# minimum-phase phase via Hilbert transform returned values

Following my previous question: HRIR Minimum phase I managed to compute the minimum-phase phase of a FIR filter (in my particular case, HRTF filters).

However I am not sure of the phase values returned by my function.

The Python function is the following. It expects an arbitrary HRIR as input. I am using a 44100 long FFT to have the freqency bins perfectly aligned at integer frequencies at 44100Hz sampling rate (so for example a sinusoid at 1000Hz will have a magnitude of exactly 1 with no spectral leakage)

import numpy as np
import matplotlib.pyplot as plt
from scipy.signal import hilbert
from scipy.fft import fft, ifft, fftfreq

def min_phase_coversion(HRIR):

HRIR_fft = fft(HRIR,44100)

xf = fftfreq(len(HRIR_fft), 1/44100)

phase=np.angle(HRIR_fft)

minimum_phase = np.imag(-hilbert(np.log(np.abs(HRIR_fft))))

plt.figure()
plt.plot(xf, phase)
plt.plot(xf,minimum_phase)
plt.plot()
plt.grid()
plt.show()


For the following plots I am using the KEMAR MIT dataset, in particular the plot have been computed for the HRIR at [azimuth=144°, elevation=30°, distance=1.4m]. The selected HRIR is the following (one per ear):  Once I perform my minimum-phase computation using the above Python function, this is what I get (phase for a single ear): However, looking at the plot, I can't understand the values returned by

np.imag(-hilbert(np.log(np.abs(HRIR_fft))))


Is the returned phase unwrapped? Or am i doing something wrong? The original phase "wraps" as expected once it gets over +3.14 or under -3.14 (+/- pi), while the minimum one doesn't.

## 1 Answer

I computed and compared the minimum phase HRIR and the original one.

This is my final code:

def min_phase_coversion(HRIR):
'''

:param HRIR: the desired HRIR impulse response to convert into minimum phase
:return: the minimum phase version of the original HRIR
'''

HRIR_fft = fft(HRIR,44100)

#computing magnitude, tested with sinusoid, it works.
xf = fftfreq(len(HRIR_fft), 1/44100)

fft_magnitude = np.abs(HRIR_fft/len(HRIR_fft))

phase=np.angle(HRIR_fft)

minimum_phase = np.imag(-hilbert(np.log(np.abs(HRIR_fft))))

plt.figure()
plt.plot(xf, fft_magnitude) #computing and plotting magnitude
plt.plot(xf, phase, label='original phase')
plt.plot(xf,minimum_phase, label='minimum phase')
plt.legend()
plt.plot()
plt.grid()
plt.show()

min_phase_HRIR = irfft(np.abs(HRIR_fft)*np.exp(1j*minimum_phase), 44100) # |H(w)|*e^(jPhi(w))

HRIR_original = irfft(HRIR_fft, 44100)

plt.figure()
plt.plot(min_phase_HRIR[0:512], label="Min_phase", linewidth=0.5, marker='o', markersize=1) #ricostruisce bene
plt.plot(HRIR_original[0:512], label="original phase", linewidth=0.5, marker='o', markersize=1) #ricostruisce bene
plt.legend()
plt.show()

plt.figure()
min_phase_fft = fft(min_phase_HRIR, 44100)
min_phase_mangnitude =  np.abs(min_phase_fft/len(min_phase_fft))
plt.plot(xf,fft_magnitude)
plt.plot(xf,min_phase_mangnitude)
plt.show()

return min_phase_HRIR


The results I get seem reasonable looking at the plots: and the magnitude is the same as I expected (the original one and the min phase one are perfectly overlapping): However, I am not totally sure of the phase behaviour (as mentioned in my question) and its returned values.