     will have a magnitude response that approximates the square
            root of the original filter's magnitude response using half
            the number of taps when ``half=True`` (default), or the
            original magnitude spectrum using the same number of taps
            when ``half=False``.

        'hilbert'
            This method [1]_ is designed to be used with equiripple
            filters (e.g., from `remez`) with unity or zero gain
            regions.

n_fft : int
    The number of points to use for the FFT. Should be at least a
    few times larger than the signal length (see Notes).
half : bool
    If ``True``, create a filter that is half the length of the original, with a
    magnitude spectrum that is the square root of the original. If ``False``,
    create a filter that is the same length as the original, with a magnitude
    spectrum that is designed to match the original (only supported when
    ``method='homomorphic'``).

    .. versionadded:: 1.14.0

Returns
-------
h_minimum : array
    The minimum-phase version of the filter, with length
    ``(len(h) + 1) // 2`` when ``half is True`` or ``len(h)`` otherwise.

See Also
--------
firwin
firwin2
remez

Notes
-----
Both the Hilbert [1]_ or homomorphic [4]_ [5]_ methods require selection
of an FFT length to estimate the complex cepstrum of the filter.

In the case of the Hilbert method, the deviation from the ideal
spectrum ``epsilon`` is related to the number of stopband zeros
``n_stop`` and FFT length ``n_fft`` as::

    epsilon = 2. * n_stop / n_fft

For example, with 100 stopband zeros and a FFT length of 2048,
``epsilon = 0.0976``. If we conservatively assume that the number of
stopband zeros is one less than the filter length, we can take the FFT
length to be the next power of 2 that satisfies ``epsilon=0.01`` as::

    n_fft = 2 ** int(np.ceil(np.log2(2 * (len(h) - 1) / 0.01)))

This gives reasonable results for both the Hilbert and homomorphic
methods, and gives the value used when ``n_fft=None``.

Alternative implementations exist for creating minimum-phase filters,
including zero inversion [2]_ and spectral factorization [3]_ [4]_.
For more information, see `this DSPGuru page
<http://dspguru.com/dsp/howtos/how-to-design-minimum-phase-fir-filters>`__.

References
----------
.. [1] N. Damera-Venkata and B. L. Evans, "Optimal design of real and
       complex minimum phase digital FIR filters," Acoustics, Speech,
       and Signal Processing, 1999. Proceedings., 1999 IEEE International
       Conference on, Phoenix, AZ, 1999, pp. 1145-1148 vol.3.
       :doi:`10.1109/ICASSP.1999.756179`
.. [2] X. Chen and T. W. Parks, "Design of optimal minimum phase FIR
       filters by direct factorization," Signal Processing,
       vol. 10, no. 4, pp. 369-383, Jun. 1986.
.. [3] T. Saramaki, "Finite Impulse Response Filter Design," in
       Handbook for Digital Signal Processing, chapter 4,
       New York: Wiley-Interscience, 1993.
.. [4] J. S. Lim, Advanced Topics in Signal Processing.
       Englewood Cliffs, N.J.: Prentice Hall, 1988.
.. [5] A. V. Oppenheim, R. W. Schafer, and J. R. Buck,
       "Discrete-Time Signal Processing," 3rd edition.
       Upper Saddle River, N.J.: Pearson, 2009.

Examples
--------
Create an optimal linear-phase low-pass filter `h` with a transition band of
[0.2, 0.3] (assuming a Nyquist frequency of 1):

>>> import numpy as np
>>> from scipy.signal import remez, minimum_phase, freqz, group_delay
>>> import matplotlib.pyplot as plt
>>> freq = [0, 0.2, 0.3, 1.0]
>>> desired = [1, 0]
>>> h_linear = remez(151, freq, desired, fs=2)

Convert it to minimum phase:

>>> h_hil = minimum_phase(h_linear, method='hilbert')
>>> h_hom = minimum_phase(h_linear, method='homomorphic')
>>> h_hom_full = minimum_phase(h_linear, method='homomorphic', half=False)

Compare the impulse and frequency response of the four filters:

>>> fig0, ax0 = plt.subplots(figsize=(6, 3), tight_layout=True)
>>> fig1, axs = plt.subplots(3, sharex='all', figsize=(6, 6), tight_layout=True)
>>> ax0.set_title("Impulse response")
>>> ax0.set(xlabel='Samples', ylabel='Amplitude', xlim=(0, len(h_linear) - 1))
>>> axs[0].set_title("Frequency Response")
>>> axs[0].set(xlim=(0, .65), ylabel="Magnitude / dB")
>>> axs[1].set(ylabel="Phase / rad")
>>> axs[2].set(ylabel="Group Delay / samples", ylim=(-31, 81),
...             xlabel='Normalized Frequency (Nyqist frequency: 1)')
>>> for h, lb in ((h_linear,   f'Linear ({len(h_linear)})'),
...               (h_hil,      f'Min-Hilbert ({len(h_hil)})'),
...               (h_hom,      f'Min-Homomorphic ({len(h_hom)})'),
...               (h_hom_full, f'Min-Homom. Full ({len(h_hom_full)})')):
...     w_H, H = freqz(h, fs=2)
...     w_gd, gd = group_delay((h, 1), fs=2)
...
...     alpha = 1.0 if lb == 'linear' else 0.5  # full opacity for 'linear' line
...     ax0.plot(h, '.-', alpha=alpha, label=lb)
...     axs[0].plot(w_H, 20 * np.log10(np.abs(H)), alpha=alpha)
...     axs[1].plot(w_H, np.unwrap(np.angle(H)), alpha=alpha, label=lb)
...     axs[2].plot(w_gd, gd, alpha=alpha)
>>> ax0.grid(True)
>>> ax0.legend(title='Filter Phase (Order)')
>>> axs[1].legend(title='Filter Phase (Order)', loc='lower right')
>>> for ax_ in axs:  # shade transition band:
...     ax_.axvspan(freq[1], freq[2], color='y', alpha=.25)
...     ax_.grid(True)
>>> plt.show()

The impulse response and group delay plot depict the 75 sample delay of the linear
phase filter `h`. The phase should also be linear in the stop band--due to the small
magnitude, numeric noise dominates there. Furthermore, the plots show that the
minimum phase filters clearly show a reduced (negative) phase slope in the pass and
transition band. The plots also illustrate that the filter with parameters
``method='homomorphic', half=False`` has same order and magnitude response as the
linear filter `h` whereas the other minimum phase filters have only half the order
and the square root  of the magnitude response.
z