1 Analysis and Implementation of the Guitar Amplifier Tone Stack David Yeh, Julius Smith Stanford University Stanford, CA

© 2006 David Yeh 2 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 Emulate behavior of classic analog gear in software As close to real thing as possible As close to real thing as possible For portability and flexibility

© 2006 David Yeh 3 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 4 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

© 2006 David Yeh 5 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

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

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

© 2006 David Yeh 8 Zeros depend on all parameters

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

© 2006 David Yeh 10 Zeros plots for parameter sweeps

© 2006 David Yeh 11 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 12 Bilinear transformation of 3 rd order system

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

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

© 2006 David Yeh 15 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 16 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

© 2006 David Yeh 17 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

© 2006 David Yeh 18 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

© 2006 David Yeh 19 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 20 Comparison of implementations DFII Table lookup Exact parameterization of tone stack behavior “ 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