nlcpy.fft.hfft

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

Computes 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. If axis is larger than the last axis of a, IndexError occurs.
norm{None, “ortho”},optional: Normalization mode. By default(None), the transforms are unscaled. It norm is set to “ortho”, the return values will be scaled by $1/\sqrt{n}$ .

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,

See also

rfft: Computes the one-dimensional discrete fourier transform for a real array.
ihfft: Computes the inverse FFT of a signal that has Hermitian symmetry.

Note

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 nlcpy as vp
>>> signal = vp.array([1, 2, 3, 4, 3, 2])
>>> vp.fft.fft(signal)     
array([15.+0.j, -4.+0.j,  0.+0.j, -1.+0.j,  0.+0.j, -4.+0.j])    # may vary
>>> vp.fft.hfft(signal[:4]) # Input first half of signal
array([15., -4.,  0., -1.,  0., -4.])
>>> vp.fft.hfft(signal, 6)  # Input entire signal and truncate
array([15., -4.,  0., -1.,  0., -4.])

>>> signal = vp.array([[1, 1.j], [-1.j, 2]])
>>> vp.conj(signal.T) - signal   # check Hermitian symmetry
array([[ 0.-0.j, -0.+0.j],
       [ 0.+0.j,  0.-0.j]])
>>> freq_spectrum = vp.fft.hfft(signal)
>>> freq_spectrum
array([[ 1.,  1.],
       [ 2., -2.]])