#
# Copyright 2005,2007,2012 Free Software Foundation, Inc.
#
# This file is part of GNU Radio
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# it under the terms of the GNU General Public License as published by
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# any later version.
#
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# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
# GNU General Public License for more details.
#
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from __future__ import print_function
from __future__ import division
from __future__ import unicode_literals
from gnuradio import gr, filter
import math
import cmath
class fm_deemph(gr.hier_block2):
"""
FM Deemphasis IIR filter
Args:
fs: sampling frequency in Hz (float)
tau: Time constant in seconds (75us in US, 50us in EUR) (float)
An analog deemphasis filter:
R
o------/\/\/\/---+----o
|
= C
|
---
Has this transfer function:
1 1
---- ---
RC tau
H(s) = ---------- = ----------
1 1
s + ---- s + ---
RC tau
And has its -3 dB response, due to the pole, at
|H(j w_c)|^2 = 1/2 => s = j w_c = j (1/(RC))
Historically, this corner frequency of analog audio deemphasis filters
been specified by the RC time constant used, called tau.
So w_c = 1/tau.
FWIW, for standard tau values, some standard analog components would be:
tau = 75 us = (50K)(1.5 nF) = (50 ohms)(1.5 uF)
tau = 50 us = (50K)(1.0 nF) = (50 ohms)(1.0 uF)
In specifying tau for this digital deemphasis filter, tau specifies
the *digital* corner frequency, w_c, desired.
The digital deemphasis filter design below, uses the
"bilinear transformation" method of designing digital filters:
1. Convert digital specifications into the analog domain, by prewarping
digital frequency specifications into analog frequencies.
w_a = (2/T)tan(wT/2)
2. Use an analog filter design technique to design the filter.
3. Use the bilinear transformation to convert the analog filter design to a
digital filter design.
H(z) = H(s)|
s = (2/T)(1-z^-1)/(1+z^-1)
w_ca 1 1 - (-1) z^-1
H(z) = ---- * ----------- * -----------------------
2 fs -w_ca -w_ca
1 - ----- 1 + -----
2 fs 2 fs
1 - ----------- z^-1
-w_ca
1 - -----
2 fs
We use this design technique, because it is an easy way to obtain a filter
design with the -6 dB/octave roll-off required of the deemphasis filter.
Jackson, Leland B., _Digital_Filters_and_Signal_Processing_Second_Edition_,
Kluwer Academic Publishers, 1989, pp 201-212
Orfanidis, Sophocles J., _Introduction_to_Signal_Processing_, Prentice Hall,
1996, pp 573-583
"""
def __init__(self, fs, tau=75e-6):
gr.hier_block2.__init__(self, "fm_deemph",
gr.io_signature(1, 1, gr.sizeof_float), # Input signature
gr.io_signature(1, 1, gr.sizeof_float)) # Output signature
# Digital corner frequency
w_c = 1.0 / tau
# Prewarped analog corner frequency
w_ca = 2.0 * fs * math.tan(w_c / (2.0 * fs))
# Resulting digital pole, zero, and gain term from the bilinear
# transformation of H(s) = w_ca / (s + w_ca) to
# H(z) = b0 (1 - z1 z^-1)/(1 - p1 z^-1)
k = -w_ca / (2.0 * fs)
z1 = -1.0
p1 = (1.0 + k) / (1.0 - k)
b0 = -k / (1.0 - k)
btaps = [ b0 * 1.0, b0 * -z1 ]
ataps = [ 1.0, -p1 ]
# Since H(s = 0) = 1.0, then H(z = 1) = 1.0 and has 0 dB gain at DC
if 0:
print("btaps =", btaps)
print("ataps =", ataps)
global plot1
plot1 = gru.gnuplot_freqz(gru.freqz(btaps, ataps), fs, True)
deemph = filter.iir_filter_ffd(btaps, ataps, False)
self.connect(self, deemph, self)
class fm_preemph(gr.hier_block2):
"""
FM Preemphasis IIR filter.
Args:
fs: sampling frequency in Hz (float)
tau: Time constant in seconds (75us in US, 50us in EUR) (float)
fh: High frequency at which to flatten out (< 0 means default of 0.925*fs/2.0) (float)
An analog preemphasis filter, that flattens out again at the high end:
C
+-----||------+
| |
o------+ +-----+--------o
| R1 | |
+----/\/\/\/--+ \
/
\ R2
/
\
|
o--------------------------+--------o
(This fine ASCII rendition is based on Figure 5-15
in "Digital and Analog Communication Systems", Leon W. Couch II)
Has this transfer function:
1
s + ---
R1C
H(s) = ------------------
1 R1
s + --- (1 + --)
R1C R2
It has a corner due to the numerator, where the rise starts, at
|Hn(j w_cl)|^2 = 2*|Hn(0)|^2 => s = j w_cl = j (1/(R1C))
It has a corner due to the denominator, where it levels off again, at
|Hn(j w_ch)|^2 = 1/2*|Hd(0)|^2 => s = j w_ch = j (1/(R1C) * (1 + R1/R2))
Historically, the corner frequency of analog audio preemphasis filters
been specified by the R1C time constant used, called tau.
So
w_cl = 1/tau = 1/R1C; f_cl = 1/(2*pi*tau) = 1/(2*pi*R1*C)
w_ch = 1/tau2 = (1+R1/R2)/R1C; f_ch = 1/(2*pi*tau2) = (1+R1/R2)/(2*pi*R1*C)
and note f_ch = f_cl * (1 + R1/R2).
For broadcast FM audio, tau is 75us in the United States and 50us in Europe.
f_ch should be higher than our digital audio bandwidth.
The Bode plot looks like this:
/----------------
/
/ <-- slope = 20dB/decade
/
-------------/
f_cl f_ch
In specifying tau for this digital preemphasis filter, tau specifies
the *digital* corner frequency, w_cl, desired.
The digital preemphasis filter design below, uses the
"bilinear transformation" method of designing digital filters:
1. Convert digital specifications into the analog domain, by prewarping
digital frequency specifications into analog frequencies.
w_a = (2/T)tan(wT/2)
2. Use an analog filter design technique to design the filter.
3. Use the bilinear transformation to convert the analog filter design to a
digital filter design.
H(z) = H(s)|
s = (2/T)(1-z^-1)/(1+z^-1)
-w_cla
1 + ------
2 fs
1 - ------------ z^-1
-w_cla -w_cla
1 - ------ 1 - ------
2 fs 2 fs
H(z) = ------------ * -----------------------
-w_cha -w_cha
1 - ------ 1 + ------
2 fs 2 fs
1 - ------------ z^-1
-w_cha
1 - ------
2 fs
We use this design technique, because it is an easy way to obtain a filter
design with the 6 dB/octave rise required of the premphasis filter.
Jackson, Leland B., _Digital_Filters_and_Signal_Processing_Second_Edition_,
Kluwer Academic Publishers, 1989, pp 201-212
Orfanidis, Sophocles J., _Introduction_to_Signal_Processing_, Prentice Hall,
1996, pp 573-583
"""
def __init__(self, fs, tau=75e-6, fh=-1.0):
gr.hier_block2.__init__(self, "fm_preemph",
gr.io_signature(1, 1, gr.sizeof_float), # Input signature
gr.io_signature(1, 1, gr.sizeof_float)) # Output signature
# Set fh to something sensible, if needed.
# N.B. fh == fs/2.0 or fh == 0.0 results in a pole on the unit circle
# at z = -1.0 or z = 1.0 respectively. That makes the filter unstable
# and useless.
if fh <= 0.0 or fh >= fs / 2.0:
fh = 0.925 * fs/2.0
# Digital corner frequencies
w_cl = 1.0 / tau
w_ch = 2.0 * math.pi * fh
# Prewarped analog corner frequencies
w_cla = 2.0 * fs * math.tan(w_cl / (2.0 * fs))
w_cha = 2.0 * fs * math.tan(w_ch / (2.0 * fs))
# Resulting digital pole, zero, and gain term from the bilinear
# transformation of H(s) = (s + w_cla) / (s + w_cha) to
# H(z) = b0 (1 - z1 z^-1)/(1 - p1 z^-1)
kl = -w_cla / (2.0 * fs)
kh = -w_cha / (2.0 * fs)
z1 = (1.0 + kl) / (1.0 - kl)
p1 = (1.0 + kh) / (1.0 - kh)
b0 = (1.0 - kl) / (1.0 - kh)
# Since H(s = infinity) = 1.0, then H(z = -1) = 1.0 and
# this filter has 0 dB gain at fs/2.0.
# That isn't what users are going to expect, so adjust with a
# gain, g, so that H(z = 1) = 1.0 for 0 dB gain at DC.
w_0dB = 2.0 * math.pi * 0.0
g = abs(1.0 - p1 * cmath.rect(1.0 / -w_0dB), (b0 * abs(1.0 - z1 * cmath.rect(1.0, -w_0dB))))
btaps = [ g * b0 * 1.0, g * b0 * -z1 ]
ataps = [ 1.0, -p1 ]
if 0:
print("btaps =", btaps)
print("ataps =", ataps)
global plot2
plot2 = gru.gnuplot_freqz(gru.freqz(btaps, ataps), fs, True)
preemph = filter.iir_filter_ffd(btaps, ataps, False)
self.connect(self, preemph, self)