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!