Module « scipy.signal »
Signature de la fonction minimum_phase
def minimum_phase(h, method='homomorphic', n_fft=None)
Description
minimum_phase.__doc__
Convert a linear-phase FIR filter to minimum phase
Parameters
----------
h : array
Linear-phase FIR filter coefficients.
method : {'hilbert', 'homomorphic'}
The method to use:
'homomorphic' (default)
This method [4]_ [5]_ works best with filters with an
odd number of taps, and the resulting minimum phase filter
will have a magnitude response that approximates the square
root of the the original filter's magnitude response.
'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).
Returns
-------
h_minimum : array
The minimum-phase version of the filter, with length
``(length(h) + 1) // 2``.
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:
http://dspguru.com/dsp/howtos/how-to-design-minimum-phase-fir-filters
Examples
--------
Create an optimal linear-phase filter, then convert it to minimum phase:
>>> 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, Hz=2.)
Convert it to minimum phase:
>>> h_min_hom = minimum_phase(h_linear, method='homomorphic')
>>> h_min_hil = minimum_phase(h_linear, method='hilbert')
Compare the three filters:
>>> fig, axs = plt.subplots(4, figsize=(4, 8))
>>> for h, style, color in zip((h_linear, h_min_hom, h_min_hil),
... ('-', '-', '--'), ('k', 'r', 'c')):
... w, H = freqz(h)
... w, gd = group_delay((h, 1))
... w /= np.pi
... axs[0].plot(h, color=color, linestyle=style)
... axs[1].plot(w, np.abs(H), color=color, linestyle=style)
... axs[2].plot(w, 20 * np.log10(np.abs(H)), color=color, linestyle=style)
... axs[3].plot(w, gd, color=color, linestyle=style)
>>> for ax in axs:
... ax.grid(True, color='0.5')
... ax.fill_between(freq[1:3], *ax.get_ylim(), color='#ffeeaa', zorder=1)
>>> axs[0].set(xlim=[0, len(h_linear) - 1], ylabel='Amplitude', xlabel='Samples')
>>> axs[1].legend(['Linear', 'Min-Hom', 'Min-Hil'], title='Phase')
>>> for ax, ylim in zip(axs[1:], ([0, 1.1], [-150, 10], [-60, 60])):
... ax.set(xlim=[0, 1], ylim=ylim, xlabel='Frequency')
>>> axs[1].set(ylabel='Magnitude')
>>> axs[2].set(ylabel='Magnitude (dB)')
>>> axs[3].set(ylabel='Group delay')
>>> plt.tight_layout()
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," 2nd edition.
Upper Saddle River, N.J.: Prentice Hall, 1999.
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