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# fft() - Signal Processing

```Y = fft(X) computes
the discrete
Fourier transform (DFT) of X using a fast
Fourier transform (FFT) algorithm.If X is a vector, then fft(X) returns
the Fourier transform of the vector.If X is a matrix, then fft(X) treats
the columns of X as vectors and returns the Fourier
transform of each column.If X is a multidimensional array,
then fft(X) treats the values along the first array
dimension whose size does not equal 1 as vectors and returns the Fourier
transform of each vector.exampleY = fft(X,n) returns
the n-point DFT. If no value is specified, Y is
the same size as X.If X is a vector and the length
of X is less than n, then X is
padded with trailing zeros to length n.If X is a vector and the length
of X is greater than n, then X is
truncated to length n.If X is a matrix, then each column
is treated as in the vector case.If X is a multidimensional array,
then the first array dimension whose size does not equal 1 is treated
as in the vector case.exampleY = 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.```

### Syntax

`Y = fft(X) exampleY = fft(X,n) exampleY = fft(X,n,dim) example`

### Example

```Noisy SignalOpen This ExampleUse Fourier transforms to find the frequency components of a signal buried in noise.Specify the parameters of a signal with a sampling frequency of 1 kHz and a signal duration of 1 second.Fs = 1000;            % Sampling frequency
T = 1/Fs;             % Sampling period
L = 1000;             % Length of signal
t = (0:L-1)*T;        % Time vector
Form a signal containing a 50 Hz sinusoid of amplitude 0.7 and a 120 Hz sinusoid of amplitude 1.S = 0.7*sin(2*pi*50*t) + sin(2*pi*120*t);
Corrupt the signal with zero-mean white noise with a variance of 4.X = S + 2*randn(size(t));
Plot the noisy signal in the time domain. It is difficult to identify the frequency components by looking at the signal X(t).plot(1000*t(1:50),X(1:50))
title('Signal Corrupted with Zero-Mean Random Noise')
xlabel('t (milliseconds)')
ylabel('X(t)')

Compute the Fourier transform of the signal.Y = fft(X);
Compute the two-sided spectrum P2.  Then compute the single-sided spectrum P1 based on P2 and the even-valued signal length L.P2 = abs(Y/L);
P1 = P2(1:L/2+1);
P1(2:end-1) = 2*P1(2:end-1);
Define the frequency domain f and plot the single-sided amplitude spectrum P1.  The amplitudes are not exactly at 0.7 and 1, as expected, because of the added noise. On average, longer signals produce better frequency approximations.f = Fs*(0:(L/2))/L;
plot(f,P1)
title('Single-Sided Amplitude Spectrum of X(t)')
xlabel('f (Hz)')
ylabel('|P1(f)|')

Now, take the Fourier transform of the original, uncorrupted signal and retrieve the exact amplitudes, 0.7 and 1.0.Y = fft(S);
P2 = abs(Y/L);
P1 = P2(1:L/2+1);
P1(2:end-1) = 2*P1(2:end-1);

plot(f,P1)
title('Single-Sided Amplitude Spectrum of S(t)')
xlabel('f (Hz)')
ylabel('|P1(f)|')

Gaussian PulseOpen This ExampleConvert 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)|')

Cosine WavesOpen This ExampleCompare cosine waves in the time domain and the frequency domain.Specify the parameters of a signal with a sampling frequency of 1kHz and a signal duration of 1 second.Fs = 1000;                    % Sampling frequency
T = 1/Fs;                     % Sampling period
L = 1000;                     % Length of signal
t = (0:L-1)*T;                % Time vector
Create a matrix where each row represents a cosine wave with scaled frequency. The result, X, is a 3-by-1000 matrix.  The first row has a wave frequency of 50, the second row has a wave frequency of 150, and the third row has a wave frequency of 300.x1 = cos(2*pi*50*t);          % First row wave
x2 = cos(2*pi*150*t);         % Second row wave
x3 = cos(2*pi*300*t);         % Third row wave

X = [x1; x2; x3];
Plot the first 100 entries from each row of X in a single figure in order and compare their frequencies.for i = 1:3
subplot(3,1,i)
plot(t(1:100),X(i,1:100))
title(['Row ',num2str(i),' in the Time Domain'])
end

For algorithm performance purposes, fft allows you to pad the input with trailing zeros.  In this case, pad each row of X with zeros so that the length of each row is the next higher power of 2 from the current length. Define the new length using the nextpow2 function.n = 2^nextpow2(L);
Specify the dim argument to use fft along the rows of X, that is, for each signal.dim = 2;
Compute the Fourier transform of the signals.Y = fft(X,n,dim);
Calculate the double-sided spectrum and single-sided spectrum of each signal.P2 = abs(Y/n);
P1 = P2(:,1:n/2+1);
P1(:,2:end-1) = 2*P1(:,2:end-1);
In the frequency domain, plot the single-sided amplitude spectrum for each row in a single figure.for i=1:3
subplot(3,1,i)
plot(0:(Fs/n):(Fs/2-Fs/n),P1(i,1:n/2))
title(['Row ',num2str(i), ' in the Frequency Domain'])
end```