DSP Practical - Experiment 1

Aim

To write and perform a MATLAB program to study the discrete systems represented with its difference equation.

Problem Statement

For a given LTI system represented with its difference equation:

          y(n) = 0.5 * y(n-1) + x(n) + x(n-1), where x(n) = u(n)

Find the system output for 0 ≤ n ≤ 50. Assume initial conditions:

          y(0) = 0, y(1) = 1

Theory

The given system is a first-order difference equation with a feedback term. The output at time n depends on the previous output value (y(n-1)), the current input value (x(n)), and the previous input value (x(n-1)).

A common method to solve this type of system is to use the recursive relationship provided in the equation and iterate over the time indices.

MATLAB Code

% MATLAB Code for Discrete System using Difference Equation

clc;
clear all;
close all;

N = 50; % Number of samples
a = 1:N+1; % Sample indices
y(1) = 1; % Initial condition y(0) = 1
u = ones(1, N+1); % Input signal (u(n) = 1 for all n)

% Iterate to calculate output y(n)
for n = 2:N+1
    y(n) = 0.5 * y(n-1) + u(n) + u(n-1); % Difference equation
end

% Plot the input and output signals
subplot(2, 1, 1);
stem(a, u);
title('System Input x(n) = u(n)');
xlabel('--> a (Sample instants)');
ylabel('--> u(n) (Amplitude)');

subplot(2, 1, 2);
stem(a, y);
title('System Output y(n) for 0 <= n <= 50');
xlabel('--> a (Sample instants)');
ylabel('--> y(n) (Amplitude)');

Expected Output

The MATLAB code generates two plots:

System Input (x(n)): A constant input signal, u(n) = 1, plotted over the time range.
System Output (y(n)): The output of the system based on the recursive difference equation, showing how the system evolves over time.

The plots will visually illustrate how the discrete-time system processes the input signal and produces an output.