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'''
Routines for designing optimal FIR filters.
For a great intro to how all this stuff works, see section 6.6 of
"Digital Signal Processing: A Practical Approach", Emmanuael C. Ifeachor
and Barrie W. Jervis, Adison-Wesley, 1993. ISBN 0-201-54413-X.
'''
import math, cmath
from gnuradio import gr
remez = gr.remez
def low_pass (gain, Fs, freq1, freq2, passband_ripple_db, stopband_atten_db,
nextra_taps=2):
"""
Builds a low pass filter.
Args:
gain: Filter gain in the passband (linear)
Fs: Sampling rate (sps)
freq1: End of pass band (in Hz)
freq2: Start of stop band (in Hz)
passband_ripple_db: Pass band ripple in dB (should be small, < 1)
stopband_atten_db: Stop band attenuation in dB (should be large, >= 60)
nextra_taps: Extra taps to use in the filter (default=2)
"""
passband_dev = passband_ripple_to_dev (passband_ripple_db)
stopband_dev = stopband_atten_to_dev (stopband_atten_db)
desired_ampls = (gain, 0)
(n, fo, ao, w) = remezord ([freq1, freq2], desired_ampls,
[passband_dev, stopband_dev], Fs)
# The remezord typically under-estimates the filter order, so add 2 taps by default
taps = gr.remez (n + nextra_taps, fo, ao, w, "bandpass")
return taps
def band_pass (gain, Fs, freq_sb1, freq_pb1, freq_pb2, freq_sb2,
passband_ripple_db, stopband_atten_db,
nextra_taps=2):
"""
Builds a band pass filter.
Args:
gain: Filter gain in the passband (linear)
Fs: Sampling rate (sps)
freq_sb1: End of stop band (in Hz)
freq_pb1: Start of pass band (in Hz)
freq_pb2: End of pass band (in Hz)
freq_sb2: Start of stop band (in Hz)
passband_ripple_db: Pass band ripple in dB (should be small, < 1)
stopband_atten_db: Stop band attenuation in dB (should be large, >= 60)
nextra_taps: Extra taps to use in the filter (default=2)
"""
passband_dev = passband_ripple_to_dev (passband_ripple_db)
stopband_dev = stopband_atten_to_dev (stopband_atten_db)
desired_ampls = (0, gain, 0)
desired_freqs = [freq_sb1, freq_pb1, freq_pb2, freq_sb2]
desired_ripple = [stopband_dev, passband_dev, stopband_dev]
(n, fo, ao, w) = remezord (desired_freqs, desired_ampls,
desired_ripple, Fs)
# The remezord typically under-estimates the filter order, so add 2 taps by default
taps = gr.remez (n + nextra_taps, fo, ao, w, "bandpass")
return taps
def complex_band_pass (gain, Fs, freq_sb1, freq_pb1, freq_pb2, freq_sb2,
passband_ripple_db, stopband_atten_db,
nextra_taps=2):
"""
Builds a band pass filter with complex taps by making an LPF and
spinning it up to the right center frequency
Args:
gain: Filter gain in the passband (linear)
Fs: Sampling rate (sps)
freq_sb1: End of stop band (in Hz)
freq_pb1: Start of pass band (in Hz)
freq_pb2: End of pass band (in Hz)
freq_sb2: Start of stop band (in Hz)
passband_ripple_db: Pass band ripple in dB (should be small, < 1)
stopband_atten_db: Stop band attenuation in dB (should be large, >= 60)
nextra_taps: Extra taps to use in the filter (default=2)
"""
center_freq = (freq_pb2 + freq_pb1) / 2.0
lp_pb = (freq_pb2 - center_freq)/1.0
lp_sb = freq_sb2 - center_freq
lptaps = low_pass(gain, Fs, lp_pb, lp_sb, passband_ripple_db,
stopband_atten_db, nextra_taps)
spinner = [cmath.exp(2j*cmath.pi*center_freq/Fs*i) for i in xrange(len(lptaps))]
taps = [s*t for s,t in zip(spinner, lptaps)]
return taps
def band_reject (gain, Fs, freq_pb1, freq_sb1, freq_sb2, freq_pb2,
passband_ripple_db, stopband_atten_db,
nextra_taps=2):
"""
Builds a band reject filter
spinning it up to the right center frequency
Args:
gain: Filter gain in the passband (linear)
Fs: Sampling rate (sps)
freq_pb1: End of pass band (in Hz)
freq_sb1: Start of stop band (in Hz)
freq_sb2: End of stop band (in Hz)
freq_pb2: Start of pass band (in Hz)
passband_ripple_db: Pass band ripple in dB (should be small, < 1)
stopband_atten_db: Stop band attenuation in dB (should be large, >= 60)
nextra_taps: Extra taps to use in the filter (default=2)
"""
passband_dev = passband_ripple_to_dev (passband_ripple_db)
stopband_dev = stopband_atten_to_dev (stopband_atten_db)
desired_ampls = (gain, 0, gain)
desired_freqs = [freq_pb1, freq_sb1, freq_sb2, freq_pb2]
desired_ripple = [passband_dev, stopband_dev, passband_dev]
(n, fo, ao, w) = remezord (desired_freqs, desired_ampls,
desired_ripple, Fs)
# Make sure we use an odd number of taps
if((n+nextra_taps)%2 == 1):
n += 1
# The remezord typically under-estimates the filter order, so add 2 taps by default
taps = gr.remez (n + nextra_taps, fo, ao, w, "bandpass")
return taps
def high_pass (gain, Fs, freq1, freq2, passband_ripple_db, stopband_atten_db,
nextra_taps=2):
"""
Builds a high pass filter.
Args:
gain: Filter gain in the passband (linear)
Fs: Sampling rate (sps)
freq1: End of stop band (in Hz)
freq2: Start of pass band (in Hz)
passband_ripple_db: Pass band ripple in dB (should be small, < 1)
stopband_atten_db: Stop band attenuation in dB (should be large, >= 60)
nextra_taps: Extra taps to use in the filter (default=2)
"""
passband_dev = passband_ripple_to_dev (passband_ripple_db)
stopband_dev = stopband_atten_to_dev (stopband_atten_db)
desired_ampls = (0, 1)
(n, fo, ao, w) = remezord ([freq1, freq2], desired_ampls,
[stopband_dev, passband_dev], Fs)
# For a HPF, we need to use an odd number of taps
# In gr.remez, ntaps = n+1, so n must be even
if((n+nextra_taps)%2 == 1):
n += 1
# The remezord typically under-estimates the filter order, so add 2 taps by default
taps = gr.remez (n + nextra_taps, fo, ao, w, "bandpass")
return taps
# ----------------------------------------------------------------
def stopband_atten_to_dev (atten_db):
"""Convert a stopband attenuation in dB to an absolute value"""
return 10**(-atten_db/20)
def passband_ripple_to_dev (ripple_db):
"""Convert passband ripple spec expressed in dB to an absolute value"""
return (10**(ripple_db/20)-1)/(10**(ripple_db/20)+1)
# ----------------------------------------------------------------
def remezord (fcuts, mags, devs, fsamp = 2):
'''
FIR order estimator (lowpass, highpass, bandpass, mulitiband).
(n, fo, ao, w) = remezord (f, a, dev)
(n, fo, ao, w) = remezord (f, a, dev, fs)
(n, fo, ao, w) = remezord (f, a, dev) finds the approximate order,
normalized frequency band edges, frequency band amplitudes, and
weights that meet input specifications f, a, and dev, to use with
the remez command.
* f is a sequence of frequency band edges (between 0 and Fs/2, where
Fs is the sampling frequency), and a is a sequence specifying the
desired amplitude on the bands defined by f. The length of f is
twice the length of a, minus 2. The desired function is
piecewise constant.
* dev is a sequence the same size as a that specifies the maximum
allowable deviation or ripples between the frequency response
and the desired amplitude of the output filter, for each band.
Use remez with the resulting order n, frequency sequence fo,
amplitude response sequence ao, and weights w to design the filter b
which approximately meets the specifications given by remezord
input parameters f, a, and dev:
b = remez (n, fo, ao, w)
(n, fo, ao, w) = remezord (f, a, dev, Fs) specifies a sampling frequency Fs.
Fs defaults to 2 Hz, implying a Nyquist frequency of 1 Hz. You can
therefore specify band edges scaled to a particular applications
sampling frequency.
In some cases remezord underestimates the order n. If the filter
does not meet the specifications, try a higher order such as n+1
or n+2.
'''
# get local copies
fcuts = fcuts[:]
mags = mags[:]
devs = devs[:]
for i in range (len (fcuts)):
fcuts[i] = float (fcuts[i]) / fsamp
nf = len (fcuts)
nm = len (mags)
nd = len (devs)
nbands = nm
if nm != nd:
raise ValueError, "Length of mags and devs must be equal"
if nf != 2 * (nbands - 1):
raise ValueError, "Length of f must be 2 * len (mags) - 2"
for i in range (len (mags)):
if mags[i] != 0: # if not stopband, get relative deviation
devs[i] = devs[i] / mags[i]
# separate the passband and stopband edges
f1 = fcuts[0::2]
f2 = fcuts[1::2]
n = 0
min_delta = 2
for i in range (len (f1)):
if f2[i] - f1[i] < min_delta:
n = i
min_delta = f2[i] - f1[i]
if nbands == 2:
# lowpass or highpass case (use formula)
l = lporder (f1[n], f2[n], devs[0], devs[1])
else:
# bandpass or multipass case
# try different lowpasses and take the worst one that
# goes through the BP specs
l = 0
for i in range (1, nbands-1):
l1 = lporder (f1[i-1], f2[i-1], devs[i], devs[i-1])
l2 = lporder (f1[i], f2[i], devs[i], devs[i+1])
l = max (l, l1, l2)
n = int (math.ceil (l)) - 1 # need order, not length for remez
# cook up remez compatible result
ff = [0] + fcuts + [1]
for i in range (1, len (ff) - 1):
ff[i] *= 2
aa = []
for a in mags:
aa = aa + [a, a]
max_dev = max (devs)
wts = [1] * len(devs)
for i in range (len (wts)):
wts[i] = max_dev / devs[i]
return (n, ff, aa, wts)
# ----------------------------------------------------------------
def lporder (freq1, freq2, delta_p, delta_s):
'''
FIR lowpass filter length estimator. freq1 and freq2 are
normalized to the sampling frequency. delta_p is the passband
deviation (ripple), delta_s is the stopband deviation (ripple).
Note, this works for high pass filters too (freq1 > freq2), but
doesnt work well if the transition is near f == 0 or f == fs/2
From Herrmann et al (1973), Practical design rules for optimum
finite impulse response filters. Bell System Technical J., 52, 769-99
'''
df = abs (freq2 - freq1)
ddp = math.log10 (delta_p)
dds = math.log10 (delta_s)
a1 = 5.309e-3
a2 = 7.114e-2
a3 = -4.761e-1
a4 = -2.66e-3
a5 = -5.941e-1
a6 = -4.278e-1
b1 = 11.01217
b2 = 0.5124401
t1 = a1 * ddp * ddp
t2 = a2 * ddp
t3 = a4 * ddp * ddp
t4 = a5 * ddp
dinf=((t1 + t2 + a3) * dds) + (t3 + t4 + a6)
ff = b1 + b2 * (ddp - dds)
n = dinf / df - ff * df + 1
return n
def bporder (freq1, freq2, delta_p, delta_s):
'''
FIR bandpass filter length estimator. freq1 and freq2 are
normalized to the sampling frequency. delta_p is the passband
deviation (ripple), delta_s is the stopband deviation (ripple).
From Mintzer and Liu (1979)
'''
df = abs (freq2 - freq1)
ddp = math.log10 (delta_p)
dds = math.log10 (delta_s)
a1 = 0.01201
a2 = 0.09664
a3 = -0.51325
a4 = 0.00203
a5 = -0.57054
a6 = -0.44314
t1 = a1 * ddp * ddp
t2 = a2 * ddp
t3 = a4 * ddp * ddp
t4 = a5 * ddp
cinf = dds * (t1 + t2 + a3) + t3 + t4 + a6
ginf = -14.6 * math.log10 (delta_p / delta_s) - 16.9
n = cinf / df + ginf * df + 1
return n