"""
Basic IIR Bilinear Transform-Based Digital Filter Design Helper
Copyright (c) March 2017, Mark Wickert
All rights reserved.
Redistribution and use in source and binary forms, with or without
modification, are permitted provided that the following conditions are met:
1. Redistributions of source code must retain the above copyright notice, this
list of conditions and the following disclaimer.
2. Redistributions in binary form must reproduce the above copyright notice,
this list of conditions and the following disclaimer in the documentation
and/or other materials provided with the distribution.
THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND
ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED
WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE
DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE LIABLE FOR
ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES
(INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND
ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
(INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
The views and conclusions contained in the software and documentation are those
of the authors and should not be interpreted as representing official policies,
either expressed or implied, of the FreeBSD Project.
"""
import numpy as np
import scipy.signal as signal
import matplotlib.pyplot as plt
from matplotlib import pylab
from matplotlib import mlab
def IIR_lpf(f_pass, f_stop, Ripple_pass, Atten_stop,
fs = 1.00, ftype = 'butter'):
"""
Design an IIR lowpass filter using scipy.signal.iirdesign.
The filter order is determined based on
f_pass Hz, f_stop Hz, and the desired stopband attenuation
d_stop in dB, all relative to a sampling rate of fs Hz.
Parameters
----------
f_pass : Passband critical frequency in Hz
f_stop : Stopband critical frequency in Hz
Ripple_pass : Filter gain in dB at f_pass
Atten_stop : Filter attenuation in dB at f_stop
fs : Sampling rate in Hz
ftype : Analog prototype from 'butter' 'cheby1', 'cheby2',
'ellip', and 'bessel'
Returns
-------
b : ndarray of the numerator coefficients
a : ndarray of the denominator coefficients
sos : 2D ndarray of second-order section coefficients
Notes
-----
Additionally a text string telling the user the filter order is
written to the console, e.g., IIR cheby1 order = 8.
Examples
--------
>>> fs = 48000
>>> f_pass = 5000
>>> f_stop = 8000
>>> b_but,a_but,sos_but = IIR_lpf(f_pass,f_stop,0.5,60,fs,'butter')
>>> b_cheb1,a_cheb1,sos_cheb1 = IIR_lpf(f_pass,f_stop,0.5,60,fs,'cheby1')
>>> b_cheb2,a_cheb2,sos_cheb2 = IIR_lpf(f_pass,f_stop,0.5,60,fs,'cheby2')
>>> b_elli,a_elli,sos_elli = IIR_lpf(f_pass,f_stop,0.5,60,fs,'ellip')
Mark Wickert October 2016
"""
b,a = signal.iirdesign(2*float(f_pass)/fs, 2*float(f_stop)/fs,
Ripple_pass, Atten_stop,
ftype = ftype, output='ba')
sos = signal.iirdesign(2*float(f_pass)/fs, 2*float(f_stop)/fs,
Ripple_pass, Atten_stop,
ftype = ftype, output='sos')
tag = 'IIR ' + ftype + ' order'
print('%s = %d.' % (tag,len(a)-1))
return b, a, sos
def IIR_hpf(f_stop, f_pass, Ripple_pass, Atten_stop,
fs = 1.00, ftype = 'butter'):
"""
Design an IIR highpass filter using scipy.signal.iirdesign.
The filter order is determined based on
f_pass Hz, f_stop Hz, and the desired stopband attenuation
d_stop in dB, all relative to a sampling rate of fs Hz.
Parameters
----------
f_stop :
f_pass :
Ripple_pass :
Atten_stop :
fs : sampling rate in Hz
ftype : Analog prototype from 'butter' 'cheby1', 'cheby2',
'ellip', and 'bessel'
Returns
-------
b : ndarray of the numerator coefficients
a : ndarray of the denominator coefficients
sos : 2D ndarray of second-order section coefficients
Examples
--------
>>> fs = 48000
>>> f_pass = 8000
>>> f_stop = 5000
>>> b_but,a_but,sos_but = IIR_hpf(f_stop,f_pass,0.5,60,fs,'butter')
>>> b_cheb1,a_cheb1,sos_cheb1 = IIR_hpf(f_stop,f_pass,0.5,60,fs,'cheby1')
>>> b_cheb2,a_cheb2,sos_cheb2 = IIR_hpf(f_stop,f_pass,0.5,60,fs,'cheby2')
>>> b_elli,a_elli,sos_elli = IIR_hpf(f_stop,f_pass,0.5,60,fs,'ellip')
Mark Wickert October 2016
"""
b,a = signal.iirdesign(2*float(f_pass)/fs, 2*float(f_stop)/fs,
Ripple_pass, Atten_stop,
ftype = ftype, output='ba')
sos = signal.iirdesign(2*float(f_pass)/fs, 2*float(f_stop)/fs,
Ripple_pass, Atten_stop,
ftype =ftype, output='sos')
tag = 'IIR ' + ftype + ' order'
print('%s = %d.' % (tag,len(a)-1))
return b, a, sos
def IIR_bpf(f_stop1, f_pass1, f_pass2, f_stop2, Ripple_pass, Atten_stop,
fs = 1.00, ftype = 'butter'):
"""
Design an IIR bandpass filter using scipy.signal.iirdesign.
The filter order is determined based on
f_pass Hz, f_stop Hz, and the desired stopband attenuation
d_stop in dB, all relative to a sampling rate of fs Hz.
Parameters
----------
f_stop1 : ndarray of the numerator coefficients
f_pass : ndarray of the denominator coefficients
Ripple_pass :
Atten_stop :
fs : sampling rate in Hz
ftype : Analog prototype from 'butter' 'cheby1', 'cheby2',
'ellip', and 'bessel'
Returns
-------
b : ndarray of the numerator coefficients
a : ndarray of the denominator coefficients
sos : 2D ndarray of second-order section coefficients
Examples
--------
>>> fs = 48000
>>> f_pass = 8000
>>> f_stop = 5000
>>> b_but,a_but,sos_but = IIR_hpf(f_stop,f_pass,0.5,60,fs,'butter')
>>> b_cheb1,a_cheb1,sos_cheb1 = IIR_hpf(f_stop,f_pass,0.5,60,fs,'cheby1')
>>> b_cheb2,a_cheb2,sos_cheb2 = IIR_hpf(f_stop,f_pass,0.5,60,fs,'cheby2')
>>> b_elli,a_elli,sos_elli = IIR_hpf(f_stop,f_pass,0.5,60,fs,'ellip')
Mark Wickert October 2016
"""
b,a = signal.iirdesign([2*float(f_pass1)/fs, 2*float(f_pass2)/fs],
[2*float(f_stop1)/fs, 2*float(f_stop2)/fs],
Ripple_pass, Atten_stop,
ftype = ftype, output='ba')
sos = signal.iirdesign([2*float(f_pass1)/fs, 2*float(f_pass2)/fs],
[2*float(f_stop1)/fs, 2*float(f_stop2)/fs],
Ripple_pass, Atten_stop,
ftype =ftype, output='sos')
tag = 'IIR ' + ftype + ' order'
print('%s = %d.' % (tag,len(a)-1))
return b, a, sos
def IIR_bsf(f_pass1, f_stop1, f_stop2, f_pass2, Ripple_pass, Atten_stop,
fs = 1.00, ftype = 'butter'):
"""
Design an IIR bandstop filter using scipy.signal.iirdesign.
The filter order is determined based on
f_pass Hz, f_stop Hz, and the desired stopband attenuation
d_stop in dB, all relative to a sampling rate of fs Hz.
Mark Wickert October 2016
"""
b,a = signal.iirdesign([2*float(f_pass1)/fs, 2*float(f_pass2)/fs],
[2*float(f_stop1)/fs, 2*float(f_stop2)/fs],
Ripple_pass, Atten_stop,
ftype = ftype, output='ba')
sos = signal.iirdesign([2*float(f_pass1)/fs, 2*float(f_pass2)/fs],
[2*float(f_stop1)/fs, 2*float(f_stop2)/fs],
Ripple_pass, Atten_stop,
ftype =ftype, output='sos')
tag = 'IIR ' + ftype + ' order'
print('%s = %d.' % (tag,len(a)-1))
return b, a, sos
def freqz_resp_list(b,a=np.array([1]),mode = 'dB',fs=1.0,Npts = 1024,fsize=(6,4)):
"""
A method for displaying digital filter frequency response magnitude,
phase, and group delay. A plot is produced using matplotlib
freq_resp(self,mode = 'dB',Npts = 1024)
A method for displaying the filter frequency response magnitude,
phase, and group delay. A plot is produced using matplotlib
freqz_resp(b,a=[1],mode = 'dB',Npts = 1024,fsize=(6,4))
b = ndarray of numerator coefficients
a = ndarray of denominator coefficents
mode = display mode: 'dB' magnitude, 'phase' in radians, or
'groupdelay_s' in samples and 'groupdelay_t' in sec,
all versus frequency in Hz
Npts = number of points to plot; default is 1024
fsize = figure size; defult is (6,4) inches
Mark Wickert, January 2015
"""
if type(b) == list:
# We have a list of filters
N_filt = len(b)
f = np.arange(0,Npts)/(2.0*Npts)
for n in range(N_filt):
w,H = signal.freqz(b[n],a[n],2*np.pi*f)
if n == 0:
plt.figure(figsize=fsize)
if mode.lower() == 'db':
plt.plot(f*fs,20*np.log10(np.abs(H)))
if n == N_filt-1:
plt.xlabel('Frequency (Hz)')
plt.ylabel('Gain (dB)')
plt.title('Frequency Response - Magnitude')
elif mode.lower() == 'phase':
plt.plot(f*fs,np.angle(H))
if n == N_filt-1:
plt.xlabel('Frequency (Hz)')
plt.ylabel('Phase (rad)')
plt.title('Frequency Response - Phase')
elif (mode.lower() == 'groupdelay_s') or (mode.lower() == 'groupdelay_t'):
"""
Notes
-----
Since this calculation involves finding the derivative of the
phase response, care must be taken at phase wrapping points
and when the phase jumps by +/-pi, which occurs when the
amplitude response changes sign. Since the amplitude response
is zero when the sign changes, the jumps do not alter the group
delay results.
"""
theta = np.unwrap(np.angle(H))
# Since theta for an FIR filter is likely to have many pi phase
# jumps too, we unwrap a second time 2*theta and divide by 2
theta2 = np.unwrap(2*theta)/2.
theta_dif = np.diff(theta2)
f_diff = np.diff(f)
Tg = -np.diff(theta2)/np.diff(w)
# For gain almost zero set groupdelay = 0
idx = np.nonzero(np.ravel(20*np.log10(H[:-1]) < -400))[0]
Tg[idx] = np.zeros(len(idx))
max_Tg = np.max(Tg)
#print(max_Tg)
if mode.lower() == 'groupdelay_t':
max_Tg /= fs
plt.plot(f[:-1]*fs,Tg/fs)
plt.ylim([0,1.2*max_Tg])
else:
plt.plot(f[:-1]*fs,Tg)
plt.ylim([0,1.2*max_Tg])
if n == N_filt-1:
plt.xlabel('Frequency (Hz)')
if mode.lower() == 'groupdelay_t':
plt.ylabel('Group Delay (s)')
else:
plt.ylabel('Group Delay (samples)')
plt.title('Frequency Response - Group Delay')
else:
s1 = 'Error, mode must be "dB", "phase, '
s2 = '"groupdelay_s", or "groupdelay_t"'
print(s1 + s2)
def freqz_cas(sos,w):
"""
Cascade frequency response
Mark Wickert October 2016
"""
Ns,Mcol = sos.shape
w,Hcas = signal.freqz(sos[0,:3],sos[0,3:],w)
for k in range(1,Ns):
w,Htemp = signal.freqz(sos[k,:3],sos[k,3:],w)
Hcas *= Htemp
return w, Hcas
def freqz_resp_cas_list(sos,mode = 'dB',fs=1.0,Npts = 1024,fsize=(6,4)):
"""
A method for displaying cascade digital filter form frequency response
magnitude, phase, and group delay. A plot is produced using matplotlib
freq_resp(self,mode = 'dB',Npts = 1024)
A method for displaying the filter frequency response magnitude,
phase, and group delay. A plot is produced using matplotlib
freqz_resp(b,a=[1],mode = 'dB',Npts = 1024,fsize=(6,4))
b = ndarray of numerator coefficients
a = ndarray of denominator coefficents
mode = display mode: 'dB' magnitude, 'phase' in radians, or
'groupdelay_s' in samples and 'groupdelay_t' in sec,
all versus frequency in Hz
Npts = number of points to plot; default is 1024
fsize = figure size; defult is (6,4) inches
Mark Wickert, January 2015
"""
if type(sos) == list:
# We have a list of filters
N_filt = len(sos)
f = np.arange(0,Npts)/(2.0*Npts)
for n in range(N_filt):
w,H = freqz_cas(sos[n],2*np.pi*f)
if n == 0:
plt.figure(figsize=fsize)
if mode.lower() == 'db':
plt.plot(f*fs,20*np.log10(np.abs(H)))
if n == N_filt-1:
plt.xlabel('Frequency (Hz)')
plt.ylabel('Gain (dB)')
plt.title('Frequency Response - Magnitude')
elif mode.lower() == 'phase':
plt.plot(f*fs,np.angle(H))
if n == N_filt-1:
plt.xlabel('Frequency (Hz)')
plt.ylabel('Phase (rad)')
plt.title('Frequency Response - Phase')
elif (mode.lower() == 'groupdelay_s') or (mode.lower() == 'groupdelay_t'):
"""
Notes
-----
Since this calculation involves finding the derivative of the
phase response, care must be taken at phase wrapping points
and when the phase jumps by +/-pi, which occurs when the
amplitude response changes sign. Since the amplitude response
is zero when the sign changes, the jumps do not alter the group
delay results.
"""
theta = np.unwrap(np.angle(H))
# Since theta for an FIR filter is likely to have many pi phase
# jumps too, we unwrap a second time 2*theta and divide by 2
theta2 = np.unwrap(2*theta)/2.
theta_dif = np.diff(theta2)
f_diff = np.diff(f)
Tg = -np.diff(theta2)/np.diff(w)
# For gain almost zero set groupdelay = 0
idx = np.nonzero(np.ravel(20*np.log10(H[:-1]) < -400))[0]
Tg[idx] = np.zeros(len(idx))
max_Tg = np.max(Tg)
#print(max_Tg)
if mode.lower() == 'groupdelay_t':
max_Tg /= fs
plt.plot(f[:-1]*fs,Tg/fs)
plt.ylim([0,1.2*max_Tg])
else:
plt.plot(f[:-1]*fs,Tg)
plt.ylim([0,1.2*max_Tg])
if n == N_filt-1:
plt.xlabel('Frequency (Hz)')
if mode.lower() == 'groupdelay_t':
plt.ylabel('Group Delay (s)')
else:
plt.ylabel('Group Delay (samples)')
plt.title('Frequency Response - Group Delay')
else:
s1 = 'Error, mode must be "dB", "phase, '
s2 = '"groupdelay_s", or "groupdelay_t"'
print(s1 + s2)
def unique_cpx_roots(rlist,tol = 0.001):
"""
The average of the root values is used when multiplicity
is greater than one.
Mark Wickert October 2016
"""
uniq = [rlist[0]]
mult = [1]
for k in range(1,len(rlist)):
N_uniq = len(uniq)
for m in range(N_uniq):
if abs(rlist[k]-uniq[m]) <= tol:
mult[m] += 1
uniq[m] = (uniq[m]*(mult[m]-1) + rlist[k])/float(mult[m])
break
uniq = np.hstack((uniq,rlist[k]))
mult = np.hstack((mult,[1]))
return np.array(uniq), np.array(mult)
def sos_cascade(sos1,sos2):
"""
Mark Wickert October 2016
"""
return np.vstack((sos1,sos2))
def sos_zplane(sos,auto_scale=True,size=2,tol = 0.001):
"""
Create an z-plane pole-zero plot.
Create an z-plane pole-zero plot using the numerator
and denominator z-domain system function coefficient
ndarrays b and a respectively. Assume descending powers of z.
Parameters
----------
sos : ndarray of the sos coefficients
auto_scale : bool (default True)
size : plot radius maximum when scale = False
Returns
-------
(M,N) : tuple of zero and pole counts + plot window
Notes
-----
This function tries to identify repeated poles and zeros and will
place the multiplicity number above and to the right of the pole or zero.
The difficulty is setting the tolerance for this detection. Currently it
is set at 1e-3 via the function signal.unique_roots.
Examples
--------
>>> # Here the plot is generated using auto_scale
>>> sos_zplane(sos)
>>> # Here the plot is generated using manual scaling
>>> sos_zplane(sos,False,1.5)
"""
Ns,Mcol = sos.shape
# Extract roots from sos num and den removing z = 0
# roots due to first-order sections
N_roots = []
for k in range(Ns):
N_roots_tmp = np.roots(sos[k,:3])
if N_roots_tmp[1] == 0.:
N_roots = np.hstack((N_roots,N_roots_tmp[0]))
else:
N_roots = np.hstack((N_roots,N_roots_tmp))
D_roots = []
for k in range(Ns):
D_roots_tmp = np.roots(sos[k,3:])
if D_roots_tmp[1] == 0.:
D_roots = np.hstack((D_roots,D_roots_tmp[0]))
else:
D_roots = np.hstack((D_roots,D_roots_tmp))
# Plot labels if multiplicity greater than 1
x_scale = 1.5*size
y_scale = 1.5*size
x_off = 0.02
y_off = 0.01
M = len(N_roots)
N = len(D_roots)
if auto_scale:
if M > 0 and N > 0:
size = max(np.max(np.abs(N_roots)),np.max(np.abs(D_roots)))+.1
elif M > 0:
size = max(np.max(np.abs(N_roots)),1.0)+.1
elif N > 0:
size = max(1.0,np.max(np.abs(D_roots)))+.1
else:
size = 1.1
plt.figure(figsize=(5,5))
plt.axis('equal')
r = np.linspace(0,2*np.pi,200)
plt.plot(np.cos(r),np.sin(r),'r--')
plt.plot([-size,size],[0,0],'k-.')
plt.plot([0,0],[-size,size],'k-.')
if M > 0:
#N_roots = np.roots(b)
N_uniq, N_mult=unique_cpx_roots(N_roots,tol=tol)
plt.plot(np.real(N_uniq),np.imag(N_uniq),'ko',mfc='None',ms=8)
idx_N_mult = np.nonzero(np.ravel(N_mult>1))[0]
for k in range(len(idx_N_mult)):
x_loc = np.real(N_uniq[idx_N_mult[k]]) + x_off*x_scale
y_loc =np.imag(N_uniq[idx_N_mult[k]]) + y_off*y_scale
plt.text(x_loc,y_loc,str(N_mult[idx_N_mult[k]]),
ha='center',va='bottom',fontsize=10)
if N > 0:
#D_roots = np.roots(a)
D_uniq, D_mult=unique_cpx_roots(D_roots,tol=tol)
plt.plot(np.real(D_uniq),np.imag(D_uniq),'kx',ms=8)
idx_D_mult = np.nonzero(np.ravel(D_mult>1))[0]
for k in range(len(idx_D_mult)):
x_loc = np.real(D_uniq[idx_D_mult[k]]) + x_off*x_scale
y_loc =np.imag(D_uniq[idx_D_mult[k]]) + y_off*y_scale
plt.text(x_loc,y_loc,str(D_mult[idx_D_mult[k]]),
ha='center',va='bottom',fontsize=10)
if M - N < 0:
plt.plot(0.0,0.0,'bo',mfc='None',ms=8)
elif M - N > 0:
plt.plot(0.0,0.0,'kx',ms=8)
if abs(M - N) > 1:
plt.text(x_off*x_scale,y_off*y_scale,str(abs(M-N)),
ha='center',va='bottom',fontsize=10)
plt.xlabel('Real Part')
plt.ylabel('Imaginary Part')
plt.title('Pole-Zero Plot')
#plt.grid()
plt.axis([-size,size,-size,size])
return M,N
