2 Amazingly Simple Example Suppose we wanted to represent the following matrix 2 itchy = To enter itchy in Matla

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1 Superlative-Laced Matlab Help for ECE155 Students Brian Kurkoski October 13, 1999 This document is an introduction to Matlab for ECE155 students. It emphasizes the aspects of Matlab that you'll need for this class, and offers some tips on good programming. 1 Amazingly Important Matlab Concepts Matlab variables are stored as matrices. Matlab matrices are analogous to arrays in other programming languages Matlab has scalars which are 1-by-1 matrices, and vectors which are 1-by-N matrices. Matlab uses double-precision floating point numbers by default. Matlab has a multitude of built-in special functions that may already do what you what to do. Try using helpdesk to find related functions. Matlab is a scripting language, with which you can either: Type in commands at the prompt (in this document, >> means the prompt), which is excellent for debugging. Put a group of commands into a batch file (called a M-file with the name FILENAME.M). You can then execute" this script file by typing FILENAME at the prompt. This is excellent for automating tasks and running programs. 1

2 2 Amazingly Simple Example Suppose we wanted to represent the following matrix 2 itchy = To enter itchy in Matlab, you would type the following: >> itchy = [ ; ; ] ; Some important things to note about this example: You don't need to declare itchy in advance You don't even need to declare the size of itchy, it is implicit in the assignment. The semicolons within the [... ;... ;...] separate the rows of the matrix. The semicolon at the end of the line means quiet". It can be omitted, which would print out the result of the operation. Once you have created a matrix, you can refer to individual portions of it, as in the following examples. The colon (:) is a special operator that refers to an entire column or entire row. matrix with index is equal to itchy(3,4) 24 itchy(2,:) [ ] itchy(2,[1 3]) [ ] In the last example, the 1st and 3rd columns of the second row have been selected. The column index is actually a 1-by-2 vector. Note that in Matlab (unlike C), indices begin at 1 (not 0). The colon can have a completely different meaning: is a shortcut for: scratchy = 1:10 scratchy = [ ] Also, you can specify the increment using the following format: is a shortcut for: scratchy = -4:2:4 scratchy = [ ] 2

3 3 Manipulating Marvelous Matrices In Matlab, applying a scalar to a matrix will change each element of the matrix, for example if a = -1: >> -scratchy >> a.* scratchy >> scratchy * -1 all produce the same thing: they take the negative of each element of the matrix scratchy. There are two different types of matrix-times-matrix multiplication: Matrix multiplication using the * operator. This is what you learned in your linear algebra class. Array multiplication using the.* operator. This multiplies two matrices of the same size on an element-by-element basis. Consider the following example. If:» 5 6 a= 4 3 Then the following results are produced: b =» >> a * b >> a.* b Of course for matrix multiplication, the inner dimension of the left-hand matrix has to agree with the outer dimension of the right-hand matrix. Array division can be accomplished using the similar./ operator. There is matrix division as well, but this topic is complicated see help slash for information on this infrequently used operator. Scalars added to a matrix will affect each element: >> scratchy

4 will subtract 10 from each element of a matrix. You can also add or subtract two matrices: >> a + b However, they must agree is size. There is no special array addition or subtraction operator (i.e. there is no.+ or.- ). The useful array power.^ operator will produce the exponent on an elementby-element basis: >> a.^ There is matrix power (^) as well, but this topic is complicated see help arith for information on this infrequently used operator. You can find the inverse of a matrix using the inv function. You can find the determinant of a matrix using the det function. You can take the transpose of a matrix using the ' apostrophe after the matrix. 4 Matlab Commands You'll be Happy You Know Commands perform system operations, as opposed to commands that operate on matrices. Useful Command Notes help The best command to know. Type help sum, for example for help with the sum command. help + Help with operators such as + - : [ ] help arith Help with multiplication and exponent operators helpdesk Run a complete help environment, includes a tutorial. who, whos Shows a list of current variables and their sizes. cd, dir, ls, mkdir,!, and Many common DOS or shell commands can other DOS/shell commands be executed directly from Matlab. Windows implementation seems to accept both DOS and UNIX commands, whereas UNIX only accepts UNIX commands.! will open an shell or DOS prompt. save, load Save and load your variables so you can come back to work later, or for storing results of long programs. 4

5 5 Matlab Functions You'll be Happy You Know The following is a list of functions that are commonly used in Matlab, along with an plain-language description of what they do. To learn more about the details of a particular function, use help <command>. Useful Function ones, zeros, eye linspace, logspace reshape sum size max, min floor, ceil, fix, round rand, randn find Plain Description ones(4,2) creates a 4-by-2 matrix filled with ones. eye(3) creates a 3-by-3 identity matrix. linspace(1,4,11) creates an 11-element vector, with linearly-spaced (i.e. evenly spaced) elements 1 is the first element, 4 is the 11th (final) element. logspace produces logarithmic spacing. Preserves the number of elements in a matrix, but changes the number of rows and columns sums the columns of a N-by-M matrix to produce a 1-by-M vector. Also works on vectors to produce a scalar. Returns height and width of a matrix (returns a vector). Use size(a,2) to get a scalar that is the length of 1-by-N vector a. Maximum and minimum by column. for rounding floating point numbers to integers. rand produces a random number, uniformly distributed between 0 and 1. randn produces a random number with normal (i.e. Gaussian) distribution, with mean 0 and variance one. Note: randn(1).* sqrt(var) + m produces a random number with normal distribution, with mean m and variance var. find searches a matrix for a particular condition, and returns the indices of the test matrix that meet a specified condition. Example: find( [ ] < 0) will produce the result [1 3] because the first and third elements in the vector are less than zero. Hint: find works better with vectors than matrices. 5

6 6 Simplest Plotting Plotting in Matlab is easy. This example is a M-file script that will generate graphs of a Lorentzian and a sine wave (not that the two have anything to do with each other, it is just an example). Two Functions, sin(ω 0 t) and 1/(1+t 2 ) f(t) sin(t) Lorenzian time (t) clear all close all time = -5:.1:5; y1 = sin(time); y2 = 1./ (1 + time.^ 2) ; plot(time,y1,'r-') hold on plot(time,y2,'g:') legend('sin(t)','lorentzian'); title('two Functions, sin( omega_0t)and 1/(1+t^2)') xlabel('time (t)'); ylabel('f(t)'); axis([ ]); The command clear all clears all variables. This line is recommended as the first line of any script file since trying to re-assign variables can lead to 6

7 unpredictable results. The command close all closes all open windows (i.e. windows containing the graph that you created the last time you ran this script) so that open windows don't accumulate on successive runs of the program. In the next three lines, the values for what will be the x-axis are defined, and the two functions (y1 and y2) that will be plotted are calculated. time is a 1-by-101 matrix, and so y1 and y2 are as well. The plot command in this case accepts the x-axis values as the first argument, the y-axis values as the second argument, and line color and type information as the third argument. The third argument is a special formatting string which can contain as many as one from each of these three columns: y yellow. point - solid m magenta o circle : dotted c cyan x x-mark -. dashdot r red + plus -- dashed g green * star b blue s square w white d diamond k black v triangle (down) ^ triangle (up) < triangle (left) > triangle (right) p h pentagram hexagram See help plot for more information on usage of the plot command. The hold on command causes the next plot to be displayed on the same graph that you just created. Omitting hold on between the two plot commands would cause the second graph to erase the first graph. To have two different windows open at the same time, see the figure command. The legend, title, xlabel and ylabel commands should be nearly self explanatory, with the following additions: Use the legend command after the plot command. The order that the plots occur in should correspond to the order that the legend labels occur in. Matlab can display Greek letters, if you put a slash before the name. For example, nlambda is and nlambda is Λ. Use ^ to display superscripts and to display subscripts. Finally, the axis command changes the y-limits of the graph to make it a bit more readable (the command keeps the x-limits of -5 and +5). 7

8 In order to print a graph (in Windows, at least) you can select File j Print on the graph's menu. However, if you want to print the graph to a postscript (or some other type) of file (like I did for this document), you can use print -deps filename.eps. Here is another example, using semilogy: 10 1 Repeat Accumuate codes with q=3, N=1024 My Results Published Results 10 2 Probability of Error P E Signal to Noise Ratio per bit (E / N ) db b 0 clear all close all mi20 =... [ ; ]; i20s20g =... [ ; ]; semilogy(i20s20g(1,:), i20s20g(2,:),'bv-') hold on semilogy(mi20(1,:), mi20(2,:),'rx-') legend('my Results','Published Results') title('repeat-accumulate codes with q=3, N=1024') xlabel('signal-to-noise Ratio per bit (E_b / N_0) db') ylabel('probability of Error P_E') axis([1 3 1E-5 0.1]) 8

9 7 Flow Control Flow control in Matlab is straight forward. Every if, while and for loop is terminated by the word end. Look at these examples: Forward for loop: for i = 1:30 end % blah blah blah Backwards for loop: for i = 30:-1:1 end % blah blah blah while loop with logical operators. Note that~= is does not equal" and you should use == for is equal" when testing (like C). Use & for logical AND", use j (the pipe) for logical OR". See help + for more logical operators: while (i == 2 & state ~= 0) end % blah blah blah You make complicated structures with ifs, elseifs and elses: if I == J A(I,J) = 2; elseif abs(i-j) == 1 A(I,J) = -1; else A(I,J) = 0; end Or simple structures: if I == J end A(I,J) = 2; 9

10 8 Toolboxes There are at least two Matlab toolboxes that you may find useful later on, the Communications Toolbox, and the Signal Processing Toolbox. These contain specific high-level functions that may do just what you are looking for. (A toolbox is nothing more than a collection of functions.) For example, the Communications Toolbox contains a viterbi function which will perform a limited form of Viterbi decoding. The Signal Processing Toolbox contains functions to help you design manytypes of filters (among other things). For example, the butter function designs a Butterworth filter. Carefully read the documentation for these higher-level functions. Although the name of the function may imply that it will perform the function that you desire, its operation may differ from you expectation. 9 Brian's Incredibly Useful Matlab Tips 9.1 for Loops Can Be Avoided Matlab is amazingly powerful in that complex operations can be performed in a single line of code. However, many beginning Matlab programmers are tempted to bring their experience from other programming languages. For example, to sum the contents in an array, most other languages would require you to write something like: addup = 0; for i = 1:size(a,2) addup = addup + a(i); end which isvalid Matlab code. However, it is far more efficient to write: addup = sum(a); The above line is also valid Matlab code, and is easier to read, and will execute faster. Often, a little bit of creative thinking, along with good knowledge of Matlab's function library can lead to very simple looking, and fast programs. Of course there are times when what you want to accomplish requires a for loop. 9.2 Functions Can Be Nested Suppose you need to generate a random sequence of +1 and -1 data, with +1 and -1 equally likely. A good way to do this would be first use the rand function to generate uniformly distributed numbers, then round numbers in the range 10

11 0-0.5 down to zero, and numbers in the range up to 1. Take the result, multiply by two, then subtract one. The following code will accomplish that: data = rand(1,20); % size is 20 data = round(data); data = data * 2-1; However, the same could be accomplished in a single line of code: data = round( rand(1,20) ) * 2-1 ; Of course you could go overboard on this technique, and it might be necessary to separate the code into more than one line to make it readable. 9.3 Common Functions Have Alternate Formats The ever-popular sum function normally will sum a matrices' columns and produce a vector, which contains the sum of each column. However, the sum function can accept a second argument which tells which direction to sum in. For example: 2 itchy = Then >>sum(itchy) is the same as >>sum(itchy,1) and both produce a sum on the columns: But >>sum(itchy,2) will produce a sum on the rows: Notice in the former case the result is a row vector, but in the later case the result is a column vector. Also commands such as ones,zeros, and rand will size their outputs according to the arguments that you pass them: 11

12 >>ones(2,4) >>x = rand(1,7) x = The size command used by itself produces an unfortunate output of a vector: >> t = ones(1,7); >> size(t) 1 7 However, you can get at the number that you really wantby using the following: >> size(t,2) 7 Actually, Matlab functions (such as size) can return multiple arguments if you use [ ] notation: >> [a,b] = size(t) a = 1 b = Always Use.* to Multiply Unless You Really Mean * There is a big difference between the array.* and matrix * operators. You should as a matter of habit use the.*,./ and.^ unless you really want todo matrix multiplication (which is less common), in which case you should use *. It is tempting to be lazy and not use the dot, particularly when working with scalars, but you'll avoid errors and unpredictable behavior when you use array operators. 12