Post on 31-Mar-2015
Analysis of plucked sound signals using the Prony methodYe Lu2011-12-15
Introduction
Physical Modelling ----Digital Waveguide Synthesis ----Formant Synthesis ----Finite element Methods
Plucked string instruments ----Karplus-Strong Algorithm
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
Prony Method
Fourier Series vs Prony Analysis
Non-parametric -- Parametric
undamped complex exponentials -- damped complex exponentials
amplitude, phase and frequency -- amplitude, phase, frequency and damping coefficients
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 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
Frequency Response
Modifications for the sound
Decay Shortening
Vibrato
Glissandi
Mathematical formulations
http://www.engr.uconn.edu/~sas03013/docs/PronyAnalysis.pdf
Mathematical formulations
Three Steps
1. Solve linear prediction model, which is constructed by the observed data set
Three steps
2. Find Roots of charactreristic polynomial formed from the linear prediction coefficients
Three steps
3. Solve the original set of linear equations to yield the estimates of the exponential amplitude and sinusoidal phase
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';
F3+F4+F5
F1
F2
Using “prony” command in Matlab
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
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
[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
Thank you!