ELEN E3084: Signals and Systems Lab Lab II: Introduction to Matlab (Part II) and Elementary Signals

Transcription

1 ELEN E384: Signals and Systems Lab Lab II: Introduction to Matlab (Part II) and Elementary Signals 1 Introduction In the last lab you learn the basics of MATLAB, and had a brief introduction on how vectors are defined and represented in MATLAB. In this lab you will learn how to further manipulate these quantities, and use them to represent signals. We will also depend our understanding on the use of complex numbers in MATLAB, which will be quite handy when we start representing signals in the frequency domain. A great feature of MATLAB is the wealth of visualization tools that are available to us. This is one of the features that makes MATLAB so appealing when compared with many programming languages. We will learn how to use some of these tools to plot various signals, and even create high-quality plots for presentations and publications. Before we begin we will start again the diary function which will capture all our commands. Go the lab2 directory using the cd command the same way as in UNIX. Then type: >> diary on; Again, note that the goal of these labs is for you to tinker and play with the concepts that are introduced, in addition to the completion of the required tasks. Show the diary file at the end of the session to a TA or LA, so that she/he can assess the extent of your work. 2 Complex numbers We have already seen that MATLAB has predefined the imaginary unit. The default variables i and j are built-in and hold the value of the imaginary unit or simply 1 (be careful not to use i or j as your own variables if you want to use complex numbers). For example all following statements define the same complex number. >> 2 + 3*i i >> 2 + 3*j i 1

2 >> 2 + 3i i >> 2 + 3j i Notice that you don t need the multiplication symbol * as MATLAB can parse the expression either way. Once we enter a complex number then pretty much all the relevant elementary functions presented in the previous lab can be used. For example even the complex logarithm is defined as we see in: >> x = 2 + 3*i; >> log(x) i Moreover, given a complex number z the functions real(z), imag(z), conj(z), abs(z) and angle(z) calculate the real part, the imaginary part, the conjugate, the absolute value and the angle of z respectively, where the angle is in radians, and defined in the interval ( π, π). For example for z = 1 j we have Re{z} = 1, Im{z} = 1, z = 1 + j, z = 2 and z = π/4. Now if we try to reproduce the same result with MATLAB we will get: >> z = 1 - j; >> disp([ real part:,num2str(real(z))]); real part: 1 >> disp([ imag part:,num2str(imag(z))]); imag part: -1 >> disp([ conjugate:,num2str(conj(z))]); conjugate: 1+1i >> disp([ abs value:,num2str(abs(z))]); abs value: >> disp([ angle:,num2str(angle(z))]); angle: The angle is in radians, so if you want it in degrees you just need to multiply by 18/π. 2

3 3 Vectors 3.1 Definition of vectors We have already seen how to create vectors in the previous lab. Here we further expand on what can be done. Recall some of the various ways you can create a vector row or column vector. Typing >> x = [ pi] or >> x = [1,2.5,3,pi] both give: x = As seen before the transpose operator {. } allows you to convert a column into a row vector, and vice-versa. >> x Vectors can be real or complex but numerical and string values cannot be mixed in the same vector 1. For example a complex vector can be defined as: >> z = [1 -j 3+2i] z = i i 1 For that we have to use cell arrays which is outside of the scope of this lab 3

4 If we want to define vectors straight in the column form instead of using the transpose operator we can concatenate the values using semicolons. >> x = [1;-12.3;pi;sqrt(2)] x = Creating vectors by using the above procedures can be quite tedious, and not practical if you need to create long vectors. The colon operator : allows you to do this quite easily. For example >> x=:1 x = gives you a row vector with all the values from up to 1, spaced by 1. You have already seen this construction when using for loops in the first lab session. You can modify the spacing between the various entries with an extra parameter >> x=:.3:3 x = Columns 1 through Columns 9 through Furthermore the spacing can be a negative number, giving you a decreasing sequence >> x=4:-.5:2 4

5 x = We will use all these constructions extensively to create and manipulate signals. 3.2 Manipulation of vectors We can select the k-th element of a vector using subscripted indexing. For example: >> a = [3;-5;12;] a = >> a(2) -5 >> a(3) 12 Notice the use of parenthesis and that elements are indexed starting at 1. The keyword end, first seen in the for and if constructs, can be used to select the last element of a vector, row or column. >> b = [-5,-2,,,12] b = >> b(end) 12 As we seen, in its simplest form the colon operator can generate an index vector. example: For >> idx1 = 3:6 idx1 =

6 This means that we can use the colon operator to slice a vector which means to select a part of it. For example: >> c = [-3,2,,2,5,7,9] c = >> c(3:6) or using the above defined idx1 >> c(idx1) Notice that when we use the colon operator for indexing we don t have to use square brackets. To Do 1 create a row vector x = (2, 3, 5, 7, 11, 13, 17, 19, 23, 29) in MATLAB. Now use the colon operator to display only the even entries of the vector, that is, the output should be the vector (3, 7, 13, 19, 29). Now do the same thing but display the output in reverse order, that is, the output should be (29, 19, 13, 7, 3) We can find the indices of the elements of a vector that match a certain logical statement using the find command. For example we can find the positive elements of the vector a using: >> a = [-2,3,-1,5,7] a = >> idx = find(a>) idx = This means the 2nd, 4th and 5th element of the vector a are positive (i.e., satisfy the statement a(i) >, for i = 2, 4, 5). Now if we use the variable idx we can see only the values of the elements that are positive: >> a(idx)

7 There is a shortcut way of doing the same thing, by simply typing a(a>). This shows you another way of indexing elements of a vector in MATLAB. In this case b=(a>) outputs a binary vector with the same length as a, and when typing a(b) you get the entries of the vector corresponding to the entries of b that are 1. Try it! Finally we can selectively delete elements of a vector using the empty square bracket operator. We can use all the methods to select elements such us subscripted indexing or slicing and just assign the empty element which is the empty square brackets. For example: >> b = [2,4,2,1,6,7,4,6] b = >> idx = find(b<=2) idx = >> b(idx) = [] b = As you see we found the indices of the elements of b that are smaller or equal to 2 and we deleted them from b. 3.3 Matrices Actually vectors are just a particular type of matrices, with either one row or one column. Matrices are extremely useful in signal processing applications. These can be used to represent images, transformations, etc. All the things that we learned so far about vectors apply to matrices as well, but instead you have to deal with two indexes instead of one. Matrices can be created as a concatenation of vectors. For example >> A=[1:4;5:8;9:12] A = or >> A=[1 2;3 4] 7

8 A = We often need to create special large vectors/matrices automatically since explicitly typing every element would be tedious. Vectors/matrices consisting of only zeros or ones turn out to be particularly useful in MATLAB. We can generate a row vector of zeros or ones using the commands: >> disp(zeros(1,5)) >> disp(ones(1,3)) If we need column versions we can get them either with transposition or by using: >> disp(zeros(5,1)) >> disp(ones(3,1)) A matrix of zeros or ones with n rows and m columns can be also easily created, for example >> A=zeros(4,3) A = yields a matrix of zeros, with 4 rows and three columns. 8

9 3.4 Arithmetic operations with vectors and matrices Arithmetic operations in MATLAB follow the usual matrix arithmetic rules. That is, you can multiply a scalar by a matrix or vector (every entry in the vector/matrix is multiplied by the scalar), and operations between vectors or matrices require that the sizes match properly, as you have learn in linear algebra (for addition and subtraction the vectors/matrices need to be of the same size, and for multiplication the number of columns of the first operand much match the number of rows of the second operand). Addition and subtraction is defined as an element-by-element operation: >> a = [1,3,-2]; >> b = [,2,1]; >> a + b >> a - b The multiplication, division and power operators, however need special attention. Normally these follow the rules of matrix multiplication, but if preceded by a dot these are performed element-by-element as illustrated below: >> a = [1,3,-2]; >> b = [,2,1]; >> b.* a 6-2 >> b./ a >> b.^ a 8 1 In the case of a scalar and a vector: >> a = [1,3,-2]; >> b = 2; >> a + b 9

10 3 5 >> a - b >> a.* b >> a./ b >> a.^ b Functions on vectors Most of the functions we have seen so far are vectorized which means that they can accept a vector as an input argument and perform their operation on each element separately. So for example: >> x = [1,3,2.1,exp(1)] x = >> log(x) >> sin(x) >> imag(x) We have to note though that there are some functions which have special meaning when operate on vectors. Assuming a vector a then the length(a) gives the number of elements whereas the sum(a) and prod(a) calculate the sum and product of its elements. The function diff(a) calculates the difference between each consecutive element. Assuming a column vector then flipud(a) reverses it upside-down and assuming a row vector then fliplr(a) reverses it left-right. For example: >> a = [1,3,2,5]; 1

11 >> disp(length(a)) 4 >> disp(sum(a)) 11 >> disp(prod(a)) 3 >> disp(diff(a)) >> disp(mean(a)) 2.75 >> disp(var(a)) >> disp(fliplr(a)) Notice that the diff command returns a vector with length one element less than the original length (3 instead of 4 elements in this case). To Do 2 Generate vectors. We will now make use of the commands we just learned in order to generate some specific vectors. Note that you shouldn t use any loops such as for or while just the colon operator and elementary functions. Show the LA or TA the code that generates each of the row vectors below: 1, 3, 5, 7, 9, 11 5, 5, 5, 5, 5 4-3i, 2-2i, - 1i, -2 4, 16, 64, Plotting and Graphics 4.1 Plots and sublots MATLAB makes it possible to create sophisticated plots. Using simple commands we are able to visualize elementary signals and even include them in our reports. Here we will only scratch the surface of the all possibilities. 11

12 Figure 1: A simple plot of a sine where the independent variables are the vector indices The basic idea of plotting in MATLAB is very simple. Assuming that the independent x and dependent y variables are stored in vectors of the same length then the plot(x,y) command plots the pairs of points in the two-dimensional plane. By default plot connects the points by straight line segments but options like dashed lines or no lines at all are possible (use the help command to learn more). If the plotting is successful, a window pops up containing the plot. Finally, using the menus on the window the plot can be printed or saved. If y is a vector, then the simplest plotting command is plot(y). In this case the vector elements are the dependent variables and the indices of the vector (in increasing order) are the independent variables. Here is an example with the resulting plot shown in figure 1: >> x = :pi/2:4*pi; >> y = sin(x); >> plot(y) Independent variables are included using the command plot(x,y) where the vector x contains the independent variables and the vector y (of the same length) contains the dependent variables. In this fashion more plots can be included in one figure by putting the corresponding pairs of vectors inside the parenthesis of the plot command. This is illustrated in the figure 2 and the code below: >> x = :pi/3:3; >> y1 = sin(x); >> y2 = cos(x); >> y3 = y1.^2; >> plot(x,y1,x,y2,x,y3) >> axis tight 12

13 Figure 2: A plot of three functions with independent variables More plots in the same figure look often neater by using the subplotting option of MATLAB. Here each plot command is preceded by the subplot command. The subplot and plot commands are separated by a comma. We will use a simple subplot command we can place plots vertically. The command is given by subplot(m,i,k) where m and k are integers and (for us) the middle number is always 1. The number m refers to the total number of plots and k refers to the indices of particular plots. Here is an example of the functions plotted in figure 2 but this time in figure 3 using subplots. >> x = linspace(,4*pi,3); >> y1 = sin(x); >> y2 = cos(x); >> y3 = y1.^2; >> subplot(3,1,1) >> plot(x,y1) >> axis tight >> subplot(3,1,2) >> plot(x,y2) >> axis tight >> subplot(3,1,3) >> plot(x,y3) >> axis tight Notice a function you haven t seen before linspace. Use the help command to learn what it does. 13

14 Figure 3: A plot of three functions with independent variables using three subplots 4.2 Miscellaneous commands Annotating your plots is essential to make then understandable (otherwise you quickly forgot what is what in the plot). Adding a title, x- and y-axis labels is easy to do using the commands title, xlabel and ylabel respectively. Adding a legend with names is also easy using the command legend. We can also manipulate the shape of the axis by using the axis command using a variety of arguments (see help axis for more). For better readability we can also make visible a grid using the grid on command. We can see those commands in action the following snippet of code which generates the Figure 4: >> t = linspace(,2,5); >> v = exp(-3*t); >> plot(t,v) >> grid on >> title( Exponential decay ) >> xlabel( Time (sec) ) >> ylabel( Voltage (mv) ) >> axis square Lastly two other useful commands are the figure command and the close command. With figure we tell MATLAB to create a new figure so that we don t overwrite any existing one. This way we can have multiple figures displayed on the screen at once. Using close we tell MATLAB to close the current figure and with close all we can close all the open figures. 14

15 1 Exponential decay Voltage (mv) Time (sec) Figure 4: Demonstrating miscellaneous plotting commands 4.3 Exporting plots for publications After using MATLAB s powerful plotting tools you often want to export your graphics to use them in reports or other publications. There are various ways of doing this for example using the print command. Another option is to use a series of specialized functions that were released with the same purpose, and are available at digest/december/export.shtml. You should download the four functions exportfig, previewfig, applytofig and restorefig and save them under the /lab2 directory. Now we will go through an example which will generate the plot on figure 5. To Do 3 Export a plot. We will now revisit the plot of figure 2 but this time we will export it using the exportfig function. Type the following code on the command prompt: >> x = linspace(,4*pi,3); >> y1 = sin(x); >> y2 = cos(x); >> y3 = y1.^2; >> plot(x,y1,x,y2,x,y3) >> grid on >> axis tight 15

16 Plot of sin, cos and sin 2 sin cos sin Figure 5: Example using the expfig function >> axis([ 4*pi ]) >> legend( sin, cos, sin^2 ); >> title( Plot of sin, cos and sin^2 ) Using the mouse resize the plot window so that it looks shorter in height as in figure 5. Then go the /lab2 directory using the cd command. Export the plot using the following commands and submit the expfig.eps file. >> set(gcf, paperunits, centimeters ); >> exportfig(gcf, expfig.eps, bounds, tight, width,8.5, linestylemap, bw ); The benefits of using this approach might not be apparent now but one can appreciate them when including plot in papers using L A TEX. For example this lab manual is written in L A TEX and figure 2 was created using the standard MATLAB exporting mechanism while figure 5 was created using exportfig. Notice that in figure 5 we managed to preserve the aspect ratio of the plot and that the line widths and font sizes are not scaled down. Also notice that the colors have been automatically substituted by different line types to make them legible in black and white. Lastly using the width parameter we were able to specify the exact width of the figure to 8.5cm which is the standard width for a two-column conference paper. 16

17 5 Elementary signals 5.1 Unit impulse, unit step and ramp functions Most of the theory presented in the Signals & Systems course examines the behavior of continuous-time signals and systems. MATLAB can use special, so-called symbolic variables to represent and manipulate continuous signals and systems (another software package called Mathematica has much more developed symbolic capabilities). When doing non-symbolic manipulations however MATLAB is limited to use discretized versions on signals. Using the vector notation and the plotting commands we introduced it is now easy to create and visualize signals. In fact we have already done that when we created sines and cosines for our plotting examples. Now we will write two simple functions to generate the unit step and ramp signals. To Do 4 Write and save in m-files under the /lab2 directory the following two functions (remember that the m-file should have the same name as the function definition plus the extension.m): The unit step function is defined as: function y = us(t) %US Unit step % y = us(t) % t: time index % y: signal y = (t >= ); us[t] = And the unit ramp function is defined as: ur[t] = { 1, t, t < { t, t, t < Write your own code to implement the ramp function and show it to your TA (LA) (the ramp function should be very similar to the unit step function, and you should be able to apply it to a vector as well). Now it is easy to create other signals using the functions we just wrote. For example: >> t = -5:.1:5; 17

18 Figure 6: A signal created using the unit step and unit ramp functions >> y = us(-t+2).* ur(t); >> plot(t,y) >> grid on >> axis([ ]) >> set(gcf, paperunits, centimeters ); >> exportfig(gcf, elem1.eps, bounds, tight, width,8.5, linestylemap, bw ); gives the plot in figure 6, and exports a.eps file. Using our functions it s now easy to visualize time shifting and time reversal. For example replacing above y=us(-t+2).* ur(t) by y=us(t+2).* ur(-t) gives the time-reversed signal. To make it even more intuitive create a vectorized function tinyramp that evaluates the function tinyramp(t) = t(u(t) u(t 2)). Now use this function to plot the following signals: tinyramp(2t 4), tinyramp(3 t) and tinyramp((t 1) 2 ). Show your code and plots in.eps format to the TA. To Do 5 We will see in class that most signals can be approximated over a finite interval using sums of sines and cosines (this is called the Fourier series). In fact the signal of Figure 6 can be expressed over the interval (, 4) as ( πn ) π 2 n 2 (( 1)n 1) cos 2 t 2 ( πn ) πn ( 1)n sin 2 t n=1. 18

19 As it is not possible to compute an infinite sum in MATLAB we will consider only the first N terms in the summation above. Write a function fourier approx that takes as argument a vector t of time indexes, and N, the number of elements in the summation to be included. Plot the true function and the approximation in the same axes, for various values of N and comment on the effect of N and the quality of the approximation. Note the ringing effect at t = 2 - This is called the Gibb s effect. What happens to the ringing when N increases? Show your code and plots to the TA. To Do 6 Average length of ones. Now we will write a function which will test our knowledge on vector manipulation. Assuming a random sequence of zeros and ones we should calculate the average length of contiguous ones. For example given a vector: >> a = [,,,,1,1,,1,,]; our function should be named avgones and give the following result: >> avgones(a) 1.5 As we see the vector has two consecutive ones at indices 5,6. This segment thus has length 2. It also has a one at index 8 which obviously has length 1. This means that the average length of consecutive ones in the vector a is (2 + 1)/2 = 1.5. The trick here is to use the diff and find commands. Obviously the cases of a vector full of ones should give as result the length of the vector and the case of a vector full of zeros should give as result zero. Lastly notice that using the same function we can calculate the average length of zeros using the command: >> a = [,,,,1,1,,1,,]; >> avgones(~a) >> (4+1+2)/ Save the function using the filename avgones.m in the directory /lab2 and submit it. 19

20 At this point you should turn off the diary function we started in the beginning of this lab session. You can do that by typing: >> diary off; This will create the diary file that contains all the commands you typed in the command prompt as well as all the output they generated. You should show the LA or TA the diary file. 2