1. Analysis and Implementation of the Guitar Amplifier Tone Stack
David Yeh, Julius Smith
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
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)
Substituted in CAPS (LADSPA plugins for guitar effects) by shelving filter

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
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

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. DFII block diagram Audio in Treble Compute DF coefs B[]
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 frequency because of asymptotic limits

17. 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.

18. White noise at different settings
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. 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