Duffing’s Equation as an Excitation Mechanism for Plucked String Instrument Models

1. Duffing’s Equation as an Excitation Mechanism for Plucked String Instrument Models by Justo A. Gutierrez Master’s Research Project Music Engineering Technology University of Miami School of Music December 1, 1999

2. Purpose • The objective of this study is to provide the basis for a new excitation mechanism for plucked string instrument models which utilizes the classical nonlinear system described in Duffing’s Equation.

3. Advantages • Using Duffing’s Equation provides a means to use a nonlinear oscillator as an excitation • A mathematical model lends itself to user control • Removes the need for saving samples in a wavetable

4. Overview • Plucked String Instrument Modeling • Excitation Modeling with Duffing’s Equation • Model Performance and Analysis

5. Wavetable Synthesis • Method of synthesis that uses tables of waveforms that are finely sampled • Desired waveform is chosen and repeated over and over producing a purely periodic signal • Algorithm written as: Yt = Yt-p • p is periodicity parameter • frequency of the tone is fs/p

6. The String Model • z-L is delay line of length L • H(z) is the loop filter • F(z) is the allpass filter • x(n) and y(n) are the excitation and output signals respectively

7. Length of String • Effective delay length determines fundamental frequency of output signal • Delay line length (in samples) is L = fs/f0

8. The Comb Filter • Works by adding, at each sample time, a delayed and attenuated version of the past output

9. Standing Wave Analogy • Poles of the comb filter occur in the z-plane at 2np/L • This is the same as the natural resonant frequencies for a string tied at both ends • Does not sound like a vibrating string because it is a perfectly periodic waveform • Does not take into account that high frequencies decay much faster than slow ones for vibrating strings

10. The Loop Filter • Idea is to insert a lowpass filter into the feedback loop of the comb filter so that high-frequency components are diminished relative to low-frequency components every time the past output signal returns • Original Karplus-Strong algorithm used a two-tap averager that was simple and effective

11. Loop Filter (continued) • Valimaki et al proposed using an IIR lowpass filter to simulate the damping characteristics of a physical string • Loop filter coefficients can be changed as a function of string length and other parameters • H1(z) = g(1+a1)/(1+a1z-1)

12. Loop Filter Signal Flowchart

13. Loop Filter Magnitude Response and Group Delay

14. Loop Filter Impulse Responses

15. The Allpass Filter • Used to fine-tune the pitch of the string model • If feedback loop were only to contain a delay line and lowpass filter, total delay would be the sum of integer delay line plus the delay of the lowpass filter • Fundamental frequency of fs/D is usually not an integer number of samples

16. Allpass Filter (continued) • Fundamental frequency is then given by f1 = fs/(D+d) where d is fractional delay • Allpass filters introduce delay but pass frequencies with equal weight • Transfer function is H(z) = (z-1+a)/(1+az-1) • a = (1-d)/(1+d)

17. Allpass Phase Response

18. Allpass Delay Response

19. Inverse Filtering • KS algorithm used a white noise burst as excitation for plucked string because it provided high-frequency content as a real pluck would provide • Valimaki et al found a pluck signal by filtering the output through the inverted transfer function of the string system

20. Inverse Filtering (continued) • The transfer function for the general string model can be given as S(z) = 1/[1-z-LF(z)H(z)] • The inverse filter is simply S-1(z) = 1/S(z)

21. Inverse Filtering Procedure • Obtain residual by inverse filtering • Truncate the first 50-100 ms of the residual • Use the truncated signal as the excitation to the string model • Run the string model

22. Steel-string Guitar Sample

23. Residual After Inverse Filtering

24. Truncated Residual Signal

25. Resynthesized Guitar

26. Duffing’s Equation • In 1918, Duffing introduced a nonlinear oscillator with a cubic stiffness term to describe the hardening spring effect in many mechanical problems • It is one of the most common examples in the study of nonlinear oscillations

27. Duffing’s Equation (continued) • The form used for this study is from Moon and Holmes, which is one in which the linear stiffness term is negative so that x” + dx - x + x3 = g cos wt. • This model was used to describe the forced oscillations of a ferromagnetic beam buckled between the nonuniform field of two permanent magnets

28. Experimental Apparatus (Moon and Holmes)

29. Modeling the Excitation • For this experiment, the coefficients in Moon and Holmes’ modification of Duffing’s Equation were adjusted to produce the desired residuals • The Runge-Kutta method was the numerical method used to calculate Duffing’s Equation

30. Procedure for manipulating Duffing’s Equation • Generate a waveform of desired frequency with (x, y). f10y is a good rule of thumb for starters. • Adjust the damping coefficient so that its envelope resembles the desired waveform’s • Adjust b, g, and w to shape the waveform, holding one constant to change the other • Normalize the waveform to digital maximum

31. Guitar Residual Synthesized by Duffing’s Equation

32. Synthesizing the Plucked String

33. Synthesized Guitar Using Duffing’s Equation as the Excitation

34. Timbral Characteristics • Synthesized guitar from Duffing’s Equation very similar to that from inverse filtering • Frequency of both residuals different from pitch of synthesized stringsinharmonicity • Sonograms of both residuals also very similar

35. Sonogram for Guitar (Inverse Filtering)

36. Sonogram for Guitar(Duffing’s Equation)

37. Tuning Performance (Harmony) • For individual pitches, the algorithm played fairly close to being in tune (perhaps slightly sharp). The allpass filter parameters can be adjusted to remedy this. • The C major chord played very well in tune, sounding very consonant with no apparent beats.

38. Tuning Performance (Range) • To test effective range of the algorithm, the lowest and highest pitches in a guitar’s range were synthesized. • Low E played in tune by itself. High E was flat. • This was more readily apparent when sounded together.

39. Summary of Tuning Performance • Algorithm performed as expected; it performed like Karplus-Strong; high frequencies tend to go flat, and this would have to be accounted for in the overall system.

40. Changing Damping Coefficient • Changing the damping coefficient can have pronounced effect on timbre of sound, specifically difference between type of pick used and type of string • The damping coefficient was adjusted to attempt to produce different sounds

41. Synthesized Residual (d = 0.2)

42. Synthesized Guitar (d = 0.2)

43. Synthesized Residual (d = 0.5)

44. Synthesized Residual (d = 0.5)

45. Summary of Damping Coefficient Adjustments • For d = 0.2, contribution of residual made for a very hard attack, as if picked • For d = 0.5, guitar tone had much softer attack, as if finger-picked • Sonograms confirm that the latter had more high-frequency content

46. Sonogram for d = 0.2

47. Sonogram for d = 0.5

48. Production of Other Waveforms • Duffing’s Equation can be used to form a variety of waveforms • User has some control over its behavior if properties of the oscillator can be controlled to obtain the desired waveform

49. Residual with Damping Only

50. Residual with Beta Only