Analysis and Implementation of the Guitar Amplifier Tone Stack

David Yeh, Julius Smith dtyeh,[email protected]

CCRMA Stanford University Stanford, CA 1

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  As close to real thing as possible For portability and flexibility 2 © 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 3 © 2006 David Yeh

Prior work Modeled by Line 6 (and others) Analyzed by Kuehnel (2005, book) Typically approximated as a bank of biquads for Low, Mid, High frequency bands 4 © 2006 David Yeh

Parameter mapping from tone controls to frequency response is very complicated Passive RC circuit   Three real poles One zero at DC, one pair of zeros with anti-resonance Circuit components are not isolated  Component values are comparable  Bridge topology Tone controls affect location of multiple poles and zeros 5 © 2006 David Yeh

Tone Stack Transfer Function Third order continuous time system Complex mapping from component values/parameters to coefficients 6 © 2006 David Yeh

Poles depend only on Bass and Mid controls 7 © 2006 David Yeh

Zeros depend on all parameters 8 © 2006 David Yeh

Poles sweeping Bass and Mid Pole 1 Pole 2 Low freq Pole 3 9 High freq © 2006 David Yeh

Zeros plots for parameter sweeps 10 © 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 11 © 2006 David Yeh

Bilinear transformation of 3 rd system order 12 © 2006 David Yeh

LADSPA plugin block diagram Audio in Component values R, C Treble Mid B[] Compute DF coefs A[] Transposed DFII core Bass 13 Audio out © 2006 David Yeh

DFII frequency response shows good match with continuous time version 14 © 2006 David Yeh

Error relative to continuous time 15 Worst case errors shown  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 frequencies due to asymptotic limits 16 © 2006 David Yeh

Table lookup implementation simplifies computation of coefficients Tabulate 25 steps of each tone control parameter = 515 kB table Lattice filter implementation for robustness to roundoff error in coefficients and to smoothly fade between coefficients as tone controls are varied Convert from z-domain transfer function to lattice coefficients by step-down algorithm 17 © 2006 David Yeh

Tone stack parameter mapping is very complicated but not computationally complex Implemented DFII and lattice filter in CAPS audio suite. Both run in real time.

 Minimal processor load (<1%) on 2.2 GHz Intel P4  Did not notice zipper noise – coefficient fade not necessary Complicated mapping – simple order system Third order filter is not computationally demanding Direct implementation is practical 18 © 2006 David Yeh

Sound samples White noise at different settings        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

19 © 2006 David Yeh

Comparison of implementations

DFII

Exact parameterization of tone stack behavior Runs in real time Arbitrary precision of tone settings Easy to change circuit component values

Table lookup

“ More efficient computation of filter coefficients Settings are quantized – can interpolate Must tabulate each circuit configuration Real time changes in tone settings not audible 20 Robust to roundoff errors in coefficients – can fade between settings © 2006 David Yeh