# scipy.signal.minimum_phase#

scipy.signal.minimum_phase(h, method='homomorphic', n_fft=None)[source]#

Convert a linear-phase FIR filter to minimum phase

Parameters:
harray

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 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_fftint

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_minimumarray

The minimum-phase version of the filter, with length `(length(h) + 1) // 2`.

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:

References

 (1,2)

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



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.



T. Saramaki, “Finite Impulse Response Filter Design,” in Handbook for Digital Signal Processing, chapter 4, New York: Wiley-Interscience, 1993.

 (1,2,3)

J. S. Lim, Advanced Topics in Signal Processing. Englewood Cliffs, N.J.: Prentice Hall, 1988.

 (1,2)

A. V. Oppenheim, R. W. Schafer, and J. R. Buck, “Discrete-Time Signal Processing,” 2nd edition. Upper Saddle River, N.J.: Prentice Hall, 1999.

Examples

Create an optimal linear-phase filter, then convert it to minimum phase:

```>>> 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_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.plot(h, color=color, linestyle=style)
...     axs.plot(w, np.abs(H), color=color, linestyle=style)
...     axs.plot(w, 20 * np.log10(np.abs(H)), color=color, linestyle=style)
...     axs.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.set(xlim=[0, len(h_linear) - 1], ylabel='Amplitude', xlabel='Samples')
>>> axs.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.set(ylabel='Magnitude')
>>> axs.set(ylabel='Magnitude (dB)')
>>> axs.set(ylabel='Group delay')
>>> plt.tight_layout()
```