introduction to matlab

© M S Ramaiah School of Advanced Studies

Introduction to MATLAB

Preetham Shankpal

Asst. Professor,

M S Ramaiah School of Advanced Studies

Email: [email protected]

1


Introduction

• What is Matlab? MATrix LABoratory.

• MATLAB is a numerical computing environment and programming language (initially written in C). MATLAB allows easy matrix manipulation, plotting of functions and data, implementation of algorithms, creation of user interfaces, and interfacing with programs in other languages.

• Powerful, extensible, highly integrated computation, programming, visualization, and simulation package

• Widely used in engineering, mathematics, and science

• It has packages with specialized functions

2


MATLAB Toolboxes

Signal & Image Processing Signal Processing Image Processing Communications Frequency Domain System Identification

Higher-Order Spectral Analysis System Identification Wavelet Filter Design

Control Design Control System Fuzzy Logic Robust Control μ-Analysis and Synthesis Model Predictive Control

Math and AnalysisOptimizationRequirements Management InterfaceStatisticsNeural NetworkSymbolic/Extended MathPartial Differential EquationsPLS ToolboxMappingSpline

Data Acquisition and ImportData AcquisitionInstrument ControlExcel LinkPortable Graph Object

HELP WINDOW - DEMOS

3


Toolboxes, Software, & Links

4


MATLAB System

• Language: arrays and matrices, control flow, I/O, data structures, user-defined functions and scripts

• Working Environment: editing, variable management, importing and exporting data, debugging, profiling

• Graphics system: 2D and 3D data visualization, animation and custom GUI development

• Mathematical Functions: basic (sum, sin,…) to advanced (fft, inv, Bessel functions, …)

• One can use MATLAB with C, Fortran, and Java, in either direction

5


Online MATLAB Resources

• www.mathworks.com/

• www.mathtools.net/MATLAB

• www.math.utah.edu/lab/ms/matlab/matlab.html

• web.mit.edu/afs/athena.mit.edu/software/matlab/

www/home.html

• www.utexas.edu/its/rc/tutorials/matlab/

• www.math.ufl.edu/help/matlab-tutorial/

• www.indiana.edu/~statmath/math/matlab/links.html

• www-h.eng.cam.ac.uk/help/tpl/programs/matlab.html

6


References

Mastering MATLAB 7, D. Hanselman and B. Littlefield,Prentice Hall, 2004

Getting Started with MATLAB 7: A Quick Introductionfor Scientists and Engineers, R. Pratap, Oxford UniversityPress, 2005.

7


Desktop Tools

• Command Window– type commands

• Workspace– view program variables– clear to clear – double click on a variable to

see it in the Array Editor

• Command History– view past commands– save a whole session using

diary

8


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 (openvar varname or 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”)

9


MATLAB Editor Window

10


MATLAB Help Window (Very Powerful)

11


Command-Line Help : List of Topic Functions

>> help matfun Matrix functions - numerical linear algebra. Matrix analysis. norm - Matrix or vector norm. normest - Estimate the matrix 2-norm. rank - Matrix rank. det - Determinant. trace - Sum of diagonal elements. null - Null space. orth - Orthogonalization. rref - Reduced row echelon form. subspace - Angle between two subspaces.

…

12


Keyword Search of Help Entries

>> lookfor whonewton.m: % inputs: 'x' is the number whose

square root we seektestNewton.m: % inputs: 'x' is the number

whose square root we seekWHO List current variables.WHOS List current variables, long form. TIMESTWO S-function whose output is two times

its input.

>> whos Name Size Bytes Class Attributes ans 1x1 8 double fid 1x1 8 double i 1x1 8 double

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Variables (Arrays) and Operators

14


Matlab as Calculator

• The basic arithmetic operators are + - * / ^ and these are used in conjunction with brackets: ( )

• The symbol ^ is used to get exponents (powers): 2^4=16

• You should type in commands shown following the prompt: >>.

>> 2+3/4*5

ans =

5.7500

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• Is this calculation 2 + 3/(4*5) or 2 + (3/4)*5? Matlab

• works according to the priorities:

1. quantities in brackets,

2. powers 2 + 3^2 ==> 2 + 9 = 11,

3. * /, working left to right (3*4/5=12/5),

4. + -, working left to right (3+4-5=7-5),

Thus, the earlier calculation was for 2 + (3/4)*5 by priority 3

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no declarations neededno declarations needed

mixed data types

mixed data types

semi-colon suppresses output of the calculation’s result

semi-colon suppresses output of the calculation’s result

>> 16 + 24ans = 40

>> product = 16 * 23.24product = 371.84

>> product = 16 *555.24;>> productproduct = 8883.8

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MATLAB Data•

The basic data type used in MATLAB is the double precision array

• No declarations needed: MATLAB automatically allocates required memory

• Resize arrays dynamically

• To reuse a variable name, simply use it in the left hand side of an assignment statement

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Numbers & Formats

• Matlab recognizes several different kinds of numbers

• The “e” notation is used for very large or very small numbers:

-1.3412e+03 = -1.3412 x103 = -1341.2

-1.3412e-01 = -1:3412 10-1 =-0.13412

• All computations in MATLAB are done in double precision, which means about 15 significant figures

• The format how Matlab prints numbers is controlled by the “format” command

19


Numbers & Formats

• Type help format for full list.

• Should you wish to switch back to the default format then format will suffice.

• The command

format compact

• is also useful in that it suppresses blank lines in the output thus allowing more information to be displayed

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Variable Names

• Legal names consist of any combination of letters and digits, starting with a letter. These are allowable:

NetCost, Left2Pay, x3, X3, z25c5

• These are not allowable:

Net-Cost, 2pay, %x, @sign

• User names that reflect the values they represent

• Special names: you should avoid using

• eps = 2.2204e-16 =2-54 (The largest number such that 1 + eps is indistinguishable from 1) and pi = 3.14159... = .

• If you wish to do arithmetic with complex numbers, both i and j have the value √-1 unless you change them

21


Variable Basicsclear all;close all;clc;product = 2 * 3^3;comp_sum = (2 + 3i) + (2 - 3i);show_i = i^2;save abc_1clearload abc_1who

complex numbers (i or j) require no special handling

save/load are used to

retain/restore workspace variables

Solve in MATLAB

22


Variables Revisited

• Variable names are case sensitive and over-written when re-used

• Basic variable class: Auto-Indexed Array– Allows use of entire arrays (scalar, 1-D, 2-D, etc…) as

operands– Vectorization: Always use array operands to get best

performance (avoiding for loops)

• Terminology: “scalar” (1 x 1 array), “vector” (1 x N array), “matrix” (M x N array)

• Special variables/functions: ans, pi, eps, inf, NaN, i, nargin, nargout, varargin, varargout, ...

• Commands who (terse output) and whos (verbose output) show variables in Workspace

23


Suppressing Output

• One often does not want to see the result of intermediate calculations terminate the assignment statement or expression with semicolon

• the value of x is hidden. Note also we can place several statements on one line, separated by commas or semicolons

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>> x=-13; y = 5*x, z = x^2+yy = -65.00z = 104.00


Trigonometric Functions

• Those known to Matlab are sin, cos, tan and their arguments should be in radians.

• e.g. to work out the coordinates of a point on a circle of radius 5 centred at the origin and having an elevation 30o = pi/6 radians.

• The inverse trig functions are called asin, acos, atan (as opposed to the usual arcsin or sin-1 etc.). The result is in radians.

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>> x = 5*cos(pi/6), y = 5*sin(pi/6)x = 4.33y = 2.50

acos(x/5), asin(y/5)ans = 0.52ans = 0.52

>> pi/6ans = 0.52


Other Elementary Functions

• These include sqrt, exp, log, log10

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>> x = 9;sqrt(x),exp(x),log(sqrt(x)),log10(x^2+6)ans = 3.00ans = 8103.08ans = 1.10ans = 1.94


Vectors – Rows and Columns

Row Vectors

• They are lists of numbers separated by either commas or spaces. The number of entries is known as the “length” of the vector and the entries are often referred to as “elements” or “components” of the vector

• The entries must be enclosed in square brackets

• Spaces can be vitally important:

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>> v = [ 1 3, sqrt(5)]v = 1.00 3.00 2.24>> length(v)ans = 3.00

>> v2 = [3+ 4 5]v2 = 7.00 5.00>> v2 = [3 +4 5]v2 = 3.00 4.00 5.00


Row Vectors

• We can do certain arithmetic operations with vectors of the same length, such as v and v3 in the previous section

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>> v, v3, v+v3v = 1.00 3.00 2.24v3 = 3.00 4.00 5.00ans = 4.00 7.00 7.24

>> v, v2, v+v2v = 1.00 3.00 2.24v2 = 7.00 5.00??? Error using ==> plusMatrix dimensions must agree.

The error is due to v and v2 having different lengths


Row Vectors

• A vector may be multiplied by a scalar (a number see v4), or added/subtracted to another vector of the same length. The operations are carried out element wise

• We can build row vectors from existing ones:

29

>> v, v4=3*vv = 1.00 3.00 2.24v4 = 3.00 9.00 6.71

>> w = [1 2 3], z = [8 9]cd = [2*z,-w], sort(cd)w = 1.00 2.00 3.00z = 8.00 9.00cd = 16.00 18.00 -1.00 -2.00 -3.00ans = -3.00 -2.00 -1.00 16.00 18.00


Row vectors

• We can also change or look at the value of particular entries

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w(2) = -2, w(3)w = 1.00 -2.00 3.00ans = 3.00


The Colon Notation

• This is a shortcut for producing row vectors:

• More generally a : b : c produces a vector of entries starting with the value a, incrementing by the value b until it gets to c (it will not produce a value beyond c). This is why 1:-1 produced the empty vector [].

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>> a= 1:4a = 1.00 2.00 3.00 4.00>> b=3:7b = 3.00 4.00 5.00 6.00 7.00>> c=-1:1c = -1.00 0 1.00>> d=1:-1d = Empty matrix: 1-by-0

Upper limitlower limit

>> 1:3:7ans = 1.00 4.00 7.00


Extracting Bits of a Vector

• To get the 3rd to 6th entries:

• To get alternate entries

• What does r5(-6:2:1) give?

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>> r5 = [1:2:6, -1:-2:-7]r5 = 1.00 3.00 5.00 -1.00 -3.00 -5.00 -7.00>> r5(3:6)ans = 5.00 -1.00 -3.00 -5.00

>> r5(1:2:7)ans = 1.00 5.00 -3.00 -7.00

>> r5(6:-2:1)ans = -5.00 -1.00 3.00


Column Vectors

• These have similar constructs to row vectors. When defining them, entries are separated by ; or “newlines”

so column vectors may be added or subtracted provided that they have the same length

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>> a= [1; 2; 3]a = 1.00 2.00 3.00>> b = [ 123]b = 1.00 2.00 3.00

>> c= 2*a -3*bc = -1.00 -2.00 -3.00


Transposing a Matrix

• We can convert a row vector into a column vector (and vice versa) by a process called transposing denoted by ‘.

• If x is a complex vector, then x' gives the complex conjugate transpose of x:

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>> a = [1 2 3]a = 1.00 2.00 3.00>> a'ans = 1.00 2.00 3.00>> c= [ 2; 3; 4]c = 2.00 3.00 4.00>> d= a'+(3*c)d = 7.00 11.00 15.00

a=[1-1i 1+1i];b=a';

1.00 - 1.00i 1.00+1.00i 1.00 + 1.00i1.00 - 1.00i


Plotting Elementary Functions

• Plot(t, sin(2*pi*100*t))

• xlabel

• ylabel

• title

• grid

• figure

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clear all;clcf=100;fs=40*f;ts=1/fs;t=0:ts:1;w=2*pi*f;y=3*sin(w*t);plot(t,y)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-3

-2

-1

0

1

2

3

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07-3

-2

-1

0

1

2

3


Multiple Plots

• plot(t,sin(wt), t,cos(wt))

• xlabel

• ylabel

• title

• hold on

• subplot

• Data cursor (marker)

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0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1-3

-2

-1

0

1

2

3

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1-4

-2

0

2

4

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1-4

-2

0

2

4


Vectorization Example*

tic;x=0.1;for k=1:199901 y(k)=besselj(3,x) +

log(x); x=x+0.001;endtoc;

Elapsed time is 17.092999 seconds.

tic;x=0.1:0.001:200;y=besselj(3,x) + log(x);toc;Elapsed time is 0.551970 seconds.

Roughly 31 times faster without use of for loop

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2D MATRIX Declaration

• M=[3 4 5; 6 7 8; 1 -1 0]

• M=magic(4)

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>> a=[1;2;3];>> b=[3;4;5];>> c=[2;3;4];>> s=[a b c]s = 1.00 3.00 2.00 2.00 4.00 3.00 3.00 5.00 4.00


Matrices: Magic Squares

This matrix is called a “magic square”

This matrix is called a “magic square”

Interestingly, Durer also dated this engraving by placing 15 and 14 side-by-side in the magic square.

Interestingly, Durer also dated this engraving by placing 15 and 14 side-by-side in the magic square.

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Durer Magic Square

>> M=magic(4);

>> [b]=[sum(M(1,:)) sum(M(2,:)) sum(M(3,:)) sum(M(4,:))]

>>[c]=[sum(M(:,1)) sum(M(:,2)) sum(M(:,3)) sum(M(:,4))]

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Matrix Manipulations

• Scalar Product (*)

• u= [ 1 2 3]; v= [2;3;4] ;

• Scalar_product= 1x2+2x3+3x4=20

• (row vector x column vector)

• (row vector x column vector)

• an error results because w is not a column vector

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1

N

i ii

uv u v

>> u= [ 1 2 3]; v= [2;3;4];>> u*vans = 20.00

>> u= [ 1 2 3]; v= [2 3 4];>> u*v??? Error using ==> mtimesInner matrix dimensions must agree.


Dot . Product• The second way of forming the product of two vectors of the same length is known

as the Hadamard product

• It is not often used in Mathematics but is an invaluable Matlab feature

• It involves vectors of the same type

• If u and v are two vectors of the same type (both row vectors or both column vectors), the mathematical definition of this product, which we shall call the dot product, is the vector having the components

• u . v = [u1v1, u2v2, ……., unvn]:

• The result is a vector of the same length and type as u and v. Thus, we simply multiply the corresponding elements of two vectors

42

>> u= [ 1 2 3]; v= [2 3 4];>> u*v??? Error using ==> mtimesInner matrix dimensions must agree. >> u.*vans = 2.00 6.00 12.00

Tabulate the function y = x sin(pi* x) forx = 0; 0:25; : : : ; 1.

plot(x,y)


Dot Division of Arrays (./)

• There is no mathematical definition for the division of one vector by another. However, in Matlab, the operator ./ is defined to give element by element division it is therefore only defined for vectors of the same size and type

43

>> a=[1 2 3];>> b=[1 2 3];>> c=a/bc = 1.00>> c=a./bc = 1.00 1.00 1.00>> a=[12 24 36];b=[6 3 9];>> c=a/bc = 3.71>> c=a./bc = 2.00 8.00 4.00


Dot Power of Arrays (.^)

• To square each of the elements of a vector we could, for example, do u.*u. However, a neater way is to use the .^ operator

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>> a=[ 1 2 3];>> a=a.^2a = 1.00 4.00 9.00

a = 1.00 4.00 9.00>> a=a.^4a = 1.00 256.00 6561.00

Draw plots for the following


Multidimensional Arrays>> r = randn(2,3,2) % create a 3 dimensional array filled

%with normally distributed random numbers

randn(2,3,2): 3 dimensions, filled with normally distributed random numbers

randn(2,3,2): 3 dimensions, filled with normally distributed random numbers

“%” sign precedes comments, MATLAB ignores the rest of the line

“%” sign precedes comments, MATLAB ignores the rest of the line

r=rand(2,3,2)

r(:,:,1) =

0.2769 0.0971 0.6948 0.0462 0.8235 0.3171

r(:,:,2) =

0.9502 0.4387 0.7655 0.0344 0.3816 0.7952

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2D Matrix Manipulations • size(A) , max, min transpose(A),

size(A)

• diag(A) fliplr(A)

• A=zeros(m,n) eye(A)

46

>> a= [ 2 3 4; 2 4 5; 5 6 7]a = 2.00 3.00 4.00 2.00 4.00 5.00 5.00 6.00 7.00>> size(a)ans = 3.00 3.00

>> diag(a)ans = 2.00 4.00 7.00

>> a= [ 2 3 4; 2 4 5]a = 2.00 3.00 4.00 2.00 4.00 5.00>> size(a)ans = 2.00 3.00>> a'ans = 2.00 2.00 3.00 4.00 4.00 5.00>> size(a')ans = 3.00 2.00

zeros(2,4)ans = 0 0 0 0 0 0 0 0

>> eye(3)ans = 1.00 0 0 0 1.00 0 0 0 1.00


Matrix Manipulation• D=[1 2 3 4]; D=diag(1:4)

• F = [0 1 8 7; 3 -2 -4 2; 4 2 1 1] (non square matrix)

• What is diag(F)?

• Tabulate the functions y = 4 sin 3x and u = 3 sin 4x for x = 0, 0.1, 0.2,….,0.5.

• Given a matrix A = [ 1 2 3; 4 5 6; 5 6 7]; find the determinant of A; Adjoint of A; inverse of A;

• Find the inverse of matrix A = [ 1 2 3 4; 1 3 5 6; 5 4 3 2];

47

>> diag(1:4)ans = 1.00 0 0 0 0 2.00 0 0 0 0 3.00 0 0 0 0 4.00


Matrix Manipulation

• Given Matrix A = [1 2 3 4; 5 3 1 7; 2 7 9 4; 2 4 6 8];

• Extract bits of A; extract locations of ( 8, 9 , 5, 1)

• Display the following

– Only row 3

– Only row 4

– Rows 2 and 4

– Cols 1 and 3

– Use the 4x 4 matrix concatenate it to make a 16x 16 Matrix

– Perform sum of diagonal elements

48


Dot product of matrices (.*)

• The dot product works as for vectors: corresponding elements are multiplied together so the matrices involved must have the same size

» A*x

49

>> A= [ 1 2 3 4; 3 4 5 6; 6 7 8 9];>> B= [ 2 3 4 5; 5 6 7 7; 8 9 3 4];>> A*B??? Error using ==> mtimesInner matrix dimensions must agree. >> A.*Bans = 2.00 6.00 12.00 20.00 15.00 24.00 35.00 42.00 48.00 63.00 24.00 36.00


Sparse Matrix

• Create a sparse 5x4 matrix S having only 3 non{zero values: S1,2 = 10, S3,3 = 11 and S5,4 =12.

50

>> m= [ 1 3 5];>> n=[2 3 4];>> v=[10 11 12];>> A=sparse(m,n,v)A = (1,2) 10.00 (3,3) 11.00 (5,4) 12.00>> A=full(A)A = 0 10.00 0 0 0 0 0 0 0 0 11.00 0 0 0 0 0 0 0 0 12.00


Systems of Linear Equations• Mathematical formulations of engineering problems often lead to

sets of simultaneous linear equations. Such is the case, for instance, when using the finite element method (FEM).

• A general system of linear equations can be expressed in terms of a coefficient matrix A, a right-hand-side (column) vector b and an unknown (column) vector x as

Ax=b

• When A is non-singular and square (n x n), meaning that the number of independent equations is equal to the number of unknowns, the system has a unique solution given by

x= A-1B51


Example

• Enter the symmetric coeffcient matrix and right hand side vector b given by

• and solve the system of equations Ax = b using the three alternative methods:

• i) x = A-1b, (the inverse A-1 may be computed in Matlab using inv(A).)

• ii) x = A \ b,

• iii) xT = btAT leading to xT = b' / A which makes use of the “slash” or “right division” operator “/”. The required solution is then the transpose of the row vector xT.

52


Example

• Use the backslash operator to solve the complex system of equations for which

53


Character Strings• close all;

• clear all;

• clc

• hi = ' hello';

• class = 'MATLAB';

• greetings = [hi class]

• vgreetings = [hi;class]semi-colon: join verticallysemi-colon: join verticallyconcatenation with blank or with “,”concatenation with blank or with “,”

54


yo =

Hello

Class

>> ischar(yo)

ans =

1

>> strcmp(yo,yo)

ans =

1

String Functions

returns 1 if argument is a characterarray and 0 otherwise

returns 1 if argument is a characterarray and 0 otherwise

returns 1 if string arguments are thesame and 0 otherwise; strcmpi ignores

case

returns 1 if string arguments are thesame and 0 otherwise; strcmpi ignores

case

55


Set FunctionsArrays are ordered sets:

>> a = [1 2 3 4 5]a = 1 2 3 4 5>> b = [3 4 5 6 7]b = 3 4 5 6 7

>> isequal(a,b)ans = 0>> ismember(a,b)ans = 0 0 1 1 1

returns true (1) if arrays are the same size and have the same values

returns true (1) if arrays are the same size and have the same values

returns 1 where a is in b and 0 otherwise

returns 1 where a is in b and 0 otherwise

56


Lab 1

• Create a row vector called X whose elements are the integers 1 through 9.• Create another row vector called Temp whose elements are:

15.6 17.5 36.6 43.8 58.2 61.6 64.2 70.4 98.8• These data are the result of an experiment on heat conduction through an iron bar. The

array X contains positions on the bar where temperature measurements were made. The array Temp contains the corresponding temperatures.

• Make a 2-D plot with temperature on the y-axis and position on the x-axis.• The data shown in your plot should lie along a straight line (according to physics) but

don’t because of measurement errors. Use the MATLAB polyfit function to fit the best line to the data (use >> hold on; for multiple plots in same figure). In other words use polyfit to determine the coefficients a and b of the equation

T = ax + b• Lastly, we can calculate a parameter called chi-square (χ2) that is a measure of how

well the data fits the line. Calculate chi-square by running the MATLAB command that does the following matrix multiplication:

>> (Temp-b-a*X)*(Temp-b-a*X)'

57


Lab 1

• close all;

• clear all;

• clc;

• x=1:9;

• temp=[15.6 17.5 36.6 43.8 58.2 61.6 64.2 70.4 98.8];

• p= polyfit(x,temp,1);

• T=p(1).*x+p(2);

• B=((temp-p(2)-p(1)).*x);

• Bd=((temp-p(2)-p(1)).*x)';

• C=B*Bd;

• figure;plot(x,temp,'-*'); hold on; plot(x,T)

•

58


Lab 2• Write a MATLAB command that will generate a column vector called

theta. theta should have values from –2π to 2π in steps of π/100.• Generate a matrix F that contains values of the following functions in the

columns indicated: Column 1: cos(θ) Column 2: cos(2θ)(1 + sin(θ2) Column 3: e -0.1|θ|

• Evaluate each of the above functions for the θ values in the theta vector from above.

• Plot each of the columns of F against theta. Overlay the three plots, using a different color for each.

• Create a new column vector called maxVect that contains the largest of the three functions above for each theta. Plot maxVect against theta.

• Create a column vector called maxIndex that has the column number of the maximum value in that row.

59


Lab 3

60

• Mesh and Contour plots

close all;clear all;clc;[X,Y] = meshgrid(2:.2:4, 1:.2:3); f =-X.*Y.*exp(-2*(X.^2+Y.^2));figure(1); mesh(X,Y,f), xlabel('x'), ylabel('y'), gridfigure (2), contour(X,Y,f)xlabel('x'), ylabel('y'), grid, hold onfmax = max(max(f));kmax = find(f==fmax);Pos = [X(kmax), Y(kmax)];plot(X(kmax),Y(kmax),'*')text(X(kmax),Y(kmax),' Maximum')


Programming

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• MATLAB m-file Editor

– To start: click icon or enter edit command in Command Window, e.g., >> edit test.m

• Scripts and Functions

• Decision Making/Looping– if/else– switch – for and while

• Running Operating System Commands

Outline

62


You can save and run the file/function/script in one step by clicking here

You can save and run the file/function/script in one step by clicking here

Tip: semi-colons suppress printing, commas (and semi-colons) allow multiple commands on one line, and 3 dots (…) allow continuation of lines without execution

Tip: semi-colons suppress printing, commas (and semi-colons) allow multiple commands on one line, and 3 dots (…) allow continuation of lines without execution

m-file Editor Window

63


Scripts and Functions

• Scripts do not accept input arguments, nor do they produce output arguments. Scripts are simply MATLAB commands written into a file. They operate on the existing workspace.

• Functions accept input arguments and produce output variables. All internal variables are local to the function and commands operate on the function workspace.

• A file containing a script or function is called an m-file

• If duplicate functions (names) exist, the first in the search path (from path command) is executed.

64


function [a b c] = myfun(x, y)b = x * y; a = 100; c = x.^2;

>> myfun(2,3) % called with zero outputs

ans = 100>> u = myfun(2,3) % called with one outputu = 100>> [u v w] = myfun(2,3) % called with all outputsu = 100v = 6w = 4

Functions – First Example

Write these two lines to a file myfun.m and save it on MATLAB’s path

Write these two lines to a file myfun.m and save it on MATLAB’s path

Any return value which is not stored in an output variable is simply discarded

Any return value which is not stored in an output variable is simply discarded

65


Fibonacci

function f = fibonacci(n)

% FIBONACCI Fibonacci sequence

% f = FIBONACCI(n) generates the first n Fibonacci numbers.

f = zeros(n,1);

f(1) = 1;

f(2) = 2;

for k = 3:n

f(k) = f(k-1) + f(k-2);

end

66


Sum

• Produce a list of the values of the sums

67

close all;clear all;clcx=1:20;s(20)=sum(1./(x.^2));for j= 21:100 s(j)=s(j-1)+(1./(j^2));ends


Fast Fourier Transform (FFT)

• fft is one of the built-in functions in MATLAB

• The fft function can compute the discrete Fourier transform of any arbitrary length sequence. fft incorporates most known fast algorithms for various lengths (e.g. power of 2)

• Not all lengths are equally fast

68


Discrete Fourier Transform Definition

•

21

0

21

0

[ ] [ ]

1[ ] [ ]

j knNN

n

j knNN

k

X k x n e

x n X k eN

69


fft and fftshift

0 2

N=11

-

1 11

0

N=11

After fftshift

70


Example: FFT of sine Wave in Noise

>> fs = 1000;>> t = [0:999]*(1/fs);>> x = sin(2*pi*250*t);>> X = fft(x(1:512));>> noise = 0.8*randn(size(x));>> xn = x + noise;>> XnMag = fftshift(20*log10(abs(fft(xn(1:512)))));>> XnMagPf = XnMag(256:512);>> frq = [0:length(XnMagPf) -

1]'*(fs/length(XnMag));>> plot(frq, XnMagPf)>> xlabel('freq. (Hz)');>> ylabel('Mag. (db)');

71


Develop code for FFT

function [ F Fr] = FCFT(fo,fl,x,fint,fs) i=1;for fr=fo:fint:fl w=2*pi*fr;u=1; for t=0:1/fs:0.5 X(u)=x(u)*exp(-1i*w*t); u=u+1; end N = length(X); F(i) = simp(X,N,1/fs); i = i + 1;endFr = fo:fint:fl; end

72


Frequency Spectrum

73


Area

74

function [A] = area(a,b,c)s = (a+b+c)/2;A = sqrt(s*(s-a)*(s-b)*(s-c));


Function Syntax Summary

• If the m-file name and function name differ, the file name takes precedence

• Function names must begin with a letter

• First line must contain function followed by the most general calling syntax

• Statements after initial contiguous comments (help lines) are the body of the function

• Terminates on the last line or a return statement

75


Function Syntax Summary (cont.)

• error and warning can be used to test and continue execution (error-handling)

• Scripts called in m-file functions are evaluated in the function workspace

• Additional functions (subfunctions) can be included in an m-file

• Use which command to determine precedence, e.g.,

>> which titleC:\MATLAB71\toolbox\matlab\graph2d\title

76


if/elseif/else Statement>> A = 2; B = 3;>> if A > B 'A is bigger' elseif A < B 'B is bigger' elseif A == B 'A equals B' else error('Something odd is happening') endans =B is bigger

77


switch Statement

>> n = 8n = 8>> switch(rem(n,3)) case 0 m = 'no remainder' case 1 m = ‘the remainder is one' case 2 m = ‘the remainder is two' otherwise error('not possible') endm =the remainder is two

78


for Loop>> for i = 2:5

for j = 3:6

a(i,j) = (i + j)^2

end

end

>> a

a =

0 0 0 0 0 0

0 0 25 36 49 64

0 0 36 49 64 81

0 0 49 64 81 100

0 0 64 81 100 121

79


A Performance Tip

Input variables are not copied into the function

workspace, unless

– If any input variables are changed, the variable will be copied

– Avoid performance penalty when using large arrays by extracting only those elements that will need modification

80


MATLAB’s Search Path

• Is name a variable?

• Is name a built-in function?

• Does name exist in the current directory?

• Does name exist anywhere in the search path?

• “Discovery functions”: who, whos, what, which, exist, help, doc, lookfor, dir, ls, ...

81


Changing the Search Path• The addpath command adds directories to the MATLAB search

path. The specified directories are added to the beginning of the search path.

• rmpath is used to remove paths from the search path

>> path

MATLABPATH

E:\MATLAB\R2006b\workE:\MATLAB\R2006b\work\f_funcsE:\MATLAB\R2006b\work\na_funcsE:\MATLAB\R2006b\work\na_scriptsE:\MATLAB\R2006b\toolbox\matlab\generalE:\MATLAB\R2006b\toolbox\matlab\ops

>> addpath('c:\');>> matlabpath

MATLABPATH

c:\E:\MATLAB\R2006b\workE:\MATLAB\R2006b\work\f_funcsE:\MATLAB\R2006b\work\na_funcsE:\MATLAB\R2006b\work\na_scriptsE:\MATLAB\R2006b\toolbox\matlab\generalE:\MATLAB\R2006b\toolbox\matlab\ops


Lab 1

• Create, perhaps using for-loops, a synthetic “image” that has a 1 in the (1,1) location, and a 255 in the (128,128) location, and i + j - 1 in the i, j location. This we'll refer to as the ”diagonal gray'' image. Can you manage to do this without using for-loops?

• Display the image using (we’ll assume you placed your image in a matrix named a) image(a); colormap(gray). (Don’t worry if this doesn’t work exactly the way you expect. Colormaps can be tricky!)

• Now convert your code to a MATLAB script

• Test your script to insure that it produces the same results as the ones obtained interactively.

83


Lab 2

• Write a MATLAB function that implements Newton’s iterative algorithm for approximating the square root of a number.

• The core of Newton’s algorithm is that if last is the last approximation calculated, the next (improved) approximation is given by

next = 0.5(last +(x/last)) where x is the number whose square root you seek.

• Two other pieces of information are needed to implement the algorithm. The first is an initial guess at the square root. (A typical starting value might be 1.0, say). The second is the accuracy required for the approximation. You might specify you want to keep iterating until you get an approximation that is good to 5 decimal places for example.

• Your MATLAB function should have three input arguments: x, the initial guess, and the accuracy desired. It should have one output, the approximate square root of x to the desired accuracy.

84


Data I/O

85


Loading and Saving Workspace Variables

• MATLAB can load and save data in .MAT format

• .MAT files are binary files that can be transferred across platforms; as much accuracy as possible is preserved.

• Load: load filename OR A = load(‘filename’)

loads all the variables in the specified file (the default name is MATLAB.MAT)

• Save: save filename variables

saves the specified variables (all variables by default) in the specified file (the default name is MATLAB.MAT)

86


ASCII File Read/Write

load and save can also read and write ASCII files with

rows of space separated values:

• load test.dat –ascii

• save filename variables

(options are ascii, double, tabs, append)

• save example.dat myvar1 myvar2 -ascii -double

87


ASCII File Read/Write (cont.)

• dlmread

M = dlmread(filename,delimiter,range);

reads ASCII values in file filename that are separated by delimiter into variable M; most useful for numerical values. The last value in a line need not have the delimiter following it.

range = [R1 C1 R2 C2] (upper-left to lower-right corner)

• dlmwrite

dlmwrite(filename,A,delimiter);

writes ASCII values in array A to file filename with values separated by delimiter

• Useful with spreadsheet data

range of data to be readrange of data to be read

88


More ASCII File Read• textread

[A, B, C, ...] = textread[‘filename’, ‘format’];[A, B, C, ...] = textread[‘filename’, ‘format’, N];[...] = textread[..., ‘param’, ‘value’, ...];

• The type of each return argument is given by format (C-style conversion specifiers: %d, %f, %c, %s, etc…)

• Number of return arguments must match number of conversion specifiers in format

• format string is reused N times or entire file is read if N not given• Using textread you can

– specify values for whitespace, delimiters and exponent characters– specify a format string to skip over literals or ignore fields

89


textread Example

• Data file, tab delimited:

• MATLAB m-file:

• Results:

‘param’,’value’ pairs

use doc textread for available param options


Import WizardImport ASCII and binary files using the Import Wizard. Type uiimport at the Command line or choose Import Data from the File menu.

91


Low-Level File I/O Functions• File Opening and Closing

– fclose: Close one or more open files – fopen: Open a file or obtain information about open files

• Unformatted I/O– fread: Read binary data from file – fwrite: Write binary data to a file

• Formatted I/O– fgetl: Return the next line of a file as a string

without line terminator(s) – fgets: Return the next line of a file as a string with line terminator(s) – fprintf: Write formatted data to file – fscanf: Read formatted data from file

92


Low-Level File I/O (cont.)

• File Positioning– feof: Test for end-of-file – ferror: Query MATLAB about errors in file input

or output – frewind: Rewind an open file – fseek: Set file position indicator – ftell: Get file position indicator

• String Conversion– sprintf: Write formatted data to a string – sscanf: Read string under format control

93


File Open (fopen)/Close (fclose)• fid = fopen(‘filename’, ‘permission’);

• status = fclose(fid);

File identifiernumber

Name offile

Permissionrequested:‘r’, ’r+’‘w’, ’w+’‘a’, ’a+’

0, if successful-1, otherwise

File identifier numberor ‘all’ for all files

94


Formatted I/O• fscanf: [A, count] = fscanf(fid,format,size);

• fprintf: count = fprintf(fid, format, A,...);

Dataarray

Numbersuccessfully

read

Fileidentifiernumber

Amount of data to read:

n, [n, m], Inf

Formatspecifier

Numbersuccessfully

read


Formatspecifier

Dataarray(s) to

write

fscanf and fprintf are similar to C version but vectorized

95


Format String Specification

%-12.5einitial %character

width andprecision

conversionspecifier

alignment flag

Specifier Description %c Single character %d Decimal notation (signed) %e Exponential notation %f Fixed-point notation %g The more compact of %e or %f %o Octal notation (unsigned) %s String of characters %u Decimal notation (unsigned) %x Hexadecimal notation ...others...

96


Other Formatted I/O Commands• fgetl: line = fgetl(fid);

reads next line from file without line terminator

• fgets: line = fgets(fid);reads next line from file with line terminator

• textread: [A,B,C,...] = textread('filename','format',N)reads N lines of formatted text from file filename

• sscanf: A = sscanf(s, format, size);reads string under format control

• sprintf: s = sprintf(format, A);writes formatted data to a string

97


Binary File I/O• [data, count] = fread(fid, num, precision);

• count = fwrite(fid, data, precision);

Dataarray

Numbersuccessfully

read


Amount to read n, [n, m],...

‘int’, ‘double’, …

array towrite

Numbersuccessfully

written


‘int’, ‘double’, …

fread and fwrite are vectorized

98


Basic Data Analysis

99


Basic Data Analysis

• Basic, and more advanced, statistical analysis is easily accomplished in MATLAB.

• Remember that the MATLAB default is to assume vectors are columnar.

• Each column is a variable, and each row is an observation.

100


Vibration Sensors Data

Each column is the raw rpm sensor data from a different sensor used in an instrumented engine test. The rows represent the times readings were made.

Each column is the raw rpm sensor data from a different sensor used in an instrumented engine test. The rows represent the times readings were made.

101


Plotting the Data

Note that in this case the plot command generates one time-series for each column of the data matrix

Note that in this case the plot command generates one time-series for each column of the data matrix

>> plot(rpm_raw)>> xlabel('sample number - during time slice');>> ylabel('Unfiltered RPM Data');>> title(‘3 sequences of samples from RPM sensor’)

102


Average of the Data:

Applying the mean function to the data matrix yields the mean of each column

Applying the mean function to the data matrix yields the mean of each column

But you can easily compute the mean of the entire matrix (applying a function to either a single row or a single column results in the function applied to the column, or the row, i.e., in both cases, the application is to the vector).

But you can easily compute the mean of the entire matrix (applying a function to either a single row or a single column results in the function applied to the column, or the row, i.e., in both cases, the application is to the vector).

1

2

>> mean(rpm_raw)

ans = 1081.4 1082.8

1002.7

>> mean(mean(rpm_raw))

ans = 1055.6

103


The mean Function

But we can apply the mean function along any dimension

But we can apply the mean function along any dimension

So we can easily obtain the row means

So we can easily obtain the row means

3

>> help mean MEAN Average or mean value. For vectors, MEAN(X) is the mean value of the elements in X. For matrices, MEAN(X) is a row vector containing the mean value of each column. For N-D arrays, MEAN(X) is the mean value of the elements along the first non-singleton dimension of X. MEAN(X,DIM) takes the mean along the dimension DIM of X. Example: If X = [0 1 2 3 4 5] then mean(X,1) is [1.5 2.5 3.5] and mean(X,2) is [1 4]>> mean(rpm_raw, 2)

ans = 1045.7 1064.7 1060.7 1055 1045

104


max and its Index

We can compute the max of the entire matrix, or of any dimension

We can compute the max of the entire matrix, or of any dimension

1 2 MAX Largest component. For vectors, MAX(X) is the largest

element in X. For matrices, MAX(X) is a row vector containing the

maximum element from each column. For N-D arrays, MAX(X) operates

along the first non-singleton dimension.

[Y,I] = MAX(X) returns the indices

of the maximum values in vector I. If the values along the first non-

singleton dimension contain more than one maximal element, the index

of the first one is returned.

>> max(rpm_raw)ans = 1115 1120 1043

>> max(max(rpm_raw))ans = 1120

>> [y,i] = max(rpm_raw)y = 1115 1120 1043i = 8 2 17

max along the columnsmax along the columns

105


min

>> min(rpm_raw)ans = 1053 1053 961

>> min(min(rpm_raw))ans = 961

>> [y,i] = min(rpm_raw)y = 1053 1053 961i = 22 1 22

min along each columnmin along each column

min of entire matrixmin of entire matrix

106


Standard Deviation, Median, Covariance

>> median(rpm_raw) % median along each columnans = 1080 1083.5 1004>> cov(rpm_raw) % covariance of the dataans = 306.4 -34.76 32.192 -34.76 244.9 -165.21 32.192 -165.21 356.25>> std(rpm_raw) % standard deviation along each

columnans = 17.504 15.649 18.875>> var(rpm_raw) % variance is the square of stdans = 306.4 244.9 356.25

107


Data Analysis: Histogram

HIST Histogram. N = HIST(Y) bins the elements of Y into 10 equally spaced containers and returns the number of elements in each container. If Y is a matrix, HIST works down the columns. N = HIST(Y,M), where M is a scalar, uses M bins. N = HIST(Y,X), where X is a vector, returns the distribution of Y among bins with centers specified by X. The first bin includes data between -inf and the first center and the last bin includes data between the last bin and inf. Note: Use HISTC if it is more natural to specify bin edges instead.

. . .

108


Histogram (cont.)>> hist(rpm_raw) %histogram of the data

109


Histogram (cont.) >> hist(rpm_raw, 20) %histogram of the data

110


Histogram (cont.)>> hist(rpm_raw, 100) %histogram of the data

111


Bin Average Filtering FILTER One-dimensional digital filter. Y = FILTER(B,A,X) filters the data in vector X with the filter described by vectors A and B to create the

filtered data Y. The filter is a "Direct Form II Transposed" implementation of the standard difference equation: a(1)*y(n) = b(1)*x(n) + b(2)*x(n-1) + ... + b(nb+1)*x(n-

nb) - a(2)*y(n-1) - ... - a(na+1)*y(n-

na)

>> filter(ones(1,3), 3, rpm_raw) ans = 359 351 335.67 719 724.33 667 1088.3 1081.7 1001 1084 1091.7 1004.7 1081 1073 1006.7

This example uses an FIR filter to compute a moving average using a window size of 3

This example uses an FIR filter to compute a moving average using a window size of 3

112


Filtered Data Plot

113


Load .dat file and carryout analysis

% MATLAB PROGRAM ecg_hfn.m

clear all % clears all active variables

close all

ecg = load('ecg_hfn.dat');

fs = 1000; %sampling rate = 1000 Hz

zero_padding = 0; % Number of samples to be added as zero padding

slen = length(ecg); %signal length

t=[1:slen]/fs;

ts=1/fs;

figure(1)

subplot(2,1,1),plot(t, ecg)

axis tight;

xlabel('Time in seconds');

ylabel('Amplitude in mV');

Title('Raw ECG');

% frequency domain

fft_samples = 2^nextpow2(slen + zero_padding);

raw_fft = fft(ecg,fft_samples);

f_axis = 0:1:(fft_samples-1)/2; % fft_samples -1 because we start at 0

avg_power = sum(abs(raw_fft(1:fft_samples/2)).^2./slen)/slen;

subplot(2,1,2)

plot( f_axis*fs/fft_samples, ...

abs(raw_fft(1:fft_samples/2))*2/slen,f_axis*fs/fft_samples,avg_power);

title('Raw ECG Frequency Domain');

ylabel('Magnitude [Normalized]');

xlabel('Frequency [Hz]');

% power spectrum in dB figure(3);psd(ecg,slen,1000)title('Power spectral density'); % high pass filter fc1= 0.5; % cut off frequencyn1=2;% order of filterwn1=(2*fc1/fs);[b1,a1] = butter(n1,wn1,'high');h1=filter(b1,a1,ecg);figure(5);plot(h1);xlabel('no. of samples');title('High pass filtered ECG at 0.5Hz');ylabel('Amplitude in mV');figure(6)freqz(b1,a1);title('High Pass Filter characteristics');

114



% frequency domain

raw_fft1= fft(h1,fft_samples);

f_axis1= 0:1:(fft_samples-1)/2; % fft_samples -1 because we start at 0

avg_power1 = sum(abs(raw_fft1(1:fft_samples/2)).^2./slen)/slen;

figure(7);

plot( f_axis1*fs/fft_samples, ...

abs(raw_fft1(1:fft_samples/2))*2/slen,f_axis1*fs/fft_samples,avg_power1);

title('High pass filtered ECG signal frequency domain');



%notch filter

f = 0:fs/2;

F0 = 60; % Center frequency of notch filter in Hz

delF = 2; % bandwidth in Hz

Theta0 = (2*pi*F0)/fs;

r = 1 - (delF*pi)/fs;

% Notch filter coefficient

b0 = (abs(1-2*r*cos(Theta0)+r^2))/(2*abs(1-cos(Theta0)));

b2 = [b0 (b0*(-2)*cos(Theta0)) b0]; % Numerator coefficients for filter

a2 = [1 (-2*r*cos(Theta0)) r^2]; % Denominator coefficients for filter

h2 = filter(b2,a2,h1); % notch filterfigure(8);plot(h2)xlabel('no. of samples');title('Notch filtered ECG at 60 Hz');ylabel('Amplitude in mV ');figure(9);freqz(b2,a2);title('Notch Filter characteristics'); % frequency domain raw_fft2= fft(h2,fft_samples);f_axis2= 0:1:(fft_samples-1)/2; % fft_samples -1 because

we start at 0avg_power2 =

sum(abs(raw_fft2(1:fft_samples/2)).^2./slen)/slen;figure(10);plot( f_axis2*fs/fft_samples, ...


title('Notch filtered signal Frequency Domain');ylabel('Magnitude [Normalized]');xlabel('Frequency [Hz]');

115



% IIR Low pass filter

fc3= 40;

n3=2;

wn3=(2*fc3/fs);

[b3,a3] = butter(n3,wn3);

h3=filter(b3,a3,h2);

figure(11);

plot(h3);

xlabel('Time in seconds');

title('Low pass filtered ECG at 40 Hz');

ylabel('LPF ECG');

figure(12)

freqz(b3,a3);

title('Low Pass Filter characteristics');

% frequency domain

raw_fft3= fft(h3,fft_samples);

f_axis3= 0:1:(fft_samples-1)/2; % fft_samples -1 because we start at 0

avg_power3 = sum(abs(raw_fft3(1:fft_samples/2)).^2./slen)/slen;

figure(13);

plot( f_axis3*fs/fft_samples, ...


title('Low pass filtered signal Frequency Domain');



% 1 epoch

figure(1)

subplot(4,1,1); plot(ecg(1:800));xlabel('samples');ylabel('Amplitude in mV');title('raw ECG');

subplot(4,1,2); plot(h1(1:800));xlabel('samples');;ylabel('Amplitude in mV');title('HPF ECG');

subplot(4,1,3); plot(h2(1:800));xlabel('samples');ylabel('Amplitude in mV');title('Notch ECG');

subplot(4,1,4); plot(h3(1:800));xlabel('samples');ylabel('Amplitude in mV');title('LPF ECG');

116


Lab 1

• Load the data_analysis_lab1.mat file into the MATLAB workspace. This will produce an array variable called grades containing grades on an exam between 0 and 100.

• Calculate the average and standard deviation of the grades.

• Plot a histogram of the grades using 100 bins.

• We want to compare the histogram with a Gaussian distribution.

• Write you own MATLAB Gaussian function M-file which returns a value y using the following formula

y=exp(-[x-m]2/2σ2)

where m is the average and σ is the standard deviation of the distribution. Your function should have input arguments x,m, and σ.

• On the histogram plot also plot a Gaussian distribution of the grades using the calculated average and standard deviation.

117


Lab 2

• Load the file data_analysis_lab2.mat. Since this is a .mat file, you should be able to load it easily using the load command.

• Your workspace should now contain a single variable x. x is 3000 points long and consists of the sum of 3 sine waves. The sampling frequency is 1000 Hz.

• Plot the first 0.33 seconds of x. You may find it convenient to create a second array (say called time) that has the time values corresponding to the samples in x.

>> Fs = 1000; %Sampling frequency >> time = [0:length(x)-1]’*(1/Fs); % time index• fft is a built-in function in MATLAB. We can compute and plot the magnitude

of the FFT of x to identify the frequencies of the sine waves. >> X = fft(x);• X is a complex valued array that is the FFT of x. We can compute the magnitude

of the FFT by taking the absolute value of X. >> Xmag = abs(X); >> plot(Xmag);

118


Lab 2 (cont.)

• The plot of Xmag shows 6 components, and also we have only index values not real frequency values along the abscissa. Six components show up because the FFT is evaluated over positive and negative frequencies. Also, the frequencies are “wrapped around”. We can take care of the wrap around using the fftshift function.

>> Xmag = fftshift(Xmag);• Next, we can generate a suitable frequency axis for plotting Xmag.

>> frq = [0:length(Xmag)-1]’*(Fs/length(Xmag)) – (Fs/2);

>> plot(frq, Xmag);• Can you see the 3 frequency components (in the positive freq. part of

the axis)? Zoom into the plot either using the axis command or the interactive zoom button on the figure’s toolbar and determine the frequencies of the 3 components.

119


Matrix Factorizations: lu

lu: factors a square matrix A into the product of a permuted lower triangular matrix L and an upper triangular matrix U such that A = LU.

Useful in computing inverses, Gaussian elimination.

>> A = [1 2 -1; 1 0 1; -1 2 1];

>> [L, U] = lu(A)L = 1 0 0 1 -0.5 1 -1 1 0U = 1 2 -1 0 4 0 0 0 2>> L * Uans = 1 2 -1 1 0 1 -1 2 1


Matrix Factorizations: chol

chol: factors a symmetric, positive definite matrix A as RTR, where R is upper triangular.

Useful in solving least

squares problems.

>> A = [2 -1; 1 1; 6 -1];>> B = A'*AB = 41 -7 -7 3>> R = chol(B)R = 6.4031 -

1.0932 0

1.3435>> R'*Rans = 41 -7 -7 3


Eigenvalues and Eigenvectors: eig

eig: computes the eigenvalues, i and eigenvectors, xi of a square matrix A..

• i and xi satisfy

Axi = i xi

• [V,D] = eig(A) returns the eigenvectors of A in the columns of V, and the eigenvalues in the diagonal elements of D.

>> A = [1 -1 0; 0 1 1; 0 0 -2];

>> [V, D] = eig(A)V = 1 1 -0.10483 0 0 -0.31449 0 0 0.94346D = 1 0 0 0 1 0 0 0 -2>> [A*V(:,3) D(3,3)*V(:,3)]ans = 0.20966 0.20966 0.62897 0.62897 -1.8869 -1.8869


Singular Value Decomposition: svdsvd: factors an n x m matrix

A as A = USVT, where U and V are orthogonal matrices, and S is a diagonal matrix with singular values of A.

• Useful in solving least squares problems.

>> A = [2 -1; 1 1; 6 -1];>> [U,S,V] = svd(A)U = -0.32993 0.47852 -

0.81373 -0.12445 -0.87653 -

0.46499 -0.93577 -0.052149

0.34874S = 6.4999 0 0 1.3235 0 0V = -0.98447 -0.17558 0.17558 -0.98447


Pseudoinverse: pinv

pinv: The pseudoinverse of an n x m matrix A is a matrix B such that

BAB = B and

ABA = A

• MATLAB uses the SVD of A to compute pinv.

• Useful in solving least squares problems.

>> A = [2 -1; 1 1; 6 -1];>> B = pinv(A)B = -0.013514 0.13514 0.14865 -0.36486 0.64865

0.013514>> A*B*Aans = 2 -1 1 1 6 -1>> B*A*Bans = -0.013514 0.13514 0.14865 -0.36486 0.64865

0.013514


Numerical Integration

• trapz: Trapezoidal integration

• quad: Adaptive, recursive Simpson’s Rule for quadrature

• quadl: Adaptive, recursive Newton-Coates

8-panel rule

• dblquad: Double integration using quad or quadl

125


Integration Example: humps Function

>> x = linspace(-1,2,150);>> y = humps(x);>> plot(x,y)>> format long>> trapz(x,y) % 5-digit accuracyans = 26.344859225225534

>> quad('humps', -1, 2) % 6-digit accuracy

ans = 26.344960501201232

>> quadl('humps', -1, 2) % 8-digit accuracy

ans = 26.344960471378968

126


Root Finding and Minimization

• roots: finds roots of polynomials

• fzero: finds roots of a nonlinear function of one

variable

• fminbnd, fminsearch: finds maxima and

minima of functions of one and several variables

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Example of Polynomial Roots

p(x)=x3+4x2-7x-10

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Example of Roots for Nonlinear Functions

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Example of Function Minimization

>> p = [1 0 -2 -5];>> x = linspace(0,2,100);>> y = polyval(p,x);>> plot(x,y)>> fminbnd('x.^3-2*x-5',0,2)ans = 0.8165

>> polyval(p,ans)ans = -6.0887

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Lab 1

• Consider the set of equations Ax=b where A is an 8x8 matrix given by

A(i,j)=1.0+(|i-j|/8.0)½

• and b is a 8x1 array given by

b(i)=i• Solve for x using:

– The \ operator

– The MATLAB pinv function

– The MATLAB inv function

– LU Decomposition

• How do your answers compare?

• For best performance, evaluate the matrix A without using any for loops

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Lab 2

• Use numerical integration to integrate 1/(1+x2) from 0 to 1. The result is analytically calculated to be /4.

• Use the following three MATLAB functions:– trap()– quad()– quadl()

and compare the accuracy of your numerical result with the exact value.

• Use quad or quadl to get the most accurate result possible with MATLAB. How accurate is it?

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Tool boxes

• Signal and Image processing

• Audio processing

– wavread

– aviread

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WAV READclose all;clear all;clc%%%%%%%%%%%%%%%%%%% READ SIGNAL [s, fs, nbits, readinfo] = wavread('ak47-1.wav'); s=s(:,1)';L=length(s);t=1:L;t=t/fs;figure; plot(t,s)s1=s+awgn(s,10);figure; plot(t,s1)% xlabel('time in seconds');% ylabel('Amplitude');% title('AK47 Signal');N=1024;X = fft(s,N);F=fs*linspace(0,1,N);figure;plot(F(1:N/2),abs(X(1:N/2))); xlim([0 2000]); title('raw signal')X1 = fft(s1,N);F=fs*linspace(0,1,N);figure;plot(F(1:N/2),abs(X1(1:N/2))); xlim([0 2000]); title('noisy signal')% xlabel('frequency');% ylabel('Magnitude');% title('FFT of input AK47 Signal');

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AVI READ• clc;• clear all;• close all;• • • %----------------Inputs---------------------• • video = '7.avi';• Frm = 17;• • • %--------Reading Frame from Video-----------• a = aviread(video,Frm)• im5 = a.cdata;• • a = aviread(video,Frm+1)• im6 = a.cdata;• • %--------Converting Grey Scale-----------• • im3=rgb2gray(im5);• im4=rgb2gray(im6);• • %----------1 st Level Pyramid-----------• • im3=impyramid(im3,'reduce');• im4=impyramid(im4,'reduce');• • %----------2 nd Level Pyramid----------- • • im3=impyramid(im3,'reduce');• im4=impyramid(im4,'reduce');• • figure; imshow(im3)• figure;imshow(im4)

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Thank You

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Documents

introduction to matlab