DSP Practical - Experiment 4

Aim

To write and perform a MATLAB program to find the circular 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) = [4 3 2 1]

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

Theory

Circular convolution is a special case of convolution in which the sequences are assumed to be periodic. It can be efficiently computed using the Discrete Fourier Transform (DFT) and the Inverse Discrete Fourier Transform (IDFT) as follows:

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

The steps to compute circular convolution using DFT & 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 Circular Convolution using DFT & IDFT

clc;
clear all;
close all;

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

% Lengths of the sequences
L1 = length(x);
L2 = length(h);

% N: The length for DFT (max of L1 and L2)
N = max(L1, L2);

% 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 circular 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 circular convolution output
subplot(3, 2, 5);
stem(abs(y));
title('Circular 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).
Circular Convolution Output: The fifth plot represents the result of the circular convolution of x(n) and h(n) using DFT & IDFT.

The plots validate the calculation of the circular convolution using the DFT and IDFT, and the effect of the periodicity assumption on the result.