DSP Practical - Experiment 2

Aim

To write and perform a MATLAB program to calculate the N-point Discrete Fourier Transform (DFT) for a given input sequence.

Problem Statement

Given the input sequence:

          x(n) = [1 2 3 4]

Calculate the 4-point DFT along with their amplitude and phase spectrum.

Theory

The Discrete Fourier Transform (DFT) is a mathematical transformation used to analyze the frequency content of a discrete signal. The N-point DFT of a sequence x(n) is given by:

          X(k) = sum (x(n) * exp(-j*2*pi*k*n/N)), where n = 0, 1, 2, ..., N-1

The amplitude spectrum represents the magnitude of the DFT coefficients, and the phase spectrum represents the phase of each coefficient.

MATLAB Code

% MATLAB Code for Calculating N-point DFT

clc;
clear all;
close all;

% Input Parameters
N = 4; % Point for DFT
x = [1 2 3 4]; % Input sequence
a = 0:N-1; % Frequency indices

L = length(x); % Length of input sequence
if N < L
    error('N should be greater than or equal to length of x');
end

% Zero-padding if necessary
x1 = [x, zeros(1, N - L)];

% Calculating DFT using the formula
for k = 0:N-1
    for n = 0:N-1
        p = exp(-1i * 2 * pi * n * k / N);
        X(k+1, n+1) = p * x1(n+1); % Multiply with input sequence
    end
end

% Calculate the DFT X[k]
X = sum(X, 2); % Sum over n to get X(k)

% Plotting the input sequence and DFT
subplot(2, 1, 1);
stem(a, x1(1:N));
title('Input Sequence x(n)');
xlabel('>a (Discrete Time)');
ylabel('x(n) (Amplitude)');

subplot(2, 1, 2);
stem(a, abs(X)); % Amplitude Spectrum of DFT
title('Amplitude Spectrum of DFT X[k]');
xlabel('>a (Discrete Frequencies)');
ylabel('|X(k)| (Amplitude)');

% Phase Spectrum Plot
subplot(2, 1, 2);
stem(a, angle(X)); % Phase Spectrum of DFT
title('Phase Spectrum of DFT X[k]');
xlabel('>a (Discrete Frequencies)');
ylabel('Phase of X(k)');

Expected Output

The MATLAB code generates two plots:

Input Sequence (x(n)): The original input sequence is displayed in the first plot.
Amplitude Spectrum of DFT (|X(k)|): The second plot shows the amplitude of the DFT coefficients.
Phase Spectrum of DFT (∠X(k)): The phase of each DFT coefficient is displayed.

The amplitude spectrum indicates the magnitude of each frequency component, while the phase spectrum shows the phase shift of these components.