Analysis and Implementation of the Guitar Amplifier Tone Stack

1. Analysis and Implementation of the Guitar Amplifier Tone Stack David Yeh, Julius Smith dtyeh,jos@ccrma.stanford.edu CCRMA Stanford University Stanford, CA

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 • 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

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

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

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

6. Tone Stack Transfer Function • Third order continuous time system • Complex mapping from component values/parameters to coefficients

7. Poles depend only on Bass and Mid controls

8. Zeros depend on all parameters

9. Poles sweeping Bass and Mid Low freq Pole 1 Pole 2 Pole 3 High freq

10. Zeros plots for parameter sweeps

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

12. Bilinear transformation of 3rd order system

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

14. DFII frequency response shows good match with continuous time version

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

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

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

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

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

20. Comparison of implementations