scipy.fft.fft

scipy.fft.fft(x, n=None, axis=-1, norm=None, overwrite_x=False, workers=None)

Compute the one-dimensional discrete Fourier Transform.

This function computes the one-dimensional n-point discrete Fourier Transform (DFT) with the efficient Fast Fourier Transform (FFT) algorithm [1].

Parameters

xarray_like

Input array, can be complex.

nint, optional

Length of the transformed axis of the output. If n is smaller than the length of the input, the input is cropped. If it is larger, the input is padded with zeros. If n is not given, the length of the input along the axis specified by axis is used.

axisint, optional

Axis over which to compute the FFT. If not given, the last axis is used.

norm{None, “ortho”}, optional

Normalization mode. Default is None, meaning no normalization on the forward transforms and scaling by 1/n on the ifft. For norm="ortho", both directions are scaled by 1/sqrt(n).

overwrite_xbool, optional

If True, the contents of x can be destroyed; the default is False. See the notes below for more details.

workersint, optional

Maximum number of workers to use for parallel computation. If negative, the value wraps around from os.cpu_count(). See below for more details.

Returns

outcomplex ndarray

The truncated or zero-padded input, transformed along the axis indicated by axis, or the last one if axis is not specified.

Raises

IndexError

if axes is larger than the last axis of x.

See also

ifft

The inverse of fft.

fft2

The two-dimensional FFT.

fftn

The n-dimensional FFT.

rfftn

The n-dimensional FFT of real input.

fftfreq

Frequency bins for given FFT parameters.

next_fast_len

Size to pad input to for most efficient transforms

Notes

FFT (Fast Fourier Transform) refers to a way the discrete Fourier Transform (DFT) can be calculated efficiently, by using symmetries in the calculated terms. The symmetry is highest when n is a power of 2, and the transform is therefore most efficient for these sizes. For poorly factorizable sizes, scipy.fft uses Bluestein’s algorithm [2] and so is never worse than O(n log n). Further performance improvements may be seen by zero-padding the input using next_fast_len.

If x is a 1d array, then the fft is equivalent to

y[k] = np.sum(x * np.exp(-2j * np.pi * k * np.arange(n)/n))

The frequency term f=k/n is found at y[k]. At y[n/2] we reach the Nyquist frequency and wrap around to the negative-frequency terms. 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.

Transforms can be done in single, double or extended precision (long double) floating point. Half precision inputs will be converted to single precision and non floating-point inputs will be converted to double precision.

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.

When overwrite_x=True is specified, the memory referenced by x may be used by the implementation in any way. This may include reusing the memory for the result, but this is in no way guaranteed. You should not rely on the contents of x after the transform as this may change in future without warning.

The workers argument specifies the maximum number of parallel jobs to split the FFT computation into. This will execute independent 1-dimensional FFTs within x. So, x must be at least 2-dimensional and the non-transformed axes must be large enough to split into chunks. If x is too small, fewer jobs may be used than requested.

References

1

Cooley, James W., and John W. Tukey, 1965, “An algorithm for the machine calculation of complex Fourier series,” Math. Comput. 19: 297-301.

2

Bluestein, L., 1970, “A linear filtering approach to the computation of discrete Fourier transform”. IEEE Transactions on Audio and Electroacoustics. 18 (4): 451-455.

Examples

>>> import scipy.fft
>>> scipy.fft.fft(np.exp(2j * np.pi * np.arange(8) / 8))
array([-2.33486982e-16+1.14423775e-17j,  8.00000000e+00-1.25557246e-15j,
        2.33486982e-16+2.33486982e-16j,  0.00000000e+00+1.22464680e-16j,
       -1.14423775e-17+2.33486982e-16j,  0.00000000e+00+5.20784380e-16j,
        1.14423775e-17+1.14423775e-17j,  0.00000000e+00+1.22464680e-16j])

In this example, real input has an FFT which is Hermitian, i.e., symmetric in the real part and anti-symmetric in the imaginary part:

>>> from scipy.fft import fft, fftfreq, fftshift
>>> import matplotlib.pyplot as plt
>>> t = np.arange(256)
>>> sp = fftshift(fft(np.sin(t)))
>>> freq = fftshift(fftfreq(t.shape[-1]))
>>> plt.plot(freq, sp.real, freq, sp.imag)
[<matplotlib.lines.Line2D object at 0x...>, <matplotlib.lines.Line2D object at 0x...>]
>>> plt.show()