Page  402 ï~~Parameter Estimation of Non-Linear Physical Models by Simulated Evolution-Application to the Flute Model Jarkko Vuori Helsinki University of Technology Laboratory of Signal Processing and Computer Technology SF-02150 ESPOO, FINLAND. Jarkko. Vuorihut. fi Vesa V"ilimiki Helsinki University of Technology Acoustics Laboratory SF-02150 ESPOO, FINLAND CARTES (Computer Arts Centre at Espoo) Ahertajankuja 4,02100 ESPOO, FINLAND vpvGvipunen.hut. fi Abstract Physical modeling of acoustic musical instruments has recently become a field of growing interest in musical acoustics. Presently several real-time DSP models of musical instruments have appeared. These models contain several tunable parameters in order to fit the model to existing real instrument sound. In general, tuning of those parameters by trial and error method is a dreadful task. Therefore, there is a finm need for making the tuning process automated. In this paper, we present a novel evolutionary computing (EC) method to this parameter optimization problem. cases [Goldberg, 1989]. Evolutionary algorithms use 1 Introduction neo-Darwinian paradigm instead. Recently a growing number of physical models of musical instruments has been emerged. Those models contain a large number of parameters whose relationship to each other is unknown and in some models there are also non-linearities that cause the estimation of the parameters to be very difficult job using conventional methods. There are many powerful methods for the estimation of parameters for linear models [Smith, 1983], but these methods are unsuitable for the current non-linear instrument models. In this paper we present a radically new concept to parameter estimation of non-linear physical models. This method uses evolutionary algorithms (EA) [Goldberg, 1989], [Back et al., 1993]. This technique has been applied to estimate the parameters of our real-time DSP flute model [Valimaki et al., 1992]. EAs can also be used for the parameter estimation of other non-linear instrument models. The ultimate goal of this approach is a model-based analysis/synthesis system that could be used for coding musical performance. In that case, the musical signal could be stored as a sequence of parameter vectors with low storage requirements instead of a direct recording of the signal samples. Standard estimation algorithms [Ljung, 1987] employ gradient-based search and are therefore subject to entrap local minima. Other methods like simulated annealing have slow convergence in some 2 Flute Model The flute model tested for parameter estimation is a slightly modified version of our earlier work [Valimaki et al., 1992]. The model is shown in Figure 1. It consists of three delay lines and appropriate digital filters between them. A sigmoid function emulates the non-linear interaction between the air jet and the tube resonator. In the model there are eight real-valued parameters which are listed in the table 1. These parameters control the sound quality of the flute model. Table 1: Flute model parameters lossFrq coefficient for the freqdependent loss filter los sGain freg-independent (DC) loss gain no iseGain gain of the white-noise signal voicedGain gain of the feedback from the tube to the nonlinearity dirFeedback gain from the nonlinearity back to itself s igmIn input gain of the sigmoid function s igmO f f set offset of the sigmoid function sigmOut output gain of the sigmoid function 4P.11 402 IC.MC Proceedings 1993

Page  403 ï~~Figure 1: Flow diagram of the flute model 3 Model Parameter Estimation Through Evolutionary Algorithms Parameter estimation problem can be seen as a search process. Modeling the search process of natural evolution can yield very robust, direct computer algorithms. The resulting evolutionary algorithms are based on the collective learning process within a population of individuals, each of which represents a search point in the space of potential solutions to a given problem. The population is arbitrarily initialized, and it evolves toward better and better regions of the search space by means of randomized process of selection, mutation, and recombination. The environment delivers quality information (fitness value) about the search points, and the selection process favours those individuals of higher fitness to reproduce more often than those of lower fitness. The recombination mechanism allows the mixing of parental information while passing it to their descendants, and mutation introduces innovation into the population. 4 Simulated Evolution Simulated evolution is one main class of the evolutionary algorithms. It has been shown to perform well in system identification problems [Fogel, 1991]. The identification model used in this case is shown in figure 2. Simulated evolution is conducted as a sequence of the following operations. 1. The initial population is determined by setting p; = P; -U(a,b)",Vi =1,...,k, where Y; is a random vector, pi is the outcome of the random vector, U(a,b)" denotes a uniform distribution ranging over [a,b] in n dimensions, and k is the number of parents. 2. Each p;,i =1,...,k, is assigned a fitness score (p)=F(pi), where F maps p -+ 91 and denotes the true fitness of p,. 3. Each p;,i =1,...,k, is altered and assigned to Pi,+k in accordance with P,+,; = '.; + iPj(p1).N(0,1),Vj = 1,...,n, where p;, denotes the jth element of the ith individual, I0 is mutation parameter and N(, a2) represent a Gaussian random variable with mean t and variance a2. 4. Each P+k,i = 1,...,k, is assigned a fitness score 5 (p+k)= F (p,+,). 5. For each p,,i =1,...,2k, a value w; is assigned m according to w; = 1w, t=1 1, ifus< wt = (1(P.) + (P1), r = tO, otherwise where = r2ku2 +11, m is the number of competitions, and u1,u2 -U(0,1). Figure 2: System identification flow diagram ICMC Proceedings 1993 403 4P.11

Page  404 ï~~6. The individuals pi,i =1,...,2k, are ranked in descending order of their corresponding value w,. The first k individuals are transcribed along with their corresponding values ($(p) to be the basis of the next generation. 7. If fitness score is over some predefimed threshold, max{CD(p )}> f, or too many iterations has been evaluated, stop. Else proceed to Step 3. First we tried to estimate the parameter values from the sound produced by the model itself. This test verifies the feasibility of the proposed method. The desired output from the flute model, sampled at 44.1 kHz, is shown in figure 3. 6000,, # I # 4000 } 0 1000 2000 3000 4000 5000 6000 7000 1000 9000 10000 Figure 3: Output from the flute model In this identification problem we want to estimate the steady state parameters of the model. In order to get rid of the phase and irrelevant components of the signal we take a 1024-point Fourier transform of the model output after steady state condition (after the 4096th sample). The result of the Fourier transform is presented in figure 4. 0. -i0! 7 w........ N." "------- w."10........y +.........%......................... "1 J -......... -4 -Figure 4: Fourier transform of the pulse in figure 3 This figure suggests that at least the eight first harmonic peaks can be detected. Therefore, in the fitness calculation of the current solution we first search the fundamental component (the first amplitude peak in the spectrum) and then the following seven harmonics. The fitness function can be thus stated as F(p,) = (Dn,,- Sn,,), where n(i) is the index vector of dominant spectral peaks, D is the spectrum bin of-the desired output and S that of the current individual parameter set. 5 Results I I 1 1 " A i tn wu wn am Ini n c r M 0 50 fOO 150 200 2% 300 350 400 450 500 Figure 5: Convergence of the algorithm Convergence of the simulated evolution algorithm is illustrated in figure 5. The error shown in the figure is the inverse of fitness function FO. It can be seen that the algorithm converges smoothly. After 500 iterations the maximum model parameter deviation from the desired is 3.4%. The identification process takes approximately ten minutes of computer time at a 33 MHz 486-based PC. References [Smith, 1983] Julius O. Smith. Techniques for Digital Filter Design and System Identification with Application to the Violin, Ph. D. Dissertation, Elec. Eng. Dept., Stanford University, June 1983. [Goldberg, 1989] David Goldberg. Genetic Algorithms in Search, Optimization & Machine Learning, AddisonWesley, 1989. [Fogel, 1991] David Fogel. System Identification Through Simulated Evolution: A Machine Learning Approach to Modeling, Ginn Press, 1991. [Valimaki et al., 1992] Vesa VtIlimaki, Matti Karjalainen, Zoltain Jtinosy, and Unto Laine. A Real-Time DSP Implementation of A Flute Model. In Proc. of IEEE Int. Conf. on Acoustics, Speech and Signal Processing (ICASSP'92), San Fransisco, March 1992. [Back et al., 1993] Thomas Back, Hans-Paul Schwefel. An Overview of Evolutionary Algorithms for Parameter Optimization. Evolutionary Computation, vol. 1, no. 1, pp. 1-23, Spring 1993. [Ljung, 1987] Lennart Ljung. System Identification, A Theory for the User, Prentice-Hall, 1987. 4P.11 404 ICMC Proceedings 1993