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Analysis of plucked sound signals using the Prony method. Ye Lu 2011-12-15. Introduction. Physical Modelling ----Digital Waveguide Synthesis ----Formant Synthesis ----Finite element Methods Plucked string instruments ---- Karplus -Strong Algorithm. Prony Method.

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Presentation Transcript
introduction
Introduction
  • Physical Modelling
  • ----Digital Waveguide Synthesis
  • ----Formant Synthesis
  • ----Finite element Methods
  • Plucked string instruments
  • ----Karplus-Strong Algorithm
prony method
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
fourier series vs prony analysis
Fourier Series vsProny Analysis
  • Non-parametric -- Parametric
  • undamped complex exponentials -- damped complex exponentials
  • amplitude, phase and frequency -- amplitude, phase, frequency and damping coefficients
karplus strong algorithm
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
implementation in matlab
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
modifications for the sound
Modifications for the sound
  • Decay Shortening
  • Vibrato
  • Glissandi
mathematical formulations
Mathematical formulations

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

three steps
Three Steps
  • 1. Solve linear prediction model, which is constructed by the observed data set
three steps1
Three steps
  • 2. Find Roots of charactreristic polynomial formed from the linear prediction coefficients
three steps2
Three steps
  • 3. Solve the original set of linear equations to yield the estimates of the exponential amplitude and sinusoidal phase
implementation in matlab1
Implementation in Matlab

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

end

C=pinv(U)*y';

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

end

problems to be aware
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
conclusion
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

references
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/docs/PronyAnalysis.pdf
  • [4] Kay and Maple, 1981, “Spectrum Analysis” Proceedings of the IEEE VOL, 69, No. 11: 1404-1406