bundles / scipy latest / scipy / signal / _fir_filter_design / minimum_phase

function

`scipy.signal._fir_filter_design:minimum_phase`

source: build-install/usr/lib/python3.14/site-packages/scipy/signal/_fir_filter_design.py:1203

Signature

def minimum_phase(h, method: Literal['homomorphic', 'hilbert'] = homomorphic, n_fft: int | None = None, *, half: bool = True)

Summary

Convert a linear-phase FIR filter to minimum phase.

Parameters

h : array

Linear-phase FIR filter coefficients.

method : {'hilbert', 'homomorphic'}

The provided methods are:

'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 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').

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.

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.

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)

✗

[<matplotlib.lines.Line2D object at 0x7f3f686352b0>]
[<matplotlib.lines.Line2D object at 0x7f3f68635160>]
[<matplotlib.lines.Line2D object at 0x7f3f68636cf0>]
[<matplotlib.lines.Line2D object at 0x7f3f68636f90>]
[<matplotlib.lines.Line2D object at 0x7f3f68636e40>]
[<matplotlib.lines.Line2D object at 0x7f3f686370e0>]
[<matplotlib.lines.Line2D object at 0x7f3f68634c20>]
[<matplotlib.lines.Line2D object at 0x7f3f68634d70>]
[<matplotlib.lines.Line2D object at 0x7f3f68635940>]
[<matplotlib.lines.Line2D object at 0x7f3f68635e80>]
[<matplotlib.lines.Line2D object at 0x7f3f68635d30>]
[<matplotlib.lines.Line2D object at 0x7f3f68634440>]
[<matplotlib.lines.Line2D object at 0x7f3f686342f0>]
[<matplotlib.lines.Line2D object at 0x7f3f686341a0>]
[<matplotlib.lines.Line2D object at 0x7f3f68637620>]
[<matplotlib.lines.Line2D object at 0x7f3f686374d0>]

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)

✗

<matplotlib.patches.Rectangle object at 0x7f3f603a6fd0>
<matplotlib.patches.Rectangle object at 0x7f3f60d85a90>
<matplotlib.patches.Rectangle object at 0x7f3f60d86350>

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.

Aliases

scipy.signal.minimum_phase