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Fast Fourier transform
Y = fft(x)
Y = fft(X,n)
Y = fft(X,[],dim)
Y
= fft(X,n,dim)
The functions Y = fft(x) and y = ifft(X) implement the transform and inverse transform pair given for vectors of length N by:
$$\begin{array}{l}X(k)={\displaystyle \sum _{j=1}^{N}x}(j){\omega}_{N}^{(j-1)(k-1)}\\ x(j)=(1/N){\displaystyle \sum _{k=1}^{N}X}(k){\omega}_{N}\end{array}$$
where
$${\omega}_{N}={e}^{(-2\pi i)/N}$$
is an Nth root of unity.
Y = fft(x) returns the discrete Fourier transform (DFT) of vector x, computed with a fast Fourier transform (FFT) algorithm.
If the input X is a matrix, Y = fft(X) returns the Fourier transform of each column of the matrix.
If the input X is a multidimensional array, fft operates on the first nonsingleton dimension.
Y = fft(X,n) returns the n-point DFT. fft(X) is equivalent to fft(X, n) where n is the size of X in the first nonsingleton dimension. If the length of X is less than n, X is padded with trailing zeros to length n. If the length of X is greater than n, the sequence X is truncated. When X is a matrix, the length of the columns are adjusted in the same manner.
Y = fft(X,[],dim) and Y = fft(X,n,dim) applies the FFT operation across the dimension dim.
A common use of Fourier transforms is to find the frequency components of a signal buried in a noisy time domain signal. Consider data sampled at 1000 Hz. Form a signal containing a 50 Hz sinusoid of amplitude 0.7 and 120 Hz sinusoid of amplitude 1 and corrupt it with some zero-mean random noise:
Fs = 1000; % Sampling frequency T = 1/Fs; % Sample time L = 1000; % Length of signal t = (0:L-1)*T; % Time vector % Sum of a 50 Hz sinusoid and a 120 Hz sinusoid x = 0.7*sin(2*pi*50*t) + sin(2*pi*120*t); y = x + 2*randn(size(t)); % Sinusoids plus noise plot(Fs*t(1:50),y(1:50)) title('Signal Corrupted with Zero-Mean Random Noise') xlabel('time (milliseconds)')
It is difficult to identify the frequency components by looking at the original signal. Converting to the frequency domain, the discrete Fourier transform of the noisy signal y is found by taking the fast Fourier transform (FFT):
NFFT = 2^nextpow2(L); % Next power of 2 from length of y Y = fft(y,NFFT)/L; f = Fs/2*linspace(0,1,NFFT/2+1); % Plot single-sided amplitude spectrum. plot(f,2*abs(Y(1:NFFT/2+1))) title('Single-Sided Amplitude Spectrum of y(t)') xlabel('Frequency (Hz)') ylabel('|Y(f)|')
The main reason the amplitudes are not exactly at 0.7 and 1 is because of the noise. Several executions of this code (including recomputation of y) will produce different approximations to 0.7 and 1. The other reason is that you have a finite length signal. Increasing L from 1000 to 10000 in the example above will produce much better approximations on average.
fft supports inputs of data types double and single. If you call fft with the syntax y = fft(X, ...), the output y has the same data type as the input X.
[1] Cooley, J. W. and J. W. Tukey, "An Algorithm for the Machine Computation of the Complex Fourier Series,"Mathematics of Computation, Vol. 19, April 1965, pp. 297-301.
[2] Duhamel, P. and M. Vetterli, "Fast Fourier Transforms: A Tutorial Review and a State of the Art," Signal Processing, Vol. 19, April 1990, pp. 259-299.
[3] FFTW (http://www.fftw.org)
[4] Frigo, M. and S. G. Johnson, "FFTW: An Adaptive Software Architecture for the FFT,"Proceedings of the International Conference on Acoustics, Speech, and Signal Processing, Vol. 3, 1998, pp. 1381-1384.
[5] Oppenheim, A. V. and R. W. Schafer, Discrete-Time Signal Processing, Prentice-Hall, 1989, p. 611.
[6] Oppenheim, A. V. and R. W. Schafer, Discrete-Time Signal Processing, Prentice-Hall, 1989, p. 619.
[7] Rader, C. M., "Discrete Fourier Transforms when the Number of Data Samples Is Prime," Proceedings of the IEEE, Vol. 56, June 1968, pp. 1107-1108.