`scipy.fftpack._basic:fft`

source: /scipy/fftpack/_basic.py :12

Signature

def fft ( x , n = None , axis = -1 , overwrite_x = False )

Summary

Return discrete Fourier transform of real or complex sequence.

Extended Summary

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

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] contains the positive-frequency terms, and A[n/2:] 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]. To rearrange the fft output so that the zero-frequency component is centered, like [-4, -3, -2, -1, 0, 1, 2, 3], use fftshift.

Both single and double precision routines are implemented. Half precision inputs will be converted to single precision. Non-floating-point inputs will be converted to double precision. Long-double precision inputs are not supported.

This function is most efficient when n is a power of two, and least efficient when n is prime.

Note that if x is real-valued, then A[j] == A[n-j].conjugate(). If x is real-valued and n is even, then A[n/2] is real.

If the data type of x is real, a "real FFT" algorithm is automatically used, which roughly halves the computation time. To increase efficiency a little further, use rfft, which does the same calculation, but only outputs half of the symmetrical spectrum. If the data is both real and symmetrical, the dct can again double the efficiency by generating half of the spectrum from half of the signal.

Examples

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

✓

Aliases

scipy.fftpack.fft

Referenced by

This package

release:0.18.0-notes