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minimum_phase(h, method='homomorphic', n_fft=None)

Notes

Both the Hilbert or homomorphic 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 and spectral factorization . For more information, see:

http://dspguru.com/dsp/howtos/how-to-design-minimum-phase-fir-filters

Parameters

h : array

Linear-phase FIR filter coefficients.

method : {'hilbert', 'homomorphic'}

The method to use:

'homomorphic' (default)

This method 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 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 .

Convert a linear-phase FIR filter to minimum phase

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()

See :

Back References

The following pages refer to to this document either explicitly or contain code examples using this.

scipy.signal._fir_filter_design.firwin scipy.signal._fir_filter_design.firwin2 scipy.signal._fir_filter_design.remez scipy.signal._fir_filter_design.minimum_phase scipy.signal._fir_filter_design.firls

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