A Very Brief Introduction to Matlab

Transcription

1 A Very Brief Introduction to Matlab by John MacLaren Walsh, Ph.D. for ECES 63 Fall 26 October 3, 26 Introduction To MATLAB You can type normal mathematical operations into MATLAB as you would in an electronic calculator. Begin by typing +3 [Enter], you should see EDU>> +3 Your display may not include the EDU before the prompt. This just indicates that you are using a version other than the student version of MATLAB. Other mathematical operations are just as simple as addition. For example, here s how to subtract two numbers EDU>> -2 - Multiplication is naturally just as simple, as is handling non-integer numbers EDU>> 2.25*2.5 To calculate a number to an exponent use the syntax EDU>> 2^2 The syntax for other binary operations implemented in MATLAB is shown in Table. Suppose we wish to store the result of an operation so we can access it later. We can do this by assigning a variable name to the result of an operation, i.e. EDU>> r=3*5; EDU>> s=2.2; EDU>> r*s 33

2 MATLAB Expression Description x+y Add x and y. x-y Subtract y from x. x*y Multiply x and y. x/y Divide x by y. x y x to the yth power ( ) Parenthesis x= y Assign the value of y to x x == y Test to see if each elements of x and y are equal Table : Arithmetic Operations in MATLAB. Here, the variable names were r and s. Variable names consist of a letter, followed by any number of letters, digits, or underscores. MATLAB uses only the first 3 characters of a variable name. MATLAB is case sensitive; it distinguishes between uppercase and lowercase letters. A and a are not the same variable. To view the matrix assigned to any variable, simply enter the variable name. Also, you should not pick variable names that are the same as the name of a function (to be discussed later). We suppressed MATLAB s output for the first two commands by including a semicolon (;) at the end of the command. Notice also that = here works differently from the usual mathematical notion of equality, instead it acts as an assignment operator. If you want to test whether or not two items are equal, you should use ==. MATLAB will return if the two items are equal and if they are not. EDU>> s=2.2; EDU>> s==2.2 EDU>> s==33. Vectors and Matrices in MATLAB MATLAB is built to naturally work with vectors and matrices (the former are a special case of the latter with one of the two dimensions having length ). To input a column vector into MATLAB, surround the vector you wish to input with brackets, and separate its elements with semicolons: EDU>> v=[ ; 2 ; 3 ; ] v = 2 3 Row vectors are input in a similar manner, only the elements are separated by commas or spaces. EDU>> v2=[,2,3,] Quote from MATLAB documentation. 2

3 v2 = 2 3 EDU>> v2=[ 2 3 ] v2 = 2 3 Note that another way to make this same vector is EDU>> v2=[:] v2 = 2 3 The second way of creating v2 exploited an operator, :, in MATLAB that counts in increments of from its left argument to its right argument. Increments other than are also possible, one just uses a : x : b, where MATLAB will produce a row vectors whose elements count in increments of x from a to b. EDU>> v3=[:.25:2] v3 = You can concatenate two row or column vectors by using them as elements in a new row or column vector, respectively. For a column vector, imitate the following syntax. EDU>> v=[;2]; EDU>> v2=[3;;5]; EDU>> v=[v;v2] v = For a row vector, imitate the following syntax EDU>> v=[,2]; EDU>> v2=[3,,5]; EDU>> v=[v,v2] v = To enter a matrix, you just combine the ideas above. Here s an example 3

4 EDU>> A=[,2,3;,5,6] A = Note that you need to be careful that the number of rows are the same for every column, and likewise the number of columns are the same for every row in your matrix. Otherwise, MATLAB will produce an error. EDU>> A=[,2;,2,3];??? Error using ==> vertcat All rows in the bracketed expression must have the same number of columns. As another example, imagine you tried to concatenate a row vector and a column vector EDU>> v=[,2]; EDU>> v2=[3;;5]; EDU>> v=[v,v2];??? Error using ==> horzcat All matrices on a row in the bracketed expression must have the same number of rows. You can access elements within a matrix or vector by just specifying their coordinates. MATLAB uses based indexing 2. Thus, the first element in any vector, v, is denoted by v(). EDU>> v=[3,,5]; EDU>> v() 3 EDU>> v(2) EDU>> v(3) 5 You can also access multiple elements of a vector at a time. EDU>> v=[,2,3,,5]; EDU>> v(2:) 2 3 You can access elements of matrices by specifying their coordinate pairs (where the row number comes first and the column number comes second). 2 This is different from the based indexing used in C, C++, JAVA and other programming languages.

5 EDU>> A=[,2;3,] A = 2 3 EDU>> A(,) EDU>> A(2,2) EDU>> A(,2) 2 MATLAB operations are made to naturally work with matrices. Matrix addition and subtraction can be performed on any two matrices with the same dimensions EDU>> A=[,2;3,]; EDU>> B=[,;,]; EDU>> C=[,2,3;,5,6]; EDU>> A+B EDU>> A-B 2 3 EDU>> A+C??? Error using ==> plus Matrix dimensions must agree. EDU>> C-A??? Error using ==> minus Matrix dimensions must agree. Matrix multiplication uses the same syntax as the multiplication of two scalars 5

6 EDU>> A=[,2;3,;5,6] A = EDU>> B=[,3;2,]; EDU>> A*B Recall from linear algebra that matrix multiplication only makes sense when the number of columns in the left matrix matches the number of rows in the right matrix. Thus, matrix multiplication is not commutative for general matrices, and while A*B might exist, B*A might not even be defined. EDU>> B*A??? Error using ==> mtimes Inner matrix dimensions must agree. Occasionally, one wishes to multiply or divide two matrices of the same dimensions element by element, this can be done using the operator.* or./ respectively EDU>> A=[,2;3,]; EDU>> B=[,;,]; EDU>> A.*B 2 3 EDU>> B./A Check Yourself: Show how to make a row vector with elements between and evenly spaced by Controlling Program Flow in MATLAB: (if, for, etc) There are eight flow control statements in MATLAB 3 : if, together with else and elseif, executes a group of statements based on some logical condition. switch, together with case and otherwise, executes different groups of statements depending on the value of some logical condition. while executes a group of statements an indefinite number of times, based on some logical condition. 3 This section taken verbatim from the MATLAB Getting Started Documentation. 6

7 Name Description Example size returns the dimensions of a matrix argument size(a) ones Creates a m n matrix whose elements are all ones ones(m,n) zeros Creates a m n matrix whose elements are all zeros zeros(m,n) max Find the largest element in each column of a matrix max(a) min Find the smallest element in each column of a matrix min(a) help Displays help about a particular command. help size Table 2: Some MATLAB commands. Note that the help command does not place parenthesis around its arguments. for executes a group of statements a fixed number of times. continue passes control to the next iteration of a for or while loop, skipping any remaining statements in the body of the loop. break terminates execution of a for or while loop. try...catch changes flow control if an error is detected during execution. return causes execution to return to the invoking function. All flow constructs use end to indicate the end of the flow control block. You can learn more about these statements in MATLAB s help system under the heading MATLABLEARNING MATLAB FLOW CON- TROL. Get there by clicking the Help menu in the command window, then clicking on MATLAB Help, then clicking on MATLAB, then clicking on LEARNING MATLAB, then clicking on PROGRAMMING, then clicking on FLOW CONTROL..3 Using Built-in MATLAB Functions The power of MATLAB lies in the thousands of functions that come with it. Functions can take arguments and produce outputs. Arguments to the function are placed in parenthesis after the function name. For example, the size function returns the dimensions of a matrix argument EDU>> A=[,2,3;,5,6]; EDU>> size(a) 2 3 Outputs from the function can be assigned to variable names as you would assign the result of any expression. EDU>> z=size(a) z = 2 3 Note that some functions will give you more information if you ask for more output arguments. For example, the max function, which finds the largest element in a vector argument, has several options for the number of output arguments. If there is one output argument, max just returns the maximum value in the vector. If there are two output arguments, then max will return the maximum value in the first output argument and the index (i.e., its location) in the second argument. These are directions for Student Version R. If you are using an older version, search the documentation for help on flow control by clicking on the search tab, typing flow control, and clicking search. 7

8 EDU>> c=[,3,2]; EDU>> z=max(c) z = 3 EDU>> [z,i]=max(c) z = 3 i = 2 A few functions that you are certain to need are shown in Table 2. However, there are way too many functions in MATLAB that you will use to list here, so in the next section we will provide you a means for determining function names and learning how to use them. Before you start to write a program to do something, check to see if there is a MATLAB routine that already does it by using a keyword search in the documentation. Oftentimes you will find that there is already a program written to do what you need, and you just need to give it the proper parameters.. Getting Help on Functions One of the most important skills for a MATLAB user is the ability to find a MATLAB function that does what you need, and then learn how to use it. To do the former, it is often helpful to do a keyword search on the MATLAB documentation. To do this, just click on the HELP menu, and select MATLAB. Then, in the window that pops up, click on the search tab, and enter a word or phrase that describes what you wish to do. This often yields results more quickly than guessing function names. If you do not want to use the java based help documentation, you can use the command lookfor at the command prompt for this same purpose. Simply type lookfor followed by some keywords describing what you want to do. Once you know the name of the function you think will do what you need, type help followed by the function name to learn the syntax for it. For example EDU>> help size SIZE Size of array. D = SIZE(X), for M-by-N matrix X, returns the two-element row vector D = [M, N] containing the number of rows and columns in the matrix. For N-D arrays, SIZE(X) returns a -by-n vector of dimension lengths. Trailing singleton dimensions are ignored. [M,N] = SIZE(X) for matrix X, returns the number of rows and columns in X as separate output variables. [M,M2,M3,...,MN] = SIZE(X) returns the sizes of the first N dimensions of array X. If the number of output arguments N does not equal NDIMS(X), then for: N > NDIMS(X), size returns ones in the "extra" variables, i.e., outputs NDIMS(X)+ through N. N < NDIMS(X), MN contains the product of the sizes of the remaining dimensions, i.e., dimensions N+ through 8

9 NDIMS(X). M = SIZE(X,DIM) returns the length of the dimension specified by the scalar DIM. For example, SIZE(X,) returns the number of rows. When SIZE is applied to a Java array, the number of rows returned is the length of the Java array and the number of columns is always. When SIZE is applied to a Java array of arrays, the result describes only the top level array in the array of arrays. See also length, ndims, numel. Overloaded functions or methods (ones with the same name in other directories) help timer/size.m help serial/size.m help gf/size.m Reference page in Help browser doc size In the next few sections, we will familiarize you with some of the functions necessary for using MATLAB for ECES 63. Check Yourself: Find a function that sums the entries of a row vector. Show the help information for this function..5 Saving and Loading Data By now, you have probably noticed that if you assign a value to a variable name, MATLAB remembers that value until you assign the variable name a new value, or exit the program. To see which variable names MATLAB is currently remembering, you can use the command who. EDU>> clear all; EDU>> A=.5; EDU>> B=.3; EDU>> who Your variables are: A B A way to make MATLAB forget that you have assigned that variable name a value is to use the clear command. EDU>> A=.5; EDU>> A A = EDU>> clear A EDU>> A??? Undefined function or variable A. You can clear several variable names by listing them after the clear command, or, if you want to clear all of the variable names you can use clear all (as in the example above). Oftentimes, you will need to save data that you have created in MATLAB to a file on your hard disk, diskette, CD, or DVD so you can use 9

10 it at a later time. This can be done using the save and load commands. The following command saves the contents of A to a file in the current directory named savedata.mat, clears the workspace, verifies that there does not exist a variable named A anymore, then loads savedata.mat into the workspace, and verifies that the contents of A have been restored. EDU>> A=.5; EDU>> A A = EDU>> save savedata.mat A EDU>> clear all EDU>> who EDU>> load savedata.mat EDU>> who Your variables are: A EDU>> A A = Saving multiple variables is just as easy, simply list the variables that you wish to save after the filename in the save command. EDU>> save savedata.mat A B If you would rather save every variable in the workspace, then do not list any variables after the save command. EDU>> save savedata.mat Note that unless you give it another path, MATLAB saves the file in the current working directory. You can determine the current directory by typing pwd. If you wish to choose another working directory, you can use the cd command followed by the name of the directory you wish to go to. You can print the contents of the current directory using the ls command..6 Plotting and Labelling 2D and 3D Data Oftentimes you will want to plot a graph using MATLAB. MATLAB s plot command takes a vector of data for the x axis of your graph and a vector of y values representing the value of the function you wish to plot at each of these points. Let s say that you wanted to plot a single period of a sinusoid. The following commands will produce the plot shown in Figure. thetas=[-pi:pi/6:pi]; yvals=sin(thetas); plot(thetas,yvals, b- ); xlabel( \theta ); ylabel( sin(\theta) ); title( The sin Function ); The first line defines the vector of evenly spaced points in increments of π/6 between -π and π to use as the x axis in the plot. The second line evaluates the function that we wish to plot at each of these

11 The sin Function.8.2 sin(θ) θ Figure : sin(θ) for θ [ π, π]. points. The next line plots the values of the function versus the values we chose for the x axis, and the b- demands that the plotter connect these data points with a blue line. To learn about other options for the line style, type help plot. The next three lines label the graph in a self explanatory way. (Note that in general, if you wish to use a Greek letter in your plot, you can precede its English name by a backslash.) The command axis tight will remove the extra white space on either side of your plot (try it). Note also that when we plot a line in this manner, we need to choose the increments in the x-axis (pi/6 in this case) carefully, so that the graph appears smooth and the essential features are visible. Oftentimes, choosing this value is a matter of guessing and testing multiple times. Now let s suppose that you wish to plot a second line, cos(θ), on your graph. You can do this by typing hold on; plot(thetas,cos(thetas), rx- ); legend( sin(\theta), cos(\theta) ); which produces the graph shown in Figure 2. The hold on command sees to it that any new lines are drawn on top of the existing graph. If you do not use it, MATLAB will erase the previous lines and labels on the graph. Note that we used a different plot style this time, rx-, to make it easy to distinguish between the two graphs. Type help plot for more line styles. Finally, notice that we labelled the lines by using a legend. Order your labels in the legend in the same order that you plotted the graphs so that the correct label is applied to the correct line style. Note that we explored just one simple type of two dimensional plot here. MATLAB is also capable of creating a number of other specialized 2D plots. Type help graph2d and help specgraph for more information. One of these other specialized 2D plots is the stem plot, which is convenient if you have a small number of discrete time data samples you would like to plot. Type help stem for more information. Check Yourself: Create a labelled graph showing the function y = e x for x between and. Make sure your graph has a title and axis labels. You will probably find the MATLAB command exp useful for this exercise. Now suppose that you want to plot a graph describing a phenomenon where there were two independent variables instead of one. Let s imagine that you wish to plot z = exp ( (x 2 + y 2 ) ) as a function of x and y. We can do this in much the same way we created the two dimensional graph. First we create a grid of evenly spaced points in x and y (this gives us a matrix of x values and a matrix of y values). The command we will use to do this is called meshgrid. We then evaluate our function on

12 .8 The sin Function sin(θ) cos(θ).2 sin(θ) θ Figure 2: cos(θ) and sin(θ) for θ [ π, π]. this grid and call a plotting routine. There are several plotting routines to choose between. The following example investigates a mesh plot (Figure 3), a surface (Figure ), and a contour plot (Figure 5). A mesh plot is a collection of lines that look like a wire mesh that has been shaped to match a perspective of the surface you plotted. In this instance, the color of the line also shows the height of a line. A surface plot is similar to mesh plot, only now a perspective of the entire surface is drawn instead of just a grid. A contour plot is a plot in which contours in x and y corresponding to a constant value of z are shown. If you are interested in using any of these plots, be sure to read the help files for each. There are also a number of other types of 3D plots, type graph3d to get a list of commands. [xx,yy]=meshgrid([-:.5:],[-:.5:]); zz=exp(-*(xx.^2+yy.^2)); figure; mesh(xx,yy,zz); xlabel( x axis ); ylabel( y axis ); zlabel( z axis ); title( exp(-(x^2+y^2)) ); figure; surf(xx,yy,zz); xlabel( x axis ); ylabel( y axis ); zlabel( z axis ); title( exp(-(x^2+y^2)) ); figure; [c,f]=contour(xx,yy,zz); clabel(c,f); xlabel( x axis ); ylabel( y axis ); title( Contours that Yield exp(-(x^2+y^2)) Constant ); Check Yourself: Create a labelled mesh plot of the surface z = x sin(x + y) for < x < π and < y < π. 2

13 exp( (x 2 +y 2 )).8 z axis.2 y axis x axis Figure 3: A mesh plot of exp( x 2 y 2 ). exp( (x 2 +y 2 )).8 z axis.2 y axis x axis Figure : A surface perspective of exp( x 2 y 2 ). 3

14 .8.2 Contours that Yield exp( (x 2 +y 2 )) Constant y axis x axis.2 Figure 5: A contour plot of exp( x 2 y 2 )..7 Using.M files as scripts and functions Instead of entering commands line by line at the MATLAB command prompt, it is often more convenient to collect them all into a single MATLAB script. A MATLAB script is a text file filled with the MATLAB commands that you wish to execute. To create a MATLAB script file, either use the MATLAB editor or your favorite text editor to type all of your MATLAB commands, then save the file with a.m extension (for example myscript.m is a valid MATLAB script file name). You can now execute all of the commands in the file at once by typing the part of the file name before the.m (in our example this is myscript) at the MATLAB prompt. Note that you need to be in the same directory in MATLAB as the script file that you wish to execute (you can do this using the cd command as you would in a DOS shell). To test this out, cut and paste the commands in the last plotting example into a text editor (e.g., the MATLAB editor), save the file under a name with a.m extension, move MATLAB to the directory you saved it in (using the cd command) and execute it. Do you get the plots shown in the figures? Using.M files allows you to assemble all of your MATLAB code into one place so that it is one clear cohesive program. You should add comments to your code (comments are text that MATLAB ignores that people reading your code can read to help understand what your program is doing/ what you want your program to do). Comments begin with %, anything after a % is ignored by MATLAB EDU>> 2+2 % MATLAB will not read this A commented version of the last plotting example might look like %create an evenly spaced grid of points [xx,yy]=meshgrid([-:.5:],[-:.5:]); %evaluate the function to plot at these points zz=exp(-*(xx.^2+yy.^2)); figure; %open a new figure window mesh(xx,yy,zz); %create a mesh plot using the data xlabel( x axis ); %give the x axis a label ylabel( y axis ); %give the y axis a label zlabel( z axis ); %give the z axis a label

15 title( exp(-(x^2+y^2)) ); %give the graph a title figure; %open a new figure window surf(xx,yy,zz);%create a surface plot using the data xlabel( x axis ); %label the x-axis ylabel( y axis ); %label the y-axis zlabel( z axis ); %label the z-axis title( exp(-(x^2+y^2)) ); %title the graph figure; %open a new figure window [c,f]=contour(xx,yy,zz); %create a contour plot using the data clabel(c,f); %label the contours according to their zz value xlabel( x axis ); %label the x axis ylabel( y axis ); %label the y axis %give the graph a title title( Contours that Yield exp(-(x^2+y^2)) Constant ); MATLAB scripts are an efficient way of organizing code, but often one wishes to write a piece of code that one can use time and time again, only perhaps with different parameters. To do this, one can write their own MATLAB function file. Functions can both take and return parameters. Here is a function that takes a parameter x and returns x 2 function y=mynewfunction(x) % This is a function that returns its argument squared. y=x^2; You must save functions in files whose name is the same as the function name and have the.m extension. For our example, the file name must be mynewfunction.m. Try copying the function above into a file, saving it, and then execute mynewfunction(2). Do you see the following? EDU>> mynewfunction(2).8 Manipulating Audio Files Among other things, MATLAB is capable of processing audio signals, provided your computer has a sound card and speakers. If your computer is not capable of playing sound you may skip this section of the lab. In this first class on signal processing, we will use this ability to give an intuitive and interactive flavor to our experiments. MATLAB already has a few audio files ready to load. Try the following example load handel; sound(y,fs); You should hear a snippet from Handel s Hallelujah Chorus. MATLAB will also work with digital audio files saved using either a Microsoft WAVE sound file (.wav) or the NeXT/SUN (.au) format. The commands to load the data are simple and easy to use. To load a.wav file, use the command wavread and to load an.au file use the command auread. Here is an example in which a wave file is loaded and played [y,fs,nbits]=wavread( \windows\media\chimes.wav ); sound(y,fs); Check Yourself: Load the audio data in laughter.mat by typing load laughter;. Play the loaded audio..9 Where to Learn More Now that you have completed this introduction, you have learned what is necessary to use MATLAB in ECES 63. If you wish to become an advanced user, however, it is recommended that you read through the Learning MATLAB section in the MATLAB Documentation. Also, a good review of many of the ideas presented in this laboratory can be found at tutorials/intropage.html. 5