SciPy

scipy.fft.hfftn

scipy.fft.hfftn(x, s=None, axes=None, norm=None, overwrite_x=False, workers=None)[source]

Compute the N-D FFT of Hermitian symmetric complex input, i.e., a signal with a real spectrum.

This function computes the N-D discrete Fourier Transform for a Hermitian symmetric complex input over any number of axes in an M-D array by means of the Fast Fourier Transform (FFT). In other words, ihfftn(hfftn(x, s)) == x to within numerical accuracy. (s here is x.shape with s[-1] = x.shape[-1] * 2 - 1, this is necessary for the same reason x.shape would be necessary for irfft.)

Parameters
xarray_like

Input array.

ssequence of ints, optional

Shape (length of each transformed axis) of the output (s[0] refers to axis 0, s[1] to axis 1, etc.). s is also the number of input points used along this axis, except for the last axis, where s[-1]//2+1 points of the input are used. Along any axis, if the shape indicated by s is smaller than that of the input, the input is cropped. If it is larger, the input is padded with zeros. If s is not given, the shape of the input along the axes specified by axes is used. Except for the last axis which is taken to be 2*(m-1) where m is the length of the input along that axis.

axessequence of ints, optional

Axes over which to compute the inverse FFT. If not given, the last len(s) axes are used, or all axes if s is also not specified.

norm{None, “ortho”}, optional

Normalization mode (see fft). Default is None.

overwrite_xbool, optional

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

Returns
outndarray

The truncated or zero-padded input, transformed along the axes indicated by axes, or by a combination of s or x, as explained in the parameters section above. The length of each transformed axis is as given by the corresponding element of s, or the length of the input in every axis except for the last one if s is not given. In the final transformed axis the length of the output when s is not given is 2*(m-1) where m is the length of the final transformed axis of the input. To get an odd number of output points in the final axis, s must be specified.

Raises
ValueError

If s and axes have different length.

IndexError

If an element of axes is larger than than the number of axes of x.

See also

ihfftn

The inverse N-D FFT with real spectrum. Inverse of hfftn.

fft

The 1-D FFT, with definitions and conventions used.

rfft

Forward FFT of real input.

Notes

For a 1-D signal x to have a real spectrum, it must satisfy the Hermitian property:

x[i] == np.conj(x[-i]) for all i

This generalizes into higher dimensions by reflecting over each axis in turn:

x[i, j, k, ...] == np.conj(x[-i, -j, -k, ...]) for all i, j, k, ...

This should not be confused with a Hermitian matrix, for which the transpose is its own conjugate:

x[i, j] == np.conj(x[j, i]) for all i, j

The default value of s assumes an even output length in the final transformation axis. When performing the final complex to real transformation, the Hermitian symmetry requires that the last imaginary component along that axis must be 0 and so it is ignored. To avoid losing information, the correct length of the real input must be given.

Examples

>>> import scipy.fft
>>> x = np.ones((3, 2, 2))
>>> scipy.fft.hfftn(x)
array([[[12.,  0.],
        [ 0.,  0.]],
       [[ 0.,  0.],
        [ 0.,  0.]],
       [[ 0.,  0.],
        [ 0.,  0.]]])

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