Department of Electronics Engineering

February 21, 2009

Introduction to

What is MATLAB History of MATLAB Why MATLAB MATLAB Applications How to start MATLAB M- File MATLAB toolboxs Keys to remember

MATrix LABoratory Calculator Interactive Programming language Scientific computation and visualization

tool Symbolic computing tool

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Developed by Cleve Moler in Stanford in 70’s

To facilitate matrix and vector computations

Easier to program than Fortran Spread by word of mouth Mathworks, Inc established in 1984

http://www.mathworks.com/

A good choice for vision program development because: Easy to do very rapid prototyping Quick to learn, and good documentation A good library for various data processing

functions Excellent display capabilities Widely used for teaching and research in

universities and industry

Easy to Learn Vector formulation instead of scaler Programmable calculator Interpreter, Not Compiler - Easy to

debug Powerful graphics Add on toolboxes

It allows quick and easy coding in a very high-level language

Data structures require minimal attention; E.g., arrays need NOT be declared

before first use; An interactive interface allows rapid

experimentation and easy debugging

High-quality graphics and visualization facilities are available;

M-files are completed portable across a wide range of platforms;

Toolboxes can be added to extend the system, giving specialized signal processing facilities and a symbolic manipulation capability.

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NASA Ames designs computer-aided joystick for spacecraft docking mechanism with MATLAB

GenCorp's Aerojet Division solves rocket fuel pressure control problem using MATLAB and Simulink.

Applied Biosystems develops a crucial DNA sequencing algorithm in MATLAB

Given Imaging developed M2A, a minimally invasive intestinal diagnostic tool, by MATLAB

B0

M(t)

y

x2D FFT-1

B1

RF

z

transmit/receive

Mag = (X2 + Y2)½

Pha = atan(Y/X)

X

Y

kx

kx

ky

ky R

I

K = R + i*I;C = ifft2(K); Z = fftshift(C);X = real(Z); Y = imag(Z);A = abs(Z); P = atan(Y ./ X);

Complex- valued k-space

MR Image formation Fourier imaging ...

MATLAB:

Matrix calculation Drawing graphs (2-D, 3-D) Symbolic/Statistic Computation Reading data from various sources

i.e. Excel file, .txt file Provide Graphical User Interface

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Define Matrix

Matrix Operation >> determinant=det(A)

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Polar plot t=0:.01:2*pi; polar(t,abs(sin(2*t).*cos(2*t))); Line plot

x=0:0.05:5; y=sin(x.^2); plot(x,y);

Stem plot: x = 0:0.1:4; y = sin(x.^2).*exp(-x);

stem(x,y)

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Mesh plot:

z=peaks(25); mesh(z);

Surface plot: z=peaks(25); surf(z);

Contour plot:

z=peaks(25); contour(z,16); Spherical harmonic: spharm2

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Differentiation:>> g=sin(x)*cos(y);

>> diff(g) % differentiate w.r.t. x

>> diff(g,y,3) % 3rd order partial derivative w.r.t. y

Integration:

>> g=(y^2-1)/(x^5+1); int(g,y) % integrate w.r.t. y

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Scilab – MATLAB emulator http://www.scilab.org/ develop jointly by Chinese and French governments

Octave – MATLAB emulator http://www.octave.org/developed by Univ. of Wisconsin

Mathematica – symbolic computation http://www.wolfram.com/

Maple – symbolic computation incorporated in MATLAB http://www.maplesoft.com/

How to start MATLAB ?

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To start MATLAB: Programs => MATLAB => MATLAB 7.4

Command Window:

where you input

MATLAB commands

Directory Window:

show the files in the directory

History Window:

record what you have

done

Matlab Window

Convenient to close all window except Command window

>> command prompt

Working in the command Window 1 Cursor must next to >> Once a command is typed, Enter key is pressed,

for Execution Several command in same line, Type comma

between the command, Enter key >> a=2,b=3 Use Up-Arrow key –command can be recall Too long command can be continued in the next

line by typing three periods …and press Enter Command continue total 4096 character

Working in the command Window 2

Semicolon(;): Typed at the end of command ,the output of

the command is not displayed(>> a=2,b=3;) Typing(%): before line-Comment Clc : clear the command window Execute the calculation according to

precedence (1.parentheses 2. Exponential, 3.Multiplication, Division 4.Addition & subtraction )

>> 27^(1/3)+32^0.2

+ addition− subtraction* multiplication^ power′ conjugate transpose\ left division[5\3=3/5]/ right divisionx = A\b the solution of A x = bx = b/A the solution of x A = b

Display format User can control Display format :

(Example:290/7)

Format short: Fixed point with 4 decimal Digit

Format long: Fixed point with 14 decimal Digit

Format short e: Scientific notation with 4 decimal Digit

Format long e: Scientific notation with 15 decimal Digit

Format short g: Best of 5 digits fixed or floting point

Format long g: Best of 15 digits fixed or floting point

Format bank: Two decimal digit>> format short>> 290/7

41.4286 41.428571428571431 4.1429e+00

4.142857142857143e+001

41.429

41.4285714285714

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Display Formats

>> format compact >> a=5 a = 5 >> format loose >> a=5

a =

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Help window Content, Index, Search, Demos

Help command (i.e. >>Help fft) Provide information about a specific function Show input and output argument

Doc command Provide information about specific function Includes example using MATLAB function

Lookfor command : Search for keyword

Illustrates the easy and powerful aspects of MATLAB

Consists of two parts Core MATLAB Toolboxes

Core MATLAB – desktop, matrices, graphics Toolboxes – financial, symbolic, Signal

Processong, Neural Network .....

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To use Programs => MATLAB => MATLAB 7.4 Inside MATLAB window, look for

MATLAB prompt “>>” Type “demo” and hit “Enter” key On left panel, click “+” on MATLAB Click “+” on any one of the folders Click on any one of the examples

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Elementary Math Built in Function

Large library of built in function Function is used, argument can be a

number, variable that has assigned a numerical value

>> squrt(81), exp(x), abs(-24), log(x), log10(x), factorial(x), sin(pi), cos(x), tan(x), cot(x), asin(x), sinh(x)

Rounding Function

Function round(x)

DescriptionRound to the nearest integer

Round(17/5)=3

Fix(x) Round towards zero Fix(13/5)=2

Ceil(x) Round towards infinity Ceil(11/5)=3

Floor(x)

Round towards -infinity Floor(-9/4)=-3

Rem(x,y)

Returns remainder Rem(13,5)=3

Sign(x) Returns 1 if x>0,-1if x<0, 0 if x=0

Sign(5)=1

Scalar Variable

Variable name= numeric value or expression Variable is name of memory location If the variable is assigned new value the

content of the memory location is replaced. Several assignment can be in the same line,

assignment must be separated with comma Once variable is assigned ,can be used

argument in function.

Example

>> x=15 x = 15 >> x=3*x-12 x = 33

>> a=12,b=4;c=(a-b)+40

a = 12 c = 48

Rules about Variable name

Variable names: Can be upto 63 character long Contain letters,digits and underscore

character Must begin with letter Matlab is case sensitive,different aa,aA,AA Avoid using the name of built in function

for a variable(avoid cos,sin,sqrt)

Predefined variable

ans Variable has the value of last expression ,not assigned specific variable

Pi Number Π

Eps The smallest difference bet two numbers

Inf Use for infinity

I Defined as

J Same as I

NaN

Not aNumber,0/0

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Useful Commands for managing Varible

Command Outcome

clear Removes all variable from memory

Clear x,y,z Removes only x,y,z from memory

who Display a list of the variables currently in the memmory

whos Displays a list of variablecurrently in memmory,bytes,class

Numbers Integers (exact): 0, 1, 2, … Real numbers (approximate): 1., 2.1,

3e-8 Complex numbers: i = j = √ -1 Special values: pi Special “non-numbers”: Inf, NaN

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Arithmetic Operators Unary: +, - Binary: +, -, *, /, \, ^ Strings: ‘Department of Electronics’

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States: true (1, non-zero), false (0) Logical Operators

Unary: ~ Binary (relational): ==, ~=, >, >=, <,

<=, &, | “Short-circuit”: &&, ||

Only evaluates second operand if required Useful for conditional evaluation subject to e.g. <test> && <operation> only evaluates <operation> if the <test> is true.

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Does not require any type declaration Automatically creates the variable and

allocate the appropriate amount of storage

Variable assignment operator : ‘=‘ A=1; a =2*pi;

At end of expression “ ; ” suppresses output.

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+ addition− subtraction* multiplication^power′ conjugate transpose\ left division/ right divisionx = A\b the solution of A x = bx = b/A the solution of x A = b

>> -5/(3.45+2.75)^2ans =-0.1301

>> (2+6i)*(2-6i)ans =40

>> sin(pi/4)ans =0.7071

>> exp(acos(0.3))ans=3.547

>> b= a* cos(.76);

A = [1 2 0; 2 5 -1; 4 10 -1];semicolon starts new row

B = [3,4,5] ;create row vector: “,” to ;separate elements

C = [A;B] ;insert new row in matrixD = [A,B’] ;insert new column in matrixE = eye(5) ;create 5 x 5 identity matrixF = ones(2,3) ;create 2 x 3 matrix of ones

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B = A’ ;transpose AC = A + 3*BD = A*BD = A.*B ;element by element

multiplicationE = A^2 ;multiply A by itselfE = A.^2 ;raise each element of A to

2ndpowerX = inv(A) ;form inverse of matrix

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Matlab Command Output

[1,2,3].*[5,6,7] [5,12,21][10:-2:0]. ^ 2 [100,64,36,16,4,0]2. ^ [1:6] [2,4,8,16,32,64][2,4,6]./ 2 [1,2,3][2,4,6] . \ 2 [1.000 0.5000

0.3333]

The Matlab command help elfun gives a list of elementary math functions

Trigonometric: sin, sinh, asin, asinh, cos,cosh, acos, acosh, tan, tanh, atan,atan2, atanh, sec, sech, asech, csc,csch, acsc, acsch, cot, coth, acot, acoth

Expotential: exp, log, log10, log2, pow2,sqrt, nextpow2

Complex: abs, angle, complex, conj, imag,real, unwrap, isreal, cplxpair

Rounding and remainder: fix, floor, ceil,round, rem, sign

Entering MATLAB Command is not practical.

Solution : M-File

Script FunctionCollections of commands use define functionExecuted in sequence when called Has input -output

Saved with extension “.m”

File – New M file

x = linspace(0,2*pi,200);

Y1= sin(x);

Y2 = cos(x);

Plot((x,Y1,x,Y2)

Save as filename.m

To execute the command,

Debug Run

Accept input argumet and generate output

dependent_variables = function_name(independent_variables)

FunctionInput Output

y = x^2-1;

function y=react_C(c,f) %react_C calculates the reactance of a capacitor. %The inputs are: capacitor value and frequency in hz

%The output is 1/(wC) and angular frequency in rad/s

y(1)=2*pi*f; w=y(1); y(2)=1/(w*c); File must be saved to a known path with filename the same as the function name and with an extension ‘.m’ (i.e react_c.m)

Call function by its name and arguments

Communications Toolbox Control System Toolbox Database Toolbox Model-Based Calibration Toolbox Neural Network Toolbox Optimization Toolbox Partial Differential Equation (PDE) Toolbox Signal Processing Toolbox Statistics Toolbox Symbolic Math Toolbox System Identification Toolbox Wavelet Toolbox

; (Semicolon) Suppress the output of results to the screen

Up-arrow scroll through previously typed commands

Esc   Clear command line

Ctrl+C Quit current operation and return control to the command line

Clear all Clear all variables

who This shows all active variables. Try the command whos

save filename This creates a file filename.mat which contains allVariables

save filename x y z

This creates a file filename.mat which containsvariables x y z

Thank You

det(A) determinant

rank(A)

inv(A) inverse

trace(A)

eig(A) eigenvalues

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Use round brackets and (row, column)

notation

Let a = [1 2 3; 4 5 6; 7 8 9]

Accessing: a(2, 1) 4

Assigning : a(2, 1) = 0

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Ranges of numbers: 1:10 [1 2 3 4 5 6 7 8 9 10] 0:0.3:2 [0 0.3 0.6 0.9 1.2 1.5 1.8]

Dimensional range: a(2, 1:2) [4 5] a(2:3, 1) [4; 7] a(2, : ) [4 5 6] row vector a(:, 1) [1; 4; 7] column vector a(:, : ) == a

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a = [1:4; 5:8; 9:12; 13:16] 4x4 array Pick out rows 1 and 3:

a([1 3], : ) 1 2 3 4 9 10 11 12

Matrices can be treated as vectors, withelements ordered columnwise:

a(1:3:16) = 0a(1:2:16) = [8 7 6 5 4 3 2 1]a(16:-4:1) = 1.1*(1:4)

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Arrays can have more than two dimensions e.g. A(l, m, n)

Can be created in several ways: Extension of 2-d array:

A = [1 2 3; 4 5 6; 7 8 9];A(:, :, 2) = [9 8 7; 6 5 4; 3 2 1];

A(:, :, 3) = 5; Using Matlab functions:

A = randn(4, 3, 2)

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u = [3; 1; 4]; 3x1 column vector

v = [2 0 -1]; 1x3 row vector Inner product:

x = v * u 2 1x3 x 3x1 1x1(also known as “scalar product” and “dot product”)

Outer product:x = u * v 6 0 -3

2 0 -18 0 -4 3x1 x 1x3

3x3 Cross product:

x = cross(u, v) [-1 11 -2]

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Here we want to solve the system of linear equations represented by A x = b

Use the “\” operator to do “left-division”:A = [1 2 3; 4 5 6; 7 8 10];b = [1; 1; 1];x = A \ b -1.0000

1.0000 0.0000

r = b – A * x small numbers! When m > n (over-determined), “\” least-squares solution. When m < n (under-determined), “\” a particular solution.

General solutions can be found using null(A).

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Arguments specified using round brackets and commas: e.g. atan2(y, x)

Return values can be assigned to a scalar, an array, or to an array of variables. e.g. a = [1 2 3; 4 5 6];

size(a) [2 3]mn = size(a)[m n] = size(a)

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Trigonometry: sin, cos, tan, atan, atan2(y, x)

Exponential: log, log10, exp, sqrt Complex: abs, conj, real, imag, complex Rounding: fix, floor, ceil, round Matrix functions

linspace, logspace, zeros, ones, eye reshape, size, length expm, logm, sqrtm sum, prod, diag, norm, det, inv, eig

Random numbers: rand, randn

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There are two kinds of M-files: Scripts

Automate a series of steps you need to perform many times

Have no input arguments or return values Operate on data in the workspace.

User-defined Functions Extend the Matlab language for your

application Can accept input arguments and return

output arguments Internal variables are local to the function 61

function f = fact(n) Definition line

% Compute a factorial value. H1 line

% fact(n) returns the factorial of n, Help text

% usually denoted by n!% Put simply, fact(n) is prod(1:n) Commentf = prod(1:n); Function body By convention, name the m-file “fact.m”

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Upper and lower case characters are not equivalent (MATLAB is case sensitive).

Typing the name of a variable will case MATLAB to display its current value.

MATLAB uses parenthesis, ( ), square bracket [ ], and curly braces { }, and characters followed by up arrow.

You can type help topics to access online help on the command, functions or symbol topic. Note that hyperlinks, indicated by underlines, are provided that will take you to related help items and the Help browser.

If you press the tab key after partially typing a function or variable name, MATLAB will attempt to complete it, offering you a selection of choices if there is more than one possible completion.

You can quit MATLAB by typing exit or quit.

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Weakness of MATLAB

Interpreter – slower than compilers

Clumsy in handling texts Finite-precision arithmetic Infinite-precision in symbolic

computations, but slow Expensive – scilab project