NumPy

numpy.fft.hfft

numpy.fft.hfft(a, n=None, axis=-1, norm=None)[source]

Compute the FFT of a signal that has Hermitian symmetry, i.e., a real spectrum.

Parameters
aarray_like

The input array.

nint, optional

Length of the transformed axis of the output. For n output points, n//2 + 1 input points are necessary. If the input is longer than this, it is cropped. If it is shorter than this, it is padded with zeros. If n is not given, it is taken to be 2*(m-1) where m is the length of the input along the axis specified by axis.

axisint, optional

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

norm{None, “ortho”}, optional

Normalization mode (see numpy.fft). Default is None.

New in version 1.10.0.

Returns
outndarray

The truncated or zero-padded input, transformed along the axis indicated by axis, or the last one if axis is not specified. The length of the transformed axis is n, or, if n is not given, 2*m - 2 where m is the length of the transformed axis of the input. To get an odd number of output points, n must be specified, for instance as 2*m - 1 in the typical case,

Raises
IndexError

If axis is larger than the last axis of a.

See also

rfft

Compute the one-dimensional FFT for real input.

ihfft

The inverse of hfft.

Notes

hfft/ihfft are a pair analogous to rfft/irfft, but for the opposite case: here the signal has Hermitian symmetry in the time domain and is real in the frequency domain. So here it’s hfft for which you must supply the length of the result if it is to be odd.

  • even: ihfft(hfft(a, 2*len(a) - 2) == a, within roundoff error,

  • odd: ihfft(hfft(a, 2*len(a) - 1) == a, within roundoff error.

The correct interpretation of the hermitian input depends on the length of the original data, as given by n. This is because each input shape could correspond to either an odd or even length signal. By default, hfft assumes an even output length which puts the last entry at the Nyquist frequency; aliasing with its symmetric counterpart. By Hermitian symmetry, the value is thus treated as purely real. To avoid losing information, the shape of the full signal must be given.

Examples

>>> signal = np.array([1, 2, 3, 4, 3, 2])
>>> np.fft.fft(signal)
array([15.+0.j,  -4.+0.j,   0.+0.j,  -1.-0.j,   0.+0.j,  -4.+0.j]) # may vary
>>> np.fft.hfft(signal[:4]) # Input first half of signal
array([15.,  -4.,   0.,  -1.,   0.,  -4.])
>>> np.fft.hfft(signal, 6)  # Input entire signal and truncate
array([15.,  -4.,   0.,  -1.,   0.,  -4.])
>>> signal = np.array([[1, 1.j], [-1.j, 2]])
>>> np.conj(signal.T) - signal   # check Hermitian symmetry
array([[ 0.-0.j,  -0.+0.j], # may vary
       [ 0.+0.j,  0.-0.j]])
>>> freq_spectrum = np.fft.hfft(signal)
>>> freq_spectrum
array([[ 1.,  1.],
       [ 2., -2.]])

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