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@endolith
Last active December 13, 2022 04:50
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A-weighting audio files in Python
#!/usr/bin/env python
# -*- coding: utf-8 -*-
"""
Translated from a MATLAB script (which also includes C-weighting, octave
and one-third-octave digital filters).
Author: Christophe Couvreur, Faculte Polytechnique de Mons (Belgium)
couvreur@thor.fpms.ac.be
Last modification: Aug. 20, 1997, 10:00am.
BSD license
http://www.mathworks.com/matlabcentral/fileexchange/69
Translated from adsgn.m to Python 2009-07-14 endolith@gmail.com
"""
from numpy import pi, polymul
from scipy.signal import bilinear
def A_weighting(fs):
"""Design of an A-weighting filter.
b, a = A_weighting(fs) designs a digital A-weighting filter for
sampling frequency `fs`. Usage: y = scipy.signal.lfilter(b, a, x).
Warning: `fs` should normally be higher than 20 kHz. For example,
fs = 48000 yields a class 1-compliant filter.
References:
[1] IEC/CD 1672: Electroacoustics-Sound Level Meters, Nov. 1996.
"""
# Definition of analog A-weighting filter according to IEC/CD 1672.
f1 = 20.598997
f2 = 107.65265
f3 = 737.86223
f4 = 12194.217
A1000 = 1.9997
NUMs = [(2*pi * f4)**2 * (10**(A1000/20)), 0, 0, 0, 0]
DENs = polymul([1, 4*pi * f4, (2*pi * f4)**2],
[1, 4*pi * f1, (2*pi * f1)**2])
DENs = polymul(polymul(DENs, [1, 2*pi * f3]),
[1, 2*pi * f2])
# Use the bilinear transformation to get the digital filter.
# (Octave, MATLAB, and PyLab disagree about Fs vs 1/Fs)
return bilinear(NUMs, DENs, fs)
import sys
from scipy.signal import lfilter
import numpy as np
from A_weighting import A_weighting
try:
import soundfile as sf
except ImportError:
from scikits.audiolab import wavread
# (scipy.io.wavfile can't read 24-bit WAV)
def rms_flat(a): # from matplotlib.mlab
"""
Return the root mean square of all the elements of *a*, flattened out.
"""
return np.sqrt(np.mean(np.absolute(a)**2))
if len(sys.argv) == 2:
filename = sys.argv[1]
else:
filename = 'noise.wav'
try:
x, fs = sf.read(filename)
except NameError:
x, fs, bits = wavread(filename)
print(filename)
print('Original: {:+.2f} dB'.format(20*np.log10(rms_flat(x))))
b, a = A_weighting(fs)
y = lfilter(b, a, x)
print('A-weighted: {:+.2f} dB'.format(20*np.log10(rms_flat(y))))
@ThomQu
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ThomQu commented Oct 8, 2021

OK I should have mentioned first that my input signal is sampled at 44 kHz,

The "weights from IEC 61400" curve is just the plotting of the norm's formula:
image
i.e. the difference in dB between between an A_weighted spectrum and the same spectrum without weighting

So when I apply the A_weighting function over my white-noise signal, I get a filtered signal in return. I would expect that when I plot its spectrum minus the original spectrum ("effects of weighting function" plot), I would get the same response as the "weights from IEC 61400", at least until 20 kHz (fs/2).

The "polynom" curve was the plotting of NUMs/DENs from this first code's version, applied to the frequencies array and 10-logged:

poly = 10*np.log10(np.polyval(NUMs,f_norm)/np.polyval(DENs,f_norm))
but forget it, I have to dig on filtering's theory (I'm not a spectialist), plotting that and expecting it to be similar to "weights from IEC 61400" might not be relevant...

I've just tested your newer version, the results are indeed equal to this one's.
image

@endolith
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endolith commented Oct 8, 2021

So I'm looking at my "improved" version from 2018, which is based on the MZTi method and it does indeed have better frequency response at high frequencies:

MZT vs MZTi vs BLT

But I think the reason I didn't push it because I wasn't sure I was doing it right. "Since the mapping wraps the s-plane's jω axis around the z-plane's unit circle repeatedly, any zeros (or poles) greater than the Nyquist frequency will be mapped to an aliased location." And I'm not sure if those need to be removed first, depending on the sampling rate, etc. At lower sample rates it gets worse, so it still needs some work. (Also the aliased Nyquist tail could be pushed down a bit to improve the overall average accuracy.)

This paper just describes the same BLT solution as this gist, and has a table of the minimum required sample rate to meet the tolerances:

  A-weighting C-weighting
IEC 61672-1 Class 1 35 kHz 35 kHz
IEC 61672-1 Class 2 20 kHz 20 kHz
ANSI S.1-43 Type 0 71 kHz 72 kHz
ANSI S.1-43 Type 1 35 kHz 35 kHz
ANSI S.1-43 Type 2 20 kHz 20 kHz

There are probably other papers that describe better methods, I should look for them.

@endolith
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@ThomQu @annezhangxue @shangyan0517 @frgos2 @masahf @mankind-nil @Zeokat

Can you all tell me how you use this code so I can improve it? I tried to fold it into a library with other related functions like THD+N calculators, etc. But people still comment on this gist, so maybe it's better for it to be self-contained? I assume you are all using the digital filter version and not the analog filter coefficients?

@ThomQu
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ThomQu commented Oct 11, 2021

@endolith

Personnaly, I am trying to build-up a library that wraps up a few key functions for quick audio files analyses. I've picked stuff from librosa, scipy, etc and got your code from some colleague's scripts. That's the reason why I went for that first version.

Thks for the info about the MZTi method, I'll look for a little help around to check what I can do with it.

Regards

@endolith
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@ThomQu

Ok, so maybe I should just make an installable weighting filter library that does nothing else, and then you could use it in yours, etc? I tried combining it with other things like frequency estimation and THD calculation, but maybe those are too unrelated to combine.

@endolith
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endolith commented Oct 15, 2021

I was asked on Stack Exchange for my MZTi code, even if crude, so here it is.

I don't think it's correct at very low sample rates, because of aliasing, but should be better at the typical 44.1 or 48 kHz. It's definitely wrong below fs = 2.5 kHz. In between, I think the MZTi zeroes are forcing it into compliance despite being poorly-transformed.

I don't think it makes much practical difference in dB levels vs the BLT method, by the way.

# Poles and zeros for analog filter from specification
z, p, k = ABC_weighting('A')

# Matched Z Transform:
T = 1/fs
z_MZT = np.exp(z*T)  # 4
p_MZT = np.exp(p*T)  # 6

k_MZT = k * ((np.prod(2*pi*1000 - z)/
              np.prod(2*pi*1000 - p)) *
             (np.prod(1 - exp(p*T - 2*pi*1000*T)))/
              np.prod(1 - exp(z*T - 2*pi*1000*T)))

# Tweak gain to actual magnitude response TODO: Why is this necessary?
w_MZT, h_MZT = signal.freqz_zpk(z_MZT, p_MZT, k_MZT, [1000/(fs/2)*pi])
k_MZT /= abs(h_MZT)


w = np.array([pi/3, 2*pi/3])
H = abs(signal.freqs_zpk(z, p, k, w*fs)[1] /
        signal.freqz_zpk(z_MZT, p_MZT, k_MZT, w)[1])  # relative magnitude error

# MZTi method of 0, 1/3, 2/3 Nyquist will not work, since response at 0 is 0.
H = np.concatenate(([1], H))

b1 = 0.5 * (H[0] - sqrt(H[0]**2 - 2*H[1]**2 + 2*H[2]**2))
b2 = (3 * (H[0] - b1) -
      sqrt(-3*H[0]**2 + 12*H[1]**2 - 6*H[0]*b1 - 3*b1**2 + 0j)
      ) / 6
b0 = H[0] - b1 - b2
b = np.array([b0, b1, b2])

z_correct = np.roots(b)

# TODO: Is this right?
z_MZTi = np.concatenate((z_MZT, z_correct))
p_MZTi = p_MZT
k_MZTi = k_MZT

# Tweak gain to actual magnitude response again TODO: Why?
w_MZTi, h_MZTi = signal.freqz_zpk(z_MZTi, p_MZTi, k_MZTi, [1000/(fs/2)*pi])
k_MZTi /= abs(h_MZTi)

And the reference is https://www.khabdha.org/wp-content/uploads/2008/03/optimizing-the-magnitude-response-of-mzt-filters-mzti-2007.pdf

@endolith
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so maybe I should just make an installable weighting filter library that does nothing else

and maybe include bilinear as an option so people can compare with the results of other software?

@endolith
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FYI, the MZTi version is accurate down to about 2.5 kHz:

redo 8 kHz
redo 8 kHz closeup

redo 3 kHz
redo 3 kHz closeup

redo 2 kHz
redo 2 kHz closeup

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