DSP Practical - Experiment 3

Aim

To write and perform a MATLAB program to find the linear convolution of two sequences using Discrete Fourier Transform (DFT) and Inverse Discrete Fourier Transform (IDFT).

Problem Statement

Given two sequences:

          x(n) = [1 2 3 4]
          h(n) = [1 1 1 1]

Find the output of the system using linear convolution, assuming that x(n) is aperiodic.

Theory

Linear convolution is the process of computing the output of a system with an input sequence and an impulse response (system response). For two sequences x(n) and h(n), the linear convolution can be calculated using the following steps:

          y(n) = IDFT(DFT(x(n)) * DFT(h(n)))

The steps involved in calculating linear convolution using DFT and IDFT are:

Compute the DFT of x(n) and h(n).
Multiply the DFTs of x(n) and h(n) element-wise.
Compute the IDFT of the product obtained in the previous step.

MATLAB Code

% MATLAB Code for Linear Convolution using DFT & IDFT

clc;
clear all;
close all;

% Input sequences
x = [1 2 3 4]; % First sequence
h = [1 1 1 1]; % Second sequence

% Length of the output sequence
N = length(x) + length(h) - 1;

% Compute DFTs of x(n) and h(n)
xk = fft(x, N); % DFT of x(n)
hk = fft(h, N); % DFT of h(n)

% Multiply DFTs element-wise
yk = xk .* hk;

% Compute IDFT of the product
y = ifft(yk, N); % Inverse DFT to get the linear convolution result

% Plotting the results

% Plot the first sequence x(n)
subplot(3, 2, 1);
stem(x);
title('First Sequence x(n)');
xlabel('Discrete Time');
ylabel('Amplitude');

% Plot the DFT of the first sequence
subplot(3, 2, 2);
stem(abs(xk));
title('DFT of First Sequence');
xlabel('Discrete Frequency');
ylabel('Amplitude');

% Plot the second sequence h(n)
subplot(3, 2, 3);
stem(h);
title('Second Sequence h(n)');
xlabel('Discrete Time');
ylabel('Amplitude');

% Plot the DFT of the second sequence
subplot(3, 2, 4);
stem(abs(hk));
title('DFT of Second Sequence');
xlabel('Discrete Frequency');
ylabel('Amplitude');

% Plot the linear convolution output
subplot(3, 2, 5);
stem(abs(y));
title('Linear Convolution Output using DFT & IDFT');
xlabel('Discrete Time');
ylabel('Amplitude');

Expected Output

The MATLAB code generates five plots:

First Sequence (x(n)): The original input sequence x(n) is displayed in the first plot.
DFT of First Sequence: The second plot shows the magnitude of the DFT of x(n).
Second Sequence (h(n)): The original input sequence h(n) is shown in the third plot.
DFT of Second Sequence: The fourth plot shows the magnitude of the DFT of h(n).
Linear Convolution Output: The fifth plot represents the result of the linear convolution of x(n) and h(n) using DFT & IDFT.

The plots validate the calculation of the linear convolution using the DFT and IDFT, showcasing the frequency-domain approach to convolution and its time-domain result.