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# numpy.fft.hfft

fft.hfft(a, n=None, axis=-1, norm=None, out=None)

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

  Parameters:
    a array_like

      The input array.

    n int, 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.

    axis int, optional

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

    norm{“backward”, “ortho”, “forward”}, optional

      Normalization mode (see `numpy.fft`). Default is “backward”. Indicates which direction of the forward/backward pair of transforms is scaled and with what normalization factor.

      Added in version 1.20.0: The “backward”, “forward” values were added.

    out ndarray, optional

      If provided, the result will be placed in this array. It should be of the appropriate shape and dtype.

      Added in version 2.0.0.

  Returns:
    out ndarray

      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 not a valid 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

      >>> import numpy as np
      >>> 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.]])