Matlab is a tool to make our life easier. Keep that in mind. The best way to learn Matlab is through examples, at the computer.

Transcription

1 Learn by doing! The purpose of this tutorial is to provide an introduction to Matlab, a powerful software package that performs numeric computations. The examples should be run as the tutorial is followed. A basic understanding of matrix algebra is assumed. Please consult a linear algebra text if necessary. Matlab is a tool to make our life easier. Keep that in mind. The best way to learn Matlab is through examples, at the computer. At startup Matlab opens several windows. You can customize the space and decide which windows to keep open. However, you will use mostly Editor window - to write m-files; Command window to type command wait for the prompt sign >>. You can start MATLAB by double clicking on the MATLAB icon that should be on the desktop of your computer. This brings up the window called the Command Window. This window allows a user to enter simple commands. To clear the Command Window type clc and next press the Enter key. To perform a simple computations type a command and next press the Enter key. For instance, >>s = s = 3 >>fun = sin(pi/4) fun = note: in this examples bold is what you type In the second example the trigonometric function sine and the constant π are used. In MATLAB they are named sin and pi, respectively. Note that the results of these computations are saved in variables whose names are chosen by the user. If they will be needed during your current MATLAB session, then you can obtain their values typing their names and pressing the Enter key. For instance, >>s s = 3 Variable name begins with a letter, followed by letters, numbers or underscores. MATLAB recognizes only the first 31 characters of a variable name. MATLAB statements are normally of the form: variable = expression ECE 140 MATLAB TUTORIAL Page 1

2 Expressions typed by the user are interpreted and immediately evaluated by the MATLAB system. If a MATLAB statement ends with a semicolon, MATLAB evaluates the statement but suppresses the display of the results. A matrix may be entered as follows: A = [1 2 3; 2 3 4; 3 4 5]; Note that the matrix entries must be surrounded by brackets [ ] with row elements separated by blanks or by commas. The end of each row, with the exception of the last row, is indicated by a semicolon. A matrix A can also be entered across three input lines as A = [ ]; In this case, the carriage returns replace the semicolons. A row vector B with five elements B = [ ] can be entered in MATLAB as B = [ ]; or B = [6,9,12,15,18] For readability, it is better to use spaces rather than commas between the elements. The row vector B can be turned into a column vector by transposition, which is obtained by typing C = B The above results in C = ECE 140 MATLAB TUTORIAL Page 2

3 Other ways of entering the column vector C are C = [ ] or C = [6;9;12;15;18] MATLAB is case sensitive in naming variables, commands and functions. Thus b and B are not the same variable. If you do not want MATLAB to be case sensitive, you can use the command casesen off. To obtain the size of a specific variable, type size(). For example, to find the size of matrix A, you can execute the following command: size(a) will yield ans = 3 3 The result will be a row vector with two entries. The first is the number of rows in A, the second the number of columns in A. HELP!!! One of the nice features of MATLAB is its help system. To learn more about a function you are to use, say svd, type in the Command Window >>help soundsc SOUNDSC Autoscale and play vector as sound. SOUNDSC(Y,...) is the same as SOUND(Y,...) except the data is scaled so that the sound is played as loud as possible without clipping. The mean of the dynamic range of the data is set to zero after the normalization. SOUNDSC(Y,...,SLIM) where SLIM = [SLOW SHIGH] linearly scales values in Y in the range [SLOW, SHIGH] to [-1, 1]. Values outside this range are not clipped. By default, SLIM is ECE 140 MATLAB TUTORIAL Page 3

4 [MIN(Y) MAX(Y)]. See also sound, wavplay, wavrecord. Reference page in Help browser doc soundsc If you do not remember the exact name of a function you want to learn more about use command lookfor followed by the incomplete name of a function in the Command Window. In the following example we use a "word" sv >>lookfor sound spectralanalysis1.m: %soundfile = eval(get(soundfilename,'string')); % gets the sound file name tony_sep.m: % READSOUNDS looks in a directory "sounds/" for sound files and use_phase_change.m: %sound mixture MagPhaSpecPlot.m: % Program to compute spectrogram of sound file and plot its magnitude and phase MagPhaSpecPlot2.m: % Program to compute spectrogram of sound file and plot its magnitude and phase BEEP Produce beep sound. SOUNDVIEW View and play sound with replay button AUREAD Read NeXT/SUN (".au") sound file. AUWRITE Write NeXT/SUN (".au") sound file. SOUND Play vector as sound. SOUNDSC Autoscale and play vector as sound. WAVPLAY Play sound using Windows audio output device. WAVREAD Read Microsoft WAVE (".wav") sound file. WAVRECORD Record sound using Windows audio input device. WAVWRITE Write Microsoft WAVE (".wav") sound file. XPSOUND Demonstrate MATLAB's sound capability. MACHNUMBER Compute Mach number using velocity and speed of sound. daqsong.m: %% Generation of Audio Using the Sound Card NNSOUND Turn Neural Network Design sounds on and off. PLAYSND Implementation for SOUND The helpwin command, invoked without arguments, opens a new window on the screen. To find information you need double click on the name of the subdirectory and next double click on a function to see the help text for that function. You can go directly to the help text of your function invoking helpwin command followed by an argument. For instance, executing the following command >>helpwin zeros ZEROS Zeros array. ZEROS(N) is an N-by-N matrix of zeros. ZEROS(M,N) or ZEROS([M,N]) is an M-by-N matrix of zeros. ZEROS(M,N,P,...) or ZEROS([M N P...]) is an M-by-N-by-P-by-... array of zeros. ECE 140 MATLAB TUTORIAL Page 4

5 ZEROS(SIZE(A)) is the same size as A and all zeros. See also ONES. generates an information about MATLAB's function zeros. MATLAB also provides the browser-based help. In order to access these help files click on Help in the main menu. To learn more about MATLAB capabilities you can execute the demo command in the Command Window or click on Help and next select Examples and Demos from the pull-down menu. Some of the MATLAB demos use both the Command and the Figure windows. All variables used in the current MATLAB session are saved in the Workspace. You can view the content of the Workspace by clicking on File in the toolbar and next selecting Show Workspace from the pull-down menu. You can also check contents of the Workspace typing whos in the Command Window. For instance, >>whos Name Size Bytes Class ans 1x1 16 double array (complex) fun 1x1 8 double array ld 1x1 8 double array rd 1x1 8 double array s 1x1 8 double array xi 1x1 8 double array xr 1x1 8 double array Grand total is 7 elements using 64 bytes shows all variables used in current session. You can also use command who to generate a list of variables used in current session >>who Your variables are: ans ld s xr fun rd xi To save your current workspace select Save Workspace as from the File menu. Chose a name for your file, e.g. filename.mat and next click on Save. Remember that the file you just created must be located in MATLAB's search path. Another way of saving your workspace is to type save filename in the Command Window. The following command save filename s saves only the variable s. Another way to save your workspace is to type the command diary filename in the Command Window. All commands and variables created from now will be saved in our file. The following command: diary off will close the file and save it as the text file. You can open this file in a text editor, by double clicking on the name of your file, and modify its contents if you wish to do so. ECE 140 MATLAB TUTORIAL Page 5

6 To load contents of the file named filename into MATLAB's workspace type load filename in the Command Window. To enter a statement that is too long to be typed in one line, use three periods,, followed by Enter. For instance, >>x = sin(1) - sin(2) + sin(3) - sin(4) + sin(5) -... sin(6) + sin(7) - sin(8) + sin(9) - sin(10) x = You can suppress output to the screen by adding a semicolon after the statement >>u = 2 + 3; Some Command Description % Comments: Everything appearing on the same line after % is not executed. clear clc clg length Matrix Operations Clears the variables or functions from the workspace Clears the command window during a session Clears the variables or functions from workspace Length of a matrix The basic matrix operations are addition (+), subtraction (-), multiplication (*), and conjugate transpose ( ) of matrices, and exponentiation ( ^). In addition to the above basic operations, MATLAB has two forms of matrix division: the left inverse operator \ or the right inverse operator /. Matrices of the same dimension may be subtracted or added. Thus if E and F are entered in MATLAB as >>E = [7 2 3;4 3 6;8 1 5]; >>F = [1 4 2;6 7 5;1 9 1]; and >>G = E F >>H = E + F then, matrices G and H will appear on the screen as G = ECE 140 MATLAB TUTORIAL Page 6

7 H = A scalar (1-by-1 matrix) may be added to or subtracted from a matrix. In this particular case, the scalar is added to or subtracted from all the elements of another matrix. For example, >>J = H + 1 gives J = Matrix multiplication is defined provided the inner dimensions of the two operands are the same. Thus if X is an n-by-m matrix and Y is a i-by-j matrix, X*Y is defined provided m = i. (the number of columns of X equals the number of rows of Y) Since E and F are 3-by-3 matrices, the product >>Q = E*F results in Q = Any matrix can be multiplied by a scalar. For example, >>2*Q gives ans = Note that if a variable name and the = sign are omitted, a variable name ans is automatically created. Matrix division can be the left division operator \ or the right division operator /. The right division a/b, for instance, is algebraically equivalent to, while the left division a\b is equivalent to. ECE 140 MATLAB TUTORIAL Page 7

8 If Z * I = V and Z is non-singular, then left division, Z\V is equivalent to the MATLAB expression I = inv(z)*v where inv is the MATLAB expression function for obtaining the inverse of a matrix. The right division denoted by V/Z is equivalent to the MATLAB expression I = V*inv(Z) There are MATLAB functions that can be used to produce special matrices. Examples are given in the following table. Function ones(n,m) eye(n) zeros(n,m) diag(a) Description Produces n-by-m matrix with all the elements being unity Gives n-by-n identity matrix Produces n-by-m matrix of zeros Produce a vector consisting of the elements on the diagonal of a square matrix A Array Operations Array operations refer to element-by-element arithmetic operations. Preceding the linear algebraic operations, * / \, by a period (.) indicates an array or element-by-element operation. Thus, the operators.*,.\,./,.^ represent element-by-element multiplication, left division, right division, and raising to the power, respectively. For addition and subtraction, the array and matrix operations are the same. Thus, + and.+ can be regarded as an array or matrix addition. If A1 and B1 are matrices of the same dimension, then A1.* B1 denotes an array whose elements are products of the corresponding elements of A1 and B1. Thus if, A1 = [ ]; B1 = [ ]; then C1 = A1.*B1 results in C1 = An array operation for left and right division also involves element-by-element operation. The expressions A1.\B1 give the quotient of element-by-element division of matrices A1 and B1. The statement D1 = A1./B1 gives the result ECE 140 MATLAB TUTORIAL Page 8

9 D1 = The array operation of raising to the power is denoted by.^. The general statement will be of the form: q = r1.^s1 If r1 and s1 are matrices of the same dimensions, then the result q is also a matrix of the same dimension. For example, if r1 = [7 3 5]; s1 = [2 4 3]; then q1 = r1.^s1 gives the result q1 = One of the operands can be scalar. For example, q2 = r1.^2 q3 = 2.^s1 will give q2 = q3 = Note that when one of the operands is a scalar, the resulting matrix will have the same dimensions as the matrix operand. Complex Numbers MATLAB allows operations involving complex numbers. Complex numbers are entered using function i or j. For example, a number z = 2 + j2 may be entered in MATLAB as z = 2+2*i or z = 2+2*j Also, a complex number can be entered in MATLAB as za = 2*sqrt(2)*exp((pi/4)*j) ECE 140 MATLAB TUTORIAL Page 9

10 It should be noted that when complex numbers are entered as matrix elements within brackets, one should avoid any blank spaces. For example, y = 3 + j4 is represented in MATLAB as y = 3+4*j If spaces exist around the + sign, such as u = 3 + 4*j MATLAB considers it as two separate numbers. If complex matrix is given as then we can represent it in MATLAB as w = [1+j 2-2*j; 3+2*j 4+3*j] which will produce the result w = i i i i If the entries in a matrix are complex, then the prime ( ) operator produces the conjugate transpose. Thus wp = w will produce wp = i i i i For the unconjugate transpose of a complex matrix, we can use the point transpose (. ) command. For example, wt = w. will yield wt = i i i i The imaginary unit is denoted either by i or j >>j ans = i ECE 140 MATLAB TUTORIAL Page 10

11 There are three kinds of numbers used in MATLAB: integers real numbers complex numbers In addition to classes of numbers mentioned above, MATLAB has three variables representing the nonnumbers: -Inf Inf NaN The Inf and Inf are the IEEE representations for the negative and positive infinity, respectively. Infinity is generated by overflow or by the operation of dividing by zero. The NaN stands for the not-a-number and is obtained as a result of the mathematically undefined operations such as 0.0/0.0 or. The Colon Symbol (:) The colon symbol (:) is one of the most important operators in MATLAB. It can be used (1) to create vectors and matrices, (2) to specify sub-matrices and vectors, and (3) to perform iterations. The statement t1 = 1:6 will generate a row vector containing the numbers from 1 to 6 with unit increment. MATLAB produces the result t1 = Non-unity (positive or negative) increments may be specified. For example, the statement t2 = 3:-0.5:1 will result in t2 = The statement t3 = [(0:2:10);(5:-0.2:4)] will result in a 2-by-4 matrix t3 = ECE 140 MATLAB TUTORIAL Page 11

12 Other MATLAB functions for generating vectors are linspace and logspace. Linspace generates linearly evenly spaced vectors, while logspace generates logarithmically evenly spaced vectors. The usage of these functions is of the form: linspace (i_value, f_value, np) logspace (i_value, f_value, np) where i_value is the initial value, f_value is the final value, and np is the total number of elements in the vector. For example, t4 = linspace (2,6,8) will generate the vector t4 = Columns 1 through Columns 4 through Individual elements in a matrix can be referenced with subscripts inside parentheses. For example, t2(4) is the fourth element of vector t2. Also, for matrix t3, t3(2,3) denotes the entry in the second row and third column. Using the colon as one of the subscripts denotes all of the corresponding row or column. For example, t3(:,4) is the fourth column of matrix t3. Thus, the statement t5 = t3(:,4) will give t5 = Also, the statement t3(2,:) is the second row of matrix t3. That is the statement t6 = t3(2,:) will result in t6 = If the colon exists as the only subscript, such as t3(:), the latter denotes the elements of matrix t3 strung out in a long column vector. Thus the statement t7 = ECE 140 MATLAB TUTORIAL Page 12

13 Graphing Signals In MATLAB both plot and stem can be used to plot a signal. In general, stem is used to plot short discrete-time signals, and plot is better suited for sampled approximations of continuous-time signals or for very long discrete-time signals. The difference between these functions can be seen by typing the following example: n1 = [-5:5]; n2 = [-5:0.1:5]; x1 = n1 >= -2 & n1 < 2; x2 = sin(n2); figure subplot(211), stem(n1,x1), subplot(212), plot(n2,x2) Example that we are going to use again! a bit later Sinusiodal signal x(t), whose frequency is 20Hz, is sampled at a sampling rate R = 1000Hz. Generate a Matlab program to plot signal x vs time. Solution R = 1000 Hz; %sampling rate in Hz t = 0:1/R:2; %chosen max time 2 seconds x = sin(2*pi*f.*t); figure, plot(t,x) M-Files and Scripts ECE 140 MATLAB TUTORIAL Page 13

14 More advanced computations often require execution of several lines of computer code. Rather than typing those commands in the Command Window you should create a file. Each time you will need to repeat computations just invoke your file. Another advantage of using files is the ease to modify its contents. Files that contain a computer code are called the m-files. There are two kinds of m-files: the script files and the function files. Both script and function files contain a sequence of commands. However, function files take arguments and return values. To list m-files in the current directory on your disk, you can use the MATLAB command what. The MATLAB command, type, can be used to show the contents of a specified file. Script files To make the m-file click on File next select New and click on M-File from the pull-down menu. You will be presented with the MATLAB Editor/Debugger screen. Here you will type your code, can make changes, etc. Once you are done with typing, click on File, in the MATLAB Editor/Debugger screen and select Save As. Chose a name for your file, e.g., firstgraph.m and click on Save. Make sure that your file is saved in the directory that is in MATLAB's search path. If you have at least two files with duplicated names, then the one that occurs first in MATLAB's search path will be executed. To open the m-file from within the Command Window type edit firstgraph and then press Enter key. You can open the m-file by selecting under the file menu select open file. Choose desired m-file and open it. You should see a new window that has your code in it. To illustrate the use of an m-file, type the following text into the Matlab Editor and save it as firstgraph. % Script file firstgraph. x = pi/100:pi/100:10*pi; y = sin(x)./x; plot(x,y) grid As another exercise, type Example 1 into the m-file text editor and save the m-file. To execute the script, under the debug menu, select run. You can also choose to press F5 (on a PC) to run your script. Verify that the output matches output from Example 1. Statements in a script file operate globally on the workspace data. Normally, when m- files are executing, the commands are not displayed on screen. The MATLAB echo command can be used to view m-files while they are executing. Function Files Function files are m-files that are used to create new MATLAB functions. Variables defined and manipulated inside a function file are local to the function, and they do not ECE 140 MATLAB TUTORIAL Page 14

15 operate globally on the workspace. However, arguments may be passed into and out of a function file. The general form of a function file is: function variable(s) = function_name(arguments) % help text in the usage of the function command lines end To illustrate the usage of function files and rules for writing m-file functions, let us look at an example. Write a function file to generate a sinusoidal sum, for given input frequencies, sampling rate, and desired length in seconds Solution function signal_add = u(f1,f2,r,tmax) % f-input frequncies, R- sampling rate, tmax max duration t = 0:1/R:tMax; signal_add = sin(2*pi*f1*t) + sin(2*pi*f2*t); end Save this m-file as signal_add.m It is very important that you give the function file the same name as the function itself. It is not necessary to compile the function file. Be sure the folder that you saved the function file to is in the same folder as the m-file that calls the function. Now suppose we want to find the sum of two sinusoids whose frequencies are 210 and 200 Hz, sampled at 2000Hz, for 2 seconds. The following statements can by typed in the MATLAB command window to reference the function x = u(210,200,2000,2); To interrupt a running program press simultaneously the Ctrl-c keys. Sometimes you have to repeat pressing these keys a couple of times to halt execution of your program. This is not a recommended way to exit a program, however, in certain circumstances it is a necessity. For instance, a poorly written computer code can put MATLAB in the infinite loop and this would be the only option you will have left. ECE 140 MATLAB TUTORIAL Page 15