fft (Signal Processing Toolbox)

Compute the one-dimensional fast Fourier transform.

Syntax

y = fft(x)
y = fft(x,n)

Description

fft computes the discrete Fourier transform of a vector or matrix. This function implements the transform given by

where W_N = e^-^j⁽²^/^N⁾ and N = length(x). Note that the series is indexed as n + 1 and k + 1 instead of the usual n and k because MATLAB vectors run from 1 to N instead of from 0 to N-1.

y = fft(x) is the discrete Fourier transform of vector x, computed with a fast Fourier transform (FFT) algorithm. If x is a matrix, y is the FFT of each column of the matrix.

y = fft(x,n) is the n-point FFT. If the length of x is less than n, fft pads x with trailing zeros to length n. If the length of x is greater than n, fft truncates the sequence x. If x is an array, fft adjusts the length of the columns in the same manner.

The fft function is part of the standard MATLAB language.

Examples

A common use of the Fourier transform is to find the frequency components of a time-domain signal buried in noise. Consider data sampled at 1000 Hz. Form a signal consisting of 50 Hz and 120 Hz sinusoids and corrupt the signal with zero-mean random noise.

randn('state',0)
t = 0:0.001:0.6;
x = sin(2*pi*50*t) + sin(2*pi*120*t);
y = x + 2*randn(1,length(t));
plot(y(1:50))

It is difficult to identify the frequency components by studying the original signal. Convert to the frequency domain by taking the discrete Fourier transform of the noisy signal y using a 512-point fast Fourier transform (FFT).

Y = fft(y,512);

The power spectral density, a measurement of the energy at various frequencies, is

Pyy = Y.*conj(Y) / 512;

Graph the first 256 points (the other 256 points are symmetric) on a meaningful frequency axis.

f = 1000*(0:255)/512;
plot(f,Pyy(1:256))

See the pwelch function for details on estimating the spectral density.

Sometimes it is useful to normalize the output of fft so that a unit sinusoid in the time domain corresponds to unit amplitude in the frequency domain. To produce a normalized discrete-time Fourier transform in this manner, use

Pn = abs(fft(x))*2/length(x)

Algorithm

fft is a built-in MATLAB function. See the fft reference page in the MATLAB documentation for details about the algorithm.

See Also

dct
Implement the discrete cosine transform (DCT).

dftmtx
Compute the discrete Fourier transform matrix.

fft2
Compute the two-dimensional fast Fourier transform.

fftshift
Rearrange the outputs of fft and fft2.

filter
Filter data.

freqz
Compute the frequency response of digital filters.

ifft
Compute the one-dimensional inverse fast Fourier transform.

pwelch
Estimate the power spectral density (PSD) of a signal using Welch's method.

`dct`	Implement the discrete cosine transform (DCT).
`dftmtx`	Compute the discrete Fourier transform matrix.
`fft2`	Compute the two-dimensional fast Fourier transform.
`fftshift`	Rearrange the outputs of `fft` and `fft2`.
`filter`	Filter data.
`freqz`	Compute the frequency response of digital filters.
`ifft`	Compute the one-dimensional inverse fast Fourier transform.
`pwelch`	Estimate the power spectral density (PSD) of a signal using Welch's method.

fdatool fft2