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

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

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### 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

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.

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

Overview

- Plucked String Instrument Modeling
- Excitation Modeling with Duffing’s Equation
- Model Performance and Analysis

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

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

Length of String

- Effective delay length determines fundamental frequency of output signal
- Delay line length (in samples) is L = fs/f0

The Comb Filter

- Works by adding, at each sample time, a delayed and attenuated version of the past output

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

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

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)

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

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)

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

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)

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

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

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

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

Procedure for manipulating Duffing’s Equation

- Generate a waveform of desired frequency with (x, y). f10y 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

Timbral Characteristics

- Synthesized guitar from Duffing’s Equation very similar to that from inverse filtering
- Frequency of both residuals different from pitch of synthesized stringsinharmonicity
- Sonograms of both residuals also very similar

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.

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.

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.

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

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

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

Algorithm Speed

- For 200 MHz Pentium Pro, Karplus-Strong with an inverse filtered residual took 57.46 s. with approximately 2500 samples saved on a wavetable
- With synthesized residual, Duffing’s Equation added only 4.057 s; total computation time increased by only about 5% with no saved samples

Conclusion

- Plucked string sounds were successfully produced
- Model plays in tune
- Different plucked string sounds can be produced by changing the damping coefficient
- Algorithm is fast

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