Question

Find the Fast Fourier Transform (FFT) of some functions by MATLAB and interpret them. Change the sampling frequency to observe the aliasing.

Answer 1

Syntax

Y = fft(X), Computes the discrete Fourier transform (DFT) of X using a fast Fourier transform (FFT) algorithm

Y = fft(X,n), Returns the n-point DFT. If no value is specified, Y is the same size as X

Y = fft(X,n,dim), Returns the Fourier transform along the dimension dim. For example, if X is a matrix, then fft(X,n,2) returns the n-point Fourier transform of each row.

Examples-

Convert a Gaussian pulse from the time domain to the frequency domain.

Define signal parameters and a Gaussian pulse, X.

Fs = 100;           % Sampling frequency
t = -0.5:1/Fs:0.5;  % Time vector 
L = length(t);      % Signal length

X = 1/(4*sqrt(2*pi*0.01))*(exp(-t.^2/(2*0.01)));

Plot the pulse in the time domain

plot(t,X)
title('Gaussian Pulse in Time Domain')
xlabel('Time (t)')
ylabel('X(t)')

To use the fft function to convert the signal to the frequency domain, first identify a new input length that is the next power of 2 from the original signal length. This will pad the signal X with trailing zeros in order to improve the performance of fft.

n = 2^nextpow2(L);

Convert the Gaussian pulse to the frequency domain.

Y = fft(X,n);

Define the frequency domain and plot the unique frequencies.

f = Fs*(0:(n/2))/n;
P = abs(Y/n);

plot(f,P(1:n/2+1)) 
title('Gaussian Pulse in Frequency Domain')
xlabel('Frequency (f)')
ylabel('|P(f)|')