Analysis of plucked sound signals using the Prony method Ye Lu 2011-12-15.
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Transcript of Analysis of plucked sound signals using the Prony method Ye Lu 2011-12-15.
![Page 1: Analysis of plucked sound signals using the Prony method Ye Lu 2011-12-15.](https://reader036.fdocuments.us/reader036/viewer/2022062318/551aced45503466b6a8b535a/html5/thumbnails/1.jpg)
Analysis of plucked sound signals using the Prony methodYe Lu2011-12-15
![Page 2: Analysis of plucked sound signals using the Prony method Ye Lu 2011-12-15.](https://reader036.fdocuments.us/reader036/viewer/2022062318/551aced45503466b6a8b535a/html5/thumbnails/2.jpg)
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);endd=zeros(1,N/2);for i=1:N/2 d(i)=y(i+N/2);endD=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); endend
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); endendC=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/docs/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!