Analysis of plucked sound signals using the Prony method

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

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

3. 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

4. Prony Method

5. Fourier Series vsProny Analysis • Non-parametric -- Parametric • undamped complex exponentials -- damped complex exponentials • amplitude, phase and frequency -- amplitude, phase, frequency and damping coefficients

6. 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

7. 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

8. Frequency Response

9. Modifications for the sound • Decay Shortening • Vibrato • Glissandi

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

11. Mathematical formulations

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

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

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

15. 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

16. F3+F4+F5

17. F1

18. F2

19. Using “prony” command in Matlab

20. 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

21. 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

22. 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

23. Thank you!