Butterworth Filter

p
```from math import cos, sin, sqrt, tau

from audio_filters.iir_filter import IIRFilter

"""
Create 2nd-order IIR filters with Butterworth design.

Code based on https://webaudio.github.io/Audio-EQ-Cookbook/audio-eq-cookbook.html
Alternatively you can use scipy.signal.butter, which should yield the same results.
"""

def make_lowpass(
frequency: int,
samplerate: int,
q_factor: float = 1 / sqrt(2),
) -> IIRFilter:
"""
Creates a low-pass filter

>>> filter = make_lowpass(1000, 48000)
>>> filter.a_coeffs + filter.b_coeffs  # doctest: +NORMALIZE_WHITESPACE
[1.0922959556412573, -1.9828897227476208, 0.9077040443587427, 0.004277569313094809,
0.008555138626189618, 0.004277569313094809]
"""
w0 = tau * frequency / samplerate
_sin = sin(w0)
_cos = cos(w0)
alpha = _sin / (2 * q_factor)

b0 = (1 - _cos) / 2
b1 = 1 - _cos

a0 = 1 + alpha
a1 = -2 * _cos
a2 = 1 - alpha

filt = IIRFilter(2)
filt.set_coefficients([a0, a1, a2], [b0, b1, b0])
return filt

def make_highpass(
frequency: int,
samplerate: int,
q_factor: float = 1 / sqrt(2),
) -> IIRFilter:
"""
Creates a high-pass filter

>>> filter = make_highpass(1000, 48000)
>>> filter.a_coeffs + filter.b_coeffs  # doctest: +NORMALIZE_WHITESPACE
[1.0922959556412573, -1.9828897227476208, 0.9077040443587427, 0.9957224306869052,
-1.9914448613738105, 0.9957224306869052]
"""
w0 = tau * frequency / samplerate
_sin = sin(w0)
_cos = cos(w0)
alpha = _sin / (2 * q_factor)

b0 = (1 + _cos) / 2
b1 = -1 - _cos

a0 = 1 + alpha
a1 = -2 * _cos
a2 = 1 - alpha

filt = IIRFilter(2)
filt.set_coefficients([a0, a1, a2], [b0, b1, b0])
return filt

def make_bandpass(
frequency: int,
samplerate: int,
q_factor: float = 1 / sqrt(2),
) -> IIRFilter:
"""
Creates a band-pass filter

>>> filter = make_bandpass(1000, 48000)
>>> filter.a_coeffs + filter.b_coeffs  # doctest: +NORMALIZE_WHITESPACE
[1.0922959556412573, -1.9828897227476208, 0.9077040443587427, 0.06526309611002579,
0, -0.06526309611002579]
"""
w0 = tau * frequency / samplerate
_sin = sin(w0)
_cos = cos(w0)
alpha = _sin / (2 * q_factor)

b0 = _sin / 2
b1 = 0
b2 = -b0

a0 = 1 + alpha
a1 = -2 * _cos
a2 = 1 - alpha

filt = IIRFilter(2)
filt.set_coefficients([a0, a1, a2], [b0, b1, b2])
return filt

def make_allpass(
frequency: int,
samplerate: int,
q_factor: float = 1 / sqrt(2),
) -> IIRFilter:
"""
Creates an all-pass filter

>>> filter = make_allpass(1000, 48000)
>>> filter.a_coeffs + filter.b_coeffs  # doctest: +NORMALIZE_WHITESPACE
[1.0922959556412573, -1.9828897227476208, 0.9077040443587427, 0.9077040443587427,
-1.9828897227476208, 1.0922959556412573]
"""
w0 = tau * frequency / samplerate
_sin = sin(w0)
_cos = cos(w0)
alpha = _sin / (2 * q_factor)

b0 = 1 - alpha
b1 = -2 * _cos
b2 = 1 + alpha

filt = IIRFilter(2)
filt.set_coefficients([b2, b1, b0], [b0, b1, b2])
return filt

def make_peak(
frequency: int,
samplerate: int,
gain_db: float,
q_factor: float = 1 / sqrt(2),
) -> IIRFilter:
"""
Creates a peak filter

>>> filter = make_peak(1000, 48000, 6)
>>> filter.a_coeffs + filter.b_coeffs  # doctest: +NORMALIZE_WHITESPACE
[1.0653405327119334, -1.9828897227476208, 0.9346594672880666, 1.1303715025601122,
-1.9828897227476208, 0.8696284974398878]
"""
w0 = tau * frequency / samplerate
_sin = sin(w0)
_cos = cos(w0)
alpha = _sin / (2 * q_factor)
big_a = 10 ** (gain_db / 40)

b0 = 1 + alpha * big_a
b1 = -2 * _cos
b2 = 1 - alpha * big_a
a0 = 1 + alpha / big_a
a1 = -2 * _cos
a2 = 1 - alpha / big_a

filt = IIRFilter(2)
filt.set_coefficients([a0, a1, a2], [b0, b1, b2])
return filt

def make_lowshelf(
frequency: int,
samplerate: int,
gain_db: float,
q_factor: float = 1 / sqrt(2),
) -> IIRFilter:
"""
Creates a low-shelf filter

>>> filter = make_lowshelf(1000, 48000, 6)
>>> filter.a_coeffs + filter.b_coeffs  # doctest: +NORMALIZE_WHITESPACE
[3.0409336710888786, -5.608870992220748, 2.602157875636628, 3.139954022810743,
-5.591841778072785, 2.5201667380627257]
"""
w0 = tau * frequency / samplerate
_sin = sin(w0)
_cos = cos(w0)
alpha = _sin / (2 * q_factor)
big_a = 10 ** (gain_db / 40)
pmc = (big_a + 1) - (big_a - 1) * _cos
ppmc = (big_a + 1) + (big_a - 1) * _cos
mpc = (big_a - 1) - (big_a + 1) * _cos
pmpc = (big_a - 1) + (big_a + 1) * _cos
aa2 = 2 * sqrt(big_a) * alpha

b0 = big_a * (pmc + aa2)
b1 = 2 * big_a * mpc
b2 = big_a * (pmc - aa2)
a0 = ppmc + aa2
a1 = -2 * pmpc
a2 = ppmc - aa2

filt = IIRFilter(2)
filt.set_coefficients([a0, a1, a2], [b0, b1, b2])
return filt

def make_highshelf(
frequency: int,
samplerate: int,
gain_db: float,
q_factor: float = 1 / sqrt(2),
) -> IIRFilter:
"""
Creates a high-shelf filter

>>> filter = make_highshelf(1000, 48000, 6)
>>> filter.a_coeffs + filter.b_coeffs  # doctest: +NORMALIZE_WHITESPACE
[2.2229172136088806, -3.9587208137297303, 1.7841414181566304, 4.295432981120543,
-7.922740859457287, 3.6756456963725253]
"""
w0 = tau * frequency / samplerate
_sin = sin(w0)
_cos = cos(w0)
alpha = _sin / (2 * q_factor)
big_a = 10 ** (gain_db / 40)
pmc = (big_a + 1) - (big_a - 1) * _cos
ppmc = (big_a + 1) + (big_a - 1) * _cos
mpc = (big_a - 1) - (big_a + 1) * _cos
pmpc = (big_a - 1) + (big_a + 1) * _cos
aa2 = 2 * sqrt(big_a) * alpha

b0 = big_a * (ppmc + aa2)
b1 = -2 * big_a * pmpc
b2 = big_a * (ppmc - aa2)
a0 = pmc + aa2
a1 = 2 * mpc
a2 = pmc - aa2

filt = IIRFilter(2)
filt.set_coefficients([a0, a1, a2], [b0, b1, b2])
return filt
```