George Mason University ECE 201: Introduction to Signal Analysis Spring 2017

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1 Assigned: January 27, 2017 Due Date: Week of February 6, 2017 George Mason University ECE 201: Introduction to Signal Analysis Spring 2017 Laboratory Project #1 Due Date Your lab report must be submitted on blackboard by 5:00 PM on the day of your lab on the week that it is due. Late lab reports will be penalized as noted in the Lab syllabus. Description The purpose of this laboratory is to get you comfortable with the types of operations that you will be using in later labs. Of particular importance is an understanding of the usefulness of the help command. By the end of this lab, you should understand that MATLAB operates on vectors (arrays of numbers). One of the most important operators that you should understand and be comfortable with is the colon operator. After completing this lab, you should also know how to make simple plots with labels. Get started early! Lab Report Your lab report for this lab will consist of answers and complete documentation of the questions and exercises in Section 3. The pre-lab, Section 2, is designed to get you ready for the problems in Section 3. Guidelines for preparing the lab report were discussed in the first week of lab. If you have any questions, talk to your lab TA. Each student must do his or her own work on this lab. However, you may ask other students or any of the teaching staff for advice. Honor Code Forgeries and plagiarism are a violation of the honor code and any reasonable suspicion of an honor code violation will be reported. You are allowed to discuss lab exercises with other students, but the submitted work should be original and it should be your own work. 1 Introduction Before going to lab, you may wish to review Appendix A in your textbook and read Appendix B. You may also want to review your lecture notes on complex numbers. Before starting on the lab exercises, you may find it helpful to read the help that is available for the MATLAB commands used in this lab. If you do not have access to MATLAB at home, you may read about the commands that you will be using on the MATLAB web page found here: Of special importance is the colon operator, :. Another command that is important is linspace. In this lab you will also be using the plot command, plot, to make some simple plots, but you should read the help command on plot to explore the power of this command.

2 2 Pre-Lab MATLAB will be used extensively in all the labs. The primary goal of this lab is to familiarize yourself with using MATLAB. Please read Appendix B: Programming in MATLAB for an overview. Here are two specific goals for this lab: 1. Learn basic MATLAB commands and their syntax, including the help system. 2. Learn how to make simple plots of functions. On the course web page some manuals are available if you need additional reading material. 2.1 Getting Started In the first week of labs, you were shown how to run MATLAB. When MATLAB will see the MATLAB prompt, first starts, you >> MATLAB is now waiting for you to run a program or issue some MATLAB commands. (a) At the MATLAB prompt, type intro. This will run the MATLAB m-file (program) intro.m, which generates some output to the screen along with a plot. In the last line of the MATLAB window you will see the following: View the published version of this example to learn more about "intro.m". You may wish to click on the link in blue to see further documentation/explanations of the program. (b) Next type edit intro.m at the MATLAB prompt, >> edit intro.m Another window should appear where you will find a listing of the program (m-file) that you just ran, intro.m. Look through the commands and see how they are related to the output that was produced in the MATLAB window. Begin to familiarize yourself with these commands - their syntax and how they work. Pay particular attention to the fact that MATLAB is a program that works with vectors, such as a = [1, 2, 3, 4, 5] and matrices, A = 1 2 π 0 0 e 1 j (c) Explore the MATLAB help capability by entering the following commands at the MATLAB prompt (Note that it is possible to force MATLAB to display only one screen-full of information at once by issuing the command more on - hitting the space key will advance the screen to the next page):

3 >> help >> help plot >> help colon %<--- This one is VERY IMPORTANT!! >> help ops >> help zeros >> help ones Note: The text following the % is a comment; it may be omitted. (d) Run the MATLAB help browser by typing doc. The help browswer is a hypertext interface to the MATLAB documentation. Especially helpful is the link under MATLAB in the contents that is entitled Getting Started with Matlab. (e) Type demos and explore a variety of basic MATLAB commands and plots. 2.2 MATLAB as a Calculator MATLAB may be used as a calculator to evaluate mathematical expressions. Note that when typing an expression that is to be evaluated, if there is no semicolon at the end of the command, then MATLAB will echo the answer. For example, compare the following: >> exp(1); %<- No answer typed (but result is computed and stored in ans) >> log10(10) Note that here, and in the rest of the lab, these answers will not be shown in the document, even though they will be displayed in your MATLAB window. Type the following expressions at the MATLAB prompt. >> pi*pi - 10 >> sin(pi/4) The result of each calculation is stored in the variable ans. When a new calculation is performed, the value of ans is replaced with the new result. What would you expect to see when typing the following command: >> ans ^2 which is a command that squares the value stored in ans? 2.3 Variable Name Assignments Type the following commands, noting that the expressions on the right of the equal sign is evaluated and assigned to the named variable on the left. The third and fourth commands simply perform the calculation and saves the result in ans until it is overwritten by another command. >> x1 = sin(pi/5); >> x2 = cos(pi/5); >> x1^2 + x2^2 >> cos(pi/5); %<--- where is this result stored? >> y = sqrt( 1 - x1*x1); >> ans It should be obvious that pi is used to denote π and that sqrt finds the square root of the quantity inside the parentheses.

4 2.4 Complex Numbers Complex numbers are natural in MATLAB. The basic operations are supported. Try the following and see if you can determine the values that you would get from each command before typing the commands. >> z = 3 + 4i, w = j >> real(z), imag(z) >> abs([z,w]) %<-- Construction of a vector >> conj(z+w) >> angle(z) %<-- What are the units? >> exp( j*pi ) >> exp(j*[ pi/4, 0, -pi/4 ]) 2.5 Plotting Basic plotting is easy in MATLAB, and built into MATLAB are some very powerful functions for creating 1-d, 2-d and 3-d plots. For example, type the following command: >> wernerboy The basic plot command will plot a vector y versus a vector x. For example, try the following: >> x = [ ] >> y = x.*x - 3*x >> plot( x, y ) The first command creates a vector of length five with the numbers shown in the list. The star * denotes multiplication, so >> z=3*x; would create a vector where each element is formed by multiplying each of the values in x by three, z = [ 9, 3, 0, 3, 9] Use help arith to learn how the operation x.*x works when x is a vector. By typing x.*x and looking at the result, you may be able to determine what this operator does. 2.6 The Colon Operator When unsure about a command, use help. Make sure that you understand the colon notation. You may learn what the colon does by reading Appendix B or consulting the MATLAB help facility. Explain what the following MATLAB code will produce, and then type the commands to confirm your answer. >> jkl = 1 : 10 >> jkl = 2 : 4 : 17 >> jkl = 99 : -1 : 88 >> ttt = 2 : (1/9) : 4 >> tpi = pi * [ 0:0.1:2 ]

5 2.7 Three More Operators The commands zeros and ones are useful. Type the following commands to see what they produce: >> zeros(1,3) >> zeros(3,1) >> ones(5,1) You may create a vector of five zeros followed by five ones with the command, >> x = [zeros(1,5), ones(1,5)] Another very useful command to become comfortable with that will be used often in plotting is linspace. Read the documentation on this command, and then experiment with it by executing the following commands (see if you can determine what the output will be before you type the command): >> xx = [linspace(0,1,5)] >> yy = [zeros(1,3), linspace(0,1,5), ones(1,4)] Question: How would you modify this command to create a vector of three zeros followed by the numbers 0, 0.1, , 1.0 and then four more ones? 3 Lab Exercises In general, your lab write-up should indicate that you have acquired an understanding of the topics treated by the laboratory assignment. Answer the questions below in your lab report as an assessment of your understanding of this lab s objective: a working knowledge of the basics of MATLAB. If you do not know the answers to these questions go back to the lab and try to figure them out in MATLAB (remember the command help). Also, consult Appendix B of your textbook. 1. What is the difference between the following commands: >> x=[ ]; >> y=[1, 2, 3, 4, 5]; >> z=[1; 2; 3; 4; 5]; >> w=[1, 2; 3, 4] 2. You saw how easy it is for MATLAB to generate and manipulate vectors (i.e., 1-dimensional arrays of numbers). For example, consider the following: nn = 2*(0:60); mm = ones(1,60); kk = 0:pi/60:pi; (a) What is the length of the vector kk? Explain. (b) Which line above will produce a sequence of ones? (c) How would you modify one of the above lines of MATLAB containing seventy-five sixes? code to create a vector

6 3. You have seen that MATLAB has no problem handling complex numbers. Consider the following line of code: yy = 2 j 2 sqrt(2); (a) How do you get MATLAB to return the magnitude of the complex number yy? (b) How do you get MATLAB to return the phase of the complex number yy? What are the units of the phase? (c) Use the relationship z 2 = z z to write a line of MATLAB magnitude-squared of the complex number yy. 4. In class we have discussed the solution to the problem, z N = 1 code that returns the The values of z that solve this equation are called the N roots of unity (you may wish to consult your text for a review of this topic). We may find the N roots of unity and plot them in the z-plane using the MATLAB functions roots and zplane. To m-file roots finds the roots of a polynomial whose coefficients are stored in a vector. For example, >> c=[ ]; >> roots(c) would find the roots of the polynomial x 4 + 2x 3 + 3x 2 + 4x + 5 = 0 Note that the coefficients of the polynomial are the values given in the vector c. As another example, if we wanted to find the roots of the polynomial we would write >> c = [ ] >> roots(c) Note that you could also type >> roots([ ]) 2x 5 + 5x 3 + 2x = 0 To plot the roots of a polynomial in the z-plane with the unit circle denoted on the plot, you can use the zplane command. For example, to plot the roots of the polynomial in the example above, you would type: >> r = roots([ ]); >> zplane(r) (a) Use roots and zplane to plot the four roots of unity, i.e., the roots of the equation z 4 = 1 Check your answer by making sure that there are four s in the plot, and that one of them is at z = 1.

7 (b) Repeat (a) by plotting the eight roots of unity. (c) Repeat (a) by plotting the twenty roots of unity. Note that here you may wish to use the zeros command to simplify your MATLAB code. 5. Here you will be making a plot of a quadratic function. (a) Create a vector of 100 equally spaced numbers between zero and one and store them in the variable x. (b) Square each number in the vector x and store them in the variable y. Hint: Use the dot operator. (c) Make a plot of y versus x, >> plot(x,y) This should produce a plot of the function y = x 2 ; 0 x 1 6. What does the following sequence of commands produce? >> x=linspace(0,2*pi,100); >> y=sin(x); >> plot(x,y); grid