MATLAB Primer

MATLAB contains a large collection of built-in functions and commands that are used in an interactive mode, when you are in the command window.. MATLAB allows us to construct new functio

MATLAB Primer

9.1 INTRODUCTION

MATLAB is a very powerful and well-known software package1 that is used

in science and engineering disciplines, for numerical computation, data analysis, and graphical visualization It is available in almost all platforms such as personal computers, and workstations running under several operating systems When you begin a session by typing the command matlab, the first window displayed on the monitor is the command window with the prompt >>, which is waiting for your command.2 Use the commandexit to end the session.

MATLAB contains a large collection of built-in functions and commands that are used in an interactive mode, when you are in the command window As soon

as the name of a function or a command is typed at the prompt in the command window, with the proper syntax, the answer is displayed immediately But there are two other windows, the edit window and graphics window, which will be dis- cussed later The software package is designed to use additional sets of functions that are more applicable in particular disciplines such as control systems, digital signal processing, communications engineering, and image processing There are more than 20 sets known as “toolboxes” (e.g., control toolbox, digital signal pro- cessing toolbox, communication toolbox, image processing toolbox) All of them run under MATLAB and implement the functions on the basis of matrix manipu-

lation of numerical data, and that is why the software is called MATLAB (mat rix laboratory) Simulink is another toolbox that is used to simulate the performance

of the systems, when the systems are built by connecting individual blocks resenting different subsystems and the output is obtained when the systems are

rep-1The software is available from The MathWorks, Inc., 3 Apple Hill Drive, Natick, MA01760-2098, phone 508-647-7000, fax 508-647-7001, emailinfo@mathworks.com, Websitehttp://www.mathworks.com

2If you are logging on a workstation connected to a computer network, you may have to set theproper environment by typingsetenv DISPLAY network number:or some other com-mand before launching MATLAB

Introduction to Digital Signal Processing and Filter Design, by B A Shenoi

Copyright© 2006 John Wiley & Sons, Inc

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subjected to different kinds of input signals MATLAB allows us to construct new functions using the enormous number of built-in functions, commands, and operations in MATLAB and in the many toolboxes, without having to know how

to compile, link, load, and create executable code, because MATLAB uses its own language to carry out all these steps, which are invisible to the user It carries out the steps and gives the answer very fast! In most versions of MATLAB, there is

a “symbol” toolbox, which performs does symbolic operations such as tiation, integration, matrix inversion, and solution of differential equations when the operations are expressed in their symbolic form In more recent versions of this software, new toolboxes such as Filter Design Toolbox and DSP Blockset, which are based on the object-oriented programming features of MATLAB, have been added These have been treated in some chapters of this book.

differen-9.1.1 Vectors, Arrays, and Matrices

Vectors and scalars are special cases of a matrix—all of which are represented

as arrays in MATLAB A scalar is an array of 1 × 1 dimension, whereas a row vector is an array of 1 × n dimension, and a column vector is an array of n × 1

dimension When elements of an array are typed in a row within square brackets, with a space between the elements, MATLAB displays it as a row vector For example, when you type

as the workspace The workspace saves the variables, data, contents of arrays

and matrices, and other elements as well as a record of the commands typed

by the user It is recommended that at the beginning of the session, you change

the directory to the disk drive a:if you have one in your computer so that the contents of the workspace are saved in the floppy disk in that drive Instead of using the semicolumn between the elements, you can type the element on the next line or by leaving one space, type three dots at the end of the line and continue on the next line as shown below; this is useful when the array is very long and extends beyond the end of the line:

When a scalar, vector, or matrix is multiplied (or divided) by a scalar c, every

element of the scalar, vector or matrix is multiplied (or divided) by c When

c is added to matrix, it is added to every element of the vector or matrix For

example,x = 5;F=x*Cgives the output

operation in application of matrix algebra (The command for the right divisionx=b/M gives the solution to the equation x*M=b, assuming that x and M are

compatible for multiplication and the solution in matrix algebra is given byx=bM−1.)

When we use the same variables used above but define them with new values, they become the current values for the variables, so we define and use them below as examples of the operations described above:

of the two compatible matrices are the three array operations.

Instead of multiplying the two matrices asD=A*B, now we type a dot before the sign for multiplication For example, the answer to>>D=A.*Bis

Now we computeU=X.^2andV=2.^Xand get the following outputs:

A matrix can be expanded by adding new matrices and column or row vectors

as illustrated by the following examples:

rot90(A) Rotates the matrix array by 90◦

fliplr(A) Flips the columns left to right

flipud(A) Flips the rows up to down

triu(A) Gives the upper triangular part of the matrix

tril(A) Gives the lower triangular part of the matrix

There are a few special matrices; we will list only three that are often found useful in manipulating matrices:

ones(m,n), which gives a matrix with the number one in all its m rows and

n columns

zeros(m,n), which gives a matrix with zeros in all its m rows and n columns

eye(m), which gives the “identity matrix” of order m × m.

We note that the inverse of a matrixAis obtained from the functioninv(A), the determinant of a matrixAis obtained from the functiondet(A)and the rank fromrank(A).

Since this is only a primer on MATLAB, it does not contain all the information

on its functions You should refer to the user’s guide that accompanies every software program mentioned above or any other books on MATLAB [1–3] When you have logged on to MATLAB or any of the subdirectories for the toolboxes, there is an online help readily available You type help functionname, wherefunctionname is the name of the function on which detailed information is desired, and immediately that information is displayed on the command window.

So there is no need to memorize the syntax and various features of the function and so on The best way to learn the use of MATLAB and the toolboxes is to try the functions on the computer, using the helputility if necessary.

9.1.3 Scalar Operations

If t = (0.1π) radians per second, the MATLAB functionsin(t)gives the answer

as 0.3090 To compute and plot v = sin(0.1π t), in the time interval [0 2 π ]

we have to choose discrete values for the continuous variable t and computev

at these values To do so, we create an arrayt=[0.0:0.1:2.0]; this gives the sequence of 21 values within t = 0.0–2.0 in increments of 0.1 Now if we type

the functionv=sin(pi*t), the result is a sequence of 21 values The commandstem(v)immediately plots these values forvin a “graphics window” as shown

in Figure 9.1.

We have used the commandfigureto create a new window and the commandplot(t,v)to get the plot as a continuous plot joining the discrete values ofv(see Fig 9.2) If we did not use the command figure, the second figure would replace the first one on the graphics window Typing the commandgridon the command window results in the plots having grid lines shown in these figures Next we use the commands for adding a title and the labels for the y and x

coordinates with the commands title, ylabel, andxlabel.

So the commands entered on the command window to get the two figures are as follows Since MATLAB chooses the scales for the x and y coordinates, depend-

ing on the range of their values, we may have to change the literal arguments in theylabelandxlabel:

Figure 9.1 Plot of sin(pi*nT)

1 Plot of sin (pi

9.1.4 Drawing Plots

Additional arguments can be added in the command plot(t,v)to specify the color of the curve; for example, plot(t,v,’g’) will show the curve in green color The arguments for other colors are as follows:

- solid line: dotted line dashed line-.- dash–dot–dash One can plot several curves in the same figure; for example, we can plot both

v and y versus t by the command plot(t,v,’g’,’-’t,y,’r’,’*’) Another way of plotting more than one variable in the same figure is to use the commandhold onafter plotting the first variable and then typing the command for plotting the second variable:

9.1.5 MATLAB Functions

The other functions, in addition tosin, that are available in MATLAB are given below:

TRIGONOMETRIC FUNCTIONS

atanh hyperbolic arctangent

MATHEMATICAL FUNCTIONS

abs absolute value or magnitudeangle phase angle of a complex numbersqrt square root

real real part of a complex numberimag imaginary part

conj complex conjugateround round toward nearest integerfix round toward zero

ceil round toward ∞sign signum function

num-If we type the MATLAB command format long, this number will be played with 16 digits as 12.09456783215532 If we declare theformat short

dis-e, the number will be displayed with five digits and an exponent in the form 1.2094 e +01, whereas the command format long e selects 16 digits and an exponent: 1.209456783215532e +01.

Remember that these formats are used for display on the monitor, the result

of commands, functions, and operations But the numerical computations that implement the functions and scripts are done by MATLAB with a higher degree

of precision if and when it is necessary, for example, when we use the functions and scripts in the Signal Processing Toolbox.

9.1.7 Control Flow

Three functions are used to control the flow of command execution that depend

on decisionmaking statements Such functions are found in other programming languages and also in MATLAB They arefor loops,If-elseif-end loopsand while loops, which will be illustrated below.

The statement

>> for n=1:10

x(n )=n^2+4*n

end

produces a vector containing 10 values of x(n) = n2+ 4n, for n = 1, 2, 3, , 10.

One can define an array such as n=3:-0.5:-1.0, in which the increment

is −0.5 and the result is an arrayn = [3.0 2.5 2.0 1.5 1.0 0.5 0.0-0.5 -1.0] The default value for the increment is 1.

To define a two-dimensional array, and a functionH(i,j)=0.1^i+0.2^j, we use the statements

If the values of x(n) when n takes the maximum value of 7 is desired, we

type xafter theendstatement and get the result

0 .5000 0 .2500 0 .1250 0 .0625 0 .0313 0 .0156 0 .0078

But this problem is solved more easily by the following two statements to get the same values for x(n), n = 1, 2, 3, , 7, but we have to insert a dot before

the exponent (^) since n is a row vector of seven elements It is very helpful to

find the order of a matrix or a vector A by using the statementS= size(A)to know when to use the dot for the term-by-term operation—particularly when we get an error message about the dimensions of the matrices:

>>n=1:7;

x(n)=(0.5).^n

9.1.8 Edit Window and M-file

So far we have introduced a few of the common functions and operations of MATLAB that are used in the command window and the graphics window When we are in the command window, we are in an interactive mode, where each command is executed immediately after it is typed and the answer is displayed If

we wish to make a change in any one of the previous statements or the input data,

we have to trace it back one line at a time using the↑key and edit the line, then use the ↓key to get to the line where we had stopped If that statement is very many lines before the current line or if we want to make major changes in the program, or if we wish to find the output of the program for different values for the input parameters, this is not a convenient procedure So we create a program

or a script by clicking File-Open-New-M-file and use a text editor that is built-in MATLAB or any other text editor and save it in the current directory

as a file with a name and an m extension We can write this script using the MATLAB functions and operations; we can even use other functions or functions that we have written Such a file is called an M-file, and after it is saved, we click the command window and type just the name of the M-file without the extension The entire script is executed if there are no bugs in it and the results displayed If there are any error messages, we go back to M-file in the edit window and make corrections, save it, get back to the command window, and then type the name

of the M-file to run it again Either we enter the values of the input variable(s)

in the M-file or add the following command in the M-file:

input(’Type in the input parameters for xyz’)

When the script is to be executed, the program displays the statement Type inthe input parameters for xyzand waits for the input from the keyboard.

We may have requests for input for several parameters, and when the data for all the parameters are entered by us from the keyboard, the program is executed.

This is helpful when we wish to find the response (output) of the program with different values for the input parameters.

Similarly, when we add the statement

disp(’Values of the parameter xx’)

disp(xx)

the program displays the values for the parameter after the script has been executed, which may not be otherwise displayed as the output from the pro- gram.

When any statement is preceded by the %character, the statement is not cuted by the program; it is used only as a comment for information or explanation

exe-of what the program does It is a good practice to add a few lines with this %character at the beginning of any script that we write and include the name of the file also.

Example 9.1

We click File, Open, Newfrom the menu bar when we are in the command window and then choose M-file An edit window appears next Now we give an example of a M-file that we write using the built-in text editor:

>>clear %clears all the variables in the Workspace

%This program we call Ration.m computes the value of a

>>x=0.0:0.01:1.0;

>>num=2*x+0.5;

>>den=x.^2+0.1*x+0.05; %Note the dot in the first term on

Plot of the function f(x)

9.2 SIGNAL PROCESSING TOOLBOX

The Signal Processing Toolbox is a collection of about 160 functions that are extensively used for the analysis, design, and realization of discrete-time sys- tems and tasks or operations such as modeling, detection, filtering, prediction, and spectral analysis in digital signal processing They run under MATLAB, which has about 330 functions and operations By typing help function in the command window, wherefunctionis the name of these functions, detailed information about them is displayed By typing help signal, we get a com- plete list of all the functions in the Signal Processing Toolbox, when this has been installed as a subdirectory of the MATLAB directory If we know the name

of the function that does the numerical processing but not the syntax and other details, we can typehelp function But when we have to carry out numerical processing but don’t know the name of the MATLAB function, we may have to

go through the list of all MATLAB functions and choose the appropriate one for the purpose The list of all MATLAB functions in the Signal Processing Toolbox

is given in Section 9.2.1, and students are encouraged to use thehelputility and become familiar with as many of the functions as possible That should improve their efficiency in calling up the appropriate function immediately when the need arises while they write and edit the script Note that we can use any of the thou- sands of functions found in all other toolboxes and in the simulation software

called Simulink that runs under MATLAB, which makes this software extremely

powerful and versatile.

9.2.1 List of Functions in Signal Processing Toolbox

angle - Phase angle

filternorm - Compute the 2-norm or inf-norm of a digitalfilter

freqs - Laplace transform frequency response

freqspace - Frequency spacing for frequency response.freqz - Z-transform frequency response

fvtool - Filter Visualization Tool

grpdelay - Group delay

impz - Discrete impulse response

phasez - Digital filter phase response

phasedelay - Phase delay of a digital filter

unwrap - Unwrap phase

zerophase - Zero-phase response of a real filter

zplane - Discrete pole-zero plot

fftfilt - Overlap-add filter implementation

filter - Filter implementation

filter2 - Two-dimensional digital filtering

filtfilt - Zero-phase version of filter

filtic - Determine filter initial conditions

latcfilt - Lattice filter implementation

medfilt1 - 1-Dimensional median filtering

sgolayfilt - Savitzky-Golay filter implementation.sosfilt - Second-order sections (biquad) filter

implementation

upfirdn - Up sample, FIR filter, down sample

Discrete-time filter object

dfilt - Construct a discrete-time, filter object

(Type ’’doc dfilt’’ for more information)