ref: 2c3c76cd093d2329faaf1f48b26fd992a54e6788
dir: /src/deemph.plt/
# 15/50us EIAJ de-emphasis filter for CD/DAT
#
# 09/02/98 (c) Heiko Eissfeldt
#
# 18/03/07 robs@users.sourceforge.net: changed to biquad for slightly
# better accuracy.
#
# License: LGPL (Lesser Gnu Public License)
#
# This implements the inverse filter of the optional pre-emphasis stage
# as defined by IEC 60908 (describing the audio cd format).
#
# Background: In the early days of audio cds, there were recording
# problems with noise (for example in classical recordings). The high
# dynamics of audio cds exposed these recording errors a lot.
#
# The commonly used solution at that time was to 'pre-emphasize' the
# trebles to have a better signal-noise-ratio. That is trebles were
# amplified before recording, so that they would give a stronger signal
# compared to the underlying (tape) noise.
#
# For that purpose the audio signal was prefiltered with the following
# frequency response (simple first order filter):
#
# V (in dB)
# ^
# |
# |~10dB _________________
# | /
# | / |
# | 20dB / decade ->/ |
# | / |
# |____________________/_ _ |_ _ _ _ _ _ _ _ _ Frequency
# |0 dB | |
# | | |
# | | |
# 3.1kHz ~10kHz
#
# So the recorded audio signal has amplified trebles compared to the
# original. HiFi cd players do correct this by applying an inverse
# filter automatically, the cd-rom drives or cd burners used by digital
# sampling programs (like cdda2wav) however do not.
#
# So, this is what this effect does.
#
# This is the gnuplot file for the frequency response of the deemphasis.
#
# The absolute error is <=0.04dB up to ~12kHz, and <=0.06dB up to 20kHz.
# First define the ideal filter:
# Filter parameters
T = 1. / 441000. # we use the tenfold sampling frequency
OmegaU = 1. / 15e-6
OmegaL = 15. / 50. * OmegaU
# Calculate filter coefficients
V0 = OmegaL / OmegaU
H0 = V0 - 1.
B = V0 * tan(OmegaU * T / 2.)
A1 = (B - 1.) / (B + 1.)
B0 = (1. + (1. - A1) * H0 / 2.)
B1 = (A1 + (A1 - 1.) * H0 / 2.)
# helper variables
D = B1 / B0
O = 2 * pi * T
# Ideal transfer function
Hi(f) = B0*sqrt((1 + 2*cos(f*O)*D + D*D)/(1 + 2*cos(f*O)*A1 + A1*A1))
# Now use a biquad (RBJ high shelf) with sampling frequency of 44100Hz
# to approximate the ideal curve:
# Filter parameters
t = 1. / 44100.
gain = -9.477
slope = .4845
f0 = 5283
# Calculate filter coefficients
A = exp(gain / 40. * log(10.))
w0 = 2. * pi * f0 * t
alpha = sin(w0) / 2. * sqrt((A + 1. / A) * (1. / slope - 1.) + 2.)
b0 = A * ((A + 1.) + (A - 1.) * cos(w0) + 2. * sqrt(A) * alpha)
b1 = -2. * A * ((A - 1.) + (A + 1.) * cos(w0))
b2 = A * ((A + 1.) + (A - 1.) * cos(w0) - 2. * sqrt(A) * alpha)
a0 = (A + 1.) - (A - 1.) * cos(w0) + 2. * sqrt(A) * alpha
a1 = 2. * ((A - 1.) - (A + 1.) * cos(w0))
a2 = (A + 1.) - (A - 1.) * cos(w0) - 2. * sqrt(A) * alpha
b2 = b2 / a0
b1 = b1 / a0
b0 = b0 / a0
a2 = a2 / a0
a1 = a1 / a0
# helper variables
o = 2 * pi * t
# Best fit transfer function
Hb(f) = sqrt((b0*b0 + b1*b1 + b2*b2 +\
2.*(b0*b1 + b1*b2)*cos(f*o) + 2.*(b0*b2)* cos(2.*f*o)) /\
(1. + a1*a1 + a2*a2 + 2.*(a1 + a1*a2)*cos(f*o) + 2.*a2*cos(2.*f*o)))
# plot real, best, ideal, level with halved attenuation,
# level at full attentuation, 10fold magnified error
set logscale x
set grid xtics ytics mxtics mytics
set key left bottom
plot [f=1000:20000] [-12:2] \
20 * log10(Hi(f)),\
20 * log10(Hb(f)),\
20 * log10(OmegaL/(2 * pi * f)),\
.5 * 20 * log10(V0),\
20 * log10(V0),\
200 * log10(Hb(f)/Hi(f))
pause -1 "Hit return to continue"