Page  433 ï~~Combined Linear and Non-Linear Periodic Prediction in Calibrating Models of Musical Instruments to Recordings Gary P. Scavone Perry R. Cook Center for Computer Research in Music and Acoustics (CCRMA) Stanford University Stanford, CA 94305 Abstract This paper summarizes an approach to periodic prediction for use in extracting linear and non-linear characteristics from musical instrument sounds. Linear prediction is performed using a standard single-input adaptive linear combiner. This same filtering scheme is modified to also perform non-linear polynomial coefficient prediction by replacing the "tap delay line" with succesive higher powers of the input signal. In each case, the systems are configured to predict one or one-half period in advance. This approach enables, in some situations, simultaneous identification of the linear and nonlinear components of waveguide models for winds and, potentially, bowed strings. 1 Introduction Moth prossae* In analyzing the time-domain behavior of musical instruments for the purpose of sound synthesis, it is common to consider the various physical elements of such instruments separately. The clarinet, for example, is usually modeled in terms of three main functional units: its reed/mouthpiece, bore, and bell/tonehole-lattice. Typically, a linear filter is employed to model the combined effects of the bell and toneholes, and either a "lookup" table or a generalized polynomial function is used to represent the non-linearity of the reed. The use of a digital delay-line to model the one dimensional traveling waves in a "loss-less" bore has been demonstrated to be highly efficient and effective [Smith, 1987]. In this way, separate signal processing elements are used to represent each physical element of an instrument. A similar but more general approach to the synthesis of musical instruments is demonstrated by the system of Figure 1. Here, single "lumped" elements account for the accumulated linear filtering and non-linear filtering throughout the instrument. The linear element then not only models the effects of the bell and toneholes, but also linear effects such as viscous and thermal loss in the bore (to linear approximations). The non-linear function accounts for the non-linear reed/bore coupling and any other non-linear effects throughout the instrument. The delay-line represents wave propagation over twice the bore length. From this perspective, the linear and nonlinear filters are each applied once or twice per Figure 1: Generalized Waveguide Instrument period. Thus, it is possible to use periodic prediction techniques to identify the linear and nonlinear features of real musical instruments, and incorporate the results in waveguide instrument simulations. A method for performing non-linear periodic prediction in woodwind instruments using an extended Kalman filter has previously been demonstrated [Cook, 19911. In that case, linear filtering characteristics were disregarded. 2 Linear and Non-Linear Characteristics For the purposes of analysis, the non-linearity of Figure 1 is considered to be memoryless and essentially time-invariant in the steady-state. Further, it is assumed that the non-linearity can be approximated with polynomial functions. An attempt to identify periodic non-linearities in steady-state tones is problematic, however, because period to period variation is generally minimal. This is a result of an equilibrium condition which exists between the non-linear excita ICMC Proceedings 1994 433 Sound SynthesisTechniques

Page  434 ï~~tion mechanism and the linear "lossy" element. Therefore, the current approach is to also extract the linear characteristics of the system, and remove these effects by inverse filtering before nonlinear prediction is performed. 3 Periodic Prediction Two adaptive periodic prediction schemes were implemented for this study. To perform linear prediction, a general single-input linear adaptive combiner was used to predict the input signal one or one-half period in advance. A non-linear periodic predictor was created by replacing the delay elements of the linear predictor with mulitplicative units, obtaining successive powers of the input signal as input to each weight. This is shown in Figure 2. For each predictor, the Least Mean Squares (LMS) algorithm was used to adapt the weights [Widrow and Stearns, 1985]. Bias Weight of the extracted linear weights is generally "lowpass" as would be expected, though the low frequency characteristics are questionable. The nonlinear characteristic is displayed in the form of a "reed reflection function". The non-linear predictor was found to be particularly sensitive to period discrepancies, and thus accurate initial determination of the period was important. The need for both wideband signal generating and measuring mechanisms became particularly obvious in attempting to obtain an accurate linear estimate. Size constraints on the excitation source likely played a part in the poor low frequency identification. Further, the impulsive excitation provided only enough energy for about six measured reflections within the tube, thus limiting the amount of data available for use in performing the linear prediction. Because of the high sound pressure levels existing inside "wind" instruments, and the need for a wide-range frequency response, accurate steady-state measurements were also difficult to obtain. Equipment capable of yielding accurate data under these conditions is expected to improve these results. Frequency Response o Extracted Weights 0 "0 02 0.4 0o6 0o 1 Frequency (times 10025 Hz) Reflection Function given by Extracted Non-rnea Weights ~0.5. C 11 -0. Dlifferential Pressure Figure 3: Functions of Extracted Weights References [Cook, 1991] Perry It Cook. "Non-Linear Periodic Prediction for On-Line Identification of Oscillator Characteristics in Woodwind Instruments," Proceedings IC MC 1991, Montreal, Canada. [Smith, 1987] J.O. Smith. Music Applications of Digital Waveguides, Report No. STAN-M-39, CORMA - Stanford University, 1987. [Widrow and Stearns, 1985] Bernard Widrow and Samuel D. Steamns. Adaptive Signal Processing. Englewood Cliffs, N.J.: PrenticeHall, Inc., 1985. Figure 2: Adaptive Non-Linear Periodic Predictor 4 Using "Real" Data An experimental reed woodwind instrument was impulsively excited and its "impulse response" measured from one end of its bore. This signal was then fed into the linear periodic predictor to obtain a linear system approximation via FIR filter coefficients. A steady-state signal was measured via a microphone inserted through the side of the instrument bore. A stable inverse fiter was generated from the extracted linear FIR coefficients and used to filter this steady-state signal, forming the input xk to the non-linear predictor. 5 Results Figure 3 shows functions of the linearity and nonlinearity extracted from the experimental reed woodwind instrument. The frequency response Sound SynthesisTechniques 434 ICMC Proceedings 1994