Analysis and Implementation of the
Guitar Amplifier Tone Stack
David Yeh, Julius Smith
dtyeh,jos@ccrma.stanford.edu
CCRMA
Stanford University
Stanford, CA
Digital audio effects that emulate analog
equipment are popular
“Modeling” amplifiers
Products by Line 6, Yamaha, Roland,
Korg, Universal Audio, etc.
CAPS open source LADSPA suite

http://quitte.de/dsp/caps.html
Emulate behavior of classic analog gear
in software
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As close to real thing as possible
For portability and flexibility
© 2006 David Yeh
Guitar amp tone stack is a unique
component in the sound of an amplifier
Almost every guitar
amplifier, solid state or
tube, has a tone control
circuit – referred to as a
tone stack
Passive RC filter to audio
signal
Located either directly
after preamp stage or
after stages of gain and
buffer
© 2006 David Yeh
Prior work
Modeled by Line 6 (and
others)
Analyzed by Kuehnel
(2005, book)
Substituted in CAPS
(LADSPA plugins for
guitar effects) by
shelving filter
© 2006 David Yeh
Parameter mapping from tone controls to
frequency response is very complicated
Passive RC circuit
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Three real poles
One zero at DC, one pair of zeros with antiresonance
Shelving filter is close: 3 poles, 3 zeros
 Frequency response depends on pole locations
 But no notch filtering
Circuit components are not isolated
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Component values are comparable
Bridge topology
Tone controls affect location of multiple poles
and zeros
© 2006 David Yeh
Tone Stack Transfer Function
Third order continuous time system
Complex mapping from component
values/parameters to coefficients
© 2006 David Yeh
Poles depend only on Bass and Mid
controls
© 2006 David Yeh
Zeros depend on all parameters
© 2006 David Yeh
Poles sweeping Bass and Mid
Low freq
Pole 1
Pole 2
Pole 3
High freq
© 2006 David Yeh
Zeros plots for parameter sweeps
© 2006 David Yeh
Digitization as third-order filter
Straightforward approach
Find continuous time transfer function
Discretize by bilinear transform
Implement as transposed Direct Form II
(DFII)
Pros: Perfect mapping of tone controls to
frequency response within limitations of
bilinear transform
Cons: Complicated formulas to compute
coefficients
© 2006 David Yeh
Bilinear transformation of 3rd order
system
© 2006 David Yeh
DFII block diagram
Audio in
Component
values R, C
Treble
B[]
Mid
Bass
Compute
DF coefs A[]
Transposed DFII
core
Audio out
© 2006 David Yeh
DFII frequency response shows good match with
continuous time version
© 2006 David Yeh
Error relative to continuous time
Worst case errors shown
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B=1, M=0, T=0
Discrete time reaches low
pass asymptote but
continuous time does not
© 2006 David Yeh
Reduced sampling rate
Commercial effects pedals commonly run at
31 kHz
Guitar amplifier system is bandlimited by
speaker response: 100–6000 Hz.
For f_s = 20 kHz, error increases but only at
high frequency because of asymptotic limits
© 2006 David Yeh
Table lookup implementation simplifies
computation of coefficients
Lattice filter implementation for robustness to
roundoff error in coefficients and to smoothly
fade between coefficients as tone controls are
varied
Tabulate 25 steps of each tone control
parameter
Convert from z-domain transfer function to
lattice coefficients by step-down algorithm
Implemented DFII and lattice filter in CAPS
audio suite. Both run in real time.
© 2006 David Yeh
Sound samples
White noise at different settings
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Original white noise (2 sec)
B=0 M=0 T=0
B=0 M=1 T=0
B=1 M=0 T=1
B=1 M=1 T=0
B=1 M=1 T=1
B=0.5 M=1 T=0.5
© 2006 David Yeh
Comparison of implementations
DFII
Exact parameterization of
tone stack behavior
Table lookup
“
Runs in real time
More efficient computation
of filter coefficients
Arbitrary precision of tone
settings
Settings are quantized –
can interpolate
Easy to change circuit
component values
Must tabulate each circuit
configuration
Real time changes in tone
settings not audible
Robust to roundoff errors
in coefficients – can fade
between settings
© 2006 David Yeh