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Analysis of plucked sound signals using the Prony method Ye Lu 2011-12-15

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Introduction Physical Modelling ----Digital Waveguide Synthesis ----Formant Synthesis ----Finite element Methods Plucked string instruments ----Karplus-Strong Algorithm

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Prony Method developed by Gaspard Riche de Prony in 1795 extracts valuable information from a uniformly sampled signal and builds a series of damped complex exponentials or sinusoids

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Prony Method

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Fourier Series vs Prony Analysis Non-parametric -- Parametric undamped complex exponentials -- damped complex exponentials amplitude, phase and frequency -- amplitude, phase, frequency and damping coefficients

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Karplus-Strong Algorithm [1] Karplus,K., and A. Strong. 1983. "Digital Synthesis of Plucked-String and Drum Timbres." Computer Music Journal 7(2) : 43-55. [2] Jaff, D., and J. Smith. 1983. "Extensions of the Karplus-Strong Plucked-String Algorithm." Computer Music Journal 7(2): 56-69

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Implementation in Matlab x=(2*rand(Time,1)-1); for i=N+1:Time x(i)=0; end for i=1:N y(i)=x(i); end y(N+1)=x(1); for i=N+2:Time y(i)=x(i)+0.5*(y(i-N)+y(i-N-1)); end

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Frequency Response

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Modifications for the sound Decay Shortening Vibrato Glissandi

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Mathematical formulations http://www.engr.uconn.edu/~sas03013/docs/PronyAnalysis.pdf

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Mathematical formulations

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Three Steps 1. Solve linear prediction model, which is constructed by the observed data set

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Three steps 2. Find Roots of charactreristic polynomial formed from the linear prediction coefficients

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Three steps 3. Solve the original set of linear equations to yield the estimates of the exponential amplitude and sinusoidal phase

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Implementation in Matlab y=zeros(1,N); for i=1:N y(i)=x(800*i); end d=zeros(1,N/2); for i=1:N/2 d(i)=y(i+N/2); end D=zeros(N/2,N/2); for i=1:N/2 for j=N/2:-1:1 D(i,-j+N/2+1)=y(i+j-1); end a=pinv(D)*d'; muhat=roots([1,-a']); U=zeros(N,N/2); for i=1:N for j=1:N/2 U(i,j)=muhat(j,1)^(i-1); end C=pinv(U)*y';

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F3+F4+F5

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F1

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F2

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Using “prony” command in Matlab

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Problems to be aware p less than N/2 Noise impacts the accuracy of the Prony pole estimation Noise can cause the damping factors to be too large

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Conclusion Prony method extracts valuable information from a uniformly sampled signal and builds a series of damped complex exponentials or sinusoids Provide information of amplitude, phase, frequency and damping coefficients Very sensitive to the noise, and behave badly when noise presents

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References [1] Karplus,K., and A. Strong. 1983. "Digital Synthesis of Plucked-String and Drum Timbres." Computer Music Journal 7(2) : 43-55. [2] Jaff, D., and J. Smith. 1983. "Extensions of the Karplus-Strong Plucked-String Algorithm." Computer Music Journal 7(2): 56-69 [3]http://www.engr.uconn.edu/~sas03013/do cs/PronyAnalysis.pdf [4] Kay and Maple, 1981, “Spectrum Analysis” Proceedings of the IEEE VOL, 69, No. 11: 1404- 1406

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Thank you!

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