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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**Synthesized Guitar Using Duffing’s Equation as the**Excitation**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