DISCRETE-TIME SIGNALS IN THE TIME DOMAIN

Program P1 1 can be used to generate and plot a unit sample sequence.

% Program P1_1

% Generation of a Unit Sample Sequence

clf;

% Generate a vector from -10 to 20

n = -10:20;

% Generate the unit sample sequence

u = [zeros(1,10) 1 zeros(1,20)];

% Plot the unit sample sequence

stem(n,u);

xlabel(’Time index n’);ylabel(’Amplitude’);

title(’Unit Sample Sequence’);

axis([-10 20 0 1.2]);

For the generation of a complex exponential sequence we use the command

x = K*exp(c*n);

For the generation of a real exponential sequence we use the command

x = K*a.^+n;

A simple example that generates a sinusoidal signal.

% Program P1_2

% Generation of a sinusoidal sequence

n = 0:40;

f = 0.1;

phase = 0;

A = 1.5;

arg = 2*pi*f*n - phase;

x = A*cos(arg);

clf; % Clear old graph

stem(n,x); % Plot the generated sequence

axis([0 40 -2 2]);

grid;

title(’Sinusoidal Sequence’);

xlabel(’Time index n’);

ylabel(’Amplitude’);

axis;

Simple Operations on Sequences

Signal Smoothing:

Program P1 3 implements the above algorithm.

% Program P1_3

% Signal Smoothing by Averaging

clf;

R = 51;

d = 0.8*(rand(R,1) - 0.5); % Generate random noise

m = 0:R-1;

s = 2*m.*(0.9.^m); % Generate uncorrupted signal

x = s + d’; % Generate noise corrupted signal

subplot(2,1,1);

plot(m,d’,’r-’,m,s,’g--’,m,x,’b-.’);

xlabel(’Time index n’);ylabel(’Amplitude’);

legend(’d[n] ’,’s[n] ’,’x[n] ’);

x1 = [0 0 x];x2 = [0 x 0];x3 = [x 0 0];

y = (x1 + x2 + x3)/3;

subplot(2,1,2);

plot(m,y(2:R+1),’r-’,m,s,’g--’);

legend(’y[n] ’,’s[n] ’);

xlabel(’Time index n’);ylabel(’Amplitude’);

DISCRETE-TIME SYSTEMS IN TIME DOMAIN

% Program P2_1

% Simulation of an M-point Moving Average Filter

% Generate the input signal

n = 0:100;

s1 = cos(2*pi*0.05*n); % A low frequency sinusoid

s2 = cos(2*pi*0.47*n); % A high frequency sinusoid

x = s1+s2;

% Implementation of the moving average filter

M = input(’Desired length of the filter = ’);

num = ones(1,M);

y = filter(num,1,x)/M;

% Display the input and output signals

clf;

subplot(2,2,1);

plot(n,s1);

axis([0, 100, -2, 2]);

xlabel(’Time index n’); ylabel(’Amplitude’);

title(’Signal # 1’);

subplot(2,2,2);

plot(n,s2);

axis([0, 100, -2, 2]);

xlabel(’Time index n’); ylabel(’Amplitude’);

title(’Signal # 2’);

subplot(2,2,3);

plot(n,x);

axis([0, 100, -2, 2]);

xlabel(’Time index n’); ylabel(’Amplitude’);

title(’Input Signal’);

subplot(2,2,4);

plot(n,y);

axis([0, 100, -2, 2]);

xlabel(’Time index n’); ylabel(’Amplitude’);

title(’Output Signal’);

axis;

Illustration Cascade of LTI Systems

% Program for Cascade Realization

clf;

x = [1 zeros(1,40)]; % Generate the input

n = 0:40;

% Coefficients of 4th-order system

den = [1 1.6 2.28 1.325 0.68];

num = [0.06 -0.19 0.27 -0.26 0.12];

% Compute the output of 4th-order system

y = filter(num,den,x);

% Coefficients of the two 2nd-order systems

num1 = [0.3 -0.2 0.4];den1 = [1 0.9 0.8];

num2 = [0.2 -0.5 0.3];den2 = [1 0.7 0.85];

% Output y1[n] of the first stage in the cascade

y1 = filter(num1,den1,x);

% Output y2[n] of the second stage in the cascade

y2 = filter(num2,den2,y1);

% Difference between y[n] and y2[n]

d = y - y2;

% Plot output and difference signals

subplot(3,1,1);

stem(n,y);

ylabel(’Amplitude’);

title(’Output of 4th-order Realization’);grid;

subplot(3,1,2);

stem(n,y2)

ylabel(’Amplitude’);

title(’Output of Cascade Realization’);grid;

subplot(3,1,3);

stem(n,d)

xlabel(’Time index n’);ylabel(’Amplitude’);

title(’Difference Signal’);grid;

Illustration Convolution

The following MATLAB program illustrates this approach.

% Program

clf;

h = [3 2 1 -2 1 0 -4 0 3]; % impulse response

x = [1 -2 3 -4 3 2 1]; % input sequence

y = conv(h,x);

n = 0:14;

subplot(2,1,1);

stem(n,y);

xlabel(’Time index n’); ylabel(’Amplitude’);

title(’Output Obtained by Convolution’);grid;

x1 = [x zeros(1,8)];

y1 = filter(h,1,x1);

subplot(2,1,2);

stem(n,y1);

xlabel(’Time index n’); ylabel(’Amplitude’);

title(’Output Generated by Filtering’);grid;

Illustration of Filtering Concept

System No. 1

System No. 2

MATLAB Program P2_4 is used to compute the outputs of the above two systems for an input

% Program

% Generate the input sequence

clf;

n = 0:299;

x1 = cos(2*pi*10*n/256);

x2 = cos(2*pi*100*n/256);

x = x1+x2;

% Compute the output sequences

num1 = [0.5 0.27 0.77];

y1 = filter(num1,1,x); % Output of System No. 1

den2 = [1 -0.53 0.46];

num2 = [0.45 0.5 0.45];

y2 = filter(num2,den2,x); % Output of System No. 2

% Plot the output sequences

subplot(2,1,1);

plot(n,y1);axis([0 300 -2 2]);

ylabel(’Amplitude’);

title(’Output of System No. 1’);grid;

subplot(2,1,2);

plot(n,y2);axis([0 300 -2 2]);

xlabel(’Time index n’); ylabel(’Amplitude’);

title(’Output of System No. 2’);grid;