scipy.fftpack.fft

scipy.fftpack.fft(x, n=None, axis=-1, overwrite_x=0)

Return discrete Fourier transform of real or complex sequence.

The returned complex array contains y(0), y(1),..., y(n-1) where

y(j) = (x * exp(-2*pi*sqrt(-1)*j*np.arange(n)/n)).sum().

Parameters :

x : array_like

Array to Fourier transform.

n : int, optional

Length of the Fourier transform. If n < x.shape[axis], x is truncated. If n > x.shape[axis], x is zero-padded. The default results in n = x.shape[axis].

axis : int, optional

Axis along which the fft’s are computed; the default is over the last axis (i.e., axis=-1).

overwrite_x : bool, optional

If True the contents of x can be destroyed; the default is False.

Returns :

z : complex ndarray

with the elements:

[y(0),y(1),..,y(n/2),y(1-n/2),...,y(-1)]        if n is even
[y(0),y(1),..,y((n-1)/2),y(-(n-1)/2),...,y(-1)]  if n is odd

where:

y(j) = sum[k=0..n-1] x[k] * exp(-sqrt(-1)*j*k* 2*pi/n), j = 0..n-1

Note that y(-j) = y(n-j).conjugate().

See also

ifft

Inverse FFT

rfft

FFT of a real sequence

Notes

The packing of the result is “standard”: If A = fft(a, n), then A[0] contains the zero-frequency term, A[1:n/2+1] contains the positive-frequency terms, and A[n/2+1:] contains the negative-frequency terms, in order of decreasingly negative frequency. So for an 8-point transform, the frequencies of the result are [ 0, 1, 2, 3, 4, -3, -2, -1].

For n even, A[n/2] contains the sum of the positive and negative-frequency terms. For n even and x real, A[n/2] will always be real.

This is most efficient for n a power of two.

Examples

>>> from scipy.fftpack import fft, ifft
>>> x = np.arange(5)
>>> np.allclose(fft(ifft(x)), x, atol=1e-15)  #within numerical accuracy.
True