Duffing

1
Duffings Equation as an Excitation Mechanism for
Plucked String Instrument Models

• by
• Justo A. Gutierrez
• Masters 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 Duffings
  Equation.

3
Advantages

• Using Duffings 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 Duffings 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

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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

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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

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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

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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

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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(1a1)/(1a1z-1)

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Loop Filter Signal Flowchart
13
Loop Filter Magnitude Response and Group Delay
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Loop Filter Impulse Responses
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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

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Allpass Filter (continued)

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

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Allpass Phase Response
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Allpass Delay Response
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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

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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)

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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

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Steel-string Guitar Sample
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Residual After Inverse Filtering
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Truncated Residual Signal
25
Resynthesized Guitar
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Duffings 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

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Duffings 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

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Experimental Apparatus (Moon and Holmes)
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Modeling the Excitation

• For this experiment, the coefficients in Moon and
  Holmes modification of Duffings Equation were
  adjusted to produce the desired residuals
• The Runge-Kutta method was the numerical method
  used to calculate Duffings Equation

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Procedure for manipulating Duffings 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 waveforms
• Adjust b, g, and w to shape the waveform, holding
  one constant to change the other
• Normalize the waveform to digital maximum

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Guitar Residual Synthesized by Duffings Equation
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Synthesizing the Plucked String
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Synthesized Guitar Using Duffings Equation as
the Excitation
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Timbral Characteristics

• Synthesized guitar from Duffings 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)
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Sonogram for Guitar(Duffings Equation)
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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.

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Tuning Performance (Range)

• To test effective range of the algorithm, the
  lowest and highest pitches in a guitars range
  were synthesized.
• Low E played in tune by itself. High E was flat.
• This was more readily apparent when sounded
  together.

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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.

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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

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Synthesized Residual (d 0.2)
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Synthesized Guitar (d 0.2)
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Synthesized Residual (d 0.5)
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Synthesized Residual (d 0.5)
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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

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Sonogram for d 0.2
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Sonogram for d 0.5
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Production of Other Waveforms

• Duffings 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

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Residual with Damping Only
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Residual with Beta Only
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Residual with Forcing Function Only
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Residual with Strong High-Frequency Forcing
Function
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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, Duffings Equation
  added only 4.057 s total computation time
  increased by only about 5 with no saved samples

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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