# scipy.fftpack.fft#

scipy.fftpack.fft(x, n=None, axis=-1, overwrite_x=False)[source]#

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:
xarray_like

Array to Fourier transform.

nint, 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]`.

axisint, optional

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

overwrite_xbool, optional

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

Returns:
zcomplex 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
```

`ifft`

Inverse FFT

`rfft`

FFT of a real sequence

Notes

The packing of the result is “standard”: If `A = fft(a, n)`, then `A` 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.
True
```