Page  157 ï~~Non-Linear Periodic Prediction for On-Line Identification of Oscillator Characteristics in Woodwind Instruments Perry R. Cook Stanford Center for Computer Research in Music and Acoustics Abstract Non-linear periodic prediction techniques are used to identify characteristics of woodwind instruments, and to investigate the non-linear oscillator elements. A Kalman filter implementation of a polynomial non-linear predictor was configured to predict the signal either one or one-half period ahead. Using knowledge about instrument structure, the resulting polynomial coefficients can be algebraically manipulated to yield forms which fit the parameters of standard digital simulation models of such instruments. Distinctions between hard and soft single clarinet reeds are easily seen, as well as distinctions between reeds and jets. A brief discussion is included on the mathematical and modeling differences in non-linearities exhibiting memory vs. memoryless non-linearities. 1 Non-Linear Elements in Musical Systems Most musical instruments capable of producing a sustained tone can be modeled by combining linear elements with non-linear oscillator elements[1][2]. Often the linear and non-linear elements do not vary with time. The acoustic tubes of the clarinet bore and vocal tract, the bell, and the vibrating string can often be modeled quite well by linear elements (filters or simple delay lines). The nonlinear oscillators model the reflection/transmission behavior of the reed, reeds, vocal folds, lips, or bow/string interface. These non-linearities can be modeled to arbitrary accuracy by polynomial functions, or state dependent polynomials in cases such as the bow/string interface where hysteresis is present. A memoryless polynomial non-linearity has a simple form which contains no delayed versions of the signal (no memory). The mathematical form of an Nth order memoryless non-linear element is: N f(x(n)) = Zaix(n)' (1) i=0 A general non-linear element can be represented as the formation, weighting, and summing of products of the signal and delayed versions of the signal. The memoryless polynomial element can be viewed as a special case of the general non-linear element. The mathematical form of the general non-linear element grows rapidly in complexity for even small orders of non-linearity and delay. This introduces practical considerations in calculating and identifying the coefficients of the polynomial representation of a high-order general non-linearity. An example of a non-linear element with 3rd order non-linearity and 2nd order delay is: ICMC 157

Page  158 ï~~(z(n)) = { ao,o + a1,ox(n) + a2,ox(n)2 + a3,ox(n)3 + al,lx(n - 1) + a2,1 x(n - 1)2+ a3,1 x(n - 1)3 + ai,2x(n - 2) + a2,2x(n - 2)2 + a3,2x(n - bo,lx(n)x(n - 1) + bo,2x(n)x(n - 2) + bi,2x(n - 1)x(n - 2)+ co,,l x(n)x(n - 1)2 +co,2,2x(n)x(n - 2)2 + ci,2,2x(n - 1)x(n - 2)2+ c1,1,2x(n- 1)2x(n - 2)+co,i,2x(n)x(n - 1)x(n - 2) (2) To investigate some mathematical and computational issues, a simplified clarinet/flute model with no tone holes will be used as an example. If the bell reflection element is 'pushed through' the bore and joined with the reed element, the system of Figure 1 results. P Non-Linearity f(x(n-m),P) x(n )nnm) Delay Line (length = m) Figure 1: A simplified clarinet/flute model. The cylindrical bore is modeled by a simple delay line, and the characteristics of bore and reed reflection/transmission are lumped into a single general two input two output non-linear element. Denote the instantaneous breath pressure as p(n), the output as x(n), and the non-linear operation as F(x,p). In the quasi-steady state of oscillation, p(n) can be considered to be a constant, P. The equation governing steady state oscillation is: x(n) = F(x(n - m),P) (3) where m is the period of oscillation in the case of a recorder or flute, and is one-half the period of oscillation in the case of the clarinet. If the behavior of the non-linear operation is reed-like and assumed to be memoryless, the output can be modeled as a memoryless polynomial function of the differential pressure operating on the reed. N F(x,P) = Z:ai(P- x) i=O (4) Often a function F'(x) is desired which models the normalized characteristics of a particular physical non-linearity, such as a reed reflection table. Relating Equation 4 to the coefficients of the physical non-linearity of a reed table can be accomplished through algebraic manipulation, and simplifies if the constant steady-state pressure is normalized to 1: x(n) = F'(x(n - m))x(n - m) + (1 - F'(x(n - m))) == N-1 F'(x) = >kiO i=0 (5) The coefficients ki and ai are related by an upper diagonal matrix whose entries are the binomial coefficients of Pascal's Triangle. For a 4th order predictor, the coefficients of the reed non-linearity are related to those of the predictor by: ICMC 158

Page  159 ï~~1 k0 kl k2 k3 1 0 0 0 0 -1 -1 0 0 0 1 -1 1 2 -3 4 1 -3 6 0 -1 4 0 0 1 ao al a2 a3 a4 (6) Various possibilities for the signs of the entries of the matrix of Equation 6 are determined by the form of the system being identified. These depend on the number of inversions in the system, the input variable P as assumed to be +1, and other mathematical factors. An indication is given as to the correctness of the assumptions, measurements, normalizations, and calculations by the first condition of Equation 6, which states that the sum of the predictor coefficients should be +1. Other forms dictate that the sum should be -1. 2 Identification Method A modified Kalman filter [3] was used to identify the non-linear function coefficients. The state equations for the memoryless non-linearity are: Xk+m ao al a2 an 1 1 0 0 0 Xk 0 1 0 0 0 0 1 0... x "o" 0 Â~. 1 Xk ao al a2 an Wk 0 0 0 0 (7) Yk-=[1 0 0 0... O] xk ao a1 a2 an +Vk (8) where Wk and Vk are zero-mean white Gaussian noise sequences, representing model and measurement noise, respectively. 3 Experimental Results for Memoryless Predictor Two tones were synthetically generated using a computer clarinet model. One tone used a reed with normal rest position (Reed 1), and the other used a reed with a large opening rest position (Reed 2). Figure 2 shows the results of identification performed on synthesized clarinet tones. Histograms are shown below each reed curve to show how many samples were used to compute each point on the non-linear function. This gives an estimate as to the confidence of regions of the curve. Two tones were recorded with a microphone mounted inside of a section of pipe joined to a clarinet mouthpiece. The two tones were generated using a soft reed and a hard reed. Figure 2 shows the ICMC 159

Page  160 ï~~results of identification performed on the clarinet tones. The two curves of the soft actual reed were computed using the acoustical signal inside and outside the instrument. The stiffness of each reed is reflected in the slope of the curve as it passes through the origin (rest position). The greater the negative slope, the softer the reed. Figure 2: Non-linear reed identification results for synthetic clarinet tones (left), and actual clarinet tones (right) performed with soft and hard reeds. As a final example, a tenor recorder was recorded while blowing the fundamental (all holes covered), then overblowing to the first overtone one octave above. Figure 3 shows the results of predicting the signal one-half period ahead (interpreting the instrument structure as that of a clarinet), and one period ahead (correct for the recorder). The half-period prediction results yield greatly different jet characteristics for the two tones, while the full-period prediction results yield quite similar jet reflection curves for the two tones. Relection Recorders F n Octavavvu -o 0.9 Diffeeti Reoodsr Jet Prdced 12 R2ecorder Jet Pmdicsed 1 Fundamentl Ahead Fundamental Ahead Figure 3: Non-linear identification results for actual recorder tones at half-period (left) prediction, and full-period (right) prediction. The two tones are the fundamental and the overblown octave above. References [1] M. E. McIntyre, R. T. Schumacher, and J. Woodhouse, "On the Oscillations of Musical Instruments." JASA, 74, 5, 1325-1345, 1983. [2] J. O. Smith, "Musical Applications of Digital Waveguides." Stanford University Center For Computer Research in Music and Acoustics, Report No. STAN-M-39, 1987. [3] T. Kailath, Lectures on Wiener and Kalman Filtering. Springer, 1981. ICMC 160