1. Signals and Systems Laboratory 2013 Teacher: Marek OssowskiTeacher: Marek Ossowski marek.ossowski@p.lodz.pl Phone: 42 6312515, 501 673231 Room 14, C3, Al.Politechniki 11 Laboratory: Room 105, C3 Department of Nonlinear Circuits and Systems, Institute of Electrical Engineering Systems (I12)

2. Signals and Systems Laboratory Teacher home page:Teacher home page: http://matel/p.lodz.pl/wee/i12zet/mosso/mos so_index.htmlhttp://matel/p.lodz.pl/wee/i12zet/mosso/mos so_index.htmlhttp://matel/p.lodz.pl/wee/i12zet/mosso/mos so_index.htmlhttp://matel/p.lodz.pl/wee/i12zet/mosso/mos so_index.html Files for today:Files for today: –Front page –Experiment 0 –Experiment 1

3. Signals and Systems Laboratory Introduction What is Matlab? A WORKSHOP. Experiment 0 Experiment 0 –Elementary Matlab functions and operations – Basic Signals Manipulations – Systems identifications

4. Signals and SystemsLaboratory Experiment 1. Discrete and analog convolution Experiment 1. Discrete and analog convolution Experiment 2. Properties of Discrete Fourier Transform. Experiment 2. Properties of Discrete Fourier Transform. Experiment 3. Spectral analysis of periodic signals. Fourier series. Experiment 3. Spectral analysis of periodic signals. Fourier series. Experiment 4. Fourier transform. Experiment 4. Fourier transform. Experiment 5. Analysis of dynamic LTI systems using state equations and Laplace transform concept. Experiment 5. Analysis of dynamic LTI systems using state equations and Laplace transform concept. Experiment 6. Z-transform. Digital filters.Experiment 6. Z-transform. Digital filters.

5. MATLAB Short Overview

6. Before getting started… Acknowledgement: –This material was almost entirely created from information compiled from various internet sources It is available for download from SBEL website in PPT format for other to be able to save, edit, and distribute as they see fit Please let me know of any mistakes you find –email me at mylastname@wisc.edu, where mylastname is: negrutmylastname@wisc.edu Purpose of the document: –You will not be able to say at the end of workshop that you know MATLAB but rather that you have been exposed to MATLAB (I don’t know MATLAB myself, I’m just using it…:-) –It is expected of you to use MATLAB’s “help” function a lot, as well as search the web for examples that come close to what you need You learn how to use MATLAB by using it, that’s why the start is maybe slow and at times frustrating

7. Contents 1.First part of workshop  What is Matlab?  MATLAB Components  MATLAB Desktop  Matrices  Numerical Arrays  String Arrays  Importing and Exporting Data  Elementary math with MATLAB

8. Contents Continued 2. Second part of workshop  M-file Programming  Functions vs. Scripts  Variable Type/Scope  Debugging MATLAB functions  Flow control in MATLAB  Other Tidbits  Function minimization  Root finding  Solving ODE’s  Graphics Fundamentals  Data Types most likely won’t have time for it

9. What is MATLAB? Integrated software environment –Computation –Visualization –Easy-to-use environment High-level language –Data types –Functions –Control flow statements –Input/output –Graphics –Object-oriented programming capabilities

10. MATLAB Parts High Level Development Environment Programming Language Graphics Toolboxes Application Program Interface

11. Toolboxes Collections of functions to solve problems from several application fields. –DSP (Digital Signal Processing) Toolbox –Image Toolbox –Wavelet Toolbox –Neural Network Toolbox –Fuzzy Logic Toolbox –Control Toolbox –Multibody Simulation Toolbox –And many others…

12. MATLAB 7x Desktop Tools Command Window Command History Help Browser Workspace Browser Editor/Debugger To select what you want to see, go under “Desktop” tab

13. MATLAB 7x DESKTOP Editor/Debugger Interpreter Command Window CommandHistory HELP BROWSER FigureWindow WokspaceBrowser

14. MATLAB 5.3 – main window= interpreter

15. MATLAB – editor (new file)

16. MATLAB – editor (opening a file)

17. MATLAB – figure editor

18. MATLAB – path browser

19. HELP - help, lookfor

20. Calculations at the Command Line »a = 2; »b = 5; »a^b ans = 32 »x = 5/2*pi; »y = sin(x) y = 1 »z = asin(y) z = 1.5708 »a = 2; »b = 5; »a^b ans = 32 »x = 5/2*pi; »y = sin(x) y = 1 »z = asin(y) z = 1.5708 Results assigned to “ans” if name not specified () parentheses for function inputs Semicolon suppresses screen output MATLAB as a calculator Assigning Variables A Note about Workspace: Numbers stored in double-precision floating point format » -5/(4.8+5.32)^2 ans = -0.0488 » (3+4i)*(3-4i) ans = 25 » cos(pi/2) ans = 6.1230e-017 » exp(acos(0.3)) ans = 3.5470 » -5/(4.8+5.32)^2 ans = -0.0488 » (3+4i)*(3-4i) ans = 25 » cos(pi/2) ans = 6.1230e-017 » exp(acos(0.3)) ans = 3.5470

21. General Functions whos: List current variables and their size clear: Clear variables and functions from memory cd: Change current working directory dir: List files in directory pwd: Tells you the current directory you work in echo: Echo commands in M-files format: Set output format (long, short, etc.) diary(filename): Saves all the commands you type in in a file in the current directory called filename

22. Getting help help command ( >>help ) lookfor command ( >>lookfor ) Help Browser ( >>doc ) helpwin command ( >>helpwin ) Search Engine Printable Documents – “ Matlabroot\help\pdf_doc\ ” Link to The MathWorks

23. Handling Matrices in Matlab

24. Matrices Entering and Generating Matrices Subscripts Scalar Expansion Concatenation Deleting Rows and Columns Array Extraction Matrix and Array Multiplication

25. »a=[1 2;3 4] a = 1 2 3 4 »b=[-2.8, sqrt(-7), (3+5+6)*3/4] b = -2.8000 0 + 2.6458i 10.5000 »b(2,5) = 23 b = -2.8000 0 + 2.6458i 10.5000 0 0 0 0 0 0 23.0000 »a=[1 2;3 4] a = 1 2 3 4 »b=[-2.8, sqrt(-7), (3+5+6)*3/4] b = -2.8000 0 + 2.6458i 10.5000 »b(2,5) = 23 b = -2.8000 0 + 2.6458i 10.5000 0 0 0 0 0 0 23.0000 Any MATLAB expression can be entered as a matrix element (internally, it is regarded as such) In MATLAB, the arrays are always rectangular Entering Numeric Arrays NOTE: 1) Row separator semicolon (;) 2) Column separator space OR comma (,) Use square brackets [ ]

26. The Matrix in MATLAB 410162 81.29425 7.257111 00.54556 238313010 1 2 Rows (m) 3 4 5 Columns (n) 1 2 3 4 5 16111621 27121722 38131823 49141924 510152025 A = A (2,4) A (17) Rectangular Matrix: Scalar:1-by-1 array Vector:m-by-1 array 1-by-n array Matrix:m-by-n array

27. »w=[1 2;3 4] + 5 w = 6 7 8 9 »x = 1:5 x = 1 2 3 4 5 »y = 2:-0.5:0 y = 2.0000 1.5000 1.0000 0.5000 0 »z = rand(2,4) z = 0.9501 0.6068 0.8913 0.4565 0.2311 0.4860 0.7621 0.0185 »w=[1 2;3 4] + 5 w = 6 7 8 9 »x = 1:5 x = 1 2 3 4 5 »y = 2:-0.5:0 y = 2.0000 1.5000 1.0000 0.5000 0 »z = rand(2,4) z = 0.9501 0.6068 0.8913 0.4565 0.2311 0.4860 0.7621 0.0185 Scalar expansion Creating sequences: colon operator (:) Utility functions for creating matrices. Entering Numeric Arrays

28. Examples (Entering Numeric Arrays) Enter the matrix: Enter the matrix enter the row vector:enter the row vector Enter the column vector :

29. Numerical Array Concatenation »a=[1 2;3 4] a = 1 2 3 4 »cat_a=[a, 2*a; 3*a, 4*a; 5*a, 6*a] cat_a = 1 2 2 4 3 4 6 8 3 6 4 8 9 12 12 16 5 10 6 12 15 20 18 24 »a=[1 2;3 4] a = 1 2 3 4 »cat_a=[a, 2*a; 3*a, 4*a; 5*a, 6*a] cat_a = 1 2 2 4 3 4 6 8 3 6 4 8 9 12 12 16 5 10 6 12 15 20 18 24 Use [ ] to combine existing arrays as matrix “elements” Row separator: semicolon (;) Column separator: space / comma (,) Use square brackets [ ] Note: The resulting matrix must be rectangular 4*a

30. Deleting Rows and Columns »A=[1 5 9;4 3 2.5; 0.1 10 3i+1] A = 1.0000 5.0000 9.0000 4.0000 3.0000 2.5000 0.1000 10.0000 1.0000+3.0000i »A(:,2)=[] A = 1.0000 9.0000 4.0000 2.5000 0.1000 1.0000 + 3.0000i »A(2,2)=[] ??? Indexed empty matrix assignment is not allowed. »A=[1 5 9;4 3 2.5; 0.1 10 3i+1] A = 1.0000 5.0000 9.0000 4.0000 3.0000 2.5000 0.1000 10.0000 1.0000+3.0000i »A(:,2)=[] A = 1.0000 9.0000 4.0000 2.5000 0.1000 1.0000 + 3.0000i »A(2,2)=[] ??? Indexed empty matrix assignment is not allowed.

31. Array Subscripting / Indexing 410162 81.29425 7.257111 00.54556 238313010 1234512345 1 2 3 4 5 16111621 27121722 38131823 49141924 510152025 A = A(3,1) A(3) A(1:5,5) A(:,5) A(21:25) A(4:5,2:3) A([9 14;10 15]) A(1:end,end) A(:,end) A(21:end)’

32. Matrix Multiplication »a = [1 2 3 4; 5 6 7 8]; »b = ones(4,3); »c = a*b c = 10 10 10 26 26 26 »a = [1 2 3 4; 5 6 7 8]; »b = ones(4,3); »c = a*b c = 10 10 10 26 26 26 [2x4] [4x3] [2x4]*[4x3] [2x3] a(2nd row).b(3rd column) »a = [1 2 3 4; 5 6 7 8]; »b = [1:4; 1:4]; »c = a.*b c = 1 4 9 16 5 12 21 32 »a = [1 2 3 4; 5 6 7 8]; »b = [1:4; 1:4]; »c = a.*b c = 1 4 9 16 5 12 21 32 c(2,4) = a(2,4)*b(2,4) Array Multiplication

33. Matrix Manipulation Functions zeros : Create an array of all zeros ones : Create an array of all ones eye: Identity Matrix rand: Uniformly distributed random numbers diag: Diagonal matrices and diagonal of a matrix size: Return array dimensions fliplr: Flip matrices left-right flipud: Flip matrices up and down repmat: Replicate and tile a matrix

34. Matrix Manipulation Functions transpose ( ’ ): Transpose matrix rot90: rotate matrix 90 tril: Lower triangular part of a matrix triu: Upper triangular part of a matrix cross: Vector cross product dot: Vector dot product det: Matrix determinant inv: Matrix inverse eig: Evaluate eigenvalues and eigenvectors rank: Rank of matrix

35. Exercise 1 (5 minutes) Define a matrix A of dimension 2 by 4 whose (i,j) entry is A(i,j)=i+j Extract two 2 by 2 matrices A1 and A2 out of the matrix A. A1 contains the first two columns of A, A2 contains the last two columns of A Compute the matrix B to be the sum of A1 and A2 Compute the eigenvalues and eigenvectors of B Solve the linear system Bx=b, where b has all the entries equal to 1 Compute the determinant of B Compute the inverse of B Compute the condition number of B NOTE: Use only MATLAB native functions for all operations

36. Character String Manipulation

37. Character Arrays (Strings) Created using single quote delimiter (') Each character is a separate matrix element (16 bits of memory per character) Indexing same as for numeric arrays »str = 'Hi there,' str = Hi there, »str2 = 'Isn''t MATLAB great?' str2 = Isn't MATLAB great? »str = 'Hi there,' str = Hi there, »str2 = 'Isn''t MATLAB great?' str2 = Isn't MATLAB great? 1x9 vector str = Hithere,

38. »str ='Hi there,'; »str1='Everyone!'; »new_str=[str, ' ', str1] new_str = Hi there, Everyone! »str2 = 'Isn''t MATLAB great?'; »new_str2=[new_str; str2] new_str2 = Hi there, Everyone! Isn't MATLAB great? »str ='Hi there,'; »str1='Everyone!'; »new_str=[str, ' ', str1] new_str = Hi there, Everyone! »str2 = 'Isn''t MATLAB great?'; »new_str2=[new_str; str2] new_str2 = Hi there, Everyone! Isn't MATLAB great? 1x19 vector 1x9 vectors String Array Concatenation Using [ ] operator: Each row must be same length Row separator: semicolon (;) Column separator: space / comma (,) For strings of different length: STRVCAT char »new_str3 = strvcat(str, str2) new_str3 = Hi there, Isn't MATLAB great? »new_str3 = strvcat(str, str2) new_str3 = Hi there, Isn't MATLAB great? 2x19 matrix (zero padded) 1x19 vectors

39. Working with String Arrays String Comparisons –strcmp: compare whole strings –strncmp: compare first ‘N’ characters –findstr: finds substring within a larger string Converting between numeric & string arrays: –num2str: convert from numeric to string array –str2num: convert from string to numeric array

40. Elementary Math

41. Logical Operators Math Functions Polynomial and Interpolation

42. Logical Operations » Mass = [-2 10 NaN 30 -11 Inf 31]; » each_pos = Mass>=0 each_pos = 0 1 0 1 0 1 1 » all_pos = all(Mass>=0) all_pos = 0 » all_pos = any(Mass>=0) all_pos = 1 » pos_fin = (Mass>=0)&(isfinite(Mass)) pos_fin = 0 1 0 1 0 0 1 » Mass = [-2 10 NaN 30 -11 Inf 31]; » each_pos = Mass>=0 each_pos = 0 1 0 1 0 1 1 » all_pos = all(Mass>=0) all_pos = 0 » all_pos = any(Mass>=0) all_pos = 1 » pos_fin = (Mass>=0)&(isfinite(Mass)) pos_fin = 0 1 0 1 0 0 1 = = equal to > greater than < less than >= Greater or equal <= less or equal ~ not & and | or isfinite(), etc.... all(), any() find Note: 1 = TRUE 0 = FALSE

43. Elementary Math Function abs, sign: Absolute value and Signum Function sin, cos, asin, acos…: Triangular functions exp, log, log10: Exponential, Natural and Common (base 10) logarithm ceil, floor: Round toward infinities fix: Round toward zero

44. Elementary Math Function round: Round to the nearest integer gcd: Greatest common devisor lcm: Least common multiple sqrt: Square root function real, imag: Real and Image part of complex rem: Remainder after division

45. Elementary Math Function max, min: Maximum and Minimum of arrays mean, median: Average and Median of arrays std, var: Standard deviation and variance sort: Sort elements in ascending order sum, prod: Summation & Product of Elements trapz: Trapezoidal numerical integration cumsum, cumprod: Cumulative sum, product diff, gradient: Differences and Numerical Gradient

46. Polynomials and Interpolation Polynomials –Representing –Roots ( >> roots ) –Evaluation ( >> polyval ) –Derivatives ( >> polyder ) –Curve Fitting ( >> polyfit ) –Partial Fraction Expansion (>> residue ) Interpolation –One-Dimensional ( interp1 ) –Two-Dimensional ( interp2 )

47. polysam=[1 0 0 8]; roots(polysam) ans = -2.0000 1.0000 + 1.7321i 1.0000 - 1.7321i polyval(polysam,[0 1 2.5 4 6.5]) ans = 8.0000 9.0000 23.6250 72.0000 282.6250 polyder(polysam) ans = 3 0 0 [r p k]=residue(polysam,[1 2 1]) r = 3 7 p = -1 -1 k = 1 -2 polysam=[1 0 0 8]; roots(polysam) ans = -2.0000 1.0000 + 1.7321i 1.0000 - 1.7321i polyval(polysam,[0 1 2.5 4 6.5]) ans = 8.0000 9.0000 23.6250 72.0000 282.6250 polyder(polysam) ans = 3 0 0 [r p k]=residue(polysam,[1 2 1]) r = 3 7 p = -1 -1 k = 1 -2 Example

48. x = [0: 0.1: 2.5]; y = erf(x); p = polyfit(x,y,6) p = 0.0084 -0.0983 0.4217 -0.7435 0.1471 1.1064 0.0004 x = [0: 0.1: 2.5]; y = erf(x); p = polyfit(x,y,6) p = 0.0084 -0.0983 0.4217 -0.7435 0.1471 1.1064 0.0004 Example interp1(x,y,[0.45 0.95 2.2 3.0]) ans = 0.4744 0.8198 0.9981 NaN interp1(x,y,[0.45 0.95 2.2 3.0]) ans = 0.4744 0.8198 0.9981 NaN

49. Exercise 2 (5 minutes) Let x be an array of values from 0 to 2, equally spaced by 0.01 –Compute the array of exponentials corresponding to the values stored in x –Find the polynomial p of degree 5 that is the best least square approximation to y on the given interval [0,2] –Evaluate the polynomial p at the values of x, and compute the error z with respect to the array y –Interpolate the (x,z) data to approximate the value of the error in interpolation at the point.9995

50. Programming and Application Development

51. Topics discussed… The concept of m-file in MATLAB Script versus function files The concept of workspace Variables in MATLAB –Type of a variable –Scope of a variable Flow control in MATLAB The Editor/Debugger

52. Before getting lost in details… Obtaining User Input –“input” - Prompting the user for input >> apls = input( ‘How many apples? ‘ ) –“keyboard” - Pausing During Execution (when in M- file) Shell Escape Functions (! Operator) Optimizing MATLAB Code –Vectorizing loops –Preallocating Arrays

53. Function M-file function r = ourrank(X,tol) % rank of a matrix s = svd(X); if (nargin == 1) tol = max(size(X)) * s(1)* eps; end r = sum(s > tol); function r = ourrank(X,tol) % rank of a matrix s = svd(X); if (nargin == 1) tol = max(size(X)) * s(1)* eps; end r = sum(s > tol); function [mean,stdev] = ourstat(x) [m,n] = size(x); if m == 1 m = n; end mean = sum(x)/m; stdev = sqrt(sum(x.^2)/m – mean.^2); function [mean,stdev] = ourstat(x) [m,n] = size(x); if m == 1 m = n; end mean = sum(x)/m; stdev = sqrt(sum(x.^2)/m – mean.^2); Multiple Input Arguments use ( ) Multiple Output Arguments, use [ ] »r=ourrank(rand(5),.1); »[m std]=ourstat(1:9);

54. Basic Parts of a Function M-File function y = mean (x) % MEAN Average or mean value. % For vectors, MEAN(x) returns the mean value. % For matrices, MEAN(x) is a row vector % containing the mean value of each column. [m,n] = size(x); if m == 1 m = n; end y = sum(x)/m; Output Arguments Input Arguments Function Name Online Help Function Code

55. Script and Function Files Script Files Work as though you typed commands into MATLAB prompt Variable are stored in MATLAB workspace Function Files Let you make your own MATLAB Functions All variables within a function are local All information must be passed to functions as parameters Subfunctions are supported

56. The concept of Workspace At any time in a MATLAB session, the code has a workspace associated with it The workspace is like a sandbox in which you find yourself at a certain point of executing MATLAB Base Workspace: the workspace in which you live when you execute commands from prompt Remarks: Each MATLAB function has its own workspace (its own sandbox) A function invoked from a calling function has its own and separate workspace (sandbox) A script does not lead to a new workspace (unlike a function), but lives in the workspace from which it was invoked

57. Variable Types in MATLAB Local Variables In general, a variable in MATLAB has local scope, that is, it’s only available in its workspace The variable disappears when the workspace ceases to exist Recall that a script does not define a new workspace – be careful, otherwise you can step on variables defined at the level where the script is invoked Since a function defines its own workspace, a variable defined in a function is local to that function Variables defined outside the function should be passed to function as arguments. Furthermore, the arguments are passed by value Every variable defined in the subroutine, if to be used outside the body of the function should be returned back to the calling workspace

58. Variable Types in MATLAB Global Variables These are variables that are available in multiple workspaces They have to be explicitly declared as being global Not going to expand on this, since using global variables are a bad programming practice Note on returning values from a function Since all variables are local and input arguments are passed by value, when returning from a function a variable that is modified in the function will not appear as modified in the calling workspace unless the variable is either global, or declared a return variable for that function

59. if ((attendance >= 0.90) & (grade_average >= 60)) pass = 1; end; if ((attendance >= 0.90) & (grade_average >= 60)) pass = 1; end; eps = 1; while (1+eps) > 1 eps = eps/2; end eps = eps*2 eps = 1; while (1+eps) > 1 eps = eps/2; end eps = eps*2 if Statement while Loops Flow Control Statements

60. a = zeros(k,k) % Preallocate matrix for m = 1:k for n = 1:k a(m,n) = 1/(m+n -1); end a = zeros(k,k) % Preallocate matrix for m = 1:k for n = 1:k a(m,n) = 1/(m+n -1); end method = 'Bilinear'; switch lower(method) case {'linear','bilinear'} disp('Method is linear') case 'cubic' disp('Method is cubic') otherwise disp('Unknown method.') end Method is linear method = 'Bilinear'; switch lower(method) case {'linear','bilinear'} disp('Method is linear') case 'cubic' disp('Method is cubic') otherwise disp('Unknown method.') end Method is linear for Loop: switch Statement: Flow Control Statements

61. Editing and Debugging M-Files The Editor/Debugger Debugging M-Files –Types of Errors (Syntax Error and Runtime Error) –Using keyboard and “ ; ” statement –Setting Breakpoints –Stepping Through  Continue, Go Until Cursor, Step, Step In, Step Out –Examining Values  Selecting the Workspace  Viewing Datatips in the Editor/Debugger  Evaluating a Selection

62. Debugging Select Workspace Set Auto- Breakpoints tips

63. Importing and Exporting Data Using the Import Wizard Using Save and Load command load fname load fname x y z load fname -ascii load fname -mat load fname load fname x y z load fname -ascii load fname -mat save fname save fname x y z save fname -ascii save fname -mat save fname save fname x y z save fname -ascii save fname -mat

64. Read formatted data, reusing the format string N times. Import and Exporting Numeric Data with General ASCII delimited files »[A1…An]=textread(filename,format,N) Input/Output for Text File » M = dlmread(filename,delimiter,range)

65. Input/Output for Binary File fopen: Open a file for input/output fclose: Close one or more open files fread: Read binary data from file fwrite: Write binary data to a file fseek: Set file position indicator » fid= fopen( ' mydata.bin ', ' wb ' ); » fwrite (fid,eye(5), ' int32 ' ); » fclose (fid); » fid= fopen( ' mydata.bin ', ' rb ' ); » M= fread(fid, [5 5], ' int32 ') » fclose (fid); » fid= fopen( ' mydata.bin ', ' wb ' ); » fwrite (fid,eye(5), ' int32 ' ); » fclose (fid); » fid= fopen( ' mydata.bin ', ' rb ' ); » M= fread(fid, [5 5], ' int32 ') » fclose (fid);

66. Exercise 3: A debug session (10 minutes) Use the function demoBisect provided on the next slide to run a debug session –Save the MATLAB function to a file called demoBisect.m in the current directory –Call once the demoBisect.m from the MATLAB prompt to see how it works >>help demoBisect >>demoBisect(0, 5, 30) –Place some breakpoints and run a debug session Step through the code, and check the values of variables Use the MATLAB prompt to echo variables Use dbstep, dbcont, dbquit commands

67. function xm = demoBisect(xleft,xright,n) % demoBisect Use bisection to find the root of x - x^(1/3) - 2 % % Synopsis: x = demoBisect(xleft,xright) % x = demoBisect(xleft,xright,n) % % Input: xleft,xright = left and right brackets of the root % n = (optional) number of iterations; default: n = 15 % % Output: x = estimate of the root if nargin<3, n=15; end % Default number of iterations a = xleft; b = xright; % Copy original bracket to local variables fa = a - a^(1/3) - 2; % Initial values of f(a) and f(b) fb = b - b^(1/3) - 2; fprintf(' k a xmid b f(xmid)\n'); for k=1:n xm = a + 0.5*(b-a); % Minimize roundoff in computing the midpoint fm = xm - xm^(1/3) - 2; % f(x) at midpoint fprintf('%3d %12.8f %12.8f %12.8f %12.3e\n',k,a,xm,b,fm); if sign(fm)==sign(fa) % Root lies in interval [xm,b], replace a a = xm; fa = fm; else % Root lies in interval [a,xm], replace b b = xm; fb = fm; end

68. Other Tidbits

69. Nonlinear Numerical Functions inline function » f = inline('3*sin(2*x.^2)','x') f = Inline function: f(x) = 3*sin(2*x.^2) » f(2) ans = 2.9681 » f = inline('3*sin(2*x.^2)','x') f = Inline function: f(x) = 3*sin(2*x.^2) » f(2) ans = 2.9681 Use char function to convert inline object to string Numerical Integration using quad » Q = quad('1./(x.^3-2*x-5)',0,2); » F = inline('1./(x.^3-2*x-5)'); » Q = quad(F,0,2); » Q = quad('myfun',0,2) » Q = quad('1./(x.^3-2*x-5)',0,2); » F = inline('1./(x.^3-2*x-5)'); » Q = quad(F,0,2); » Q = quad('myfun',0,2) function y = myfun(x) y = 1./(x.^3-2*x-5); Note : quad function use adaptive Simpson quadrature

70. Nonlinear Numerical Functions fzero finds a zero of a single variable function –fun is inline function or m-function fminbnd minimize a single variable function on a fixed interval. x 1 <x<x 2 fminsearch minimize a several variable function Use optimset to determine options parameter. [x,fval]= fzero(fun,x0,options) [x,fval]= fminbnd(fun,x1,x2,options) [x,fval]= fminsearch(fun,x0,options) options = optimset('param1',value1,...)

71. Ordinary Differential Equations (Initial Value Problem) An explicit ODE with initial value: Using ode45 for non-stiff functions and ode23t for stiff functions. [t,y] = solver(odefun,tspan,y0,options) [initialtime finaltime] Initialvlue function dydt = odefun(t,y) Use odeset to define options parameter

72. ODE Example: » [t,y]=ode45( ' myfunc ',[0 20],[2;0]) function dydt=myfunc(t,y) dydt=zeros(2,1); dydt(1)=y(2); dydt(2)=(1-y(1)^2)*y(2)-y(1); Note: Help on odeset to set options for more accuracy and other useful utilities like drawing results during solving.

73. Example 4: Using ODE45 (5 minutes) Use the example on the previous page to solve the slightly different IVP on the interval [0,20] seconds:

74. Graphics Fundamentals

75. Graphics and Plotting in MATLAB Basic Plotting –plot, title, xlabel, grid, legend, hold, axis Editing Plots –Property Editor Mesh and Surface Plots –meshgrid, mesh, surf, colorbar, patch, hidden Handle Graphics

76. 2-D Plotting Syntax: Example: plot(x1, y1, 'clm1', x2, y2, 'clm2',...) x=[0:0.1:2*pi]; y=sin(x); z=cos(x); plot(x,y,x,z,'linewidth',2) title('Sample Plot','fontsize',14); xlabel('X values','fontsize',14); ylabel('Y values','fontsize',14); legend('Y data','Z data') grid on x=[0:0.1:2*pi]; y=sin(x); z=cos(x); plot(x,y,x,z,'linewidth',2) title('Sample Plot','fontsize',14); xlabel('X values','fontsize',14); ylabel('Y values','fontsize',14); legend('Y data','Z data') grid on color line marker

77. Sample Plot TitleYlabelXlabelGridLegend

78. Displaying Multiple Plots Nomenclature: –Figure window – the window in which MATLAB displays plots –Plot – a region of a window in which a curve (or surface) is displayed Three typical ways to display multiple curves in MATLAB (other combinations are possible…) –One figure contains one plot that contains multiple curves Requires the use of the command “hold” (see MATLAB help) –One figure contains multiple plots, each plot containing one curve Requires the use of the command “subplot” –Multiple figures, each containing one or more plots, each containing one or more curves Requires the use of the command “figure” and possibly “subplot”

79. Subplots Syntax: » subplot(2,2,1); » … » subplot(2,2,2) »... » subplot(2,2,3) »... » subplot(2,2,4) »... » subplot(2,2,1); » … » subplot(2,2,2) »... » subplot(2,2,3) »... » subplot(2,2,4) »... subplot(rows,cols,index)

80. The “figure” Command Use if you want to have several figures open for plotting The command by itself creates a new figure window and returns its handle >> figure If you have 20 figures open and want to make figure 9 the default one (this is where the next plot command will display a curve) use >> figure(9) >> plot(…) Use the command close(9) if you want to close figure 9 in case you don’t need it anymore

81. Surface Plot Example x = 0:0.1:2; y = 0:0.1:2; [xx, yy] = meshgrid(x,y); zz=sin(xx.^2+yy.^2); surf(xx,yy,zz) xlabel('X axes') ylabel('Y axes') x = 0:0.1:2; y = 0:0.1:2; [xx, yy] = meshgrid(x,y); zz=sin(xx.^2+yy.^2); surf(xx,yy,zz) xlabel('X axes') ylabel('Y axes')

82. 3-D Surface Plotting contourf-colorbar-plot3-waterfall-contour3-mesh-surf

83. Specialized Plotting Routines bar-bar3h-hist-area-pie3-rose

84. Handle Graphics Graphics in MATLAB consist of objects: –root, figure, axes, image, line, patch, rectangle, surface, text, light Creating Objects Setting Object Properties Upon Creation Obtaining an Object’s Handles Knowing Object Properties Modifying Object Properties –Using Command Line –Using Property Editor

85. Root object Figure object UIControl objects UIMenu objects Axes object Figure object Surface object Line objects Text objects UIControl objects UIMenu objects Axes object Figure object Graphics Objects

86. 1. Upon Creation 2. Utility Functions 0- root object handle gcf- current figure handle gca- current axis handle gco- current object handle 3. FINDOBJ Obtaining an Object’s Handle h_obj = findobj(h_parent, 'Property', 'Value',...) h_line = plot(x_data, y_data,...) What is the current object? Last object created OR Last object clicked Default = 0 (root object)

87. Modifying Object Properties Obtaining a list of current properties: Obtaining a list of settable properties: Modifying an object’s properties  Using Command Line  Using Property Editor get(h_object) set(h_object) set(h_object,'PropertyName','New_Value',...)

88. Graphical User Interface What is GUI? What is figure and *.fig file? Using guide command GUI controls GUI menus

89. Push Buttons Radio Buttons Frames Checkbox Slider Edit text static text Axes

90. Guide Editor Property Inspector Result Figure

91. Data Types

92. Numeric Arrays Multidimensional Arrays Structures and Cell Arrays

93. Multidimensional Arrays » A = pascal(4); » A(:,:,2) = magic(4) A(:,:,1) = 1 1 1 1 1 2 3 4 1 3 6 10 1 4 10 20 A(:,:,2) = 16 2 3 13 5 11 10 8 9 7 6 12 4 14 15 1 » A(:,:,9) = diag(ones(1,4)); » A = pascal(4); » A(:,:,2) = magic(4) A(:,:,1) = 1 1 1 1 1 2 3 4 1 3 6 10 1 4 10 20 A(:,:,2) = 16 2 3 13 5 11 10 8 9 7 6 12 4 14 15 1 » A(:,:,9) = diag(ones(1,4)); 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 Page N Page 1 0 0 0 0 16 2 3 13 5 11 10 8 9 7 6 12 4 14 15 1 1 1 1 1 1 2 3 4 1 3 6 10 1 4 10 20 The first references array dimension 1, the row. The second references dimension 2, the column. The third references dimension 3, The page.

94. Structures Arrays with named data containers called fields. » patient.name='John Doe'; » patient.billing = 127.00; » patient.test= [79 75 73; 180 178 177.5; 220 210 205]; » patient.name='John Doe'; » patient.billing = 127.00; » patient.test= [79 75 73; 180 178 177.5; 220 210 205]; Also, Build structure arrays using the struct function. Array of structures » patient(2).name='Katty Thomson'; » Patient(2).billing = 100.00; » Patient(2).test= [69 25 33; 120 128 177.5; 220 210 205]; » patient(2).name='Katty Thomson'; » Patient(2).billing = 100.00; » Patient(2).test= [69 25 33; 120 128 177.5; 220 210 205];

95. Cell Arrays Array for which the elements are cells and can hold other MATLAB arrays of different types. » A(1,1) = {[1 4 3; 0 5 8; 7 2 9]}; » A(1,2) = {'Anne Smith'}; » A(2,1) = {3+7i}; » A(2,2) = {-pi:pi/10:pi}; » A(1,1) = {[1 4 3; 0 5 8; 7 2 9]}; » A(1,2) = {'Anne Smith'}; » A(2,1) = {3+7i}; » A(2,2) = {-pi:pi/10:pi}; Using braces {} to point to elements of cell array Using celldisp function to display cell array

96. Conclusion  Matlab is a language of technical computing.  Matlab, a high performance software, a high- level language  Matlab supports GUI, API, and …  Matlab Toolboxes best fits different applications  Matlab …

97. Getting more help Contact http://www.mathworks.com/support You can find more help and FAQ about mathworks products on this page. Contact comp.soft-sys.matlab Newsgroup Use google to find more information (like the content of this presentation, in the first place)

98. Example 1a

99. Example 1b

100. Example 1c