MORE MATLAB. MATLAB variables

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1 MORE MATLAB This lab experience assumes that you have a basic grasp of the principles in the first tutorial. You will gain some further hands-on experience with some of the core functionality of MATLAB and learn how to use it to solve a few more basic engineering problems. Remember, if you need help with a specific function, you can always type help < function > at the MATLAB prompt. If you don't know the function name, you can just type help help or use the lookfor function. MATLAB variables... 1 Complex numbers... 4 M-file scripts and functions... 5 Homework... 7 MATLAB variables Using vectors in MATLAB can greatly simplify coding and speed computation. To make some interesting comparisons, you can use timing functions in MATLAB % starts a stopwatch timer % reports the elapsed time since was performed Note that you must enter instantly, or it will include the time you took to type it. You can do this by opening the editor. Click the New Script button (or type Ctrl-N) and type the code into the editor. Then highlight the code you want to run, and press F9. This will execute the highlighted code snippet. Create a long sine wave using a for loop, and time the whole process. for i=1: , t = (i-1)/16000;

2 s(i) = sin(2*pi*t); end Now do the same thing without a for loop, and time this. i = 1: ; t = (i-1)/16000; s = sin(2*pi*t); The vectorized version above can be implemented more succinctly as: s = sin(2*pi/16000*[0:999999]); The third one is most efficient in terms of both multiplies and memory accesses. Try running the for-loop version again and comparing the time to the first time you ran it. Report the speed the first and second times you ran this code snippet. The reason the for-loop code runs faster the second time through is that the vector variable s has already been created the second time through instead of being created on the fly. MATLAB can generate a vector at once much more efficiently than it can build a vector by adding to the end of an existing vector. To avoid slowing down a loop that uses a vector, create the vector before the for-loop using the zeros function. First, clear the s variable to force MATLAB to recreate it:

3 clear s Then create the variable and run the loop snippet again: s = zeros(1, ); for i=1: , t = (i-1)/16000; s(i) = sin(2*pi*t); end Revisit the Fibonacci sequence generator from the first MATLAB lab. Calculate the first 1,000,000 Fibonacci numbers. Then speed up this code by creating the f vector and then calculate the first 1,000,000 Fibonacci numbers. Report the timing comparison. You can view or modify an array value by referring to its indices. Create a 2x3 random array: x = randn(2,3) To see only one value, at location (2,2): x(2,2) or change the value like this: x(2,2) = 12; and view the entire array:

4 x Colon notation : can be used to specify a range of indices in an array. To see all elements on the second row: x(2,:) To see elements 2 and 3 on the second row: x(2,2:3) Complex numbers MATLAB can handle complex-valued variables and arrays conveniently. The letter i and the letter j denote the imaginary unit. (Note, however, that if you set either i or j to a parular value, it will not act as the imaginary unit any longer in that session. This is a parular pitfall when using loops, as the letters i and j are often inadvertently chosen as loop index variables. Determine what each of these commands does: z = 3 + 4i real(z) imag(z) y = 2i w = z*y z2 = z*conj(z) z2b = abs(z)^2 za = angle(z) zm = abs(z) zb = zm*exp(i*za) Note that the transpose ' operator has a hidden function. It can transpose a matrix:

5 x' % compare to x above However, it also takes the complex conjugate of a matrix or scalar. In the scalar case, it is equivalent to conj(): conj(z) z' M file scripts and functions MATLAB allows user-defined scripts and functions to extend its capability. A sequence of commands can be placed in a file and stored with a.m extension. This file can then be executed in MATLAB using the filename (without the.m) as a command. In the Command History window, you can highlight commands you want to store as a script and then right-click to Create Script. This will bring up the MATLAB Editor, where you can modify the commands and then save to a file. Click on the New Script icon in the Editor Toolbar (see figure below), and enter the following commands: t = [0:1/100:2]; wave = sin(2*pi*t); plot(t,wave) Save to a file called plotwave.m. Then run plotwave MATLAB also allows users to create functions -- special scripts in which arguments can be passed in and results returned. In this section, we will produce two functions that will allow us to model the behavior of a simple system. Use the section on functions in the MATLAB chapter of the text as a resource. Study the sinewav function example.

6 We want to create a function that creates a different signal. Instead of creating a sine wave, we will create a signal that has a constant term plus a decaying exponential term. Here is an example for a constant 2, a decaying exponential of magnitude 3, and a time constant of 4: dur = 10; t = [0:1:dur]; xout = 2 + 3*exp(-t/4); Feel free to plot this to verify its behavior. Now write a function called expc that takes an input argument con for constant, mag for exponential magnitude, and tc for time constant and returns the output signal xout and the time base t. Use sinewav as an example to follow. Save the function to expc.m. Run your function with the values in the previous example to verify that it works properly. You should be able to call the function like this: xout_function = expc(2,3,4); Now we will create and save a function that calculates the error (sum of squared errors) between the function and a set of data. function err = my_err(vars) data1 = [ ]; dur = 10; t = [0:1:dur]; xout = vars(1) + vars(2)*exp(-t/vars(3)); err = sum((data1 - xout).^2);

7 We can use various functions in MATLAB to minimize the error between the signal model and the data by making the best choice of the variables that control the signal model. One of these functions is fminunc -- minimize a function that is unconstrained. fmincon has two arguments: one is the name of the function be minimized, expressed as a string -- 'my_err'; the other is a vector variable containing the unknowns to be determined in the model. An initial guess for all the values is required. In the case above, we guess con=3.5, mag=2, and tc=5. So we let the initial vector be X0 = [ ]; and call the function: X = fminunc('my_err',x0) Using the returned values, create the signal evaluated at the defined t values. Plot this on the same plot with the data points to compare the two. Homework (to be done individually and turned in before the beginning of the next lab): Write a short memo to Dr. Reeves using MS Word: 1. Create an m-file script of your lab work. After appropriately editing and commenting your script, use the File->Publish to publish a Word version of your lab work with headings and code. Save to PDF and submit. Note that comments can be included by preceding them with a percent symbol.