following

example we use a "word" sv

lookfor sv

ISVMS True for the VMS version of MATLAB.

HSV2RGB Convert hue-saturation-value colors to red-green-blue.

RGB2HSV Convert red-green-blue colors to hue-saturation-value.

GSVD Generalized Singular Value Decomposition.

SVD Singular value decomposition.

SVDS Find a few singular values and vectors.

HSV Hue-saturation-value color map.

JET Variant of HSV.

CSVREAD Read a comma separated value file.

CSVWRITE Write a comma separated value file.

ISVARNAME Check for a valid variable name.

RANDSVD Random matrix with pre-assigned singular values.

Trusvibs.m: % Example: trusvibs

SVD Symbolic singular value decomposition.

RANDSVD Random matrix with pre-assigned singular values.

The helpwin command, invoked without arguments, opens a new window on the screen. To find an

information you need double click on the name of the subdirectory and next double click on a

function to see the help text for that function. You can go directly to the help text of your function

invoking helpwin command followed by an argument. For instance, executing the following

command.

helpwin zerosZEROS Zeros array.

ZEROS(N) is an N-by-N matrix of zeros.

ZEROS(M,N) or ZEROS([M,N]) is an M-by-N matrix of zeros.

ZEROS(M,N,P,...) or ZEROS([M N P ...]) is an M-by-N-by-P-by-...

array of zeros.

ZEROS(SIZE(A)) is the same size as A and all zeros.

See also ONES.

generates an information about MATLAB's function zeros.

MATLAB also provides the browser-based help. In order to access these help files click on Help and

next select Help Desk (HTML). This will launch your Web browser. To access information you need

click on a highlighted link or type a name of a function in the text box. In order for the Help Desk to

work properly on your computer the appropriate help files, in the HTML or PDF format, must be

installed on your computer. You should be aware that these files require a significant amount of the

disk space.

1.3 DemosTo learn more about MATLAB capabilities you can execute the demo command in the Command

Window or click on Help and next select Examples and Demos from the pull-down menu. Some of

the MATLAB demos use both the Command and the Figure windows.



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To learn about matrices in MATLAB open the demo window using one of the methods described

above. In the left pane select Matrices and in the right pane select Basic matrix operations then

click on Run Basic matrix … . Click on the Start >> button to begin the show.

If you are familiar with functions of a complex variable I recommend another demo. Select

Visualization and next 3-D Plots of complex functions. You can generate graphs of simple power

functions by selecting an appropriate button in the current window.

1.3.1 Interrupting a running program

To interrupt a running program press simultaneously the Ctrl-c keys. Sometimes you have to repeat

pressing these keys a couple of times to halt execution of your program. This is not a recommended

way to exit a program, however, in certain circumstances it is a necessity. For instance, a poorly

written computer code can put MATLAB in the infinite loop and this would be the only option you

will have left.

1.3.2 Long Command Lines

To enter a statement that is too long to be typed in one line, use three periods, … , followed by Enter

or Return. For instance,

x = sin(1) - sin(2) + sin(3) - sin(4) + sin(5) -...

sin(6) + sin(7) - sin(8) + sin(9) - sin(10)

x =

0.7744

You can suppress output to the screen by adding a semicolon after the statement

u = 2 + 3;

1.3.3 Matlab Resources on internet

If your computer has an access to the Internet you can learn more about MATLAB and also download

user supplied files posted in the public domain. We provide below some pointers to information

related to MATLAB.

The MathWorks Web site: http://www.mathworks.com/Here you can find information about new

products, MATLAB related books, user supplied files and much more.

http://dir.yahoo.com/science/mathematics/software/matlab/

http://www.cse.uiuc.edu/cse301/matlab.html

The Mastering Matlab Web site: http://www.eece.maine.edu/mm

Recommended link for those who are familiar with the book Mastering Matlab 5.

A Comprehensive Tutorial and Reference, by D. Hanselman and B. Littlefield (see [2].)

1.3.4 References[1] Getting Started with MATLAB, Version 5, The MathWorks, Inc., 1996.

[2] D. Hanselman and B. Littlefield, Mastering MATLAB 5. A Comprehensive Tutorial and

Reference, Prentice Hall, Upper Saddle River, NJ, 1998.

[3] K. Sigmon, MATLAB Primer, Fifth edition, CRC Press, Boca Raton, 1998.

[4] Using MATLAB, Version 5, The MathWorks, Inc., 1996.

[5] Matlab tutorials Edward Neuman

http://www.mathworks.com/


http://www.eece.maine.edu/mm




http://www.mathworks.com/



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Lab 2: Introduction to Matrix

2.1 IntroductionMatrices are the basic elements of the MATLAB environment. A matrix is a two-dimensional

array consisting of m rows and n columns. Special cases are column vectors (n = 1) and rowvectors (m = 1).

In this section we will illustrate how to apply different operations on matrices. The following

topics are discussed: vectors and matrices in MATLAB, the inverse of a matrix, determinants,

and matrix manipulation.

MATLAB supports two types of operations, known as matrix operations and array

operations. Matrix operations will be discussed first.

2.2 Matrix generationMatrices are fundamental to MATLAB. Therefore, we need to become familiar with matrix

generation and manipulation. Matrices can be generated in several ways.

2.2.1 Entering a vector

A vector is a special case of a matrix. The purpose of this section is to show how to create

vectors and matrices in MATLAB. As discussed earlier, an array of dimension 1xn is called a

row vector, whereas an array of dimension m£ 1 is called a column vector. The elements of

vectors in MATLAB are enclosed by square brackets and are separated by spaces or by

commas. For example, to enter a row vector, v, type

>> v = [1 4 7 10 13]

v =1 4 7 10 13

Column vectors are created in a similar way, however, semicolon (;) must separate

thecomponents of a column vector,

>> w = [1;4;7;10;13]

w =

1

4

7

10

13

On the other hand, a row vector is converted to a column vector using the transpose operator.

The transpose operation is denoted by an apostrophe or a single quote (').

>> w = v'

w =

1

4

710



10

13

Thus, v(1) is the first element of vector v, v(2) its second element, and so forth. Furthermore,

to access blocks of elements, we use MATLAB's colon notation (:). For example, to access

the first three elements of v, we write,

>> v(1:3)

ans =

1 4 7

Or, all elements from the third through the last elements,

>> v(3,end)

ans =

7 10 13

where end signifies the last element in the vector. If v is a vector, writing

>> v(:)

produces a column vector, whereas writing

>> v(1:end)

produces a row vector.

2.2.2

Entering a matrixA matrix is an array of numbers. To type a matrix into MATLAB you must

begin with a square bracket, [

separate elements in a row with spaces or commas (,)

use a semicolon (;) to separate rows

end the matrix with another square bracket, ].

Here is a typical example. To enter a matrix A, such as,

A =1 2 34 5 67 8 9

Type,

>> A = [1 2 3; 4 5 6; 7 8 9]

MATLAB then displays the 3 x3 matrix as follows,

A =

1 2 3

4 5 6



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7 8 9

Note that the use of semicolons (;) here is different from their use mentioned earlier

tosuppress output or to write multiple commands in a single line.Once we have entered the

matrix, it is automatically stored and remembered in theWorkspace. We can refer to it simply

as matrix A. We can then view a particular element ina matrix by specifying its location. We

write,

>> A(2,1)

ans =

4

A(2,1) is an element located in the second row and first column. Its value is 4.

2.2.3 Matrix indexing

We select elements in a matrix just as we did for vectors, but now we need two indices.Theelement of row i and column j of the matrix A is denoted by A(i,j). Thus, A(i,j)in MATLAB

refers to the element Aij of matrix A. The first index is the row number andthe second index

is the column number. For example, A(1,3) is an element of first row andthird column.

Here,A(1,3)=3.

Correcting any entry is easy through indexing. Here we substitute A(3,3)=9 by

A(3,3)=0. The result is

>> A(3,3) = 0

A =

1 2 3

4 5 6

7 8 0

Single elements of a matrix are accessed as A(i,j), where i ¸ 1 and j ¸ 1. Zero or

negativesubscripts are not supported in MATLAB.

2.2.4 Colon operator

The colon operator will prove very useful and understanding how it works is the key to

efficient and convenient usage of MATLAB. It occurs in several different forms.

Often we must deal with matrices or vectors that are too large to enter one element at a time.

For example, suppose we want to enter a vector x consisting of points(0; 0:1; 0:2; 0:3; … ;

5). We can use the command

>> x = 0:0.1:5;

The row vector has 51 elements.



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2.2.5 Linear spacing

On the other hand, there is a command to generate linearly spaced vectors: linspace. Itis

similar to the colon operator (:), but gives direct control over the number of points.

Forexample,

y = linspace(a,b)

generates a row vector y of 100 points linearly spaced between and including a and b.

y = linspace(a,b,n)

generates a row vector y of n points linearly spaced between and including a and b. This

isuseful when we want to divide an interval into a number of subintervals of the same

length.For example,

>> theta = linspace(0,2*pi,101)

divides the interval [0; 2π ] into 100 equal subintervals, then creating a vector of 101elements.

2.2.6 Colon operator in a matrix

The colon operator can also be used to pick out a certain row or column. For example,

thestatement A(m: n,k:l ) specifies rows m to n and column k to l . Subscript expressions

referto portions of a matrix. For example,

>> A(2,:)

ans =

4 5 6

is the second row elements of A.The colon operator can also be used to extract a sub-matrix

from a matrix A.

>> A(:,2:3)

ans =

2 3

5 6

8 0

A(:,2:3) is a sub-matrix with the last two columns of A.A row or a column of a matrix can be

deleted by setting it to a null vector, [ ].

>> A(:,2)=[]

ans =

1 3



13

4 6

7 0

2.2.7 Creating a sub-matrix

To extract a submatrix B consisting of rows 2 and 3 and columns 1 and 2 of the matrix A,do

the following

>> B = A([2 3],[1 2])

B =

4 5

7 8

To interchange rows 1 and 2 of A, use the vector of row indices together with the

colonoperator.

>> C = A([2 1 3],:)

C =

4 5 6

1 2 3

7 8 0

It is important to note that the colon operator (:) stands for all columns or all rows. Tocreatea vector version of matrix A, do the following

>> A(:)

ans =

1

2

3

4

56

7

8

0

The submatrix comprising the intersection of rows p to q and columns r to s is denoted

byA(p:q,r:s).As a special case, a colon (:) as the row or column specifier covers all entries in

that row orcolumn; thus

A(:,j) is the jth column of A, while

A(i,:) is the ith row, and A(end,:) picks out the last row of A.



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The keyword end, used in A(end,:), denotes the last index in the specified dimension. Hereare

some examples.

>> A

A =

1 2 3

4 5 67 8 9

>> A(2:3,2:3)

ans =

5 6

8 9

>> A(end:-1:1,end)

ans =

9

6

3

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>> A([1 3],[2 3])

ans =

2 3

8 9

2.2.8

Deleting row or columnTo delete a row or column of a matrix, use the empty vector operator, [ ].

>> A(3,:) = []

A =

1 2 3

4 5 6

Third row of matrix A is now deleted. To restore the third row, we use a technique

forcreating a matrix

>> A = [A(1,:);A(2,:);[7 8 0]]

A =

1 2 3

4 5 6

7 8 0

Matrix A is now restored to its original form.

2.2.9 Dimension

To determine the dimensions of a matrix or vector, use the command size. For example,

>> size(A)

ans =

3 3means 3 rows and 3 columns.

Or more explicitly with,



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>> [m,n]=size(A)

2.2.10 Continuation

If it is not possible to type the entire input on the same line, use consecutive periods, calledan

ellipsis …, to signal continuation, then continue the input on the next line.

B = [ 4/5 7.23*tan(x) sqrt(6); ...

1/x^2 0 3/(x*log(x)); ...x-7 sqrt(3) x*sin(x)];

Note that blank spaces around +, - , = signs are optional, but they improve readability.

2.2.11 Transposing a matrix

The transpose operation is denoted by an apostrophe or a single quote ('). It flips a

matrixabout its main diagonal and it turns a row vector into a column vector. Thus,

>> A'

ans =

1 4 7

2 5 8

3 6 0

By using linear algebra notation, the transpose of m £ n real matrix A is the n £ m matrixthat

results from interchanging the rows and columns of A. The transpose matrix is denoted AT .

2.2.12 Concatenating matrices

Matrices can be made up of sub-matrices. Here is an example. First, let's recall our

previousmatrix A.

A =

1 2 3

4 5 6

7 8 9

The new matrix B will be,

>> B = [A 10*A; -A [1 0 0; 0 1 0; 0 0 1]]

B =

1 2 3 10 20 30

4 5 6 40 50 60

7 8 9 70 80 90

-1 -2 -3 1 0 0

-4 -5 -6 0 1 0

-7 -8 -9 0 0 1

2.3 Matrix generatorsMATLAB provides functions that generates elementary matrices. The matrix of zeros,

thematrix of ones, and the identity matrix are returned by the functions zeros, ones, and

eye,respectively.

Table 2.4: Elementary matriceseye(m,n) Returns an m-by-n matrix with 1 on the main diagonal

eye(n) Returns an n-by-n square identity matrix



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zeros(m,n) Returns an m-by-n matrix of zeros

ones(m,n) Returns an m-by-n matrix of ones

diag(A) Extracts the diagonal of matrix A

rand(m,n) Returns an m-by-n matrix of random numbers

For a complete list of elementary matrices and matrix manipulations, type help elmator doc

elmat. Here are some examples:1. >> b=ones(3,1)

b =

1

1

1

Equivalently, we can define b as >> b=[1;1;1]

2. >> eye(3)

ans =

1 0 0

0 1 0

0 0 1

3. >> c=zeros (2, 3)

c =

0 0 0

0 0 0

In addition, it is important to remember that the three elementary operations of addition (+),

subtraction (¡), and multiplication (¤ ) apply also to matrices whenever thedimensions are

compatible.

Two other important matrix generation functions are rand and randn, which generatematricesof (pseudo-)random numbers using the same syntax as eye.In addition, matrices can be

constructed in a block form. With C defined by C = [12; 3 4], we may create a matrix D as

follows

>> D = [C zeros(2); ones(2) eye(2)]

D =

1 2 0 0

3 4 0 0

1 1 1 0

1 1 0 1

2.3.1 Special matrices

MATLAB provides a number of special matrices (see Table 2.5). These matrices have

interesting properties that make them useful for constructing examples and for testing

algorithms.For more information, see MATLAB documentation.

Table 2.5: Special matrices

hilb Hilbert matrix

invhilb Inverse Hilbert matrix

magic Magic square

pascal Pascal matrixtoeplitz Toeplitz matrix

vander vandermonde matrix



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wilkinson Wilkinson's eigenvalue test matrix

2.3.2 Exercise:INTELLIGENT CRICKET SCORECARD

This is an exercise in matrix manipulation. Suppose that two teams A and B score the

following runs in a game of ten over. Below the number of runs scored and no of wickets

fallen in each over are displayed.

TEAM A

Runs : 8 6 9 0 11 13 6 0 10 3

Wickets : 1 0 2 0 0 0 1 0 0 0

TEAM B

Runs : 9 5 3 0 0 8 13 15 9 0

Wickets : 1 1 0 0 0 0 1 2 0 0

These are to be declared as two 2 x 10 matrices in MATLAB. Write a routine that takes these

matrices as input and generates an intelligent score card for both the teams considering that

team A played first.

TEAM X

Over

No

Runs in

this over

Total

score

Run

average

Wickets in

this over

Total no of

wickets

Run rate

required to

win

1

2

3

4

56

7

8

9

10

This table should be returned in the form of a MATRIX for team A and B both. Of course the

last column in the table would hold true only for team playing second.



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Lab 3: Programing in MATLABThis Lab is intended for those who want to learn basics of MATLAB programming language. Even

with a limited knowledge of this language a beginning programmer can write his/her own computer

code for solving problems that are complex enough to be solved by other means. Numerous examples

included in this text should help a reader to learn quickly basic programming tools of this language.

Topics discussed include the m-files, inline functions, control flow, relational and logical operators,

strings, cell arrays, rounding numbers to integers and MATLAB graphics.

3.1 The m-filesFiles that contain a computer code are called the m-files. There are two kinds of m-files: the script files and the

function files. Script files do not take the input arguments or return the output arguments. The function

files may take input arguments or return output arguments.

To make the m-file click on File next select New and click on M-File from the pull-down menu. You

will be presented with the MATLAB Editor/Debugger screen. Here you will type your code, canmake changes, etc. Once you are done with typing, click on File, in the MATLAB

Editor/Debuggerscreen and select Save As…. Chose a name for your file, e.g., firstgraph.m and

click on Save. Make sure that your file is saved in the directory that is in MATLAB's search path.

If you have at least two files with duplicated names, then the one that occurs first in MATLAB's

search path will be executed.

To open the m-file from within the Command Window type edit firstgraph and then press Enter or

Returnkey.

Here is an example of a small script file

% Script file firstgraph.

x = pi/100:pi/100:10*pi; y = sin(x)./x; plot(x,y) grid

Let us analyze contents of this file. First line begins with the percentage sign %. This is a comment.

All comments are ignored by MATLAB. They are added to improve readability of the code. In the

next two lines arrays x and y are created. Note that the semicolon follows both commands. This

suppresses display of the content of both vectors to the screen (see Lab 1 for more details). Array xholds 1000 evenly spaced numbers in the interval [/100 10/ ] while the array y holds the values of

the sinc function y = sin(x)/x at these points. Note use of the dot operator . before the right division

operator /. This tells MATLAB to perform the component wise division of two arrays sin(x) and x.

Special operators in MATLAB and operations on one- and two dimensional arrays are discussed in

detail in Lab 3, Section 3.2. The command plot creates the graph of the sinc function using the points

generated in two previous lines. For more details about command plot see Section 2.8.1 of this Lab.

Finally, the command grid is executed. This adds a grid to the graph. We invoke this file by typing its

name in the Command Window and next pressing the Enter or Return key

firstgraph



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3.2 FunctionsWhen you use “sin” or “log” or “exp” in Matlab you are using “function m-files”. They are different

from “script m-files” primarily because they have inputs and outputs. To specify which variables in

the m-file are the inputs, and which are the outputs, the first line of the m-file should be in this form:

function output=function_name(input)

For example, an m-file that begins with the line:

function L=vec_length(V)

takes a column vector V as its input, and returns its length L as its output. Here is the complete file

vec_length.m :

function L=vec_length(V)

% returns the length of column vector V

L=sqrt(V' * V); % square root of the dot product of V with itself

You would invoke on the command line this way:

>> W=[1; 2; -3]

>>Length_of_W=vec_length(W)

Several more examples are given. We recommend you make the name of the function given on the

first line match the file name of the m-file, i.e. function vec_length is in file vec_length.m, not in

sillyname.m

We also recommend you put only one function in each m-file (though under some circumstances it is

allowed to group a set of functions together into a single m-file.) If you store all your m-files in the

same directory, any function can call any other function. If you get a message saying that Matlab

cannot find your m-file, on the command line type

>> pwd

to make sure Matlab is looking in the correct folder, and then

>> dir

or

>> ls



20

to make sure that the m-file is really in that folder. You may have accidentally saved it in the wrong

folder.

Here is an example of the function file

function[b, j] = descsort(a)

% Function descsort sorts, in the descending order, a real array a.

% Second output parameter j holds a permutation used to obtain % array b from the array a.

[b ,j] = sort(-a);

b = -b;

To demonstrate functionality of the function under discussion leta = [pi – 10 35 0.15];

[b, j] = descsort(a) b =

35.0000 3.1416 0.1500 -10.0000 j =

3 1 4 2

You can execute function descsort without output arguments. In this case an information about a

permutation used will be lostdescsort(a)ans =

35.0000 3.1416 0.1500 -10.0000

Since no output argument was used in the call to function descorder a sorted array a is assignedto the default variable ans.

3.2.1 Inline function and feval command

Sometimes it is handy to define a function that will be used during the current MATLAB session

only. MATLAB has a command inline used to define the so-called inline functions in the Command

Window.

Let

f = inline('sqrt(x.^2+y.^2)','x','y')

f =Inline function:f(x,y) = sqrt(x.^2+y.^2)

You can evaluate this function in a usual way

f(3,4) ans =

5

Note that this function also works with arrays. Let

A = [1 2;3 4]

A =

1 2

3 4

and

B = ones(2)

B =

1 1



21

1 1

ThenC = f(A, B)

C =

1.4142 2.2361

3.1623 4.1231

For the later use let us mention briefly a concept of the string in MATLAB. The character string is a

text surrounded by single quotes. For instance,

str = 'programming in MATLAB is fun'

str =

programming in MATLAB is fun

is an example of the string. Strings are discussed in Section 2.5 of this Lab.

In the previous section you have learned how to create the function files. Some functions take as the

input argument a name of another function, which is specified as a strin g . In order to execute function

specified by string you should use the command feval as shown below

feval('functname', input parameters of function functname)

Consider the problem of computing the least common multiple of two integers. MATLAB has a built-

in function lcm that computes the number in question. Recall that the least common multiple and the

greatest common divisor (gcd) satisfy the following equation

ab = lcm(a, b)gcd(a, b)

MATLAB has its own function, named gcd, for computing the greatest common divisor.

To illustrate the use of the command feval let us take a closer look at the code in the m-filemylcm

functionc = mylcm(a, b)

% The least common multiple c of two integers a and b.

If feval('isint',a) & feval('isint',b) c = a.*b./gcd(a,b);

else error('Input arguments must be integral numbers')

end

Command feval is used twice in line two (I do do not count the comment lines and the blank lines). Itchecks whether or not both input arguments are integers. The logical and operator &used here is

discussed in Section 2.4. If this condition is satisfied, then the least common multiple is computed

using the formula mentioned earlier, otherwise the error message is generated. Note use of the

command error,which takes as the argument a string. The conditional if - else – end used here is

discussed in Section 2.4 of this tutorial. Function that is executed twice in the body of the function

mylcm is named isint

functionk = isint(x);

% Check whether or not x is an integer number. % If it is, function isint returns 1 otherwise it returns 0.

If abs(x - round(x)) < realmin k = 1;



22

else k = 0;

end

New functions used here are the absolute value function (abs) and the round function (round). The

former is the classical math function while the latter takes a number and rounds it to the closest

integer. Other functions used to round real numbers to integers are discussed in Section 2.7. Finally,

realmin is the smallest positive real number on your computer

format long

realmin

ans =2.225073858507201e-308

format short

The Trapezoidal Rule with the correction term is often used to numerical integration of functions that

are differentiable on the interval of integration

where h = b – a. This formula is easy to implement in MATLAB

functiony = corrtrap(fname, fpname, a, b)

% Corrected trapezoidal rule y. % fname - the m-file used to evaluate the integrand,

% fpname - the m-file used to evaluate the first derivative % of the integrand, % a,b - endpoinds of the interval of integration.

h = b - a; y = (h/2).*(feval(fname,a) + feval(fname,b))+ (h.^2)/12.*(...

feval(fpname,a) - feval(fpname,b));

The input parameters aand bcan be arrays of the same dimension. This is possible because the dot

operator proceeds certain arithmetic operations in the command that defines the variable y.

In this example we will integrate the sine function over two intervals whose end points are stored in

the arrays aand b, where

a = [0 0.1]; b = [pi/2 pi/2 + 0.1];

y = corrtrap('sin', 'cos', a, b)

y =0.9910 1.0850

Since the integrand and its first order derivative are both the built-in functions, there is no need to

define these functions in the m-files.



23

3.3 Control FlowTo control the flow of commands, the makers of MATLAB supplied four devices a programmer

can use while writing his/her computer code

the for loops

the whileloops

the if -else-end constructions the switch-case constructions

3.3.1 Repeating with for loop

Syntax of the for loop is shown below

for k = array

commands

end

The commands between the for and end statements are executed for all values stored in the array.

Suppose that one-need values of the sine function at eleven evenly spaced points πn/10, for n = 0, 1,

…, 10. To generate the numbers in question one can use the for loop

for n=0:10 x(n+1) = sin(pi*n/10);

end

x

x =Columns 1 through 7

0 0.3090 0.5878 0.8090 0.9511 1.0000

0.9511Columns 8 through 110.8090 0.5878 0.3090 0.0000

The for loops can be nested

H = zeros(5); for k=1:5

for l=1:5 H(k,l) = 1/(k+l-1);

end end

H

H =1.0000 0.5000 0.3333 0.2500 0.20000.5000 0.3333 0.2500 0.2000 0.16670.3333 0.2500 0.2000 0.1667 0.14290.2500 0.2000 0.1667 0.1429 0.12500.2000 0.1667 0.1429 0.1250 0.1111

Matrix H created here is called the Hilbert matrix. First command assigns a space in computer's

memory or the matrix to be generated. This is added here to reduce the overhead that is required by

loops in MATLAB.

The for loop should be used only when other methods cannot be applied. Consider the following

problem. Generate a 10-by-10 matrix A = [akl], where akl = sin(k)cos(l). Using nested loops one can

compute entries of the matrix A using the following code.



24

A = zeros(10);

for k=1:10 for l=1:10

A(k,l) = sin(k)*cos(l); end

end

A loop free version might look like this

k = 1:10; A = sin(k)'*cos(k);

First command generates a row array k consisting of integers 1, 2, … , 10. The command sin(k)'

creates a column vector while cos(k) is the row vector. Components of both vectors are the values of

the two trig functions evaluated at k . Code presented above illustrates a powerful feature of

MATLAB called vectorization. This technique should be used whenever it is possible.

3.3.2 Repeating with while loopsSyntax of the while loop is

while expression

statements

end

This loop is used when the programmer does not know the number of repetitions a priori.

Here is an almost trivial problem that requires a use of this loop. Suppose that the number π is divided

by 2. The resulting quotient is divided by 2 again. This process is continued till the current quotient is

less than or equal to 0.01. What is the largest quotient that is greater than 0.01?

To answer this question we write a few lines of code

q = pi; while q > 0.01 q = q/2;

end

q

q =0.0061

3.3.3 The if-else-end constructionSyntax of the simplest form of the construction under discussion is

if expression commands

end

This construction is used if there is one alternative only. Two alternatives require the construction

if expression

commands (evaluated if expression is true)

else

commands (evaluated if expression is false)

end

Construction of this form is used in functions mylcm and isint (see Section 2.3).

If there are several alternatives one should use the following construction



25

if expression1

commands (evaluated if expression 1 is true)

elseif expression 2

commands (evaluated if expression 2 is true)

elseif …

.

.

.

else

commands (executed if all previous expressions evaluate to false)

end

Chebyshev polynomialsTn(x), n = 0, 1, … of the first kind are of great importance in numerical

analysis. They are defined recursively as follows

Tn(x) = 2xTn – 1(x) – Tn – 2(x), n = 2, 3, … , T0(x) = 1, T1(x) = x.

Implementation of this definition is easy

functionT = ChebT(n)

% Coefficients T of the nth Chebyshev polynomial of the firstkind. % They are stored in the descending order of powers.

t0 = 1; t1 = [1 0]; if n == 0

T = t0; elseif n == 1;

T = t1; else

for k=2:n

T = [2*t1 0] - [0 0 t0]; t0 = t1; t1 = T;

end end

Coefficients of the cubic Chebyshev polynomial of the first kind are

coeff = ChebT(3)

coeff =4 0 -3 0

Thus T3(x) = 4x3 – 3x.

3.3.4 The switch-case construction

Syntax of the switch-case construction is

switch expression (scalar or string)

case value1 (executes if expression evaluates to value1)

commands

case value2 (executes if expression evaluates to value2)

commands .

.



26

.

otherwise

statements

end

Switch compares the input expression to each case value. Once the match is found it executes the

associated commands.

In the following example a random integer number x from the set {1, 2, … , 10} is generated. If x = 1

or x= 2, then the message Probability = 20% is displayed to the screen. If x = 3 or 4 or 5, then the

message Probability = 30% is displayed, otherwise the message Probability = 50% is generated. The

script file fswitch utilizes a switch as a tool for handling all cases mentioned above.

% Script file fswitch.

x = ceil(10*rand); % Generate a random integer in {1,2,...,10} switch x

case {1,2} disp('Probability = 20%');

case {3,4,5}

disp('Probability = 30%'); otherwise disp('Probability = 50%');

end

Note use of the curly braces after the word case. This creates the so-called cell array rather than the

one-dimensional array, which requires use of the square brackets.

Here are new MATLAB functions that are used in file fswitch

rand – uniformly distributed random numbers in the interval (0, 1)

ceil – round towards plus infinity infinity (see Section 2.5 for more details)

disp – display string/array to the screen

Let us test this code ten times

for k = 1:10 fswitch

end

Probability = 50%

Probability = 30%

Probability = 50%

Probability = 50%

Probability = 50%

Probability = 30%

Probability = 20%Probability = 50%

Probability = 30%

Probability = 50%

3.3.5 Relational and logical operators

Comparisons in MATLAB are performed with the aid of the following operators

Operator Description

< Less than

<= Less than or equal to

> Greater

>= Greater or equal to== Equal to

~= Not equal to



27

Operator ==compares two variables and returns ones when they are equal and zeros otherwise.

Let

a = [1 1 3 4 1]

a =

1 1 3 4 1

Then

ind = (a == 1)

ind =

1 1 0 0 1

You can extract all entries in the array a that are equal to 1 using

b = a(ind)

b =

1 1 1

This is an example of so-called logical addressing in MATLAB. You can obtain the same result using

function find

ind = find(a == 1)

ind =

1 2 5

Variable indnow holds indices of those entries that satisfy the imposed condition. To extract all ones from the

array a use

b = a(ind)

b =

1 1 1

There are three logical operators available in MATLAB

Logical operator Description

| Or

& And

~ Not

Suppose that one wants to select all entries xthat satisfy the inequalities x>= 1or x < -0.2 where

x = randn(1,7)

x =

-0.4326 -1.6656 0.1253 0.2877 -1.1465 1.1909 1.1892

is the array of normally distributed random numbers. We can solve easily this problem using

operators discussed in this section

ind = (x >= 1) | (x < -0.2)

ind =

1 1 0 0 1 1 1

y = x(ind)

y =



28

-0.4326 -1.6656 -1.1465 1.1909 1.1892

Solve the last problem without using the logical addressing.

In addition to relational and logical operators MATLAB has several logical functions designed for

performing similar tasks. These functions return 1 (true) if a specific condition is satisfied and 0(false)

otherwise. A list of these functions is too long to be included here. The interested reader is referred to

[1], pp. 85-86 and [4], Chapter 10, pp. 26-27. Names of the most of these functions begin with the

prefix is. For instance, the following command

isempty(y)

ans =

0

returns 0 because the array y of the last example is not empty. However, this command

isempty([ ])

ans =1

returns 1 because the argument of the function used is the empty array [ ].

Here is another example that requires use of the isempty command

functiondp = derp(p)

% Derivative dp of an algebraic polynomial that is % represented by its coefficients p. They must be stored % in the descending order of powers.

n = length(p) - 1; p = p(:)'; % Make sure p is a row array. dp = p(1:n).*(n:-1:1); % Apply the Power Rule. k = find(dp ~= 0); if ~isempty(k)

dp = dp(k(1):end); % Delete leading zeros if any. else

dp = 0; end

In this example p(x) = x3 + 2x2 + 4. Using a convention for representing polynomials in MATLAB as

the array of their coefficients that are stored in the descending order of powers, we obtain

dp = derp([1 2 0 4])

dp =

3 4 0

3.3.6 Rounding to integers. Function ceil, Floor, Round

We have already used two MATLAB functions round and ceil to round real numbers to integers.

They are briefly described in the previous sections of this tutorial. A full list of functions designed for

rounding numbers is provided below

Function Description

Floor Round towards minus infinityCeil Round towards plus infinity

Fix Round towards zero



29

Round Round towards nearest integer

To illustrate differences between these functions let us create first a two-dimensional array of random

numbers that are normally distributed (mean = 0, variance = 1) using another MATLAB function

randn

randn('seed', 0) % This sets the seed of the random numbers generator to zero

T = randn(5)

T =

1.1650 1.6961 -1.4462 -0.3600 -0.0449

0.6268 0.0591 -0.7012 -0.1356 -0.7989

0.0751 1.7971 1.2460 -1.3493 -0.7652

0.3516 0.2641 -0.6390 -1.2704 0.8617

-0.6965 0.8717 0.5774 0.9846 -0.0562

A = floor(T)

A =1 1 -2 -1 -1

0 0 -1 -1 -1

0 1 1 -2 -1

0 0 -1 -2 0

-1 0 0 0 -1

B = ceil(T)

B =

2 2 -1 0 0

1 1 0 0 0

1 2 2 -1 01 1 0 -1 1

0 1 1 1 0

C = fix(T)

C =

1 1 -1 0 0

0 0 0 0 0

0 1 1 -1 0

0 0 0 -1 0

0 0 0 0 0

D = round(T)

D =

1 2 -1 0 0

1 0 -1 0 -1

0 2 1 -1 -1

0 0 -1 -1 1

-1 1 1 1 0

In the following m-file functions floor and ceil are used to obtain a certain representation of a

nonnegative real number



30

function [m, r] = rep4(x)

% Given a nonnegative number x, function rep4 computes an integer m % and a real number r, where 0.25 <= r < 1, such that x = (4^m)*r.

If x == 0 m = 0; r = 0;

return end u = log10(x)/log10(4); if u < 0 m = floor(u)

else m = ceil(u);

end r = x/4^m;

Command returncauses a return to the invoking function or to the keyboard. Function log10 is the

decimal logarithm.

[m, r] = rep4(pi)

m =

1

r =

0.7854

We check this result

format long

(4^m)*r

ans =

3.14159265358979

format short

3.3.7 Reference

[1] D. Hanselman and B. Littlefield, Mastering MATLAB 5. A Comprehensive Tutorial

and Reference, Prentice Hall, Upper Saddle River, NJ, 1998.

[2] P. Marchand, Graphics and GUIs with MATLAB, Second edition, CRC Press, BocaRaton,1999.



[5] Using MATLAB Graphics, Version 5, The MathWorks, Inc., 1996.

3.3.8 Exercise problemsIn Problems 1- 4 you cannot use loops foror while.

1. Write MATLAB function sigma = ascsum(x) that takes a one-dimensional array xof real

numbers and computes their sum sigma in the ascending order of magnitudes.Hint: You maywish to use MATLAB functions sort, sum, and abs.



31

2. In this exercise you are to write MATLAB function d = dsc(c) that takes a one-dimensional

array of numbers cand returns an array d consisting of all numbers in the array cwith all

neighboring duplicated numbers being removed. For instance, if c = [1 2 2 2 3 1], then d = [1

2 3 1].

3. Write MATLAB function p = fact(n) that takes a nonnegative integer n and returns value of

the factorial function n! = 1*2* … *n. Add an error message to your code that will be

executed when the input parameter is a negative number

4. Write MATLAB function [in, fr] = infr(x)that takes an array xof real numbers and returns arrays in

and fr holding the integral and fractional parts, respectively, of all numbers in the array x.

5. Given an array band a positive integer mcreate an array d whose entries are those in the array

beach replicated m-times. Write MATLAB function d = repel(b, m) that generates array das

described in this problem.

6. In this exercise you are to write MATLAB function d = rep(b, m) that has more functionality

than the function repel of Problem 5. It takes an array of numbers band the array m of

positive integers and returns an array dwhose each entry is taken from the array b and is

duplicated according to the corresponding value in the array m. For instance, if b = [ 1 2] and

m = [2 3], then d = [1 1 2 2 2].

7. A checkerboard matrix is a square block diagonal matrix, i.e., the only nonzero entries are in

the square blocks along the main diagonal. In this exercise you are to write MATLAB

function A = mysparse(n)that takes an odd number n and returns a checkerboard matrix as

shown below

A = mysparse(3)

A =1 0 0

0 1 2

0 3 4

A = mysparse(5)

A =

1 0 0 0 0

0 1 2 0 0

0 3 4 0 0

0 0 0 2 3

0 0 0 4 5

A = mysparse(7)

A =

1 0 0 0 0 0 0

0 1 2 0 0 0 0

0 3 4 0 0 0 0

0 0 0 2 3 0 0

0 0 0 4 5 0 0

0 0 0 0 0 3 4

0 0 0 0 0 5 6

First block in the upper-left corner is the 1-by-1 matrix while the remaining blocks are all 2- by-2.



32

8. The Legendre polynomials Pn(x), n = 0, 1, … are defined recursively as follows

nPn(x) = (2n-1)xPn -1 – (n-1)Pn-2(x), n = 2, 3, … , P0(x) = 1, P1(x) = x.

Write MATLAB function P = LegendP(n) that takes an integer n – the degree of Pn(x)

and returns its coefficient stored in the descending order of powers.

9. In this exercise you are to implement Euclid's Algorithm for computing the greatest common divisor

(gcd) of two integer numbers aand b:gcd(a, 0) = a, gcd(a, b) = gcd(b, rem(a, b)).

Here rem(a, b) stands for the remainder in dividing a by b. MATLAB has function rem.

Write MATLAB function gcd = mygcd(a,b) that implements Euclid's Algorithm.

10. The Pascale triangle holds coefficients in the series expansion of (1 + x)n, where n = 0, 1, 2,

… . The top of this triangle, for n = 0, 1, 2, is shown here

1

1 1

1 2 1

Write MATLAB function t = pasctri(n) that generates the Pascal triangle t up to the level n.

Remark . Two-dimensional arrays in MATLAB must have the same number of columns ineach row. In order to aviod error messages you have to add a certain number of zero entries to

the right of last nonzero entry in each row of t but one. This

t = pasctri(2) t =

1 0 01 1 01 2 1

is an example of the array t for n = 2.

11. This is a continuation of Problem 10. Write MATLAB function t = binexp(n) that computes

an array t with row k+1 holding coefficients in the series expansion of (1-x)^k , k = 0, 1, ... ,

n, in the ascending order of powers. You may wish to make a call from within your function

to the function pasctri of Problem 10. Your output sholud look like this (case n = 3)t = binexp(3)t =

1 0 0 01 -1 0 01 -2 1 0

1 -3 3 -1

12. MATLAB come with the built-in function mean for computing the unweighted arithmetic

mean of real numbers. Let x = {x1, x2, … , xn} be an array of n real numbers. Then

In some problems that arise in mathematical statistics one has to compute the weighted

arithmetic mean of numbers in the array x. The latter, abbreviated here as wam, is defined as

follows

Here w = {w1, w2, … , wn} is the array of weights associated with variables x. The weights

are all nonnegative with w1 + w2 + … + wn > 0. In this exercise you are to write MATLAB

function y = wam(x, w) that takes the arrays of variables and weights and returns theweighted arithmetic mean as defined above. Add three error messages to terminate

prematurely execution of this file in the case when:



33

arrays x and w are of different lengths

at least one number in the array w is negative

sum of all weights is equal to zero.

13. Let w = {w1, w2, … , wn} be an array of positive numbers. The weighted geometric mean,

abbreviated as wgm, of the nonnegative variables x = {x1, x2, … , xn} is defined as follows

Here we assume that the weights w sum up to one. Write MATLAB function y = wgm(x, w)

that takes arrays x and w and returns the weighted geometric mean y of x with weights stored

in the array w. Add three error messages to terminate prematurely execution of this file in the

case when:

arrays x and w are of different lengths

at least one variable in the array x is negative

at least one weight in the array w is less than or equal to zero

Also, normalize the weights w, if necessary, so that they will sum up to one.

14. Write MATLAB function [nonz, mns] = matstat(A) that takes as the input argument a real

matrix A and returns all nonzero entries of A in the column vector nonz. Second output parameter mns holds values of the unweighted arithmetic means of all columns of A.

15. Solving triangles requires a bit of knowledge of trigonometry. In this exercise you are to write

MATLAB function [a, B, C] = sas(b, A, c) that is intended for solving triangles given two

sides b and c and the angle A between these sides. Your function should determine remaining

two angels and the third side of the triangle to be solved. All angles should be expressed in

the degree measure.

16. Write MATLAB function [A, B, C] = sss(a, b, c) that takes three positive numbers a, b, and

c. If they are sides of a triangle, then your function should return its angles A, B, and C, in the

degree measure, otherwise an error message should be displayed to the screen.

17. In this exercise you are to write MATLAB function dms(x) that takes a nonnegative number

x that represents an angle in the degree measure and converts it to the form x deg. y min. z

sec.. Display a result to the screen using commands disp and sprintf .

Example:dms(10.2345)Angle = 10 deg. 14 min. 4 sec.

18. In this exercise you are to model one of the games known as Lottery. Three numbers, with

duplicates allowed, are selected randomly from the set {0,1,2,3,4,5,6,7,8,9} in the game Pick3

and four numbers are selected in the Pick4 game. Write MATLAB function winnumbs =

lotto(n) that takes an integer n as its input parameter and returns an array winnumbs

consisting of n numbers from the set of integers described in this problem. Use MATLAB

function rand together with other functions to generate a set of winning numbers. Add an

error message that is displayed to the screen when the input parameter is out of range.

19. Write MATLAB function t = isodd(A) that takes an array A of nonzero integers and returns

1 if all entries in the array A are odd numbers and 0 otherwise. You may wish to use

MATLAB function rem in your file.

20. Given two one-dimensional arrays a and b, not necessarily of the same length. Write

MATLAB function c = interleave(a, b) which takes arrays a and b and returns an array c

obtained by interleaving entries in the input arrays. For instance, if a = [1, 3, 5, 7] and b = [-2,

– 4], then c = [1, – 2, 3, – 4, 5, 7]. Your program should work for empty arrays too.You cannotuse loops for or while.



34



35

Lab 4: Introduction to Graphics

4.1 Plotting MATLAB contains numerous commands for creating two- and three-dimensional plots.

Themost basic of these commands is ‘plot’ which can have multiple optional arguments.Asimple example of this command is to plot a function of time.

t = linspace(0, 8, 401); %Define a vector of times from 0 to 8 s with 401 points

x = t.*exp(-t).*cos(2*pi*4*t); %Define a vector of x values

plot(t,x); %Plot x vs. t

xlabel(’Time (s)’); %Label time axis

ylabel(’Amplitude’); %Label amplitude axis

4.1.1

Simple plotting commandsThe simple 2D plotting commands include

plot Plot in linear coordinates as a continuous function

stem Plot in linear coordinates as discrete samples

loglog Logarithmic x and y axes

semilogx Linear y and logarithmic x axes

semilogy Linear x and logarithmic y axes

bar Bar graph

errorbar Error bar graph

hist Histogram

polar Polar coordinates

4.1.2 Customization of plots

There are many commands used to customize plots by annotations, titles, axes labels, etc.

A few of the most frequently used commands are

xlabel Labels x-axis

ylabel Labels y-axis

title Puts a title on the plot

grid Adds a grid to the plot

axis Allows changing the x and y axes

figure Create a figure for plotting

hold on Allows multiple plots to be superimposed on the same

axes

hold off Release hold on current plot

close(n) Close figure number n

subplot(a,b,c) Create an a × b matrix of plots with c the current

figureorient Specify orientation of a figure.

4.2 MATLAB Graphics

MATLAB has several high-level graphical routines. They allow a user to create various graphicalobjects including two- and three-dimensional graphs, graphical user interfaces (GUIs), movies, to



36

mention the most important ones. For the comprehensive presentation of the MATLAB graphics the

interested reader is referred to [2].

Before we begin discussion of graphical tools that are available in MATLAB I recommend that you

will run a couple of demos that come with MATLAB. In the Command Window click on Help and

next select Examples and Demos. Chose Visualization, and next select 2-D Plots. You will be

presented with several buttons. Select Line and examine the m-file below the graph. It should give

you some idea about computer code needed for creating a simple graph. It is recommended that youexamine carefully contents of all m-files that generate the graphs in this demo.

4.2.1 2-D Graphics

Basic function used to create 2-D graphs is the plot function. This function takes a variable num ber of

input arguments. For the full definition of this function type help plot in the Command Window.

In this example the graph of the rational function () = + ,-2 ≤ x ≤ 2 will be plotted using a variable

number of points on the graph of f(x).

% Script file graph1.

% Graph of the rational function y = x/(1+x^2).

for n=1:2:5

n10 = 10*n;x = linspace(-2,2,n10);

y = x./(1+x.^2);

plot(x,y,'r')

title(sprintf('Graph %g. Plot based upon n = %g points.' ...

, (n+1)/2, n10))

axis([-2,2,-.8,.8])

xlabel('x')

ylabel('y')

grid

pause(3)

end Let us analyze contents of this file. The loop for is executed three times. Therefore, three graphs of

the same function will be displayed in the Figure Window. A MATLAB function linspace(a, b, n)

generates a one-dimensional array of n evenly spaced numbers in the interval [a b]. The y-ordinates

of the points to be plotted are stored in the array y. Command plot is called with three arguments: two

arrays holding the x- and the y-coordinates and the string 'r', which describes the color (red) to be

used to paint a plotted curve. You should notice a difference between three graphs created by this file.

There is a significant difference between smoothness of graphs 1 and 3. Based on your visual

observation you should be able to reach the following conclusion: "more points you supply the

smoother graph is generated by the function plot". Function title adds a descriptive information to the

graphs generated by this m-file and is followed by the command sprintf . Note that sprintf takes herethree arguments: the string and names of two variables printed in the title of each graph. To specify

format of printed numbers we use here the construction %g, which is recommended for printing

integers. The command axis tells MATLAB what the dimensions of the box holding the plot are. To

add more information to the graphs created here, we label the x- and the y-axes using commands

xlabel and the ylabel, respectively. Each of these commands takes a string as the input argument.

Function grid adds the grid lines to the graph. The last command used before the closing end is the

pause command. The command pause(n) holds on the current graph for n seconds before continuing,

where n can also be a fraction. If pause is called without the input argument, then the computer waits

to user response. For instance, pressing the Enter key will resume execution of a program. Function

subplot is used to plot of several graphs in the same Figure Window. Here is a slight modification of

the m-file graph1




37

% Several plots of the rational function y = x/(1+x^2)

% in the same window.

k = 0;

for n=1:3:10

n10 = 10*n;

x = linspace(-2,2,n10);

y = x./(1+x.^2);

k = k+1;subplot(2,2,k)

plot(x,y,'r')

title(sprintf('Graph %g. Plot based upon n = %g points.' ...

, k, n10))

xlabel('x')

ylabel('y')

axis([-2,2,-.8,.8])

grid

pause(3);

end

The command subplot is called here with three arguments. The first two tell MATLAB that a 2-by-2

array consisting of four plots will be created. The third parameter is the running index telling

MATLAB which subplot is currently generated.

graph2

Using command plot you can display several curves in the same Figure Window.

We will plot two ellipses( 3 )36 ( 2 )

81 = 1 ( 7 )4 ( 8 )

36 = 1 using command plot


% Graphs of two ellipses

% x(t) = 3 + 6cos(t), y(t) = -2 + 9sin(t) as x=a+rcos(theta) and

% y=a+rsin(theta)are

% and

% x(t) = 7 + 2cos(t), y(t) = 8 + 6sin(t).

t = 0:pi/100:2*pi;x1 = 3 + 6*cos(t);

y1 = -2 + 9*sin(t);



38

x2 = 7 + 2*cos(t);

y2 = 8 + 6*sin(t);

h1 = plot(x1,y1,'r',x2,y2,'b');

set(h1,'LineWidth',1.25)

axis('square')

xlabel('x')

h = get(gca,'xlabel');

set(h,'FontSize',12)set(gca,'XTick',-4:10)

ylabel('y')

h = get(gca,'ylabel');

set(h,'FontSize',12)

set(gca,'YTick',-12:2:14)

title('Graphs of (x-3)^2/36+(y+2)^2/81 = 1 and (x-7)^2/4+(y-8)^2/36 =1.')

h = get(gca,'Title');


grid

In this file we use several new MATLAB commands. They are used here to enhance the readability of

the graph. Let us now analyze the computer code contained in the m-file graph3. First of all, theequations of ellipses in rectangular coordinates are transformed to parametric equations. This is a

convenient way to plot graphs of equations in the implicit form. The points to be plotted, and

smoothed by function plot, are defined in the first five lines of the file. I do not count here the

comment lines and the blank lines. You can plot both curves using a single plot command. Moreover,

you can select colors of the curves. They are specified as strings (see line 6). MATLAB has several

colors you can use to plot graphs:

y yellow

m magenta

c cyan

r red

g green

b blue

w white

k black

Note that the command in line 6 begins with h1 = plot… Variable h1 holds an information about the

graph you generate and is called the handle graphics. Command set used in the next line allows a user

to manipulate a plot. Note that this command takes as the input parameter the variable h1. We change

thickness of the plotted curves from the default value to a width of our choice, namely 1.25. In the

next line we use command axis to customize plot. We chose option 'square' to force axes to have

square dimensions. Other available options are:

'equal', 'normal', 'ij', 'xy', and 'tight'. To learn more about these options use MATLAB's help.

If function axis is not used, then the circular curves are not necessarily circular. To justify this let us

plot a graph of the unit circle of radius 1 with center at the origin

t = 0:pi/100:2*pi;

x = cos(t);

y = sin(t);

plot(x,y)



39

Another important MATLAB function used in the file under discussion is named get (see line 10). Ittakes as the first input parameter a variable named gca = get current axis. It should be obvious to you,

that the axis targeted by this function is the x-axis. Variable h = get(gca, … ) is the graphics handle

of this axis. With the information stored in variable h, we change the font size associated with the x-

axis using the 'FontSize' string followed by a size of the font we wish to use. Invoking function set in

line 12, we will change the tick marks along the x-axis using the 'XTick' string followed by the array

describing distribution of marks. You can comment out temporarily line 12 by adding the percent sign

% before the word set to see the difference between the default tick marks and the marks generated

by the command in line 12. When you are done delete the percent sign you typed in line 12 and click

on Save from the File

menu in the MATLAB Editor/Debugger. Finally, you can also make changes in the title of your

plot. For instance, you can choose the font size used in the title. This is accomplished here by using

function set. It should be obvious from the short discussion presented here that two MATLAB

functions get and set are of great importance in manipulating graphs.

Graphs of the ellipses in question are shown

graph3

MATLAB has several functions designed for plotting specialized 2-D graphs. A partial list of these

functions is included here fill, polar, bar, barh, pie, hist, compass, errorbar, stem, and feather.

In this example function fill is used to create a well-known object

n = -6:6;

x = sin(n*pi/6);

y = cos(n*pi/6);

fill(x, y, 'r')

axis('square')

title('Graph of the n-gone')

text(-0.45,0,'What is a name of this object?')

Function in question takes three input parameters - two arrays, named here x and y. They hold the x-and y-coordinates of vertices of the polygon to be filled. Third parameter is the user-selected color to

be used to paint the object. A new command that appears in this short code is the text command. It is



40

used to annotate a text. First two input parameters specify text location. Third input parameter is a

text, which will be added

to the plot.

Graph of the filled object

that is generated by

this code is displayed

below

4.2.2 3-D Graphics

MATLAB has several built-in functions for plotting three-dimensional objects. In this subsection wewill deal mostly with functions used to plot curves in space (plot3), mesh surfaces (mesh), surfaces

(surf ) and contour plots (contour). Also, two functions for plotting special surfaces, sphere and

cylinder will be discussed briefly. I recommend that any time you need help with the 3-D graphics

you should type help graph3d in the Command Window to learn more about various functions that

are available for plotting three-dimensional objects.

Let r(t) = < t cos(t), t sin(t), t >, -10≤t≤10, be the space curve. We plot its graph over the indicated

interval using function plot3


% Curve r(t) = < t*cos(t), t*sin(t), t >.

t = -10*pi:pi/100:10*pi;

x = t.*cos(t);y = t.*sin(t);

h = plot3(x,y,t);

set(h,'LineWidth',1.25)

title('Curve u(t) = < t*cos(t), t*sin(t), t >')



xlabel('x')



ylabel('y')

h = get(gca,'ylabel');set(h,'FontSize',12)

zlabel('z')



41

h = get(gca,'zlabel');


grid

Function plot3 is used in line 4. It takes three input parameters – arrays holding coordinates of points

on the curve to be plotted. Another new command in this code is the zlabel command (see line 4 from

the bottom). Its meaning is self-explanatory.

graph4

Function mesh is intended for plotting graphs of the 3-D mesh surfaces. Before we begin to work

with this function, another function meshgrid should be introduced. This function generates two two-

dimensional arrays for 3-D plots. Suppose that one wants to plot a mesh surface over the grid that is

defined as the Cartesian product of two sets

x = [0 1 2];

y = [10 12 14];

The meshgrid command applied to the arrays x and y creates two matrices

[xi, yi] = meshgrid(x,y)

xi =

0 1 20 1 2

0 1 2

yi =

10 10 10

12 12 12

14 14 14

Note that the matrix xi contains replicated rows of the array x while yi contains replicated columns of

y. The z-values of a function to be plotted are computed from arrays xi and yi. In this example we will plot the hyperbolic paraboloid z = y2 – x2 over the square – 1 ≤ x ≤ 1,-1 ≤ y ≤ 1

x = -1:0.05:1;



42

y = x;

[xi, yi] = meshgrid(x,y);

zi = yi.^2 – xi.^2;

mesh(xi, yi, zi)

axis off

To plot the graph of the mesh surface together with the contour plot beneath the plotted surface use

function meshc

meshc(xi, yi, zi)

axis off

Function surf is used to visualize data as a shaded surface. Computer code in the m-file graph5

should help you to learn some finer points of the 3-D graphics in MATLAB

% Script file graph5.% Surface plot of the hyperbolic paraboloid z = y^2 - x^2

% and its level curves.

x = -1:.05:1;

y = x;

[xi,yi] = meshgrid(x,y);

zi = yi.^2 - xi.^2;

surfc(xi,yi,zi)

colormap copper

shading interp

view([25,15,20])

grid off

title('Hyperbolic paraboloid z = y^2 – x^2')




43


xlabel('x')



ylabel('y')



zlabel('z')h = get(gca,'zlabel');


pause(5)

figure

contourf(zi), hold on, shading flat

[c,h] = contour(zi,'k-'); clabel(c,h)

title('The level curves of z = y^2 - x^2.')



xlabel('x')



ylabel('y')



graph5



44

There are several new commands used in this file. On line 5 (again, I do not count the blank lines andthe comment lines) a command surfc is used. It plots a surface together with the level lines beneath.

Unlike the command surfc the command surf plots a surface only without the level curves. Command

colormap is used in line 6 to paint the surface using a user-supplied colors. If the command

colormap is not added, MATLAB uses default colors. Here is a list of color maps that are available in

MATLAB

hsv - hue-saturation-value color map

hot - black-red-yellow-white color map

gray - linear gray-scale color map

bone - gray-scale with tinge of blue color map

copper - linear copper-tone color map

pink - pastel shades of pink color map

white - all white color map

flag - alternating red, white, blue, and black color map

lines - color map with the line colors

colorcube - enhanced color-cube color map

vga - windows colormap for 16 colors

jet - variant of HSV

prism - prism color map

cool - shades of cyan and magenta color map

autumn - shades of red and yellow color map

spring - shades of magenta and yellow color map

winter - shades of blue and green color map

summer - shades of green and yellow color map

Command shading (see line 7) controls the color shading used to paint the surface. Command in

question takes one argument. The following

shading flat sets the shading of the current graph to flat

shading interp sets the shading to interpolated

shading faceted sets the shading to faceted, which is the default.

are the shading options that are available in MATLAB.

Command view (see line 8) is the 3-D graph viewpoint specification. It takes a three-dimensional

vector, which sets the view angle in Cartesian coordinates.

We will now focus attention on commands on lines 23 through 25. Command figure prompts

MATLAB to create a new Figure Window in which the level lines will be plotted. In order to

enhance the graph, we use command contourf instead of contour. The former plots filled contour

lines while the latter doesn't. On the same line we use command hold on to hold the current plot and

all axis properties so that subsequent graphing commands add to the existing graph. First command on

line 25 returns matrix c and graphics handle h that are used as the input parameters for the functionclabel, which adds height labels to the current contour plot.



45

Due to the space limitation we cannot address here other issues that are of interest for programmers

dealing with the 3-D graphics in MATLAB. To learn more on this subject the interested reader is

referred to [1-3] and [5].

4.2.3 Reference

[6] D. Hanselman and B. Littlefield, Mastering MATLAB 5. A Comprehensive Tutorial

andReference, Prentice Hall, Upper Saddle River, NJ, 1998.

[7] P. Marchand, Graphics and GUIs with MATLAB, Second edition, CRC Press, Boca

Raton,1999.



[10] Using MATLAB Graphics, Version 5, The MathWorks, Inc., 1996.

[11] Matlab tutorial by Edward Neuman

4.2.4

Problems

1. Write a script file Problem 1 to plot, in the same window, graphs of two parabolas y = x2and x =

y2, where – 1 ≤ x ≤ 1. Label the axes, add a title to your graph and use command grid. To improve

readability of the graphs plotted add a legend. MATLAB has a command legend. To learn more

about this command type help legend in the Command Window and press Enter or Return key.

2. Write MATLAB function eqtri(a, b) that plots the graph of the equilateral triangle with two

vertices at (a,a) and (b,a). Third vertex lies above the line segment that connects points (a, a) and

(b, a). Use function fill to paint the triangle using a color of your choice.

3. In this exercise you are to plot graphs of the Chebyshev polynomial Tn(x) and its first order

derivative over the interval [-1, 1]. Write MATLAB function plotChT(n) that takes as the input

parameter the degree n of the Chebyshev polynomial. Use functions ChebT and derp, included in

pervious Lab, to compute coefficients of Tn(x) and T'n(x), respectively. Evaluate both, the

polynomial and its first order derivative at x = linspace(-1, 1) using MATLAB function polyval.

Add a meaningful title to your graph. In order to improve readability of your graph you may wish

to add a descriptive legend. Here is a sample output and formula of polynomial is given in

previous Lab

plotChT(5)



46

4. Use function sphere to plot the graph of a sphere of radius r with center at (a, b, c). Use

MATLAB function axis with an option 'equal'. Add a title to your graph and save your computer

code as the MATLAB function sph(r, a, b,

c).

5. Write MATLAB function ellipsoid(x0,

y0, z0, a, b, c) that takes coordinates (x0,y0, z0) of the center of the ellipsoid with

semiaxes (a, b, c) and plots its graph. Use

MATLAB functions sphere and surf . Add

a meaningful title to your graph and use

function axis('equal').

6. In this exercise you are to plot a graph of the

two-sided cone, with vertex at the origin,

and the-axis as the axis of symmetry. Write

MATLAB function cone(a, b), where the input parameters a and b stand for the radius of the

lower and upper base, respectively. Use MATLAB functions cylinder and surf to plot a cone in

question. Add a title to your graph and use function shading with an argument of your choice. A

sample output is shown below

cone(1, 2)

7. The space curve r (t) = < cos(t)sin(4t), sin(t)sin(4t), cos(4t) >, 0 ≤ t ≤ 2π, lies on the surface of

the unit sphere x2 + y2 + z2 = 1. Write MATLAB script file curvsph that plots both the curve and

the sphere in the same window. Add a meaningful title to your graph. Use MATLAB functions

colormap and shading with arguments of your choice. Add the view([150 125 50]) command.



47

Lab 5: Linear Algebra using MATLAB

One of the nice features of MATLAB is its ease of computations with vectors and matrices. In this

tutorial the following topics are discussed: vectors and matrices in MATLAB, solving systems of

linear equations, the inverse of a matrix, determinants, vectors in n-dimensional Euclidean space,

linear transformations, real vector spaces and the matrix eigenvalue problem. Applications of linear

algebra to the curve fitting, message coding and computer graphics are also included.

5.1 Special character and MATLAB functions used in LAB4For the student's convenience we include lists of special characters and MATLAB functions that are

used in this Lab.

Function Description

acos Inverse cosine

axis Control axis scaling and appearance

char Create character array

chol Cholesky factorization

cos Cosine function

cross Vector cross product

det Determinantdiag Diagonal matrices and diagonals of

a matrix

double Convert to double precision

eig Eigenvalues and eigenvectors

eye Identity matrix

fill Filled 2-D polygons

fix Round towards zero

fliplr Flip matrix in left/right direction

flops Floating point operation count

grid Grid lines

hadamard Hadamard matrix

hilb Hilbert matrix

hold Hold current graph



48

inv Matrix inverse

isempty True for empty matrix

legend Graph legend

length Length of vector

linspace Linearly spaced vector

logical Convert numerical values to logical

magic Magic square

max Largest componentmin Smallest component

norm Matrix or vector norm

null Null space

num2cell Convert numeric array into cell

array

num2str Convert number to string

ones Ones array

pascal Pascal matrix

plot Linear plot

poly Convert roots to polynomial

polyval Evaluate polynomialrand Uniformly distributed random

numbers

randn Normally distributed random

numbers

rank Matrix rank

reff Reduced row echelon form

rem Remainder after division

reshape Change size

roots Find polynomial roots

sin Sine function

size Size of matrix

sort Sort in ascending order

subs Symbolic substitution

sym Construct symbolic numbers and

variables

tic Start a stopwatch timer

title Graph title

toc Read the stopwatch timer

toeplitz Tioeplitz matrix

tril Extract lower triangular part

triu Extract upper triangular part

vander Vandermonde matrix

varargin Variable length input argument listzeros Zeros array

5.1.1 Vector and matrices in MATLAB

The purpose of this section is to demonstrate how to create and transform vectors and matrices in

MATLAB.

This command creates a row vector

a = [1 2 3]

a =

1 2 3

Column vectors are inputted in a similar way, however, semicolons must separate the components of a

vector b = [1;2;3]

b =



49

1

2

3

The quote operator ' is used to create the conjugate transpose of a vector (matrix) while the dotquote

operator .' creates the transpose vector (matrix). To illustrate this let us form a complex vector a +

i*b' and next apply these operations to the resulting vector to obtain

(a+i*b')'

ans =

1.0000 - 1.0000i

2.0000 - 2.0000i

3.0000 - 3.0000i

While

(a+i*b').'

ans =

1.0000 + 1.0000i

2.0000 + 2.0000i

3.0000 + 3.0000i

Command length returns the number of components of a vector length(a)

ans =

3

The dot operator . plays a specific role in MATLAB. It is used for the component wise application of

the operator that follows the dot operator

a.*a

ans =

1 4 9

The same result is obtained by applying the power operator ^ to the vector a

a.^2

ans =1 4 9

Componentwise division of vectors a and b can be accomplished by using the backslash operator \

together with the dot operator .

a.\b'

ans =

1 1 1

For the purpose of the next example let us change vector a to the column vector

a = a'

a =

1

23

The dot product and the outer product of vectors a and b are calculated as follows

dotprod = a'*b

dotprod =

14

outprod = a*b'

outprod =

1 2 3

2 4 6

3 6 9

The cross product of two three-dimensional vectors is calculated using command cross. Let the vector

a be the same as above and let



50

b = [-2 1 2];

Note that the semicolon after a command avoids display of the result. The cross product of a and b is

cp = cross(a,b)

cp =

1 -8 5

The cross product vector cp is perpendicular to both a and b

[cp*a cp*b']

ans =0 0

We will now deal with operations on matrices. Addition, subtraction, and scalar multiplication are

defined in the same way as for the vectors.

This creates a 3-by-3 matrix

A = [1 2 3;4 5 6;7 8 10]A =

1 2 3

4 5 6

7 8 10

Note that the semicolon operator ; separates the rows. To extract a sub matrix B consisting of rows 1

and 3 and columns 1 and 2 of the matrix A do the followingB = A([1 3], [1 2])

B =

1 2

7 8

To interchange rows 1 and 3 of A use the vector of row indices together with the colon operator

C = A([3 2 1],:) C =

7 8 10

4 5 6

1 2 3

The colon operator : stands for all columns or all rows. For the matrix A from the last example the

following command A(:)ans =

1

4

7

2

5

8

3

6

10

creates a vector version of the matrix A. We will use this operator on several occasions. To delete a

row (column) use the empty vector operator [ ] A(:, 2) = []A =

1 3

4 6

7 10

Second column of the matrix A is now deleted. To insert a row (column) we use the technique for

creating matrices and vectors

A = [A(:,1) [2 5 8]' A(:,2)]

A =

1 2 3

4 5 67 8 10

Matrix A is now restored to its original form.



51

Using MATLAB commands one can easily extract those entries of a matrix that satisfy an imposed

condition. Suppose that one wants to extract all entries of that are greater than one. First, we define a

new matrix A

A = [-1 2 3;0 5 1]A =

-1 2 3

0 5 1

Command A > 1 creates a matrix of zeros and ones

A > 1ans =

0 1 1

0 1 0

with ones on these positions where the entries of A satisfy the imposed condition and zeros

everywhere else. This illustrates logical addressing in MATLAB. To extract those entries of the

matrix A that are greater than one we execute the following command

A(A > 1)ans =

2

53

The dot operator . works for matrices too. Let now

A = [1 2 3; 3 2 1] ;

The following command A.*A ans =

1 4 9

9 4 1

computes the entry-by-entry product of A with A. However, the following command

A*A ¨??? Error using ==> *

Inner matrix dimensions must agree.generates an error message.

Function diag will be used on several occasions. This creates a diagonal matrix with the diagonal

entries stored in the vector dd = [1 2 3];D = diag(d)D =

1 0 0

0 2 0

0 0 3

To extract the main diagonal of the matrix D we use function diag again to obtain

d = diag(D)d =

1

2

3

What is the result of executing of the following command?diag(diag(d));

In some problems that arise in linear algebra one needs to calculate a linear combination of several

matrices of the same dimension. In order to obtain the desired combination both the coefficients and

the matrices must be stored in cells. In MATLAB a cell is inputted using curly braces{ }. Thisc = {1,-2,3}c =

[1] [-2] [3]is an example of the cell. Function lincomb will be used later on in this tutorial.

function M = lincomb(v,A)



52

% Linear combination M of several matrices of the same size.

% Coefficients v = {v1,v2,…,vm} of the linear combination and the

% matrices A = {A1,A2,...,Am} must be inputted as cells.

m = length(v);

[k, l] = size(A{1});

M = zeros(k, l);

for i = 1:m

M = M + v{i}*A{i};End

5.2 Solving system of linear equationsMATLAB has several tool needed for computing a solution of the system of linear equations. Let A

be an m-by-n matrix and let b be an m-dimensional (column) vector. To solve the linear system Ax =

b one can use the backslash operator \ , which is also called the left division.

1. Case m = n

In this case MATLAB calculates the exact solution (modulo the roundoff errors) to the system in

question.

Let A = [1 2 3;4 5 6;7 8 10]A =

1 2 3

4 5 6

7 8 10

and let b = ones(3,1);

Thenx = A\bx =

-1.0000

1.0000

0.0000

In order to verify correctness of the computed solution let us compute the residual vector rr = b - A*xr =

1.0e-015 *

0.1110

0.6661

0.2220

Entries of the computed residual r theoretically should all be equal to zero. This example illustrates an

effect of the roundoff errors on the computed solution.

2. Case m > n

If m > n, then the system Ax = b is overdetermined and in most cases system is inconsistent. Asolution to the system Ax = b, obtained with the aid of the backslash operator \ , is the leastsquares

solution.

Let now A = [2 --1; 1 10; 1 2];

and let the vector of the right-hand sides will be the same as the one in the last example. Thenx = A\bx =

0.5849

0.0491

The residual r of the computed solution is equal tor = b - A*xr =

-0.1208

-0.0755



53

0.3170

Theoretically the residual r is orthogonal to the column space of A. We haver'*A ans =

1.0e-014 *

0.1110

0.6994

3. Case m < n

If the number of unknowns exceeds the number of equations, then the linear system is

underdetermined . In this case MATLAB computes a particular solution provided the system is

consistent. Let now A = [1 2 3; 4 5 6]; b = ones(2,1);

Then

x = A\bx =

-0.5000

0

0.5000

A general solution to the given system is obtained by forming a linear combination of x with thecolumns of the null space of A. The latter is computed using MATLAB function null

z = null(A)z =

0.4082

-0.8165

0.4082

Suppose that one wants to compute a solution being a linear combination of x and z, with coefficients

1 and – 1. Using function lincomb we obtain

w = lincomb({1,-1},{x,z})w =

-0.9082

0.81650.0918

The residual r is calculated in a usual way

r = b - A*wr =

1.0e-015 *

-0.4441

0.1110

5.2.1 Function rref and its applications

The built-in function rref allows a user to solve several problems of linear algebra. In this section we

shall employ this function to compute a solution to the system of linear equations and also to find the

rank of a matrix. Other applications are discussed in the subsequent sections of this lab.Function rref takes a matrix and returns the reduced row echelon form of its argument. Syntax of the

rref command is

B = rref(A) or [B, pivot] = rref(A)

The second output parameter pivot holds the indices of the pivot columns.

Let A = magic(3); b = ones(3,1);

A solution x to the linear system Ax = b is obtained in two steps. First the augmented matrix of

the system is transformed to the reduced echelon form and next its last column is extracted[x, pivot] = rref([A b])x =

1.0000 0 0 0.06670 1.0000 0 0.0667

0 0 1.0000 0.0667



54

pivot =

1 2 3

x = x(:,4)x =

0.0667

0.0667

0.0667

The residual of the computed solution is b - A*xans =

0

0

0

Information stored in the output parameter pivot can be used to compute the rank of the matrix A

length(pivot)ans =

3

5.3 The inverse of a MatrixMATLAB function inv is used to compute the inverse matrix.

Let the matrix A be defined as follows A = [1 2 3;4 5 6;7 8 10]A =

1 2 3

4 5 6

7 8 10

ThenB = inv(A)B =

-0.6667 -1.3333 1.0000-0.6667 3.6667 -2.0000

1.0000 -2.0000 1.0000

In order to verify that B is the inverse matrix of A it sufficies to show that A*B = I and B*A = I,

where I is the 3-by-3 identity matrix. We have A*Bans =

1.0000 0 -0.0000

0 1.0000 0

0 0 1.0000

In a similar way one can check that B*A = I.

The Pascal matrix, named in MATLAB pascal, has several interesting properties. Let A = pascal(3)A =

1 1 1

1 2 3

1 3 6

Its inverse B

B = inv(A)B =

3.0000 -3.0000 1.0000

-3.0000 5.0000 -2.0000

1.0000 -2.0000 1.0000

is the matrix of integers. The Cholesky triangle of the matrix A is



55

S = chol(A)S =

1 1 1

0 1 2

0 0 1

Note that the upper triangular part of S holds the binomial coefficients. One can verify easily that

A = S'*S.

Function rref can also be used to compute the inverse matrix. Let A is the same as above. We create

first the augmented matrix B with A being followed by the identity matrix of the same size as A.

Running function rref on the augmented matrix and next extracting columns four through six of the

resulting matrix, we obtainB = rref([A eye(size(A))]);B = B(:, 4:6)B =

3 -3 1

-3 5 -2

1 -2 1

To verify this result, we compute first the product A *B A*Bans =

1 0 0

0 1 0

0 0 1

and next B*A

B*A ans =

1 0 0

0 1 0

0 0 1

This shows that B is indeed the inverse matrix of A.

5.3.1 Determinants

In some applications of linear algebra knowledge of the determinant of a matrix is required.

MATLAB built-in function det is designed for computing determinants.

Let A = magic(3);

Determinant of A is equal todet(A)

ans =-360

One of the classical methods for computing determinants utilizes a cofactor expansion. For more

details, see e.g., [2], pp. 103-114.

Function ckl = cofact(A, k, l) computes the cofactor ckl of the akl entry of the matrix A

function ckl = cofact(A,k,l)

% Cofactor ckl of the a_kl entry of the matrix A.

[m,n] = size(A);

if m ~= n

error('Matrix must be square')

end

B = A([1:k-1,k+1:n],[1:l-1,l+1:n]);ckl = (-1)^(k+l)*det(B);

Function d = mydet(A) implements the method of cofactor expansion for computing determinants



56

function d = mydet(A)% Determinant d of the matrix A. Function cofact must be% in MATLAB's search path.[m,n] = size(A);if m ~= nerror('Matrix must be square')end a = A(1,:);

c = [];for l=1:nc1l = cofact(A,1,l);c = [c;c1l];end d = a*c;

Let us note that function mydet uses the cofactor expansion along the row 1 of the matrix A. Method

of cofactors has a high computational complexity. Therefore it is not recommended for computations

with large matrices. Its is included here for pedagogical reasons only. To measure a computational

complexity of two functions det and mydet we will use MATLAB built-in function flops. It counts

the number of floating-point operations (additions, subtractions, multiplications and divisions). Let A = rand(25);

be a 25-by-25 matrix of uniformly distributed random numbers in the interval ( 0, 1 ). Using

function det we obtainflops(0)det(A)ans =

-0.1867

flopsans =

10100

For comparison, a number of flops used by function mydet isflops(0)

mydet(A)ans =

-0.1867

flopsans =

223350

The adjoint matrix adj(A) of the matrix A is also of interest in linear algebra (see, e.g., [2], p.108).

function B = adj(A)% Adjoint matrix B of the square matrix A.[m,n] = size(A);if m ~= nerror('Matrix must be square')end B = [];for k = 1:nfor l=1:nB = [B;cofact(A,k,l)];end end B = reshape(B,n,n);

The adjoint matrix and the inverse matrix satisfy the equation

A-1= adj(A)/det(A)

(see [2], p.110 ). Due to the high computational complexity this formula is not recommended for

computing the inverse matrix.



57

5.4 The matrix Eigen value problemMATLAB function eig is designed for computing the eigenvalues and the eigenvectors of the matrix

A. Its syntax is shown below

[V, D] = eig(A)

The eigenvalues of A are stored as the diagonal entries of the diagonal matrix D and the associated

eigenvectors are stored in columns of the matrix V.

Let A = pascal(3);

Then[V, D] = eig(A)V =

-0.5438 -0.8165 0.1938

0.7812 -0.4082 0.4722

-0.3065 0.4082 0.8599

D =

0.1270 0 0

0 1.0000 0

0 0 7.8730

Clearly, matrix A is diagonalizable. The eigenvalue-eigenvector decomposition A = VDV -1of A iscalculated as follows V*D/V

ans =

1.0000 1.0000 1.0000

1.0000 2.0000 3.0000

1.0000 3.0000 6.0000

Note the use of the right division operator / instead of using the inverse matrix function inv. This is

motivated by the fact that computation of the inverse matrix takes longer than the execution of the

right division operation.

The characteristic polynomial of a matrix is obtained by invoking the function poly.

Let A = magic(3);

be the magic square. In this example the vector chpol holds the coefficients of the characteristic

polynomial of the matrix A. Recall that a polynomial is represented in MATLAB by its

coefficients that are ordered by descending powerschpol = poly(A)chpol =

1.0 -15.0000 -24.0000 360.0000

The eigenvalues of A can be computed using function roots

eigenvals = roots(chpol)eigenvals =

15.0000

-4.8990

4.8990

This method, however, is not recommended for numerical computing the eigenvalues of a matrix.

There are several reasons for which this approach is not used in numerical linear algebra.

The Caley-Hamilton Theorem states that each matrix satisfies its characteristic equation, i.e.,

chpol(A) = 0, where the last zero stands for the matrix of zeros of the appropriate dimension. We use

function lincomb to verify this result

Q = lincomb(num2cell(chpol),

{A^3, A^2, A, eye(size(A))})Q =

1.0e-012 *

-0.5684 -0.5542 -0.4832



58

-0.5258 -0.6253 -0.4547

-0.5116 -0.4547 -0.6821

5.5 Application of linear algebraList of applications of methods of linear algebra is long and impressive. Areas that relay heavily on

the methods of linear algebra include the data fitting, mathematical statistics, linear programming,

computer graphics, cryptography, and economics, to mention the most important ones. Applications

discussed in this section include the data fitting, coding messages, and computer graphics.

5.5.1 Data fitting

In many problems that arise in science and engineering one wants to fit a discrete set of points in the

plane by a smooth curve or function. A typical choice of a smoothing function is a polynomial of a

certain degree. If the smoothing criterion requires minimization of the 2-norm, then one has to solve

the least-squares approximation problem. Function fit takes three arguments, the degree of the

approximating polynomial, and two vectors holding the x- and the y- coordinates of points to be

approximated. On the output, the coefficients of the least-squares polynomials are returned. Also, its

graph and the plot of the data points are generated.

function c = fit(n, t, y)

% The least-squares approximating polynomial of degree n (n>=0).

% Coordinates of points to be fitted are stored in the column vectors

% t and y. Coefficients of the approximating polynomial are stored in

% the vector c. Graphs of the data points and the least-squares

% approximating polynomial are also generated.

if ( n >= length(t))

error('Degree is too big')

end

v = fliplr(vander(t));v = v(:,1:(n+1));

c = v\y;

c = fliplr(c');

x = linspace(min(t),max(t));

w = polyval(c, x);

plot(t,y,'ro',x,w);

title(sprintf('The least-squares polynomial of degree n = %2.0f',n))

legend('data points','fitting polynomial')

To demonstrate functionality of this code we generate first a set of points in the plane. Our goal is to

fit ten evenly spaced points with the y-ordinates being the values of the function y = sin(2t) at these points

t = linspace(0, pi/2, 10); t = t';y = sin(2*t);

We will fit the data by a polynomial of degree at most three

c = fit(3, t, y)c =

-0.0000 -1.6156 2.5377 -0.0234



59

5.5.2 Coded messages

Some elementary tools of linear algebra can be used to code and decode messages. A typical message

can be represented as a string. The following 'coded message' is an example of the string in

MATLAB. Strings in turn can be converted to a sequence of positive integers using MATLAB's

function double. To code a transformed message multiplication by a nonsingular matrix is used.

Process of decoding messages can be viewed as the inverse process to the one described earlier. This

time multiplication by the inverse of the coding matrix is applied and next MATLAB's function char

is applied to the resulting sequence to recover the original message. Functions code and decode

implement these steps.

function B = code(s, A)

% String s is coded using a nonsingular matrix A.% A coded message is stored in the vector B.

p = length(s);

[n,n] = size(A);

b = double(s);

r = rem(p,n);

if r ~= 0

b = [b zeros(1,n-r)]';

end

b = reshape(b,n,length(b)/n);

B = A*b;

B = B(:)';

function s = dcode(B, A)% Coded message, stored in the vector B, is% decoded with the aid of the nonsingular matrix A % and is stored in the string s.[n,n]= size(A); p = length(B);B = reshape(B,n,p/n);d = A\B;s = char(d(:)');

A message to be coded is

s = 'Linear algebra is fun';As a coding matrix we use the Pascal matrix

A = pascal(4);



60

This codes the message s

B = code(s,A)B =

Columns 1 through 5

392 1020 2061 3616 340

Columns 6 through 10

809 1601 2813 410 1009

Columns 11 through 152003 3490 348 824 1647


2922 366 953 1993 3603


110 110 110 110

To decode this message we have to work with the same coding matrix A

dcode(B,A)ans =

Linear algebra is fun

5.5.3 Computer graphics

Linear algebra provides many tools that are of interest for computer pr ogrammers especially for thosewho deal with the computer graphics. Once the graphical object is created one has to transform it to

another object. Certain plane and/or space transformations are linear. Therefore they can be realized

as the matrix-vector multiplication. For instance, the reflections, translations, rotations all belong to

this class of transformations. A computer code provided below deals with the plane rotations in the

counter clockwise direction. Function rot2d takes a planar object represented by two vectors x and y

and returns its image. The angle of rotation is supplied in the degree measure.

function [xt, yt] = rot2d(t, x, y)

% Rotation of a two-dimensional object that is represented by two

% vectors x and y. The angle of rotation t is in the degree measure.

% Transformed vectors x and y are saved in xt and yt, respectively.t1 = t*pi/180;

r = [cos(t1) -sin(t1);sin(t1) cos(t1)];

x = [x x(1)];

y = [y y(1)];

hold on

grid on

axis equal

fill(x, y,'b')

z = r*[x;y];

xt = z(1,:);

yt = z(2,:);fill(xt, yt,'r');

title(sprintf('Plane rotation through the angle of %3.2f degrees',t))

hold off

Vectors x and y

x = [1 2 3 2]; y = [3 1 2 4];

are the vertices of the parallelogram. We will test function rot2d on these vectors using as the

angle of rotation t = 75.

[xt, yt] = rot2d(75, x, y)xt =

-2.6390 -0.4483 -1.1554 -3.3461 -2.6390

yt =

1.7424 2.1907 3.4154 2.9671 1.7424



61

The right object is the original parallelogram while the left one is its image.

5.5.4 References

[1] B.D. Hahn, Essential MATLAB for Scientists and Engineers, John Wiley & Sons, New York,

NY, 1997.

[2] D.R. Hill and D.E. Zitarelli, Linear Algebra Labs with MATLAB, Second edition, Prentice

Hall, Upper Saddle River, NJ, 1996.

[3] B. Kolman, Introductory Linear Algebra with Applications, Sixth edition, Prentice Hall,

Upper Saddle River, NJ, 1997.

[4] R.E. Larson and B.H. Edwards, Elementary Linear Algebra, Third edition, D.C. Heath and

Company, Lexington, MA, 1996.

[5] S.J. Leon, Linear Algebra with Applications, Fifth edition, Prentice Hall, Upper Saddle River,

NJ, 1998.

[6] G. Strang, Linear Algebra and Its Applications, Second edition, Academic Press, Orlando,

FL, 1980.

5.5.5 Problems

In Problems 1 – 12 you cannot use loops for and/or while.

1. Create a ten-dimensional row vector whose all components are equal 2. You cannot enter number 2 more than once.

2. Given a row vector a = [1 2 3 4 5]. Create a column vector b that has the same components as the

vector a but they must bestored in the reversed order.

3. MATLAB built-in function sort(a) sorts components of the vector a in the ascending order.Use

function sort to sort components of the vector a in the descending order.

4. To find the largest (smallest) entry of a vector you can use function max (min). Suppose that

these functions are not available. How would you calculate

a. the largest entry of a vector ?

b. the smallest entry of a vector?

5. Suppose that one wants to create a vector a of ones and zeros whose length is equal to 2n ( n = 1,2, … ). For instance, when n = 3, then a = [1 0 1 0 1 0]. Given value of n create a vector a with

the desired property.



62

6. Let a be a vector of integers.

a. Create a vector b whose all components are the even entries of the vector a.

b. Repeat part (a) where now b consists of all odd entries of the vector a.

c. Hint: Function logical is often used to logical tests. Another useful function you may

consider to use is rem(x, y) - the remainder after division of x by y.

7. Given two nonempty row vectors a and b and two vectors ind1and ind2 with length(a) =

length(ind1) and length(b) = length(ind2). Components of ind1 and ind2 are positive integers.Create a vector c whose components are those of vectors a and b. Their indices are determined by

vectors ind1 and ind2, respectively.

8. Using function rand, generate a vector of random integers that are uniformly distributed in the

interval (2, 10). In order to insure that the resulting vector is not empty begin with a vector that

has a sufficient number of components.

Hint: Function fix might be helpful. Type help fix in the Command Window to learn more about

this function.

9. Let A be a square matrix. Create a matrix B whose entries are the same as those of A except the

entries along the main diagonal. The main diagonal of the matrix B should consist entirely of

ones.

10. Let A be a square matrix. Create a tridiagonal matrix T whose subdiagonal, main diagonal, and

the superdiagonal are taken from the matrix A.

Hint: You may wish to use MATLAB functions triu and tril. These functions take a second

optional argument. To learn more about these functions use MATLAB's help.

11. In this exercise you are to test a square matrix A for symmetry. Write MATLAB function s =

issymm(A) that takes a matrix A and returns a number s. If A is symmetric, then s = 1, otherwise

s = 0.

12. Let A be an m-by-n and let B be an n-by-p matrices. Computing the product C = AB requires

mnp multiplications. If either A or B has a special structure, then the number of multiplications

can be reduced drastically. Let A be a full matrix of dimension m-by-n and let B be an upper

triangular matrix of dimension n-by-n whose all nonzero entries are equal to one. The product AB

can be calculated without using a single multiplication. Write an algorithm for computing the

matrix product C = A*B that does not require multiplications.

Test your code with the following matrices A = pascal(3) and B = triu(ones(3)).

13. Given square invertible matrices A and B and the column vector b. Assume that the matrices A

and B and the vector b have the same number of rows. Suppose that one wants to solve a linear

system of equations ABx = b. Without computing the matrix-matrix product A*B, find a solution

x to to this system using the backslash operator \.

14. Find all solutions to the linear system Ax = b, where the matrix A consists of rows one through

three of the 5-by-5 magic square

A = magic(5); A = A(1:3,: )A =

17 24 1 8 15

23 5 7 14 16

6 13 20 22

and b = ones(3; 1).

15. Determine whether or not the system of linear equations Ax = b, where A = ones(3, 2); b = [1; 2; 3];

possesses an exact solution x.

16. The purpose of this exercise is to demonstrate that for some matrices the computed solution to Ax

= b can be poor. Define

A = hilb(50); b = rand(50,1);



63

Find the 2-norm of the residual r = A*x – b. How would you explain a fact that the computed

norm is essentially bigger than zero?

17. In this exercise you are to use MATLAB function rref to compute the rank of the

followingmatrices:

(a) A = magic(3)

(b) A = magic(4)

(c) A = magic(5)(d) A = magic(6)

Based on the results of your computations what hypotheses would you formulate about the

rank(magic(n)), when n is odd, when n is even?

18. Use MATLAB to demonstrate that det(A + B) ≠det(A) + det(B) for matrices of your choice.

19. The inverse matrix of a symmetric nonsingular matrix is a symmetric matrix. Check this property

using function inv and a symmetric nonsingular matrix of your choice.

20. Let A be a square matrix. A matrix B is said to be the square root of A if B^2 = A. In MATLAB

the square root of a matrix can be found using the power operator ^. In this exercise you are to

use the eigenvalue-eigenvector decomposition of a matrix find the squareroot of A = [3 3;-2 -2].



64

Lab 6: Introduction to Signals

6.1 IntroductionIn this modern world we are surrounded by all kinds of signals in various forms.

Some of the signals are natural, but most of the signals are manmade. Some signalsare necessary (speech), some are pleasant (music), while many are unwanted or

unnecessary in a given situation. In an engineering context, signals are carriers of

information, both useful and unwanted. Therefore extracting or enhancing the useful

information from a mixof conflicting information is a simplest form of signal

processing. More generally, signal processing is anoperation designed for extracting,

enhancing, storing, and transmitting useful information. The distinction between

useful and unwanted information is often subjective as well asobjective. Hence

signal processing tends to be application dependent.

The signals that we encounter in practice are mostly analog signals. Thesesignals,

which vary continuously in time and amplitude, areprocessedusing electrical

networks containing active and passive circuit elements.This approach is known

asanalog signal processing (ASP)--for example,radio and television receivers.

Analog signal: q,(t) Analog signal processor ya(t) :Analog signal

They can also be processed using digital hardware containing adders,multipliers,

and logic elements or using special-purpose microprocessors.However, one needs

toconvert analog signals into a form suitable fordigital hardware. This form of the

signal is called a digital signal. It takesone of the finite numbers of valuesat specific

instances in time, and henceit can be represented by binary numbers, or bits. The

processing of digitalsignals is called DSP;in block diagram form it is represented as:

It appears from the above two approaches to signal processing, analog and digital,

that the DSP approach is the more complicated, containing more components than



65

the “simpler looking” ASP. Therefore one might ask a question: Why process signals

digitally? The answer lies in many advantages offered by DSP.

6.2 ADVANTAGESOF DSP OVERASPA major drawback of ASP is its limited scope for performing complicatedsignal

processing applications. This translates into non-flexibility in processingand

complexity in system designs. All of these generally lead toexpensive products. On

the other hand, using a DSP approach, it is possibleto convert an inexpensive

personal computer into a powerful signalprocessor. Some important advantages of

DSP are these:

1. Systems using the DSP approach can be developed using software running on a

general-purpose computer. Therefore DSP is relatively convenient to develop

and test, and the software is portable.

2. DSP operations are based solely on additions and multiplications, leading to

extremely stable processing capability-for example, stability independent of

temperature.

3. DSP operations can easily be modified in real time, often by simple programming

changes, or by reloading of registers.

4. DSP has lower cost due to VLSI technology, which reduces costs of memories,

gates, microprocessors, and so forth.

The principal disadvantage of DSP is the speed of operations, especiallyat very high

frequencies. Primarily due to the above advantages,DSP is nowbecoming a first

choice in many technologies and applications,such asconsumer electronics,

communications, wireless telephones,and medical imaging.

6.2.1 TWOIMPORTANTCATEGORIESOF DSP

Most DSP operations can be categorized as being either signal analysis tasks or

signal filtering tasks as shown below.

6.2.2 Signal analysis

This task deals with the measurement of signal properties. It is generally a frequency-domain operation. Some of its applications are

spectrum (frequency and/or phase) analysis



66

speech recognition

speaker verification

target detection

6.2.3 Signal filtering

This task is characterized by the “signal in-signal out”situation. The systems that perform

this task are generally called filters. It is usually (but not always) a time domain operation.Some of the applications are

removal of unwanted background noise

removal of interference

separation of frequency bands

shaping of the signal spectrum

6.2.4 Signals Representation in MATLAB

Signal Processing in MATLAB is all about processing a discrete-time input signal to

get discrete-time output signal that has more desirable properties. In some

applications the purpose of this processing is merely to extract certain pieces of

information contained in the input signal by applying some processing algorithms.

In other applications the prime objective may be to scrutinize the properties of a

discrete-time system by observing the output of the system against some specific

inputs. Its, therefore, important to first lean how to generate fundamental discrete-

time signals inMATLAB and apply some very basic and primitive operations on

them. In this lab we will generate and plot these signals in time domain. In the

forthcoming lab we will proceed to study their frequency domain representation. Soa discrete time signal is just a sequence of numbers which can be denoted by the

following notation.

where the up-arrow indicates the sample at n = 0.

In MATLAB we can represent a finite-duration sequence by a ROW vector of

appropriate values. However, such a vector does not have any information aboutsample position n. Therefore a correct representation of x(n) would require two

vectors, one each for x and n. For example, a sequence

x(n) = {2,1, -l,0, 1,4,3,7} can be represented in MATLAB by

>>n=[-3,-2,-1,0,1,2,3,4]; x=[2,1,-1,0,1,4,3,7];

Generally, we will use the x-vector representation alone when the sample positioninformation is not required or when such information is trivial(e.g. when the



67

sequence begins at n = 0). An arbitrary infinite-duration sequence cannot be

represented in MATLAB due to the finite memory limitations

6.3 Generation of SequencesThe purpose of this section is to familiarize you with the basic commands inMATLABfor signal generation and for plotting the generated signal. MATLAB has been

designed to operate on data stored as vectors or matrices. For our purposes, sequenceswill be stored as vectors – which are actually matrices with single row. Therefore, allsignals are limited to being causal and of finite length.

6.3.1 Unit sample/Impulse sequence

One of the simplest discrete time signals is the unit impulse or unit sample, which is defined

as

[] = {1 = 0

0 ≠ 0

}

In Matlabthe function zeros(1,N) generates a row vector of N zeros, which can be used to

implement[]over a finite interval.

>>u = [1 zeros(1, N-1)] where N is the length of sequence.

Command Explained:

The function zeros() is the built-in function in MATLAB that generates zeros. It hasvarious

versions some of which are discussed here.

zeros(n): generates an n×n square matrix of zeros.

zeros(m,n): generates an m×n matrix of zeros.

zeros(m,n,p): generates an m×n×p matrix of zeros. Three dimensional arrays can be

assumed as pages of a book.

NOTE: m, n, and p must be non-negative integers. Negative integers are treated aszeros.The

above command actually appends two vectors into one. 1st vector is ‘1’ and thesecond one is

generated by zeros() command. A unit sample sequence of length N delayed in time by M

samples where M˂ N can begenerate in MATLAB by the command

ud = [zeros(1,M) 1 zeros(1, N-M-1)]

Program 1.1 helps understand the implementation of commands discussed above.

% Program 1.1% Generation of a Unit Sample Sequence

clf; % Clear any existing figure from the current axes

n = -10:20; % Generate a vector from -10 to 20

u = [zeros(1,10) 1 zeros(1,20)]; % Generate the unit sample

sequence

stem(n,u); % Plot the unit sample sequence

xlabel(’Time index n’);

ylabel(’Amplitude’);

title(’Unit Sample Sequence’);

axis([-10 20 0 1.2]);

Q1.1 Run Program 1.1 to generate the unit sample sequence u[n] and sketch it.



68

However, thelogical relation n==0 is an elegant way of implementing []. For example,to

implement

0 n no[ ]

1 n=non no

over the n1 ≤ n0 ≤ n2interval, we will use the following MATLAB function.

function [x, n] =impseq(no,n1,n2)

n= n1:n2;

x=[(n-no)==0]

6.3.2 Unit step sequence

A unit step sequence, denoted by μ[n], and defined mathematically as

μ[n] =

1 n>=0

0 n<0

can be generated in the same fashion using ones() command that works in the sameway as

zeros() do with the exception that it fills the generated matrix with 1s insteadof 0s.

Q1.2 What are the purposes of the commands?

clf ------------------------------------------------------------------------------------------------------------axis ------------------------------------------------------------------------------------------------------------title ------------------------------------------------------------------------------------------------------------xlabel ------------------------------------------------------------------------------------------------------------ylabel ------------------------------------------------------------------------------------------------------------

Q1.3 Modify Program 1.1 to generate a delayed unit sample sequence ud[n] of length80with a delay of 21 samples. Run the modified program and display the sequence. showthe output plot along with the code to your instructor.

Q1.4 Modify Program 1.1 to generate a unit step sequence s[n]. Run themodifiedprogram and display the sequence generated. Mention the modification youdone?



69

Q1.5 Modify Program 1.1 to generate a delayed unit step sequence sd[n] with anadvanceof 7 samples. Write done your modified code here.

Q1.6 One more approach to implement this sequence is given below. Explain the difference

between these two codes.

function [x, n] =stepseq(no,n1,n2)

n= n1:n2;x=[(n-no)>=0]

6.3.3 Sinusoidal sequence

The general form of a periodic trigonometric function is

y = B + A∗ function (+)

where

A: Amplitude (plotted on y-axis)

B: Offset in amplitude

function: It may be sine, cosine or tangent etc.

θ: Angular frequency (plotted on x-axis)φ: Offset in angular frequency normally known as phase shift.

Program 1.2 generates four sine sequences with different combinations of offsets butthe

same frequency and amplitude.

% Program 1.2

% Generation of sinusoidal sequences

clf;

n = 0:60; % time index

f = 0.1; % linear frequency

A = 2; % Amplitude

dc = 0; % Offset in Amplitude

phase = 0; % Phase Shift

arg = 2*pi*f*n + phase;x = dc + A*sin(arg);

stem(n,x);



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axis([0 60 -3 3]);

title('Pure Sinusoidal Sequence');

xlabel('Time index n');

ylabel('Amplitude');

grid;

Q1.6 Modify Program 1.2 to generate four sinusoidal signal on a single screen usingsubplot()command. The plots must show sinusoidal signals with

1. dc = 0, offset = 0

2. dc = non-zero, offset = zero

3. dc = 0, offset = non-zero

4. dc = non-zero, offset = non-zero

Sketch your output plot

Q1.7 What is the purpose of the command subplot ()? What do its argumentsrepresent?

6.3.4 Exponential sequence

Another basic discrete-time sequence is the exponential sequence. Such a

sequencecan be generated using the MATLAB operators .^ and exp.

Program 1.3a generates a real-valued exponential sequence

% Program 1.3a% Generation of a real exponential sequence

clf;



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n = 0:40;

a = +1.2;

K = 0.2;

x = K*a.^+n;

stem(n,x);

xlabel('Time index n');

ylabel('Amplitude');

To generate a complex-value exponential sequence, use program 1.3b.

% Program 1.3b

% Generation of a complex exponential sequence

clf;

phase = -0.1 + 1*i;

K = 2;

n = 0:80;x = K*exp(phase*n);

subplot(2,2,2);

stem(n,real(x));

xlabel('Time index n');ylabel('Amplitude');

title('Real part');

axis ([0 60 -2.5 2.5])

subplot(2,2,4);

stem(n,imag(x));


title('Imaginary part');axis ([0 60 -2.5 2.5])

subplot(2,2,[1 3]);

stem(n,abs(x));


title('Magnitude');

axis ([0 60 0 2.5])

Q1.8 Run Program 1.3a and generate the real-valued exponential sequence and

sketch it.

Q1.9 Which parameter controls the rate of growth or decay of this sequence?

Whichparameter controls the amplitude of this sequence?

Q1.10 What will happen if the parameter a is



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1. less than 1 and greater than 0

2. less than 0 and greater than -1

3. less than -1

Modify the program and show all four graphs on a single screen. Show the output

withyour code to instructor.

Q1.11 Run Program 1.3b and generate the complex-valued exponential sequence and

sketch it.

Q1.12 What happens if the parameter c is changed to (1/12)+(pi/6)*i?

Q1.13 What are the purposes of the operators real() and imag()?

Q1.14 What is the difference between the arithmetic operators ^ and .^?

Q1.15 You can use the MATLAB command sum(s.*s) to compute the energy of a

realsequence s[n] stored as a vector s. Evaluate the energy of the real-valued

exponentialsequences x[n] generated in Questions Q1.8 and Q1.10. Give your

remarks on theenergies of all four signals.



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6.3.5 Random sequence

Many practical sequences cannot be described by mathematical expressions like those

above. These sequences are called random (or stochastic) sequences and are characterized by

parameters of the associated probability density functions or their statistical moments.

InMATLAB two types of (pseudo) random sequences are available. Therand(1,N) generates

a length N random sequence whose elements are uniformly distributed between [0,1]. The

randn(1,N) generates a lengthN Gaussian random sequence with mean 0 and variance 1.

Other random sequencescan be generated using transformations of the above functions. As

naturally occurring noise is random in nature (it may follow any statistical distribution

though) these commands are normally used to model noise in the MATLAB.

Q1.16 Write a MATLAB code to generate and display a random signal of length 100whose

elements are uniformly distributed in the interval [− 2 , 2].

Q1.17 Write a MATLAB code to generate and display a Gaussian random signal of length 75

whose elements are normally distributed with mean 12 and a variance of 3.

Q1.18 Write a MATLAB program to generate and display four sample sequences of a

random sinusoidal signal of length 31

X [n] = A · cos(ωn + φ) ,

where the amplitude A and the phase φare statistically independent random variables

withuniform probability distribution in the range 0 ≤ A ≤ 4 for the amplitude and in the range

0 ≤φ≤ 2πfor the phase and sketch it down.



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6.4 SIMPLE OPERATIONS ON SEQUENCESAs indicated earlier, the purpose of digital signal processing is to generate a signal withmore desirable properties from one or more given discrete-time signals. The processingalgorithm consists of performing a combination of basic operations such as addition,scalar multiplication, time-reversal, delaying, and product operation. We consider here acouple of very simple examples to illustrate the application of such operations.

6.4.1 Signal addition

This is a sample-by-sample addition given by{x1(n)+x2(n)}= {x1(n )}+{x2(n)}

It is implemented in MATLAB by the arithmetic operator “+”. However, the lengths of x1(n) and 22 (n) must be the same. If sequences are of unequal lengths, or if the samplepositions are different for equal-length sequences, then we cannot directly use the operator+. We have to first augment x1 (n) and 22 (n) so that they have the same position vectorn(and hence the same length). This requires careful attention to MATLAB’Sindexing operations. In particular, logical operation of intersection ‘ &’ , relational operations like “<=”

and “==”, and the find function are required to make x1 (n) and x2(n) of equal length. The

followingfunction, called the sigadd function, demonstrates these operations.function [y,n] = sigadd(xl,nl,x2,n2)

% implements y(n) = xi(n)+x2(n)% [y,n] = sigadd(x1,n1,x2,n2)%y = sum sequence over n, which includes n1 and n2 initialization% x1 = first sequence over nl% x2 - second sequence over n2 (n2 can be different from nl)n = min(min(n1) ,min(n2)) :max(max(n1) ,max(n2)) ; % duration ofy(n)y1= zeros(1,length(n));y2 = y1;y1(find((n>=min(n1))&(n<max(n1))==1))=x1; % x1 with duration of yy2(find((n>=min(n2))&(n<=max(n2))==1))=x2; % x2 with duration of yy = yl+y2; % sequence addition



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6.4.2 Signal multiplication

This is a sample-by-sample multiplication

{x1(n)x2(n)}= {x1(n )} . {x2(n)}

It is implemented in MATLAB by the array operator ‘‘.*”. Once again the similar restrictions

apply for the .* operator as for the + operator. Therefore we have developed the sigmultfunction, which is similar to the sigadd function.

function [y,n] = sigmult(xl,nl,x2,n2)% implements y(n) = xi(n)+x2(n)% [y,n] = sigmult(x1,n1,x2,n2)%y = sum sequence over n, which includes n1 and n2 initialization% x1 = first sequence over nl% x2 - second sequence over n2 (n2 can be different from nl)n = min(min(n1) ,min(n2)) :max(max(n1) ,max(n2)) ; % duration ofy(n)y1= zeros(1,length(n));

y2 = y1;y1(find((n>=min(n1))&(n<max(n1))==1))=x1; % x1 with duration of yy2(find((n>=min(n2))&(n<=max(n2))==1))=x2; % x2 with duration of yy = yl.*y2; % sequence multiplication

6.4.3 Signal smoothing

A common example of a digital signal processing application is the removal of the

noise component from a signal corrupted by additive noise. Let s[n] be the signal

corrupted bya random noise d[n] resulting in the noisy signal x[n] = s[n] + d[n]. Our

objective is to operate on x[n] to generate a signal y[n] which is a reasonable

approximation to s[n]. To this end, a simple approach is to generate an output

sample by averaging a number of input samples around the sample at instant n. For

example, a three-point moving average algorithm is given by

y[n] =1/ 3(x[n− 1] + x[n] + x[n + 1])

% Program 1.4

% Signal Smoothing by Averaging

clf;

N = 60;

d = 0.8*(rand(1,N) - 0.5); % Generate random noise of length N

m = 1:N;

s = 2*m.*(0.9.^m); % Generate uncorrupted signal

x = s + d; % Generate noise corrupted signal

subplot(2,1,1);

plot(m,d,'r-',m,s,'g--',m,x,'b-.');


legend('d[n] ','s[n] ','x[n]');

x1 = [0 0 x];x2 = [0 x 0];x3 = [x 0 0];

y = (x1 + x2 + x3)/3;

subplot(2,1,2); plot(m,y(2:N+1),'r-',m,s,'g--');



76

legend('y[n] ','s[n] ');


Q1.19Run Program 1.4 and generate all pertinent signals. show the output to your

instructor.

Q1.20 What is the form of the uncorrupted signal s[n]? What is the form of the

additive noise d[n]?

Q1.22In the second plot command, why the range of vector y is taken from 2:

N+1?Why didn’t we plot it on its whole range?

Q1.23What is the purpose of the legend command?



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Lab 7: Signals Transformation

7.1 Time Shifting Consider a continues signal x(t). A time shifted version of this signal is

Y(t) = x(t – t0)

Where ‘to’ is a constant

Note that y(t0) = x(0) – Hence if ‘t0’ is positive the shifted signal is delayed in time (shifted to

the right relative to x(t). If to is negative , y(t) is advanced in time (shifted to the left).

Matlab Implementation:

Inc=0.01;

t=-10:inc:10;

f=0.1;

t0=2;x=sin(2*pi*f*t);

y=sin(2*pi*f*(t-t0));

hold on;

Plot(t,x);

Plot(t,y,’r’);

Grid on

Q1.1 Explain the above mention code.

Now for a discrete time signal each sample of x(n) is shifted by an amountk to obtain a

shifted sequence y(n).

y(n)= x(n- k)

If we letm = n - k, then n = m + k and the above operation is given by

Y (m + k) = {x(m)}

Hence this operation has no effect on the vector x, but the vector nischanged by adding k to

each element. This is shown in the functionsigshift.

function [y,n]= sigshift(x,m,n0)

% implements y(n) = x(n-n0)

% [y,n]= sigshift(x,m,nO)

%

n = m+n0;

y = x;

7.2 Amplitude and Time Scaling In amplitude scaling each sample is multiplied by a scalar a.

a {x(n)}= {ax(n)}

An arithmetic operator "*" is used to implement the scaling operation inMATLAB



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For Time Scaling, Given a signal x(t) a time scaled version of this signal is

Y(t) = x(at) i

Where ‘a’ is a real constant

Consider a = 2 such that y(t) = x(2t) i

For any particular value of time t = t0, y(t0) = x(2t0) and x(t0) = (yt0/2). It is seen that y(t0) =

x(2t0) is a time compressed (speed up) version of x(t0).

Consider a = ½ such that y(t) = x(t/2)

For any particular value of time t = t0 y(t0) = x(t0/2) and x(t0) = y(2t0). It is observed that

y(t0) = x(t0/2) is a time (slowed down) version of x(t0)

Explanation

This coding plots the function x = sin(2πft) and also plots the function

Y(t) = x(2t);Here a = 2 so the function y(t) will be compressed in time (speed up) version of the signal

x(t). Here the frequency will be doubled.

t=-10:0.01:10;

f=0.1;a=2;

x=sin(2*pi*f*t);

y=sin(2*pi*f*(a*t));

hold on

plot(x,t);

plot(y,t,’r’);

grid on

Q1.2 Explain the above mention code by taking a=0.5 and -0.5.

7.3 Folding or time reversal:In this operation each sample of x(n) is flipped aroundn= 0 to obtain a folded sequence y(n).

y(n)= x(-n)

In MATLABthis operation is implemented by fliplr (x) function for sample values and by

-fliplr(n) function for sample positions as shown in thesigfold function.

function [y,n]= sigfold(x,n)% implements y(n) = x(-n)% [y,n]= sigfold(r,n)

%y = fliplr(x); n = -fliplr(n);



79

Q1.3 Take a sinusoidal signal of frequency 1Hz and find out its time reversal and

plot these two signals.

Q1.4 Generate and plot each of the following sequences over the indicated interval.



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EXAMPLE: Generate the complex-valued signal

() = (−.+.), -10 < n < 10

and plot its magnitude, phase, the real part, and the imaginary part in four separate

subplots.solution

MATLAB Script>>n = [-10:1:10]; alpha = -0.1+0.3j;

>>x=exp(alpha*n);

>> subplot (2,2,1) ; stem(n,real(x)) ;title( 'real part ') ; xlabel( 'n' )

>> subplot (2,2,2) ; stem(n, imag(x)) ; title('imaginary part' ) ;xlabel( 'n' )

>> subplot (2,2,3) ; stem(n, abs (x)) ; title ('magnitude part ' ) ;xlabel( 'n')

>> subplot (2,2,4) ; stem(n,(180/pi)*angle (x)); title ( 'phase part');xlabel( 'n')

7.4 Even odd synthesisA real-valued sequence x(n) is called even symmetric if

x(-n) = x(n)

Similarly, a real-valued sequence x(n) is called odd (antisymmetric) if

x(-n) = -x(n)

Then any arbitrary real-valued sequence z(n) can be decomposed into its even and odd

components

X(t) = Xe(t) + Xo (t) (2.2)

where the even and odd parts are given by

Xe(t) = ½ [x(t) + x(-t)] and Xo(t) = ½ [x(t) – x(-t0] (2.3)



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respectively. We will use this decomposition in studying properties of the Fourier transform.

Therefore it is a good exercise to develop a simpleMATLABfunction to decompose a given

sequence into its even and odd components. Using MATLAB operations discussed so far,

we can obtain the following evenodd function.function [xe, xo, m] = evenodd(x,n) % Real signal decomposition into even and odd parts xif any(imag(x) ~= 0)

error('x is not a real sequence') end

m = -fliplr(n) ; m1 = min([m,n]); m2 = max([m,n]); m = m1:m2; nm = n(1)-m(1); n1 = 1:length(n); X1 = zeros(1,length(m)); X1(n1+nm) = x; x = X1; xe = 0.5*(x + fliplr(x)); xo = 0.5*(x - fliplr(x)) The sequence and its support are supplied in x and n arrays, respectively. It first checks if

the given sequence is real and determines the support of the even and odd components in m

array. Itthen implements (2.3)with special attention to the MATLAB indexing operation.

Theresultingcomponents are stored in xe and xo arrays.

EXAMPLE Let s(n) = u(n) - u(n - 10). Decompose s(n) into even and odd components.

Solution

The sequence s(n), which is nonzero over 0<n<10, is called a rectangular pulse. We will use

MATLABto determine and plot its even and odd parts.

>>n = [0:10]; x = stepseq(0,0,10)-stepseq(10,0,l0);

>> [xe,xo,m] = evenodd(x,n);

>> figure(1);

>> subplot(2,2,1); stem(n,x); title(’Rectaugu1ar pulse’) Q1.5 Plot the original sequence with there even odd combination in the same window and

sketch it down.

Example: Amplitude Modulation

An amplitude modulated signal can be generated by modulating a high-frequency

sinusoidal signal xH [n] = cos(ωHn) with a low-frequency modulating signal

xL[n] =cos(ωLn). The resulting signal y[n] is of the form

y[n] = A(1 + m. cos(wLn)) cos (wHn)



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where m, called the modulation index , is a number chosen to ensure that (1 + m ·

xL[n])is positive for all n. Program 1.5 can be used to generate an amplitude

modulated signal.% Program 1.5

% Generation of amplitude modulated sequence

clf;

n = 0:200; m = 0.4;fH = 0.1; fL = 0.01;

xH = sin(2*pi*fH*n);

xL = sin(2*pi*fL*n);

y = (1+m*xL).*xH;

plot(n,y);grid;


Q1.6 Run the program 1.5 and provide its output.

7.4.1 Problems

1. Generate and plot the samples (use the stem function) of the following sequences

using MATLAB

2. Consider the following sequence.

Generate and plot the samples (use the stem function) of the following sequences.



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3. Decompose the sequences given in Problem 1 into their even and odd

components. Plot these components using the stem function.

4. The operation of signal dilation (or decimation or down-sampling) is defined by

in which the sequence x(n) is down-sampled by an integer factor M. For example,

if

then the down-sampled sequences by a factor 2 are given by

a. Develop a MATLAB function dnsample that has the form

function y= dnsample(x,M)

to implement the above operation. Use the indexing mechanism of MATLAB

with careful attention to the origin of the time axis n = 0.

b. Generate x(n) = sin(0.125*pi*n), -50 <n < 50. Decimate x(n) by a factor of 4 to

generate y(n). Plot both x ( n) and y(n) using subplot and comment on the results.c. Repeat the above using z(n) = sin(0.5m),50 <n < 50. Qualitatively discuss the

effectof down-sampling on signals.



84

Lab 8: Sampling, periodicity, Harmonicsand Sound Manipulation

8.1 PERIODICITY & HARMONICSA very common class of signals that we encounter is the class of periodic signals. A

periodic continuous time signal has the property that there is a positive value of ‘t’

for which

X(t) = x(t + aT) A (continuous time)

Where a = 0,1,2,3,4 ……

For all values of ‘t’. In other words a periodic signal has the property that it is

unchanged by a time shift of ‘T’. In this case we say that x(t) is periodic signal is the

sine signal.X(t) = sin (wt) Where w = 2πf; therefore

X(t) = sin (2πft)

Since sin(Q) = sin(Q + 2π) therefore the signal is periodic with the period ‘2π’.

A periodic signal in discrete time is defined similarly to continuous time periodic

signal.

Specifically a discrete time signal x[n] is periodic with period ‘N’ where ‘N’ is

positive integer, i.e. if it is unchanged by a time shift of ‘N’ i.e.

X[n] = x[n + bN] B (discrete time)

Where b = 0,1,2,3,4,……

8.1.1 FUNDAMENTAL PERIOD

The fundamental period is the smallest positive value of ‘T’(continuous) or

‘N’(discrete) for which the equations of periodicity held. Consider the example of a

sine function. In that case the signal is periodic on 2π, 4π, 6π and so on but the

fundamental time period of the signal is 2π.

8.1.2 MATLAB EXPLANATION OF PERIODIC SIGNALS

Example # 1

t=0:0.01:6;

f=0.5;

x=sin(2*pi*f*t);

hold on;

plot(t,x);

xL=[1/f 1/f];

yL=[1 1];

plot(xL,yL.’r:’)

hold off

Note that in the above example

x (2) = x(2 + 2/0.01) = x(202)



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8.1.3 HARMONICS:

Harmonics are integral multiples of a signal. Consider the signal

cos (wot). its harmonics are cos(2 wot), cos (3 wot)is the fundamental harmonic.

8.1.4 EVEN, ODD HARMONICS:

If the harmonic of a signal function has an even co – efficient it is called an even harmonic.

Otherwise if the co – efficient is odd it is called an odd harmonic.X(t) = Cos (wot)

Cos (2 wot), Cos (4 wot) , Cos (6 wot) …………………….. Even Harmonics

Cos (3 wot), Cos (5 wot) , Cos (7 wot) …………………….. Odd Harmonics

Even, odd harmonic play an important role in signal construction. Different waveforms can

be obtained by adding one type of harmonic e.g adding infinite odd harmonics of

sine/cosine where generates a square wave. Both will have a 90 phase shift from each other.

Practically it is not possible to add all the (infinite) number of harmonics to obtain a perfect

square wave but as the number of harmonics is increased from 10 onwards we obtain a

decent square wave.

Example # 2

0.01;

0 : : 4;

0.25;

2* * ;

fundamental=sin(w*t);

harmaonic2=sin(2*w*t);



hold on;

plot(t,fundamental,'ro');

plot(t,harmonic2,'k--');

plot(t,harmonic3,'m:'

inc

t inc

f

w pi f

);

plot(t,harmonic4,'y');

hold off

8.1.5 SQUARE WAVE GENERATION BY COS FUNCTIONS:

For square wave generation by ‘cos’ functions odd harmonics we use.

X(t)=4/π; (Cos (wt) – 1/3 Cos (3 wt) + 1/5 Cos (5 wt)+…….

Example#3

0:0.01:6

0.5

2* *

% function x(t)

t

f

w pi f

plot



86

8.2 SAMPLING, PLAYBACK & SOUND MANIPULATION

8.2.1 SAMPLING:

A continuous time signal x (t) has its value defined for every value of time ‘t’. on the other

hand a discrete time signal has its value at regular fired intervals. A continuous time signal

can be represented by its values or samples at points equally spaced in time.Consider a

continuous time signal x(t). This signal can be represented as a discrete time signal having

values of x(t)at ‘n’ number of intervals spaced by a time interval ‘Ts’. We can’t obtain all the

data in the signal by sampling it. Even if we sample at a very high rate some data will still be

lost but the results of course will be very good.

8.2.2 NYQUIST CRITERIA:

If the greater the number of samples the greater the accuracy, then why we don’t sample at a

very high rate. There are two reasons for it. First of all high number of samples require a lot

of manipulation on the behalf of machine software that we are using which results in move

time. Secondly most of the vital data that we need can easily be obtained from low sampling

rates.

Therefore we need some criteria to define the rate at which the signals should be sampled.

This criterion is called the Nyquist criteria. According to it, the sampling frequency must (at

least) be twice the signal frequency. When the sampling rate doesn’t satisfy the Nyquist.

When the sampling get the correct information about the signal. Reducing the number of

samples reduces information above original signal.

In MATLAB choosing a very high sampling rate simulates analog signals. Normally the

sampling frequency chooses to be 10 times higher than the maximum frequency component

in a given signal.

Example#01:F=50;

Fs=10*50;

Ts=1/Fs;

T=1:Ts:0.1;

X=sin(2*pi*f*t);

stem(T,X)

EXPLANATION:

This example generates a sinusoidal signal x(t) = sin wt = sin 2πft will f = 50 Hz. The

sampling frequency ‘fs’ is chosen to be 10 times the signal frequency. Note that in one

complete cycle it contains 10 samples.

Example#02:



87

f=1;

w=2*pi*f;

a=4;

fs1=2*f; fs2=10*f; fs3=20*f; fs4=40*f;

ts1=1/fs1; ts2=1/fs2; ts3=1/fs3; ts4=1/fs4;

t1=0:ts1:a; t2=0:ts2:a; t3=0:ts3:a; t4=0:ts4:a;x1=cos(w*t1); x2=cos(w*t2);

x13cos(w*t3); x4=cos(w*t4);

subplot(221); stem(t1,x1);




EXPLANATION:

The above example samples a cos(wt) signal of frequency 1 Hz With the sampling

frequencies 2Hz, 10Hz, 20Hz & to Hz and displays the four sampled signals in &

independent plots using the stem command. We can easily see that as the closer and signal

accuracy increases.

Lab work

The purpose of this lab is to familiarize students with the concepts of sampling, quantization

and digitization to bits. Firstly a simple sine wave with high sampling frequency is

generated assuming it to be an analog signal. Secondly this assumed analog signal issampled. This sampled signal is then quantized to certain levels. After quantization bits are

assigned to each level.

Lab tasks:

Step 1:

Generate the sine wave with

Frequency = 3Hz

Sampling Frequency = 1000 Hz

Amplitude = 1

Number of cycles = 3

Note: Here the sine signal generated is assumed to be an analog signal.

Step 2:

Sample the assumed analog signal. To do so, write the code in following steps

Generate a vector matrix having zeros; the length of this zero vector should be same

as that of assumed analog signal.

Assign the value one ‘1’ to every 40th sample in the zero vector. Now it is a pulse

train.

Perform element by element multiplication between the assumed analog signal and

pulse train, to obtain the sampled signal.



88

Note: If plot of assumed analog signal and stem of sampled signal is in the same figure the

sampled values can be seen quite evidently from the figure, as shown below.

Step 3:Quantize the sampled signals. To do so, write the code in following steps

Remove zeros from the sampled signals. (Hint: Apply ~= , == or see help find).

Assign levels to the sampled signal as shown in the figure below.

Assign bits to the levels as according to the given table.

Amplitude level (A) Assigned value Assigned bitsA < -0.5 -2 0 0

-0.5 <A< 0 -1 0 1

0 <A< 0.5 1 1 0

A> 0.5 2 1 1

The quantization done based on levels defined is explained through the figure below.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1

-0.5

0

0.5

1

1.5

Time

A m p

l i t u d e

Sampling of an Analog signal

Analog signal

Sampled values



89

8.2.3 SOUND MANIPULATION IN MATLAB

PLAYBACK An analog signal input to the sound card is sampled and digitized by sound

software, such as the “sound recorder” in windows. The recorded sound signal is saved in a

wav file. This file can be retrieved in MATLAB and played back.

Mono/Stereo Matlab Understands The Mono & Stereo Sounds As Single Column And 2

Column Vectors respectively. In STEREO different data using 2 hardware channels. STEREO

was different data for left and right speakers. Mono is represented by single column vector

and stereo by two column vectors in MATLAB.

8.2.4 MANIPULATION:

We know that MATLAB treats all the inputs and outputs as matrices/vector. Therefore for

the manipulation of an audio signal, is no different than altering the elements of a matrix.

Avoid using loops in manipulating a signal.

Example#03:

Make a folder in "c:" directory with name "soundlab" copy the file "testsound.wav" to the

"soundlab" folder.

[soundsig, fs, nbits]=wavread('c:\soundlab\testsound.wav');

left=soundsig(:,1);

right=soundig(:,2);

L=length(left);

half(left(1:2:L);

reverse=flipud(left);

wavplay(soundsig,fs);

EXPLANATION:

0 5 10 15 20 25 30 35 40

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Samples

Quantized Signal

Sampled Signal



90

The ‘wavread’ function reads the “wav” file and returns the sampled audio signal in

variable “soundsig”, “fs” is the sampling frequency & “nbits” represents the number of bits.

The “wavplay” function plays the audio signal with sampling frequency “fs”.

ECHO

Consider the following diagram input signal = “sig”.In echo generator the original signal isattenuated, delayed and then added to the original signal. This is the signal that we hear and

experience an echo effect.

Example#04:

function[sigx, echosig, reflectedsig]=echobt031018(sig,fs,td,attn);

zer=zeros(td*fs,2);

sigx=[sig; zer];refectedsig=[zer; sig*attn];

echosig=sigx+reflectedsig;

Example#05:

att=0.2;

td=1;

[signal,fs,nbits]=waveread(‘c:\soundlab\testsound.wav’);

[signal,echo,ref]=echobt031018(signal,fs,td,att);

wavplay(signal,fs);

wavplay(echo,fs);

The coding in example # 5 produces a “achosignal” giving the reflected sound anattenuation of 0.2 and time delay of 1 second. Next the “warplay(signalx, fs” plays the

original signal and “wavplayecho,fs” plays the echo version of the original signal with

sampling frequency “fs”.

Example # 6:

Replace the last line in “example # 3” “wavplay (soundsignal,fs)” with the following values.

Run the “example # 3” for each value given below.

1 wavplay(sound signal, fs);

2 wavplay (sound signal, 0.5* fs);3 wavplay (sound signal, 2 *fs);

4 wavplay (half, fs);

5 wavplay (half, 0.5*fs);

6 wavplay (half, 2*fs);

7 wavplay ( reverse, fs);

8 wavplay (reverse, o.5*fs);

9 wavplay (reverse, 2*fs);

Turn on your “speaker/headphones” and hear the result for all above values numbered 1

through 9 by running the example # 3 & changing last line to the above values separately &

hearing the output.

Execute the following code and correctly answer the following question



91

Fs = 100; % Sampling frequency

Freq = 10; % Frequency of the signal

time = 0:1/Fs:(3/Freq); % Time vector

A = 4; % Amplitude of the signal

signal = A * sin(2*pi*Freq*time); % Signal it self

stem(time,signal)

Number of samples in the signal: _____________________

Number of cycles in the plotted signal: ______________

Number of samples in one cycle: ______________________

Time period of signal: _______________________________

Frequency of the signal: _____________________________

Power of the signal is defined by the equation

Power for a discrete time signal can be computed as

N is total number of discrete values in the signal.

Write the code to determine the power of the signal

Will the power of the signal going to change if Fs (sampling frequency) of the signal is

increased in this case.

a. Yes b. No

SQUARE Square wave generation.

SQUARE(T) generates a square wave with period 2*Pi for the elements of time vector T.

SQUARE (T) is like SIN(T), only it creates a square wave with peaks of +1 to -1 instead of a

sine wave.

For example, generate a 30 Hz square wave:

t = 0:.0001:.0625;



92

y = SQUARE(2*pi*30*t);, plot(t,y)

SAWTOOTH Sawtooth and triangle wave generation.

SAWTOOTH(T) generates a sawtooth wave with period 2*pi for the elements of time vector

T. SAWTOOTH(T) is like SIN(T), only it creates a sawtooth wave with peaks of +1 to -1

instead of a sine wave.

SINC Sin(pi*x)/(pi*x) function.

SINC(X) returns a matrix whose elements are the sinc of the elements of X, i.e.

y = sin(pi*x)/(pi*x) if x ~= 0

= 1 if x == 0

where x is an element of the input matrix and y is the resultant output element.

With the execution of the code

Fs = 1000;

freq = 10;

time = -(2/freq) : 1/Fs : (2/freq);

plot(time,sinc(2*pi*freq*time))

sinc function is generated and the output looks like

Generate the square wave with

Frequency = 10Hz

Sampling Frequency = 30 Hz Amplitude = 10


Generate the sawtooth wave

Frequency = 1 Hz

Sampling Frequency = 10 Hz

Amplitude = 1


DC shift = 2

Generate a full wave rectified waveform of the sine wave

-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

Sinc function



93

Hint: use the element by element multiplication of sine and square waves having the same

frequencies.

8.2.5 EXERCISE:

E.1. Write MATLAB function for plotting the 5 harmonics of a signal x(t)=cos(wt) with a

fundamental frequency of f = 0.5 Hz. Take t = 0:0.01:2

E.2. What are the frequencies and Time periods at the all 5 harmonics in questions E.1.

E.3. Consider the general form of an equation

F(t) = 4/nπ sin(nwt) 1 ≤ n ≤ ∞

Take the case of 1 ≤ n ≤ 5, w = 2πf, f=2Hz

t = -2:0.01: 2. Plot all the 5 harmonics of the above function in same figure using the subplot

command. Subplot (511), subplot(512) --------

Subplot (513), Subplot(514), Subplot (515),

E.4. If the frequency of x(t) = sin(2πft) is f = o.2hz what will be the time period of the 4rth

harmonic of the above signal x(t). plot x(t) and its 4rth harmonic and check the time period

of 4rth harmonic in MATLAB. Do result agree with your theoretical calculation of time

period of 4rth harmonic?

E.5. Refer to general function f(t) in question 3 Nowinf

1

(4 /( 8 )) * sin( * * )n

n pi n w t

Plot the function f(t) in Matlab upto n=5 take f=1Hz, t=0:0.01:4

E.6. Implement example # 4 example # 5. Run the example # 5 for the following values of

‘att’ and ‘td’ and explain what is the difference between the following. Speakers

/Headphones should be on to hear the output.

1) att =0.5 2) att =0.5 3) att =0.3 4) att=0.1td=.5 td=1 td=1.3 td=2

E.7. Do the example # 6 and hear the output for all the values numbered 1 to 9. Give

explanation for each from 1 to 9 that what we hear? 3 what is the effect of 0.5fs and 2fs?

E.8. Consider a signal y(t) = sin(wt) where “w=2πf” Take the case when

f = 23 (Hertz). What should be the sampling frequency (fs) according to Nyquist Criteria?

E.9. If we use fs = 460 (sampling frequency) in above question then does it satisfy the

Nyquist Criteria?

For fs = 460 how many samples are there in each cycle of “sin(wt)”.



94

E.10. Consider a signal s(t) = 50 cos (20πt) + 10 cos (100πt) what is the maximum frequency

component in the above signal (Remember the relation cos wt = cos 2πft)

Fmax = Hz

What should be the sampling frequency according to the Nyquist Criteria in the above

signal “s(t)”?

E.11. Refer to the echo block diagram given at back of page # 2. we want to modify &

implement a new echo generator with its block diagram below. Modify the codes of example

# 4 and example # 5 so that it implements, the system given below

Where = attenuation

& td = Time Delay

2 td

+

2

Input Sig

Reflected Sig

Output Sig



95

Lab 9: Convolution and LTI SystemsStudents often have a difficult time understanding what convolution is. Students

canoften evaluate the convolution integral (continuous time case), convolution sum

(discrete-time case), or perform graphical convolution but may not have a good

grasp of what is happening. In other words, students can solve the formula but oftendo not understand the result or why they get that result. Most engineering texts

explain convolution by giving the convolution integral (and/or convolution sum)

and doing some mathematical and graphical examples. They often do not attempt to

explain how convolution corresponds with what is happening between the system

and the input to give the output response. In this paper, a more intuitive explanation

of convolution is given and MATLAB and SIMULINK simulations of physical

systems are used to give a more intuitive approach to understanding convolution

from a systems perspective.

9.1 IntroductionStudents are often introduced to convolution before they see the use for it. In most

Electrical Engineering curriculums, convolution is introduced in sophomore or

junior level signals and systems courses. Convolution is often performed

numerically and students have a tendency to blindly accept the results their

calculator or computer provides. Thus it is important for students to understand the

use, along with the theory of convolution, so they can better evaluate the results they

get from convolution. If convolution is explained from a systems perspective withgood examples, they will see the use, and hopefully understand the theory and be

able to determine if their results make sense. This lab proposes explaining

convolution from a systems perspective using simulations of familiar systems. It is

hoped that this will give students insight into what is happening with convolution.

9.2 SystemA system is a plant or a process that produces a response called an output in response

to an excitation called an input.

If a system’s input and output signals are scalars, the system is called a single-input

single-output (SISO) system. If a system’s input and output signals are vectors, the



96

system is called a multiple-input multiple-output (MIMO) system. Asingle-input

multiple-output (SIMO) system and a multiple-input single-output (MISO) system can

also be defined in a similar way.

For example, a spring-damper-mass system is a mechanical system whose output to

an input force is the displacement and velocity of the mass. Another example is an

electric circuit whose inputs are voltage/current sources and whose outputs arevoltages/currents/charges in the circuit. A mathematical operation or a computer

program transforming input argument(s) (together with the initial conditions) into

output argument(s) as a model of a plantor a process may also be called a system. A

system is called a continuous-time/discrete-time system if its input and output

arebothcontinuous-time/discrete-time signals.

9.2.1 DISCRETE SYSTEMS

Mathematically, a discrete time system (or discrete system for short) is describedas

an operator T[.] that takes a sequence x(n) (called excitation)and transforms it into

another sequence y(n) (called response). That is,

y[n]=T[x[n]]

we will say that the system processes an input signal into an output signal. Discrete

systems are broadly classified into linear and nonlinear systems. We will deal mostly

with linear systems.

9.2.1.1 LINEARSYSTEMS

A system is said to be linear if the superposition principle holds in the sense that it satisfies

the following properties:

- Additivity:The output of the system excited by more than one independent

input is the algebraic sum of its outputs to each of the inputs applied

individually.

- Homogeneity: The output of the system to a single independent input is

proportional to the input.

This superposition principle can be expressed as follows:



97

9.2.1.2 Linear time-invariant (LTI) system

A linear system in which an input-output pair, x(n) and y(n), is invariant to a shift n

in time is called a linear time invariant system. For an LTI system the linear system

L[.] and the shifting operators are reversible as shown below.

9.2.1.3 Stability

This is a very important concept in linear system theory. The primary reason for

considering stability is to avoid building harmful systems or to avoid burnout or

saturation in the system operation. A system is said to be bounded-input bounded-

output (BIBO) stable if every bounded input produces a bounded output.

An LTI system is BIBO stable if and only if its impulse response is

absolutelysummable.

9.2.1.4 Causality

This important concept is necessary to make sure that systemscan be built. A system

is said to be causal if the output at index nodepends only on the input up to and

including the index no; that is, theoutput does not depend on the future values of the

input. An LTI systemis causal if and only if the impulse response

h(n) =o, n < O (2.13)

Such a sequence is termed a causal sequence. In signal processing, unlessotherwise

stated, we will always assume that the system is causal.

9.3 Convolution Definition and Systems BackgroundWe begin with the definition of convolution and a brief discussion of some system

theory.

9.3.1 Definition of Convolution

Convolution is usually defined via the convolution integral for continuous timefunctions or the convolution sum for discrete-time functions. The convolution of two

continuous time functions h(t ) and x(t ) writtenh(t )* x(t ) and is given by the

convolution integral

The convolution integral exists if h(t ) and x(t ) are absolutely integrable.

The convolution of two discrete-time functions h[n] and x[n] written h[n]* x[n]can be

found using the convolution sum



98

Note that convolution is both commutative and distributive.

9.3.2

Matlab ImplementationIn general , the convolution operation is used to describe the response of an LTI system. In

DSP it is an important operation and has many other uses.

Exercise 1

Convolve the following by hand

x1 =[4 2 6 3 8 1 5];

x2=[3 8 6 9 5 7];

4 2 6 3 8 1 5

3 8 6 9 6 7

If arbitrary sequences are of infinite duration, then MATLAB cannot be used directly to

compute the convolution. MATLAB does provide a built-in function called conv that

computes the convolution between two finite-duration sequences. The conv function

assumes that the two sequences begin at n = 0 and is invoked by

>> y = conv(x, h);

Exercise 2

Log on to Matlab and enter:

x1 =[2.4 3.6 0.2 5.3 1.4 4.4 3.6];

x2=[3.1 3.8 5.2 7.7 2.5 4.6];

y=conv(x1,x2)

Write down the output



99

You will note that there are no marker arrows in this sequence, which mean that the

sequences start from x = 0.

However if the sequences do not start at x = 0 then the problem is a little more complicated.

Consider the following sequence:

x1[n] * x2[n] where:

x1[n]={2.4 3.6 0.2 5.3 1.4 4.4 3.6}

x2[n]={3.1 3.8 5.2 7.7 2.5 4.6}

Note the * in this context means ‘convolve with’

As these are different lengths and the starting points are not the first number they need to be

padded with zero’s as shown below

0 0 2.4 3.6 0.2 5.3 1.4 4.4 3.6

3.1 3.8 5.2 7.7 2.5 4.6 0 0 0

To solve this we need to pad these values out with zero’s to make them all the same length.

This can be solved by writing a MATLAB function.

However, the conv function neither provides nor accepts any timing information if the

sequences have arbitrary support. To solve this inconvenience, a simple extension of the

conv function, called conv_m , which performs the convolution of arbitrary support

sequences can now be designed as

function [y, ny] = conv_m(x, nx, h, nh)

% Modified convolution routine for signal processing

% --------------------------------------------------------------% [y, ny] = convolution result

% [x, nx] = first signal

% [h, nh] = second signal

%

nyb = nx(1) + nh(1);

nye = nx(length(x)) + nh(length(h));

ny = [nyb:nye];

y = conv(x, h);% End of the function

Exercise 3

Copy the MATLAB program from the handout and then using this function, plot the

sequences for



100

x1[n] = {0, 0, .8, .8, .8, .8, .8, .8, }

x2[n] = { 0.7,0.7,0.7,0.7,0.7,0.7,0.7,0.7,0.7,0.7}

and their convolution sum

x1=[.8 .8 .8 .8 .8 .8];n1=(2:7);x2=[.7 .7 .7 .7 .7 .7 .7 .7 .7 ];n2=(-4:4);

convstem(x1,n1,x2,n2)

9.3.3 Correlation of sequences

Correlation is an operation used in many applications in digital signal processing. It is a

measure of the degree to which two sequences are similar. Given two real-valued sequences

x(n) and y(n) of finite energy, the crosscorrelation of x(n) and y(n) is a sequence )(l r xy

defined as

( ) ( ) ( ) xy

n

r l x n y n l

The index l is called the shift or lag parameter. The special case when ( ) ( ) y n x n is called

autocorrelation and is defined by

( ) ( ) ( ) xx

n

r l x n x n l

If we compare the convolution operation with the crosscorrelation of two sequences, we

observe a close resemblance. The crosscorrelation ( ) xyr l can be put in the form

( ) ( ) ( ) xyr l x l y l

with the autocorrelation )(l r xx

in the form

( ) ( ) ( ) xx

r l x l x l

Therefore these correlations can be computed using the conv function if sequences are offinite duration.

The signal-processing toolbox MATLAB provides a function called xcorr for sequence

correlation computations. However, the function does not provide the timing (or lag)

information, which then must be obtained by some other means.

A. Convolution of Sequences

Given the following two sequences

x1(n) = [ 0, 0, 1, 1, 1, 1, 1] and x2(n) =[0, 0, 0, 1, 1, 1]



101

1. For –10 k 10, plot x1(k) and x2(k) using the stem function.

2. Plot x1(k) and x2(-k) and then with a loop control structure, write a program to determine

y(0) =

10

1 2

10

( ) ( )k

x k x k

3. Plot x1(k) and x2(1-k) and then, with a loop control structure, write a program todetermine

y(1) =

10

1 2

10

( ) (1 )k

x k x k

4. Plot x1(k) and x2(-1-k) and then with a loop control structure, write a program todetermine

y(-1) =

10

1 2

10

( ) ( 1 )k

x k x k



102

5. Develop a program to determine

1 2 1 2( ) ( )* ( ) ( ) ( ) 15 15k

y n x n x n x k x n k n

6. For –15 n 15, plot the sequence y(n) obtained in (5). What are the beginning point andend point of y(n) that are non-zero values?

7. Invoke the following MATLAB commands and plot the result (y versus ny)>> x1 = [0, 0, 1, 1, 1, 1, 1]; nx1 = [-3:3];

>> x2 = [0, 0, 0, 1, 1, 1]; nx2 = [-3:2];

>> [y, ny] = conv_m(x1, nx1, x2, nx2);

8. Compare the plot in (7) to the plot in (6). Briefly explain what you get.

B. Correlation of Sequences

1. Create a MATLAB function called xcorr_m to perform the correlation operation basedon the convolution operation.

2. Let x(n) = [3, 11, 7, 0, -1, 4, 2] be a prototype sequence and y(n) = x(n-2)+w(n) be itsnoise-corrupted and shifted version. Write the following code and save as a M-file.

% noise sequence 1

x =[3, 11, 7, 0, -1, 4, 2]; nx = [-3:3]; % given signal x(n)

[y, ny] = sigshift(x, nx, 2); % obtain x(n-2)

w = randn(1, length(y)); nw= ny; % generate w(n)

[y, ny] = sigadd(y, ny, w, nw); % obtain y(n) = x(n-2) + w(n)

[rxy, nrxy] = xcorr_m(x, nx, y, ny); % crosscorrelation

subplot(2, 1, 1); stem(nrxy, rxy);axis([-5, 10, -50, 250]); xlabel(‘lag variable l’);

ylabel(‘rxy’); title(‘Crosscorrelation: noise sequence 1’);



103

%

% noise sequence 2

x =[3, 11, 7, 0, -1, 4, 2]; nx = [-3:3]; % given signal x(n)

[y, ny] = sigshift(x, nx, 2); % obtain x(n-2)

w = randn(1, length(y)); nw = ny; % generate w(n)

[y, ny] = sigadd(y, ny, w, nw); % obtain y(n) = x(n-2) + w(n)

[rxy, nrxy] = xcorr_m(x, nx, y, ny); % crosscorrelation

subplot(2, 1, 2); stem(nrxy, rxy);

axis([-5, 10, -50, 250]); xlabel(‘lag variable l’);

ylabel(‘rxy’); title(‘Crosscorrelation: noise sequence 2’);

3. Run the script saved in (2) and Sketch the output figure.



104

Lab 10: Laplace-Transform

10.1 Laplace TransformLaplace transform of a function f(t) is defined as

Where

Laplace transform of a function f(t) can be obtained with Matlab's function Laplace.

Syntax: L =laplace(f)

The usage is demonstrated in the following examples.

a) x(t) = t, you can use the following MATLAB program.

>> syms f t

>> f=t;

>> laplace(f)

ans =1/s^2

where f and t are the symbolic variables, f the function, t the time variable.

b)

>> syms t s

>> f=-1.25+3.5*t*exp(-2*t)+1.25*exp(-2*t);

>> F=laplace(f,t,s)

>>simplify(F)

>>pretty(ans)

Alternatively, one can write the function f(t) directly as part of the laplace command:

>>F2=laplace(-1.25+3.5*t*exp(-2*t)+1.25*exp(-2*t))

C)

Matlab performs Laplace transform symbolically. Thus, you need to first define the variable

t as a "symbol".

>> syms t

Next, enter the function f(t):

>> f=5*exp(-2*t);

Finally, enter the following command:

>> L=laplace(f)

Matlab yields the following answer:



105

L =

5/(s+2)

You may want to carry out the transformation by hand (or using Laplace transform table) to

verify this result.

10.1.1 Inverse Laplace Transform with MATLAB:

The command one uses now is ilaplace. One also needs to define the symbols t and s.

Lets calculate the inverse of the previous function F(s),

1)

>> syms t s

>> F=(s-5)/(s*(s+2)^2);>> ilaplace(F)

ans =

-5/4+(7/2*t+5/4)*exp(-2*t)

>> simplify(ans)

ans =

-5/4+7/2*t*exp(-2*t)+5/4*exp(-2*t)

>> pretty(ans)

- 5/4 + 7/2 t exp(-2 t) + 5/4 exp(-2 t)

Which corresponds to f(t)

Alternatively one can write

>> ilaplace ((s-5)/(s*(s+2)^2))

2)

>> F=10*(s+2)/(s*(s^2+4*s+5));

>> ilaplace(F)

ans =

-4*exp(-2*t)*cos(t)+2*exp(-2*t)*sin(t)+4

Which gives f(t),

3)

>> syms F S

>> F=24/(s*(s+8));



106

>> ilaplace(F)

ans =

3-3*exp(-8*t)

4) Find the inverse Laplace transform of

In Matlab Command window:

>> ilaplace(1/s-2/(s+4)+1/(s+5))

Matlab result:

ans =

1-2*exp(-4*t)+exp(-5*t)

or

which is the solution of the differential equation

10.1.2 Zeros and poles

To find the zero, poles and gain of a transfer function from the vectors ‘num’ and ‘den’

which contain the co efficients of the numerator and denominator polynomials type.

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

The zeros are stored in z, 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]

10.1.3 Frequency response

To compute the frequency response it (w) of a transfer function, store the numeratror and

denominator of the transfer function in the vectors numand den define a vector w that

contains the frequencies for which H(w) is to be computed for example W = a:b:c where ‘a’ is

the lowest frequency ‘c’ is the highest frequency and ‘b’ is the increment n frequency.

The command is = freqs (num, den, w)

Returns a complex vector ‘H’ that contains the value of ‘H(w)’ for each frequency in ‘w’. Todraw a bode plot of a transfer function which has been stored in the vect ors ‘num’ and ‘den’

type bode (num, den).

If denominator >= Numerator than the polynomial is not in a proper form. To convert an

improper polynomial to a proper one we require long division. After long division and the

partial fraction expansion we can get the required inverse Laplace Transformation.

The command used to covert an improper H to proper H is

[Q,R]=deconv(B,A)

This deconvolves vector A out of vector B. The result is returned in vector Q and the

remainder in vector R such thatB= conv (A,Q) + R



107

If A and B are vectors of polynomial coefficients deconvolutions is equivalent to division.

The result of dividing B/A is Quotient Q and the remainder R.

The command

[R,P,k] =residues (B,A)

Returns the residues in column vector R, poles in column vector P and direct terms in row

vector K. K vector is empty if length (B) < length (A) I;e if H is in the proper form.

Example

= 5 4 3

In this case the num = [1 -5] den = [1 -4 3]

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

Example # 2 gives r = [-1 2], p= [3 1], k=[ ]

The general expression is given below

B(S) = K(S) + R(1) + R(2) + …. + R(n) --------------------- (A1)

A(S) S-P(1) S-P(2) S-(n)

Putting values of r, p and k in eq A1, we have

B(S) = 0 + -1 + 2 ----------------------- (B1)

A(S) S-3 S-1

Therefore Matlab makes the task of creating partial fractions much easier.

10.1.4 Lab Task:

Use “laplace” to find the Laplace transforms

Use “ilaplace” to find the Inverse Laplace transforms



108

Lab 11: Z-transform and Inverse Z-transform Analysis

11.1 Description:In mathematics and signal processing, the Z-transform converts a discrete time-domain

signal, which is a sequence of real or complex numbers, into a complex frequency-domain

representation.

The Z-transform, like many other integral transforms, can be defined as either a one-sided or

two-sided transform.

11.1.1 Bilateral Z-transform

The bilateral or two-sided Z-transform of a discrete-time signal x[n] is the function X(z) defined as

11.1.2 Unilateral Z-transform

Alternatively, in cases where x[n] is defined only for n ≥ 0, the single-sided or unilateral Z-

transform is defined as

In signal processing, this definition is used when the signal is causal. As analog filters are

designed using the Laplace transform, recursive digital filters are developed with a parallel technique

called the z-transform. The overall strategy of these two transforms is the same: probe the impulse

response with sinusoids and exponentials to find the system's poles and zeros. The Laplace transforms

deals with differential equations, the s-domain, and the s-plane. Correspondingly, the z-transform

deals with difference equations, the z-domain, and the z-plane. However, the two techniques are nota

mirror image of each other; the s-plane is arranged in a rectangular coordinate system, while the z-

plane uses a polar format. Recursive digital filters are often designed by starting with one of the

classic analog filters, such as the Butterworth, Chebyshev, or elliptic. A series of mathematical

conversions are then used to obtain the desired digital filter. The Z transform of a discrete time system

X[n] is defined as Power Series.

11.1.2.1 Rational Z-transform to factored Z-transform:

Example:

Let the given transfer function be in the rational form,

2z4+16z3+44z2+56z+32

G(z)= --------------------------------3z4+3z3-15z2+18z-12

http://en.wikipedia.org/wiki/Signal_processing

http://en.wikipedia.org/wiki/Causal_system

http://en.wikipedia.org/wiki/Causal_system

http://en.wikipedia.org/wiki/Signal_processing



109

It is required to convert it into factored form, so that we can find the poles and zeros

mathematically by applying quadratic equation.

Matlab command required for converting rational form to factored form be

‘Zp2sos’

The factored form of G(z) as evaluated by ‘zp2sos’ be,

G(z)=( 0.6667 + 0.4z-1 + 0.5333 z-2) (1.000 + 2.000 z-1 +2.000 z-2)(1.000 + 2.000z-1 -4.000z-2 )(1.000 - 1.000 z-1 + 1.000 z-2)

11.1.2.2 Factored Z-transform / zeros,poles to rational Z-transform:

It is the inverse of the above case, when the transfer function is given in fa ctored form and it

is required to convert in rational form then a single ‘matlab’ command can serve the

purpose.

Example:

Lets use the above result i-e;transfer function in factored for,

G(z)= ( 0.6667 + 0.4z-1 + 0.5333 z-2) (1.000 + 2.000 z-1 +2.000 z-2)

(1.000 + 2.000z-1 -4.000z-2 )(1.000 - 1.000 z-1 + 1.000 z-2)

For building up transfer function in rational form we find the poles and zeros of above

system simply by using Matlab ‘root’ command or by hand. Or simply we have poles and

zeros of the given system we can find the transfer function in factored form.

Matlab command that converts poles and zeros of the system in to transfer function is

‘zp2tf’.

11.1.2.3 Rational Z-transform to partial fraction form:

This technique is usually used, while taking the inverse Z-transform and

when the order ‘H(z)’ is high so that it is quite difficult to solve it mathematically.

Example:

Consider the transfer function in the rational form i-e;

18z3

G(z)= ------------------

18z3+3z2-4z-1 We can evaluate the partial fraction form of the above system using matlab command. The

partial fraction form be,

G(z)= 0.36__ + __0.24__ + _0.4____

1 – 0.5z-1 1+0.33 z-1 (1+0.33 z-1)

Matlab command that converts rational z-transform in to partial fraction form is

‘residuez’.



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11.1.2.4 Partial fraction form to Z-transform:

This technique is used when it is required to convert partial fraction expression in to rational

Z-transform.

Example:

Take the partial fraction form of above ,

The partial fraction form be,G(z)= 0.36__ + __0.24__ + _0.4____

1 – 0.5z-1 1+0.33 z-1 (1+0.33 z-1)

Matlab command that converts partial fraction form into rational z-transform is‘residuez’

Zplane:

Zero-pole plot

zplane(b,a)

This function displays the poles and zeros of discrete-time systems.

MATLAB:

syms z n

a=ztrans(1/16^n)

11.2 Inverse Z-Transform:MATLAB:

syms Z n

iztrans(3*Z/(Z+1))

11.3 FUNCTIONS RELATED TO Z-TRANSFORMFollowing are the MATLAB commands which are related to the Z transform

1- freqz used for calculating / displaying frequency response2- impz used for calculating / displaying impulse response3- z place plots the zeros and poles with unit circle4- tf2zp finds zeros, poles and gain from H = B/A5- ZP2tf Transforms from zero, poles , gain back to t = B/A6- Residuez Finds residues, poles, direct terms of partial fraction7- Poly convert roots to polynomial8- Roots computes roots of a polynomial9- conv used for multiplying 2 polynomials A & B.

11.3.1 Pole Zero Diagrams For A Function In Z Domain:

Z plane command computes and display the pole-zero diagram of Z function.The Command isZplane(b,a)

To display the pole value, use root(a)

To display the zero value, use root(b)

Matlab Code:

b=[0 1 1 ]

a= [1 -2 +3]

roots(a)

roots(b)



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zplane(b,a);

ans =

1.0000 + 1.4142i

1.0000 - 1.4142i

ans=

-1

Frequency Response:

The Freqz function computes and display the frequency response of given Z- Transformof the

function

freqz(b,a,Fs)

b= Coeff. Of Numerator

a= Coeff. Of Denominator

Fs= Sampling Frequency

Matlab Code:

b=[2 5 9 5 3]

a= [5 45 2 1 1]

freqz(b,a);



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Example:

Plot the magnitude and phase of the frequency response of the given digital filter Using freqz

function:

Y(n) = 0.2x(n) + 0.52y(n-1) – 0.68(y(n-2)

Matlab Code:

b = [0.2];

a= [1, -0.52, 0.68];

w = [0:1:500]*pi/500;

H=freqz(b,a,w);

magH = abs(H);

phaH = angle(H)*180/pi;

subplot(2,1,1);

plot(w/pi,magH);

title('Magnitude Response');

xlabel('frequency in pi units');

ylabel('│H│');subplot(2,1,2);

plot(w/pi,phaH);

title('Phase Response');

xlabel('frequency in pi units');ylabel('Degrees');

LAB TASK:



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Task#1: Express the following z-transform in factored form , plot its poles and zeros, and

then determine its ROCs.

2z4+16z3+44z2+56z+32

G(z)= --------------------------------

3z4+3z3-15z2+18z-12

Task#2:Determine the partial fraction expansion of the z-transform G(z) given by

18z3

G(z)= ------------------

18z3+3z2-4z-1

Task#2: Evaluate the z-transform of the following a) nb) n2 c) an d) n2 an

e) sin(bn)f) cos(bn)g) ansin(bn)h) ancos(bn)



114

Lab 12: Fourier series and the GibbsPhenomenon

12.1 The Fourier series of the Square WaveLet the function sq(t) be defined by

The Fourier series associated with this function is

The Fourier coefficients of this 1-periodic function are given by

Because sq(t) is odd, it is clear that bk = 0 for all k≥0. From symmetry itis clear that:

Thus, we find that the Fourier series associated with sq(t) is

We would now like to examine the extent to which this series truly represents the square

wave.

12.2 A Quick CheckBefore proceeding to analyze the series we have found, it behooves us to check that the

calculations were performed correctly. For a 1-periodic function like sq( t), Parseval's

equation states that

In our case this means that



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We can check that this sum is correct using MATLAB. One way to perform a quick check isto give MATLAB the commands

L = [1:2:10001];(8/pi^2) * sum(1./L.^2)

MATLAB responds withans =

1.0000

which is an indication that we may have performed all of the calculationscorrectly.A secondway to have MATLAB check this computation is to give MATLAB the commandssyms k

(sym('8')/sym('pi')^2) * symsum(1/(2*k+1)^2,k,0,Inf)

MATLAB responds to these commands withans =

1

indicating that the series does, indeed, sum to 1.

Figure 5.1: Summing the Fourier series. The “ringing" associated with Gibb'sphenomenon is

clearly visible.

12.3 “Seeing" the Sum

It is not difficult to have MATLAB calculate the Fourier series and displayits values.

Conisder the following code:

t = [-500:500] * 0.001;

total = zeros(size(t));

for k = 1 : 2 : 101

total = total + (4/pi) * sin(2*pi*k*t) / k;

end



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plot(t,total)

This code defines a “time" vector, t, with 1001 elements. It then defines avector to hold the

sum, total, and proceeds to sum the first 51 terms inthe Fourier series. The code causes

MATLAB to produce the plot shown inFigure 5.1. The “ringing" produced by Gibb's

phenomenon is clearly visible.

12.4 The ExperimentLetsaw(t) = tfor -1/2 < t< 1/2 and continue saw(t) periodically outside of this region.

1. Calculate the Fourier coefficients associated with saw(t).

2. Check Parseval's equation for saw(t) both numerically and symbolically.

3. Sum the first 3 terms in the Fourier series and plot the Fourier seriesas a function

of time.


of time.


of time.When summing the Fourier series, make sure that the time samples you take are sufficiently

closely spaced that you clearly see Gibb's phenomenon.

Example # 1 (MATLAB IMPLEMENTATION)

“Squarewavefourierseries.m”

function [t, fourierseries]= squarwavefourierseries(har,tmax);

fo= 1/(2*pi);

wo= 2*pi*fo;

t=-tmax : 0.05 : tmax;

n=1: har;

ao=1/2;an=(2./(n*pi)).* sin(n*pi/2);

bn= zeros (1, length(n));

y= zeros (har,length(t));

for n=1:har

y(n,:) = an(n) * cos( n* wo*t) + bn (n) * sin(n*wo*t);

end

fourierseries = ao + sum (y,1)

Save this function with the some name i:e “squarewavefourierseries”. This function takes the

harmonics in variable “har”. These are the number of harmonics to be computed in the

fourier series expansion. The variable “tmax” contains the maximum time limit of the signal.

The function returned “fourierseries” variable which contains fourier series expansion and

‘t’ contains the time.

Example # 2:

Close All;

tmax= 2*Pi;

[tx1,fx1] =squarewavefourierseries(2000, tmax);

Figure;

Plot(tx1,fx1); grid;[tx2,fx2] =squarewavefourierseries(50, tmax);

Figure;



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Plot(tx2,fx2); grid;


Figure;



Figure;



Figure;



Figure;


This coding calls the function “squarewavefourierseries” with different values for “har”

variable and plots the fourier series representation for each. Note that as “har” values

increases the waveform becomes more closer to the ideal one.

Example # 3:

function[t, fourierseries]= sawtoothfourierseries(har,tmax);

a=5;

fo=1/(2*pi);

wo= 2*pi*fo;

t= -tmax : 0.05 : tmax;

n=1: har;

ao=0;

an= zeros(1,length(n));

bn=(2*a)./(n.*n.*pi.*pi).*(sin(n*pi)-(n*pi).*cos(n*pi));

y=zeros(har,length(t));

for n=1:hary(n,:)= an(n) * cos(n* wo*t)+ bn(n) * sin( n*wo*t);

end

fourierseries= ao+ sum(y,1);

Save this function with the some name i:e “sawtoothfourierseries”. This function takes the

harmonics in variable “har” and the max time limit in variable tmax. These are the number

of harmonics to be computed in the Fourier series expansion. The variable “tmax” contains

the maximum time limit of the signal. The function returned “fourierseries” variable which

contains Fourier series representation for the sawtooth waveform. The ‘t’ variable is the axis

for that signal. As ‘n’ approaches to infinity, the waveform approaches to an ideal wave

form of sawtooth.

Example # 4: Close All;

tmax= 2*Pi;

[tx1,fx1] = sawtoothfourierseries (2000, tmax);

Figure;



Figure;


[tx3,fx3] = sawtoothfourierseries (20, tmax);Figure;



118



Figure;



Figure;



Figure;


This coding calls the function “squarewavefourierseries” with different values for “har”

variable and plots the Fourier series representation for each. Note that as “har” values

increases the waveform becomes closer to the ideal one.



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Lab 13: Discrete-Time Signals in theFrequency Domain

13.1 IntroductionIn the previous two exercises you dealt with the time-domain representation of discrete-time

signals and systems, and investigated their properties. Further insight into the properties of

such signals and systems is obtained by their representation in the frequency-domain. To

this end three commonly used representations are the discrete-time Fourier transform

(DTFT), the discrete Fourier transform (DFT), and the z -transform. In this exercise you will

study all three representations of a discrete-time sequence.

13.2 Background Review

R3.1 The discrete-time Fourier transform (DTFT) X (e jω) of a sequence x[n] is defined by

The quantity |X(e jω)| is called the magnitude function and the quantity θ (ω) is called the phase

function , with both functions again being real functions of ω. In many applications, the

Fourier transform is called the Fourier spectrum and, likewise, |X(e jω)| andθ (ω) are referred to

as the magnitude spectrum and phase spectrum, respectively.

R3.2 The DTFT X (e jω) is a periodic continuous function in ω with a period 2π .

R3.3 For a real sequence x[n], the real part X re(e jω) of its DTFT and the magnitude Function

|X(e jω)|are even functions of ω, whereas the imaginary part X im(e jω) and thephase function θ (ω) are

odd functions of ω.

R3.4 The inverse discrete-time Fourier transform x[n] of X(e jω) is given by

R3.5 The Fourier transform X (e jω) of a sequence x[n] exists if x[n] is absolutely summable, that is,



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R3.6 The DTFT satisfies a number of useful properties that are often uitilized in a number of

applications. A detailed listing of these properties and their analytical proofs can be found in

any text on digital signal processing. These properties can also be verified using MATLAB.We list below a few selected properties that will be encountered later in this exercise.

Time-Shi fti ng Property – If G(e jω) denotes the DTFT of a sequence g [n], then the DTFT of the

time-shifted sequence g [n − no] is given by e− jωnoG(e jω).

Frequency-Shi fti ng Property – If G(e jω) denotes the DTFT of a sequence g [n], then the DTFT of

the sequence e jωon g [n] is given by G(e j(ω− ωo)).

Convolu tion Property – If G(e jω) and H (e jω) denote the DTFTs of the sequences g [n] and h[n],

respectively, then the DTFT of the sequence g [n] h[n] is given by G(e jω) H (e jω).

Modulation Property – If G(e jω) and H (e jω) denote the DTFTs of the sequences g [n] and h[n],

respectively, then the DTFT of the sequence g [n]h[n] is given by

Time-Reversal Property – If G(e jω) denotes the DTFT of a sequence g [n], then the DTFT of the

time-reversed sequence g [− n] is given by G(e− jω).

R3.7 The N -point discrete Fourier transform (DFT) of a finite-length sequence x[n],defined for

0 ≤ n ≤ N − 1, is given by

R3.8 The N -point DFT X [k ] of a length- N sequence x[n] , n = 0 , 1 , . . . , N − 1, is simply the

frequency samples of its DTFT X (e jω) evaluated at N uniformly spaced frequency points,

ω = ωk = 2πk/N, k = 0 , 1 , . . . , N − 1, that is,

R3.9 The N -point circular convolution of two length- N sequences g [n] and h[n], 0 ≤ n ≤ N − 1, is

defined by

where _n _ N = n modulo N . The N -point circular convolution operation is usually denoted as



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R3.10 The linear convolution of a length- N sequence g [n], 0 ≤ n ≤ N − 1, with alength- M

sequence h[n], 0 ≤ n ≤ M − 1, can be obtained by a ( N + M − 1)-point circular convolution of two

length-( N + M − 1) sequences, g e[n] and he[n],

where g e[n] and he[n] are obtained by appending g [n] and h[n] with zero-valued samples:

R3.11 The DFT satisfies a number of useful properties that are often utilized in a number of

applications. A detailed listing of these properties and their analytical proofs can be found in

any text on digital signal processing. These properties can also be verified using MATLAB.

We list below a few selected properties that will be encountered later in this exercise.

Circular Time-Shifting Property– IfG[k] denotes theN -point DFT of a length-N sequence g[n],

then the N -point DFT of the circularly time-shifted sequence g[<n – no>N ] is given by W knoN

G[k] where W N = e− j2 π/N .

Circular Frequency-Shifting Property– If G[k] denotes the N -point DFT of a length-N

sequence g[n], then the N -point DFT of the sequence W − konN g[n] is given byG[<k− ko>N ].

Circular Convolution Property– If G[k] and H [k] denote the N -point DFTs of the length-N

sequences g[n] and h[n], respectively, then the N -point DFT of the circularly convolved

sequence g[n] with h[n] is given by G[k]H [k].

Parseval’s Relation – If G[k ] denotes the N -point DFT of a length- N sequence g [n], then

13.3 Discrete-Time Fourier Transform

The discrete-time Fourier transform (DTFT) X (e jω) of a sequence x[n] is a continuous function

of ω. Since the data in MATLAB is in vector form, X (e jω) can only be evaluated at a

prescribed set of discrete frequencies. Moreover, only a class of the DTFT that is expressedas a rational function in e− jω in the form

can be evaluated. In the following two projects you will learn how to evaluate and plot the

DTFT and study certain properties of the DTFT using MATLAB.

Project 3.1 DTFT Computation

The DTFT X (e jω) of a sequence x[n] of the form of Eq. (3.31) can be computed easily at a

prescribed set of L discrete frequency points ω = ωl

using the MATLAB function freqz. SinceX (e jω) is a continuous function of ω, it is necessary to make L as large as possible so that the

plot generated using the command plot provides a reasonable replica of the actual plot of



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the DTFT. In MATLAB, freqz computes the L-point DFT of the sequences {p0 p1 . . . pM} and

{d0 d1 . . . dM}, and then forms their ratio to arrive atX (e jωl) , l= 1 , 2 , . . . , L. For faster

computation, L should be chosen as a power of 2, such as 256 or 512.

Program P3 1 can be used to evaluate and plot the DTFT of the form of Eq. (3.31).

% Program P3_1

% Evaluation of the DTFTclf;

% Compute the frequency samples of the DTFT

w = -4*pi:8*pi/511:4*pi;

num = [2 1];den = [1 -0.6];

h = freqz(num, den, w);

% Plot the DTFT

subplot(2,1,1)

plot(w/pi,real(h));grid title(’Real part of H(e^{j\omega})’) xlabel(’\omega /\ pi’);

ylabel(’Amplitude’); subplot(2,1,2) plot(w/pi,imag(h));grid title(’Imaginary part of H(e^{j\omega})’) xlabel(’\omega /\ pi’); ylabel(’Amplitude’); pausesubplot(2,1,1) plot(w/pi,abs(h));grid title(’Magnitude Spectrum |H(e^{j\omega})|’) xlabel(’\omega /\ pi’); ylabel(’Amplitude’);

subplot(2,1,2) plot(w/pi,angle(h));grid title(’Phase Spectrum arg[H(e^{j\omega})]’) xlabel(’\omega /\ pi’); ylabel(’Phase, radians’); Questions:

Q3.1 What is the expression of the DTFT being evaluated in Program P3_1? What is the

function of the MATLAB command pause?

Q3.2 Run Program P3 1 and compute the real and imaginary parts of the DTFT, and the

magnitude and phase spectra . Is the DTFT a periodic function of ω? If it is, what is the

period? Explain the type of symmetries exhibited by the four plots.



123

Q3.3 Modify Program P3 1 to evaluate in the range 0 ≤ ω ≤ π the following DTFT:

and repeat Question Q3.2. Comment on your results. Can you explain the jump in the phase

spectrum ? The jump can be removed using the MATLAB command unwrap. Evaluate thephase spectrum with the jump removed.

Q3.4 Modify Program P3 1 to evaluate the DTFT of the following finite-length sequence: g[n]=[1 3 5 7 9 11 13 15 17] ,

and repeat Question Q3.2. Comment on your results. Can you explain the jumps in thephase spectrum?

Q3.5 How would you modify Program P3 1 to plot the phase in degrees?

Project 3.2 DTFT Properties

Most of the properties of the DTFT can be verified using MATLAB. In this project you shall

verify the properties listed in R3.6. Since all data in MATLAB have to be finite-length

vectors, the sequences being used to verify the properties are thus restricted to be of finite

length.

Program P3 2 can be used to verify the time-shifting property of the DTFT.

% Program P3_2

% Time-Shifting Properties of DTFT

clf;

w = -pi:2*pi/255:pi; wo = 0.4*pi; D = 10;

num = [1 2 3 4 5 6 7 8 9];

h1 = freqz(num, 1, w);

h2 = freqz([zeros(1,D) num], 1, w);



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subplot(2,2,1)

plot(w/pi,abs(h1));grid

title(’Magnitude Spectrum of Original Sequence’)

subplot(2,2,2)


title(’Magnitude Spectrum of Time-Shifted Sequence’)

subplot(2,2,3)

plot(w/pi,angle(h1));grid

title(’Phase Spectrum of Original Sequence’)

subplot(2,2,4)


title(’Phase Spectrum of Time-Shifted Sequence’)

Questions:

Q3.6Modify Program P3_2 by adding appropriate comment statements and program

statements for labeling the two axes of each plot being generated by the program. Which

parameter controls the amount of time-shift?

Q3.7Run the modified program and comment on your results.

Q3.8Repeat Question Q3.7 for a different value of the time-shift.

Q3.9Repeat Question Q3.7 for two different sequences of varying lengths and two differenttime-shifts.

Program P3 3 can be used to verify the frequency-shifting property of the DTFT.

% Program P3_3

% Frequency-Shifting Properties of DTFT

clf; w = -pi:2*pi/255:pi; wo = 0.4*pi;

num1 = [1 3 5 7 9 11 13 15 17];



125

L = length(num1);

h1 = freqz(num1, 1, w);

n = 0:L-1;

num2 = exp(wo*i*n).*num1;

h2 = freqz(num2, 1, w);

subplot(2,2,1)


title(’Magnitude Spectrum of Original Sequence’)

subplot(2,2,2)


title(’Magnitude Spectrum of Frequency-Shifted Sequence’)

subplot(2,2,3)



subplot(2,2,4)


title(’Phase Spectrum of Frequency-Shifted Sequence’)

Questions:

Q3.10 Modify Program P3_3 by adding appropriate comment statements and program

statements for labeling the two axes of each plot being generated by the program. Which

parameter controls the amount of frequency-shift?

Q3.11 Run the modified program and comment on your results.

Q3.12 Repeat Question Q3.11 for a different value of the frequency-shift.

Q3.13 Repeat Question Q3.11 for two different sequences of varying lengths and two

different frequency-shifts.

Program P3 4 can be used to verify the convolution property of the DTFT.

% Program P3_4

% Convolution Property of DTFT

clf;



126

w = -pi:2*pi/255:pi;

x1 = [1 3 5 7 9 11 13 15 17];

x2 = [1 -2 3 -2 1];

y = conv(x1,x2);

h1 = freqz(x1, 1, w);

h2 = freqz(x2, 1, w);

hp = h1.*h2;

h3 = freqz(y,1,w);

subplot(2,2,1)

plot(w/pi,abs(hp));grid

title(’Product of Magnitude Spectra’)

subplot(2,2,2)


title(’Magnitude Spectrum of Convolved Sequence’)

subplot(2,2,3)

plot(w/pi,angle(hp));grid

title(’Sum of Phase Spectra’)

subplot(2,2,4) plot(w/pi,angle(h3));grid

title(’Phase Spectrum of Convolved Sequence’)

Questions:

Q3.14Modify Program P3 4 by adding appropriate comment statements and program

statements for labeling the two axes of each plot being generated by the program.

Q3.15Run the modified program and comment on your results.

Q3.16Repeat Question Q3.15 for two different sets of sequences of varying lengths.

Program P3 5 can be used to verify the modulation property of the DTFT.

% Program P3_5

% Modulation Property of DTFT

clf; w = -pi:2*pi/255:pi;

x1 = [1 3 5 7 9 11 13 15 17];



127

x2 = [1 -1 1 -1 1 -1 1 -1 1];

y = x1.*x2;

h1 = freqz(x1, 1, w);

h2 = freqz(x2, 1, w);

h3 = freqz(y,1,w);

subplot(3,1,1)


title(’Magnitude Spectrum of First Sequence’)

subplot(3,1,2)


title(’Magnitude Spectrum of Second Sequence’)

subplot(3,1,3)


title(’Magnitude Spectrum of Product Sequence’)

Questions:

Q3.17 Modify Program P3 5 by adding appropriate comment statements and program

statements for labeling the two axes of each plot being generated by the program.


Q3.19 Repeat Question Q3.18 for two different sets of sequences of varying lengths.

Program P3 6 can be used to verify the time-reversal property of the DTFT.

% Program P3_6

% Time-Reversal Property of DTFTclf;

w = -pi:2*pi/255:pi;

num = [1 2 3 4];

L = length(num)-1;

h1 = freqz(num, 1, w);

h2 = freqz(fliplr(num), 1, w);

h3 = exp(w*L*i).*h2;

subplot(2,2,1)


title(’Magnitude Spectrum of Original Sequence’) subplot(2,2,2)




128

title(’Magnitude Spectrum of Time-Reversed Sequence’)

subplot(2,2,3)



subplot(2,2,4)


title(’Phase Spectrum of Time-Reversed Sequence’)

Questions:


statements for labeling the two axes of each plot being generated by the program. Explain

how the program implements the time-reversal operation.


Q3.22Repeat Question Q3.21 for two different sequences of varying lengths.



129

Lab 14: Discrete Fourier Transform

The discrete Fourier transform (DFT) X [k] of a finite-length sequence x[n] can be easily

computed in MATLAB using the function fft. There are two versions of this function.fft(x)

computes the DFT X [k] of the sequence x[n] where the length of X [k] is the same as that of

x[n]. fft(x,L) computes the L-point DFT of a sequence x[n] of lengthN whereL ≥ N . IfL > N ,

x[n] is zero-padded with L− N trailing zero-valued samples before the DFT is computed. The

inverse discrete Fourier transform (IDFT) x[n] of a DFT sequence X [k] can likewise be

computed using the function ifft, which also has two versions.

Project 3.3 DFT and IDFT Computations

Questions:

Q3.23 Write a MATLAB program to compute and plot the L-point DFT X [k] of a sequence

x[n] of length N with L ≥ N and then to compute and plot the L-point IDFT of X [k]. Run the

program for sequences of different lengths N and for different values of the DFT length L.Comment on your results.

Q3.24 Write a MATLAB program to compute the N -point DFT of two length-N realsequences using a single N -point DFT and compare the result by computing directly the two

N -point DFTs.

Q3.25 Write a MATLAB program to compute the 2N -point DFT of a length-2N real sequence

using a single N -point DFT and compare the result by computing directly the2N -point DFT.

Project 3.4 DFT Properties

Two important concepts used in the application of the DFT are the circular-shift of a

sequence and the circular convolution of two sequences of the same length. As these



130

operations are needed in verifying certain properties of the DFT, we implement them as

MATLAB functions circshift1 and circonv as indicated below:

function y = circshift1(x,M)

% Develops a sequence y obtained by

% circularly shifting a finite-length

% sequence x by M samples

if abs(M) > length(x) M = rem(M,length(x));

end

if M < 0

M = M + length(x);

End

y = [x(M+1:length(x)) x(1:M)];}

function y = circonv(x1,x2)

L1 = length(x1); L2 = length(x2);

if L1 ~= L2, error(’Sequences of unequal lengths’), end

y = zeros(1,L1);

x2tr = [x2(1) x2(L2:-1:2)];

for k = 1:L1

sh = circshift1(x2tr,1-k);

h = x1.*sh;

y(k) = sum(h);

end

Questions:

Q3.26 What is the purpose of the command rem in the function circshift1?

Q3.27 Explain how the function circshift1 implements the circular time-shifting operation.

Q3.28 What is the purpose of the operator ~= in the function circonv?

Q3.29 Explain how the operation of the function circonv implements the circular

convolution operation.



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Program P3 7 can be used to illustrate the concept of circular shift of a finite-length

sequence. It employs the function circshift1.

% Program P3_7

% Illustration of Circular Shift of a Sequence

clf;

M = 6;

a = [0 1 2 3 4 5 6 7 8 9]; b = circshift1(a,M);

L = length(a)-1;

n = 0:L;

subplot(2,1,1);

stem(n,a);axis([0,L,min(a),max(a)]);

title(’Original Sequence’);

subplot(2,1,2);

stem(n,b);axis([0,L,min(a),max(a)]);

title([’Sequence Obtained by Circularly Shifting by

’,num2str(M),’Samples’]);

Questions:


statements for labeling each plot being generated by the program. Which parameter

determines the amount of time-shifting? What happens if the amount of time-shift is greater

than the sequence length?

Q3.31 Run the modified program and verify the circular time-shifting operation.

Program P3 8 can be used to illustrate the circular time-shifting property of the DFT. It

employs the function circshift1.

% Program P3_8

% Circular Time-Shifting Property of DFT

clf;x = [0 2 4 6 8 10 12 14 16];

N = length(x)-1; n = 0:N;

y = circshift1(x,5);

XF = fft(x);

YF = fft(y);

subplot(2,2,1)

stem(n,abs(XF)); grid

title(’Magnitude of DFT of Original Sequence’);

subplot(2,2,2)

stem(n,abs(YF)); grid title(’Magnitude of DFT of Circularly Shifted Sequence’);

subplot(2,2,3)



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stem(n,angle(XF)); grid

title(’Phase of DFT of Original Sequence’);

subplot(2,2,4)

stem(n,angle(YF)); grid

title(’Phase of DFT of Circularly Shifted Sequence’);

Questions:

Q3.32 Modify Program P3 8 by adding appropriate comment statements and programstatements for labeling each plot being generated by the program. What is the the amount of

time-shift?

Q3.33 Run the modified program and verify the circular time-shifting property of the DFT.

Q3.34 Repeat Question Q3.33 for two different amounts of time-shift.

Q3.35 Repeat Question Q3.33 for two different sequences of different lengths.

Program P3 9 can be used to illustrate the circular convolution property of the DFT. It

employs the function circonv.

% Program P3_9

% Circular Convolution Property of DFTg1 = [1 2 3 4 5 6]; g2 = [1 -2 3 3 -2 1];

ycir = circonv(g1,g2);

disp(’Result of circular convolution = ’);disp(ycir)

G1 = fft(g1); G2 = fft(g2);

yc = real(ifft(G1.*G2));

disp(’Result of IDFT of the DFT products = ’);disp(yc)

Questions:

Q3.36 Run Program P3 9 and verify the circular convolution property of the DFT.



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Q3.37 Repeat Question Q3.36 for two other different sets of equal-length sequences.

Program P3 10 can be used to illustrate the relation between circular and linear convolutions

(see R3.10).

% Program P3_10

% Linear Convolution via Circular Convolution

g1 = [1 2 3 4 5];g2 = [2 2 0 1 1];

g1e = [g1 zeros(1,length(g2)-1)];

g2e = [g2 zeros(1,length(g1)-1)];

ylin = circonv(g1e, g2e);

disp(’Linear convolution via circular convolution = ’);disp(ylin);

y = conv(g1, g2);disp(’Direct linear convolution = ’);disp(y)

Questions:

Q3.38 Run Program P3 10 and verify that linear convolution can be obtained via circular

convolution.

Q3.39 Repeat Question Q3.38 for two other different sets of sequences of unequal lengths.

Q3.40 Write a MATLAB program to develop the linear convolution of two sequences via the

DFT of each. Using this program verify the results of Questions Q3.38 and Q3.39.

Program P3 11 can be used to verify the relation between the DFT of a real sequence, and the

DFTs of its periodic even and the periodic odd parts (see R3.12).

% Program P3_11

% Relations between the DFTs of the Periodic Even

% and Odd Parts of a Real Sequence

x = [1 2 4 2 6 32 6 4 2 zeros(1,247)];

x1 = [x(1) x(256:-1:2)];xe = 0.5 *(x + x1);

XF = fft(x);



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XEF = fft(xe);

clf;

k = 0:255;

subplot(2,2,1);

plot(k/128,real(XF)); grid

ylabel(’Amplitude’);

title(’Re(DFT\{x[n]\})’);

subplot(2,2,2);

plot(k/128,imag(XF)); grid ylabel(’Amplitude’);

title(’Im(DFT\{x[n]\})’);

subplot(2,2,3);

plot(k/128,real(XEF)); grid

xlabel(’Time index n’); ylabel(’Amplitude’);

title(’Re(DFT\{x_{e}[n]\} )’);

subplot(2,2,4);

plot(k/128,imag(XEF)); grid

xlabel(’Time index n’);ylabel(’Amplitude’);

title(’Im(DFT\{x_{e}[n]\})’); Questions:

Q3.41 What is the relation between the sequences x1[n] and x[n]?

Q3.42 Run Program P3 11. The imaginary part of XEF should be zero as the DFT of the

periodic even part is simply the real part of XEF of the original sequence. Can you verify

that? How can you explain the simulation result?

Q3.43 Modify the program to verify the relation between the DFT of the periodic odd part

and the imaginary part of XEF.



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Parseval’s relation (Eq. (3.15)) can be verified using the following program.

% Program P3_12

% Parseval’s Relation

x = [(1:128) (128:-1:1)];

XF = fft(x);a = sum(x.*x)

b = round(sum(abs(XF).^2)/256})

Questions:

Q3.44 Run Program P3 12. Do you get the same values for a and b?

Q3.45 Modify the program in such a way that you do not have to use the command abs(XF).

Use the MATLAB command conj(x) to compute the complex conjugate of x.



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Sine Wave1

Sine WaveScope

Product

2

Constant

Message Signal

Carrier Signal

Amplitude Modulation (AM) SIgnal

Lab 15: Introduction to SIMULINK

15.1 AMPLITUDE MODULATION

MATLAB Implementation (Coding) Of Amplitude ModulationAmplitude Modulationts=1e-3;a=2;t=-2:ts:2;fm=1;fc=15; wm=2*pi*fm; wc=2*pi*fc; message=sin(wm*t);carrier=cos(wc*t);am=(a+message).*carrier;

subplot(311); plot(t,message);xlabel('Message Signal');subplot(312); plot(t,carrier);xlabel('Carrier Signal');subplot(313); plot(t,am);xlabel('Amplitude Modulated (AM) Signal');

15.1.1 SIMULINK IMPLEMENTATION OF AMPLITUDE MODULATION

Example # 1



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AM = (A + message) * Carrier A

The above is the block diagram for AM (amplitude modulation) in SIMULINK. We have toimplement equation AMessage Signal: fm=1 Hz, wm=2fm, phase = 0Carrier Signal: fc=1 Hz, wm=2fc, phase = 0

Sample Time: 1e-3 for both message and carrier signalScope: Number of axis = 3Simulation Parameters: Start time = -2, Stop time = 2Sine Type: Time based, Amplitude = 1;

15.2 TRIGNOMETRIC FOURIER SERIES (TFS)Example # 1:



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In previous Labs we implemented above square wave using Fourier series expansion. Inthis lab we will do it in SIMULINK but we will add up to 9 harmonics.The general equation of above square wave was found in Lab 4 to be

1 2 1 1 1 1( ) 3 5 7 9 ...cos cos cos cos cos

2 3 5 7 9O O O O Ow t w t w t w t w t w t

Wo = 1

We have to implement the above equation up till 9Wot and see the output in simulink.So we require 5 sine wave blocks and 1 constant block of value 1/2. “Cosine” isimplemented by giving “sine” a phase shift of

15.2.1 SIMULINK IMPLEMENTATION

Simulink parameters

Start time = -15Stop time = +15

Phase = pi/2 for all sine blocks

Sample time 1e^-3 for all sine blocks

Frequency (rad/sec) 1,3,5,7,9

Amplitude 2/pi, -2/3pi, 2/5pi, -2/7pi, 2/9pi

signals system lab manual

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