Matlab Tutorial: 1. Basics: 1.) Scalar computation: d = 36*12 + 120; b=2^5; b+d 2.) Matrices: A=[1,2,3; 4,5,6;7,8,9] A(1,1) A(3,4) A(2,1)=0 A’ inv(A) 3.) Arrays: x=[1 2 3] y=[4; 5; 6] y(3) y(3)=0 A*y

4.) Element-wise operations: a=[1 2 3] b=[2 3 4] a.*b a./b a*b’ 5.) Colon notation: a=1:10 a=1:2:10 6.) Some special matrices: A=eye(6); B=ones(5,3); C=zeros(4,5); Exercise:

Find the solution to the linear equation: [4 9 6; 1 5 7; 8 6 3 ] x = [ 1; 5; 2] 2. Programming 1.) for loop Compute the i^2 for i=1:10 x=[]; for i=1:10 x=[x; i^2]; end 2.) while loop i=1; x=[]; while i<=10 x=[x;i^2]; i=i+1; end 3.) if... else check whether x is even or not if (mod(x,2)==0) disp(‘x is even’); else disp(‘x is odd’); end 4.) Write a .m file and define a function Now we can write a function U(t) = 1 function u=unit_step(t) if t>=0 u=1; else u=0; end

Exercise 2: Using Matlab to compute 1^3 + 3^3 +… + (2k+1)^3 + 99^3 2. Signals and Systems 1.) How to define a signal. Signal is a function of time. Example, x(t)=sin(t). Define a vector for time t, and then compute the vector for x. t=[-10:0.01:10]; x=sin(t); y=cos(t); figure; hold on; plot(t,x); plot(t,y,’r’); grid; xlabel(‘t’); ylabel(‘x and y’); title(‘sin(t) and cos(t)’); legend(‘sin(t)’, ‘cos(t)’); 2.) Give an input and a system, find the output using Matlab y(t)=[x(t)]^2 x(t)=U(t) t=-1:0.01:1 x=[]; for index=1:length(t) x=[x 2*unit_step(t(index))]; end y=x.^2; figure; hold on; plot(t, x); plot(t,y,’r’); Exercise:

Define a signal x1(t)=U(t-1)-U(t-2) and x2(t)=U(2-t)U(t-1) for -5 < t <5, and plot the signal. You may use the function we write for u(t). 3.) Using Matlab to compute the output when the impulse function of the system is known. x(t)=1, 0 < t < 1 h(t)=1, 0 < t < 2 % Define a function for x(t) function x=fun_x(t) if (t > 0 && t<1) x=1; else x=0; end % Define a function for h(t) function h=fun_h(t) if (t>0 && t<2) h=1; else h=0; end % Main Program clear all; close all; t=[-5:0.001:5]; x=[]; for index=1:length(t) x=[x fun_x(t(index))]; end h=[]; for index=1:length(t) h=[h fun_h(t(index))]; end figure; subplot(3,1,1); plot(t,x); axis([-5 5 -2 2]);

grid; xlabel('t'); ylabel('x(t)'); subplot(3,1,2); plot(t,h); axis([-5 5 -2 2]); grid; xlabel('t'); ylabel('h(t)'); y=0.001*conv(x,h); subplot(3,1,3); plot([-10:0.001:10], y); axis([-5 5 -2 2]); grid; xlabel('t'); ylabel('y(t)');

Exercise: Please change x(t) to x(t)=t for 0< t<1, 2-t for 1<t<2, and 0 otherwise. Plot the output. 4.) Frequency domain analysis and filtering

s1(t)=sin(100t), s2(t)=cos(200t), x(t)=s1(t)+s2(t)=sin(100t)+cos(200t). Define a low pass filter h(t), and get y(t)=sin(100t) from x(t). clear all; close all; Ts=0.001; t=[-10:Ts:10]; s1=sin(100*t); s2=cos(300*t); figure; subplot(3,1,1); hold on; plot([0:99]*Ts,s1(1:100)); plot([0:99]*Ts,s2(1:100),'r'); %axis([-2 2 -2 2]); grid; xlabel('t'); %ylabel('s1 (s2)'); legend('s1(t)', 's2(t)'); x=s1+s2; subplot(3,1,2); plot([0:99]*Ts,x(1:100)); grid; xlabel('t'); ylabel('x(t)=s1(t)+s2(t)'); %axis([-2 2 -3 3]); % Fourier Transform X_w=Ts*fft(x); X_w=fftshift(X_w); w1=[-(length(X_w)-1)/2:(length(X_w)-1)/2]/length(X_w)*2*pi*1/Ts; %figure; subplot(3,1,3) plot(w1,abs(X_w)); axis([-400 400 0 20]); grid; xlabel('\omega'); ylabel('|X(i\omega)|'); % Filter n=[-100:1:100]; h=sinc(0.2/pi*n)*0.2/pi; figure; subplot(2,1,1); plot(n*Ts, h);

xlabel('t'); ylabel('h(t)'); h_w=fft(h); H_w=fftshift(h_w); w2=[-(length(H_w)-1)/2:(length(H_w)-1)/2]/length(H_w)*2*pi*1/Ts; subplot(2,1,2); plot(w2,abs(H_w)); axis([-400 400 0 2]); grid; xlabel('\omega'); ylabel('|H(i\omega)|'); % After filtering figure; subplot(3,1,1) hold on; plot(w1,abs(X_w)); plot(w2,abs(H_w),'r'); legend('|X(i\omega)|', '|H(i\omega)|'); axis([-400 400 0 10]); y=conv(x,h); subplot(3,1,2) plot([0:99]*Ts, y(1000:1000+99)); axis tight; Y_w=fft(y); Y_w=fftshift(Y_w); xlabel('t'); ylabel('y(t)'); w3=[-(length(Y_w)-1)/2:(length(Y_w)-1)/2]/length(Y_w)*2*pi*1/Ts; subplot(3,1,3); plot(w3,abs(Y_w)/800); axis([-400 400 0 10]); xlabel('\omega'); ylabel('|Y(i\omega)|');