Analysis and Implementation of Analysis and Implementation of the Guitar Amplifier Tone Stackthe Guitar Amplifier Tone Stack

David Yeh, Julius SmithDavid Yeh, Julius Smith

dtyeh,jos@ccrma.stanford.edudtyeh,jos@ccrma.stanford.edu

CCRMACCRMA

Stanford UniversityStanford University

Stanford, CAStanford, CA

© 2006 David Yeh

Digital audio effects that emulate analog equipment are popular

““Modeling” amplifiersModeling” amplifiers

Products by Line 6, Yamaha, Products by Line 6, Yamaha, Roland, Korg, Universal Audio, etc.Roland, Korg, Universal Audio, etc.

CAPS open source LADSPA suiteCAPS open source LADSPA suite http://quitte.de/dsp/caps.htmlhttp://quitte.de/dsp/caps.html

Emulate behavior of classic analog Emulate behavior of classic analog gear in softwaregear in software As close to real thing as possibleAs close to real thing as possible

For portability and flexibilityFor 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 stackPassive RC filter to audio signalLocated 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 Three real poles One zero at DC, one pair of zeros with anti-

resonance Shelving filter is close: 3 poles, 3 zeros

Frequency response depends on pole locations But no notch filtering

Circuit components are not isolated 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 systemComplex 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

Pole 1

Pole 2

Pole 3

Poles sweeping Bass and Mid Low freq

High freq

© 2006 David Yeh

Zeros plots for parameter sweeps

© 2006 David Yeh

Digitization as third-order filter

Straightforward approachFind continuous time transfer functionDiscretize by bilinear transformImplement as transposed Direct Form II (DFII)Pros: Perfect mapping of tone controls to frequency response within limitations of bilinear transformCons: Complicated formulas to compute coefficients

© 2006 David Yeh

Bilinear transformation of 3rd order system

© 2006 David Yeh

DFII block diagram

Mid

Treble

Bass

Compute DF coefs

Transposed DFII core

Audio out

Audio in

B[]

A[]

Component values

R, C

© 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 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 kHzGuitar 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 variedTabulate 25 steps of each tone control parameterConvert from z-domain transfer function to lattice coefficients by step-down algorithmImplemented DFII and lattice filter in CAPS audio suite. Both run in real time.

© 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

© 2006 David Yeh

Comparison of implementationsComparison of implementations

DFIIDFII Table lookupTable lookup

Exact parameterization Exact parameterization of tone stack behaviorof tone stack behavior

““

Runs in real timeRuns in real time More efficient More efficient computation of filter computation of filter coefficientscoefficients

Arbitrary precision of Arbitrary precision of tone settingstone settings

Settings are quantized Settings are quantized – can interpolate– can interpolate

Easy to change circuit Easy to change circuit component valuescomponent values

Must tabulate each Must tabulate each circuit configurationcircuit configuration

Real time changes in Real time changes in tone settings not tone settings not audibleaudible

Robust to roundoff Robust to roundoff errors in coefficients – errors in coefficients – can fade between can fade between settingssettings