Signals & systems Assignment
NAME:K.ALEKYA
ROLLN0:22211A0224
FOURIER TRANSFORM OF NON PERIODIC SIGNALS
INTRODUCTION
The Fourier Transform is a mathematical technique that decomposes a function or signal into its
constituent frequencies. It transforms signals between the time domain and the frequency domain,
making it a powerful tool .
There are two main types of Fourier Transform:
1. Forward Fourier Transform: Converts a time-domain signal into its frequency-domain
representation.
2. Inverse Fourier Transform: Converts a frequency-domain representation back into its
time-domain form
MATLAB Simulation
The matlab script that simulates the fourier transform of a Gaussian pulse.It
also plots the time domain signal and its frequency….
MATLAB Code implementation:
A Gaussian pulse in the time domain:
x(t) = e^-t^2 / (2σ^2)Analytical Fourier Transform:
X(f) = σ√(2π)· e^-2π^2 σ^2 f^2
```matlab
% Clear environment
clear; clc; close all;
% Time domain parameters
t = linspace(-1, 1, 1000); % Time vector
sigma = 0.1; % Width of Gaussian
x = exp(-t.^2 / (2 * sigma^2)); % Gaussian pulse
% Plot Time Domain Signal
figure;
plot(t, x, 'b', 'LineWidth', 2);
title('Gaussian Pulse (Time Domain)');
xlabel('Time (s)');
ylabel('Amplitude');
grid on;
% Numerical Fourier Transform (FFT)
dt = t(2) - t(1); % Time step
Fs = 1 / dt; % Sampling frequency
n = length(x); % Number of points
f = (-n/2:n/2-1)*(Fs/n); % Frequency vector
X_fft = fftshift(fft(x)); % Center FFT
X_fft_mag = abs(X_fft); % Magnitude
[17/05, 12:00 am] ChatGPT: X_fft_mag = X_fft_mag / max(X_fft_mag); % Normalize
% Plot Frequency Spectrum - Numerical
figure;
plot(f, X_fft_mag, 'b', 'LineWidth', 2);
title('Frequency Spectrum of Gaussian Pulse (Numerical FFT)');
xlabel('Frequency (Hz)');
ylabel('Normalized Magnitude');
grid on;
% Analytical Fourier Transform
X_analytic = sigma * sqrt(2*pi) * exp(-2*pi^2 * sigma^2 * f.^2);
X_analytic = X_analytic / max(X_analytic); % Normalize
% Plot Comparison
figure;
plot(f, X_fft_mag, 'b', 'LineWidth', 2); hold on;
plot(f, X_analytic, 'r--', 'LineWidth', 2);
title('Comparison: Numerical FFT vs Analytical Fourier Transform');
xlabel('Frequency (Hz)');
label('Normalized Magnitude')
('Numerical FFT ', 'Analytical');
grid on;
RESULT AND ANALYSIS OF OUTPUT
Graph 1: Gaussian Pulse in Time Domain*
- This is the original signal.
- You used a narrow Gaussian (σ = 0.1), centered at t = 0.
- The signal is non-periodic and decays rapidly outside the center.
Graph 2: Frequency Spectrum (Numerical FFT)
- Shows the magnitude of the frequency content.
- The peak is at 0 Hz, meaning the signal is centered in frequency.
- The shape is also Gaussian — matching the expected behavior from theory.
Graph 3: Analytical vs Numerical Comparison
- Overlays the analytical transform with the FFT result.
- Both curves match extremely well — validating your numerical FFT implementation.
CONCLUSION:
* The analytical and numerical methods agree, demonstrating the accuracy of your simulation.
- The Gaussian pulse in the time domain has a Gaussian frequency spectrum.
- The narrower the pulse in time, the wider its spectrum in frequency.
- The FFT output matches the analytical result, proving accuracy.
- Time-Frequency Duality: Narrow time-domain signals lead to wide frequency-domain
representations, and vice versa.
- The FFT output matches the analytical solution, validating both methods.
- This demonstrates the efficacy of numerical techniques (FFT) in signal analysis.
