EEM Simple use of Matlab

Transcription

1 EEM Simple use of Matlab 0) Starting Matlab and basic operations Matlab is a powerful engineering tool, specifically designed to allow engineers to make simulations closest to the real world while eliminating the comlexity of code writing. After installing Matlab sucessfully in your computer and starting it we get the follwing screen. The middle portion is called Command Window where we can perform simple operations. On the left we see the directory structure, on the right we have the numeric values and dimensions of our variables %%% Simple addition, where the result is assigned to the parameter 'ans' 6

2 ; %%% - is used for subtraction, use of semicolon means, do not print the answer in w a = 2.3;b = 1.9; %%% Assigning numeric values to parameters c = a + b %%% Addition. Note that the absence of semicolon allows the answer to be printed in c = C = a*b %%% * denotes multiplication C = D = a/b %%% / denotes division D = As seen, mathematical operation are basically represented by the operators used in hand writing. The other notations are shown with comment lines starting with percentage sign. These operation can also be performed by pasting them in an m file. This way they can be saved in a file which we can use later on. Show an example. 1) Scalar, array and matrix operations In this section, we are going to learn about the simple use of Matlab for scalar, array and matrix operations Matlab is desiged to handled array (also called vactorial) and matrix, i.e. dimensional operations more efficiently than the (single valued) scalar ones. Below we exemplify what we mean by these terms scalar operation (1.1) array array operation (1.2), matrix operation (1.3) These equations can be implemented in Matlab with tha same complexity and simplicity as shown below Note that black font is the actual code, The writings in green color (starting with percentage sign, %) are reserved for the comments (explanations) In the code, multiplication is represented by star, *, raising to the power is indicated by ^. The others have usual meanings of the hand written versions of the eqautions clc;clear;close all %%%% clear workspace screen, clear the variables in work space, %%% close all open graphs x = 3;y = 6 - x^2 %%% The absence of semicolon, i.e., ; y = -3 %%% allows the computed value to be printed (in workspace) x = [2 3];y = 6 - x.^2 %%% Note the use square bracket to imply array y = 2-3

3 %%% and also the use of dot '.' for the x array operation x = 2:3;y = 6 - x.^2 %%% Equivalent operation, so : is an alternative definition of array y = 2-3 x = [2:3;1 4];y = 6 - x.^2 %%% Matrix operation. Note the way of writing the matrix y = %%% Note that the above can also be copied and pasted into command window or %%% an individual m file to run %%% In such a case, the outputs will be displayed on the screen of the command window. Show an The same is applicaple to specific inbuilt functions. A sinusoidal example is given below clc;clear;close all %%%% clear command window screen, clear the variables in workspace, %%% close all open graphs t = 0.2; %%%% Define a time instance of 0.2 (seconds, sec) f1 = 1; %%% Define a frequency of one unit (1 Hertz, 1 Hz) xt = cos(2*pi*f1*t) %%% Define x(t), where x(t) is going to be scalar, i.e. single valued xt = t = 0:0.1:2;%%%% Define a time array that ranges from t = 0 upto t = 2 in increaments of 0.1 xt = cos(2*pi*f1*t); %%% Redefine x(t), where x(t) is going to be multi valued, %%% i.e. an array plot(t,xt,'linewidth',2); % add axis labels and title xlabel('\itt','fontsize',14);ylabel('\itx\rm (\it t\rm )','FontSize',14); title('graph of x(t) = cos(2\pit)');

4 %%%% Increment of 0.1 seems too large, so the cosine curve appears bloken, as a solution, %%%% let's try a smaller (finer) one t = 0:0.01:2;%%%% A finer (smaller) increament of 0.01 instead of 0.1 xt = cos(2*pi*f1*t); %%% Redefine x(t), where x(t) is going to be multi valued, %%% i.e. an array plot(t,xt,'linewidth',2); % add axis labels and title xlabel('\itt','fontsize',14);ylabel('\itx\rm (\it t\rm )','FontSize',14); title('graph of x(t) = cos(2\pit) with increments of 0.01');

5 2) Functions, tools to solve simple equations, finding roots Let us assume that we have following simple equation and we wish to find the roots (2.1) Applying simple analytic rules, we find that (2.2) The above analytic operation can be implemented in Matlab as follows %%%% Code for finding roots of 2x^2 + 3x clc;clear;close all syms x %%% Declearing x to be symbolic, i.e. it can take on any numeric value f(x) = 2*x^2 + 3*x; solve(f(x),x)

6 It is possible to solve other complex equations by using the intrinsic function "solve". Such examples are given below %%%% Code for finding roots of complex equations clc;clear;close all syms x f(x) = 2*x^ *x; solve(f(x),x) The famous quadratic equation can be solved symbolically in Matlab as follows syms a b c x f(x) = a*x^2 + b*x + c; solve(f(x),x) The same operation can be extended to other equations involving sinusoidal functions %%%% Code for finding roots of complex equations clc;clear;close all syms x f(x) = (1 + 2*j)*sin(x) + 0.3*cos(2*x); solve(f(x),x)

7 The set of linear equations can also be solved in Matlab. Consider the following (2.3) (2.4) Sum of (2.3) and (2.4) will give (2.5) Substituting the result of (2.5) into either (2.3) or (2.4) will lead to (2.6) The representation of the above in Matlab is shown below A = %%% B = sol [2 1;6.5-1]; %%% Organization of the coefficients of LHS of (2.3) and (2.4) into a matrix form [4;4.5]; %%% RHS of (2.3) and (2.4) = A\B %%% Roots x and y sol = 1 2 sol = inv(a)*b %%%% Alternative expression sol = The alternative is

8 syms x y eqn1 = 2*x + y == 4; %%% Defining equation (2.4), note the use of double equal sign eqn2 = 6.5*x - y == 4.5; %%%% Defining equation (2.4) [A,B] = equationstomatrix([eqn1, eqn2], [x, y]) %%% To convert the equations A = B = %%% into the form AX = B X = linsolve(a,b) %%% Use linsolve to solve AX = B for the vector of unknowns X. X = 3) Plotting and analysing different functions In our dicipline, exponential (exp), sinusoidal (sine and cosine) with real and complex arguments are quite important Let's define the following exponential functions, Simple negative exponential (3.1), Simple positive exponential (3.2), Gaussian exponential (3.3), Gaussian exponential with amplititude coeffcient of, Gaussian exponential with a variance of (3.4) (3.5) The five functions listed in (3.1) to (3.5) can be represented and plotted in Matlab as shown below %%%% Observing different forms of exponential function t = -1:0.1:1; %%% Define a time range xt1 = exp(-t); %%%% Simple negative exponential xt2 = exp(t); %%%% Simple positive exponential xt3 = exp(-t.^2); %%%% Gaussian exponential Ac = 2.5;sig2 = 0.25; %%% Define some coefficients xt4 = Ac*exp(-t.^2); %%%% Gaussian exponential with some amplitude coefficient xt5 = exp(-t.^2/sig2); %%%% Gaussian exponential with a different variance (exponential decay) %%%%% Plot the above functions on the same graph to see their differences plot(t,xt1,'-k','linewidth',2);hold on;plot(t,xt2,'--r','linewidth',2); plot(t,xt3,'-.g','linewidth',2);hold on;plot(t,xt4,':b','linewidth',2); hold on;plot(t,xt5,'-c','linewidth',2);hold off

9 legend('\itx\rm_1 ( \itt \rm)','\itx\rm_2 ( \itt \rm)','\itx\rm_3 ( \itt \rm)',... '\itx\rm_4 ( \itt \rm)','\itx\rm_5 ( \itt \rm)','location','northwest') Now we turn to sinusoidal functions %%%% Observing different forms of exponential function t = -1:0.01:1; %%% Define a time range f1 = 1;f2 = 2; %%% Define two different frequencies of 1 Hz and 2 Hz xt1 = sin(2*pi*f1*t); %%%% Sine function, sine time waveform with frequency f1 xt2 = cos(2*pi*f1*t); %%%% Cosine function, cosine waveform with frequency f1 xt3 = sin(2*pi*f2*t); %%%% Sine function, sine time waveform with frequency f2 Ac = 2; %%% Define an amplitude coefficient xt4 = Ac*sin(2*pi*f1*t); %%%% Sine waveform with amplitude coefficient Ac t0 = 0.5; %%% Define some time difference xt5 = cos(2*pi*f1*(t - t0)); %%%% Cosine waveform with frequency f1 and time delay t0 %%%%% Plot the above functions on the same graph to see their differences plot(t,xt1,'-k','linewidth',2);hold on;plot(t,xt2,'--r','linewidth',2); plot(t,xt3,'-.g','linewidth',2);hold on;plot(t,xt4,':b','linewidth',2); hold on;plot(t,xt5,'-c','linewidth',2);hold off legend('\itx\rm_1 ( \itt \rm)','\itx\rm_2 ( \itt \rm)','\itx\rm_3 ( \itt \rm)',... '\itx\rm_4 ( \itt \rm)','\itx\rm_5 ( \itt \rm)','location','northwest')

10 %%% From the graph it is possible to make several interesting observations, state them one by The above sinusoidals can be listed as, Sine function, sine time waveform with frequency (3.6), Cos (cosine) function, cos time waveform with frequency, Sine function, sine time waveform with frequency, Sine waveform with amplitude coefficient (3.7) (3.8) (3.9), Cos (cosine) function, cos time waveform with a time delay of (3.10) 4) Symbolic operations As already introduced above partially, in Matlab, it possible to perform numeric and symbolic operations. Symbolic operations refer to the analytic tratment of mathematical expressions, that is without assigning any numeric vaules to the parameters that we define Symbolic operations have some attractive features such that it is possible to obtain the equivalent expressions from symbolic operations. Consider the following relationship (4.1) It is possible to find the same in Matlab by applying the following

11 %%% Cos summation relationship using symbolic facility of Matlab syms A B expand(cos(a + B)) %%% Expanding cos(a + B) or finding the equivalent relationship %%%% Other examples syms x A = (x + 5)*(2*x^ )*(0.7*x^1.4 +2); expand(a) simplify(a) collect(a) syms a b x = (a^2 - b^2)/(a - b); simplify(x) x = (a^2 + b^2)/(a - j*b); %%% Note that j (or i) is the imaginary number, %%%% i.e., j = sqrt(-1) simplify(x) %%% The last example shows that symbolic operations can handle complex functions as well Analytic differentiation and integration are quite efficient and easy when symbolic facility is used. Cosider the following operations (4.2) (4.3)

12 Indefinite integration (4.4) Definite integration (4.5) Now we implement the same in Matlab using symbolic notation %%% Performning symbolic differentiation and integration syms x f(x) = x^ *x; difx1 = diff(f(x),x,1) %%% Differentiate f(x) with respect to x once (default is once anyway difx1 = % difx2 = diff(f(x),x,2) %%% Differentiate f(x) with respect to x twice intfxi = int(f(x),x) %%% Integrate f(x) indefinitely with respect to x (without limits) intfxi = intfxd = int(f(x),x,0,1) %%% Integrate f(x) definitely with respect to x from 0 to 1 intfxd = %%% It easy to verify that the results delivered by Matlab agree perfectly with %%% the ones listed in (4.2) to (4.5) Of course, the objective in the use of Matlab should be to solve relatively difficult cases. Such examples will be examined next. To this end, we take two second order differential eqautions Simple second order differential equation (DE) (4.3) Solution is Solution with two unknown constants, A and B to be determined by applying boundary conditions (4.4) More complicated second order differential equation (4.5) Solution is

13 where and are Bessel functions of first and second kinds (4.6) Now we test these solutions in the following m file % define functions syms x A B nu %%%%% Testing the solution of DE in (4.3) f(x) = A*cos(x) + B*sin(x); %%% Seting f(x) to the solution of the first DE in (4.3) dfx2 = diff(f(x),x,2); %%% Finding the second order derivative of f(x), %%% i.e. the first term of (4.3) dfx2 + f(x) == 0 %%% Test if DE of (4.3) is zero simplify(dfx2 + f(x)) %%%% Alternative expression %%%%% Testing the solution of DE in (4.5) f(x) = A*besselj(nu,x) + B*bessely(nu,x); %%% Seting f(x) to %%% the solution of the second DE in (4.5) dfx1 = diff(f(x),x,1); %%% Finding the first order derivative of f(x) %%% in the second term of (4.5) dfx2 = diff(f(x),x,2); %%% Finding the second order derivative of f(x) %%% in the first term of (4.5) DE2 = x^2*dfx2 + x*dfx1 + (x^2 - nu^2)*f(x); DE2 == 0 %%% Test if DE of (4.5) is zero simplify(de2) When running the above code, it is quite interesting that DE2, i.e. (4.5) is verified only after applying the simplify function. 5) Polar coordinates representation and polar graphs

14 Polar coordinates are important, since in addition to being the counterpart of Cartesioan coordinates, they can also handle complex functions. We start with the simple definition of a circle, that is, below we show how to plot this in Matlab % define values th = 0:pi/50:2*pi; %%%% Define a whole angle range of 2pi R = 3; %%%% Define the circle radius x = R*cos(th); %%%%% Find the Cartesian coordinate x y = R*sin(th); %%%%% Find the Cartesian coordinate y % plot circle (xvalues,yvalues) and line ([ ],[-3 3]) plot(x,y,[ ],[-3 3],'LineWidth',2); % make axis square and place grid axis square; grid on; box on; % add axis labels and title xlabel('\itx','fontsize',14);ylabel('\ity','fontsize',14); title('a circle with x^2 + y^2 = 9'); %%%% The line shows that for a given x value, there are two coressponding y values An alternative for ploting a circle is the function, "ploarplot". Some examples are shown below th = 0:0.01:2*pi; %%%% Define the angle polar coordinates R = 5; %%%% Define radius

15 rho = R*sqrt(sin(th).^2 + cos(th).^2); %%%% Define the radial axis of the polar coordinates polarplot(th,rho,'linewidth',2) %%% Ploting in polar coordinates th = 0:0.01:2*pi; %%%% Define the angle polar coordinates z = exp(-j*th); %%%% Define cirle as complex exponential polarplot(z,'--r','linewidth',2) %%% Ploting in polar coordinates

16 th = 0:0.01:2*pi; %%%% Define the angle polar coordinates (in radians) z = exp(-j*th) + 2*exp(-j*2*th); %%%% Define some profile as sum of complex exponential polarplot(z,'--g','linewidth',2) %%% Ploting in polar coordinates

17 6) Limits (bounds) of Matlab and points to be taken into consideration to arrive at correct results Both for numeric and symbolic operations, there are several limitations that we have to careful about, otherwise we are likely to obtain misleading results. The simplest is the following. We know that theoretically (6.1) Let's now test this in Matlab tan(pi/2) %%% Evaluating the tangent of pi / e+16 So we ask the question why do we get comments to this question, these are listed below ; instead of (6.1). There are several answers and 1. Matlab is not designed to deliver (correct) theoretical result for this specific case 2. Matlab uses series expansions to evaluate tangent values, in this specific case, the number of terms used in this series expansion is insufficient, on the other hand we can envisage that the result of infinity could only be achieved by taking an infinite number of terms anyway 3. The best way is to recognize that such a limitation (bound) exists and take precautions to cope with it.

18 From Matlab documentations and help files, we understand that Matlab uses 16 digits for its numeric evaluations. In this manner when scaled to unity (1), the result given above corresponds nearly to infinity. Let s now perform the following in Matlab cot(pi/2) %%% Evaluating the cotangent of pi / 2, theoretically we expect to get zero e-17 The smallest number in Mattab is called eps and can be found simply by typing eps, thus eps e-16 This way we that the result of will fail is well appriximated by (which is below eps. Normally such numeric values do not pose any problems, but a test like the following tan(pi/2) == 0 logical 0 Correct answer is obtained by the following arrangement th = [0 pi/4 pi/2]; %%% Set the angle to 0, pi/4, pi/2 y = tan(th) %%% Evaluate tangent at these angular settings, correct execpt the one at pi/2 y = 1.0e+16 * %%%%%% Correct the misleading results yloc = y > 1/eps; %%%% Finding the element of y which is greater than 1/eps [r,c] = find(yloc == 1); %%% Locating the true answer (column) y(r,c) = inf; %%% Replacing 1/eps by infinity y %%%% Printing out the correct answer y = Inf Now we turn to another case. We try to define a delta function and test through graphical illustration if it actaully looks like a delta function. %%%% Define a delta function numerically y = [zeros(1,50) ones(1,1) zeros(1,50)]; %%%% Plot to see if it looks like a delta function plot(y,'linewidth',2)

19 %%%% Define delta (driac) function with grater number of zeros y = [zeros(1,500) ones(1,1) zeros(1,500)]; plot(y,'linewidth',2)

20 Obviously, the second plot approximates more to the theoretical definition of a delta function. In most cases we do not need to add so mant zeros in order to approach the theoretical limit and perform the correct operation. Cosider the following rectangular time waveform Fig. 6.1 Rectangular time waveform This waveform can be defined as (6.2) (6.2) can be written in several ways in Matlab %%%%% Defining the ractangular waveform of Fig. 6.1 A = 1; xt = [0 A 0]; %%% Defining one one sample for the intervals t < 0, 0 < t < T and t > T xt = [0 0 A A 0 0]; %%% Defining two samples for the intervals t < 0, 0 < t < T and t > T xt = [zeros(1,100) A*ones(1,100) zeros(1,100)]; %%% Defining one hundered sample %%%% for the intervals t < 0, 0 < t < T and t > T

21 Normally, there are three basic intervals in Fig. 6.1, hence the representation xt = [0 A 0] is sufficient. But if we want to reach the same (correct) result in each case, then each case must be handled by considering the horizontal time axis in the correct manner. Let us assume that we wish to compute the energy in x (t) of Fig Initailly we carry out the analytic part * is complex conjugate (6.3) Now we verify the result of (6.3) in Matlab by several methods syms t A T assume(a,'real');assume(t,'real') %%% First method xt = [0 A 0]; %%% Shortest version of x(t) Ex = int(xt.^2,t,0,t);sum(ex) %%% Second method xt*xt'*t %%% ' indicates the transpose operation %%% Third method sum(xt.^2)*t %%% T corresponds to dt in the integral %%% Fourth method - with many samples xt = [zeros(1,100) A*ones(1,100) zeros(1,100)]; xt*xt'*t/100 %%% T/100 corresponds to one hundred samples of the interval from 0 to T 7) Help documentation of Matlab and Mathworks site In order to get help on Matlab, we can initially use the help menu as shown below (after pressing help and searching for function sine)

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23 Additionally from the workspace screen you can press, "LearnMatlab" or you can go to the site