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Page 00000001 A fractional delay application: time-varying propagation speed in waveguides S. Tassart R. Msallam, Ph. Depalle, S. Dequidt IRCAM, PARIS, FRANCE firstname.lastname@example.org, http://www.ircam.fr/equipes/analyse-synthese/tassart Abstract Much work has been done in the field of physical modelling in order to digitally simulate the propagation of acoustical plane waves in waveguides. In most of the work, the acoustical assumptions taken into account result in the linear behaviour of the waveguide; the acoustical wave speed can be considered as constant and independent from the local acoustical pressure. But for some wind instruments (e.g. the trombone), the acoustic pressure is so high that the linear assumption is no longer valid. In this case the model must include the nonlinear propagation. This paper presents a new nonlinear filter for simulating the nonlinear propagation in digital waveguides. 1 Introduction In the first section, a nonlinear wave equation is proposed. From this equation and in the case of a plane wave propagation in a waveguide, we derive a continuous-time input-output relation. Then we propose an explicit discrete-time system which implements the nonlinear propagation in a digital waveguide. Finally we demonstrate some of the features of nonlinear propagation in digital waveguides for modelling musical instruments. 2 Nonlinear Wave Equation Let us consider the first order nonlinear wave equation describing the propagation in the positive direction of the x-axis of a wave p(x, t) where propagation speed c(yp) is a given function of p: Px + C(P)pt = 0 (1) This nonlinear equation, studied in , is known to "propagate" in space an initial profile p(O, t) with velocity c(p): Figure 1: Successive profiles at position x = 0, x = L input wave and on the length L of the guide: p(L,t + d+(t)) = ýp(0, t) with d+(t) = L c(p(0,t)) (3) 3 Continuous-time System From Section 2 we see that an input wave e(t) p(O, t) is delayed by a time-varying input-delay d+(t) during its nonlinear propagation. The output wave s(t) = (L,t) is observed d+(t) after it enters the waveguide: s(t + d+(t)) = e(t). Using now a signal processing description, we express the output s at time t as a function of the present and past of the input e, which leads to s(t) = e(t - d-(t)), where d-(t) is the output-delay. c(p(0, t)) p(0,t) The propagation time throught a waveguide from input to output d+ depends on the magnitude of the
Page 00000002 3.1 Input-Delay to Output-Delay Conversion In the linear approximation, the speed c is constant, so that the physical and signal processing points of view become identical and d- = d+. But when the speed c varies in time, no trivial relationship remains between d- and d+. Since the waveguide is to be modelled by a fractional delay filter, we have to design an efficient system for converting the input delay d+ into the output delay d-, which controls the filter delay. 3.1.1 Formal Description Let us define two time scales r7 and 7-, as: Sr- (t) def I + d (t) te - d () 4) These two time-scales being defined, we may recast the two primary relations describing the propagation as: Figure 2: Graphical interpretation horizontal distance between the axis and the curve The graphical representation clearly proves that the d+ to d- conversion is causal, since the horizontal line measuring d- intersects the 7+ curve on its left, corresponding to the past of the signal. In other words, knowing the past of d (t) is sufficient to determine d- (t). s 0 s e eoy (5) Under the constraint that r7 or r- is reversible1, we deduce that the two time-scales are mutually reciprocals: + o07+ 0+07 - Id Id By introducing the definitions of rwe obtain the final relations: or 7+ in Eq. 6, d- o r+ = d+ d+ o 7- = d which may be recast as implicit equations using traditional notation: 3.1.3 System Interpretation The input-output generic relation s(t) = e(t - d- (t)) is to be interpreted as the linear filtering of the input e(t) by a delay filter of delay d- (t), but we get Eq. 8 by formally substituting e by d+ and s by din this generic input-output relation. This is to be interpreted as a filtering relation between d+ and d(see Fig. 3). Thus the d+ to d- convertion must be understood as a delay filter whose delay is controlled by its output. The feedback loop is the origin of the nonlinearity of the system. d+ (t) sd- (t) d- (t) Figure 3: Formal interpretation 3.2 Complete Delay-line Model In Eq. 3 we note that the input-delay depends on the amplitude of the input wave. Provided that there exists an instantaneous one-to-one relation g between the input wave and the input-delay, d+ = g o e, we iThe reversibility assumption means physically that no shock-wave occurs along the propagation. It also means that the derivatives d+'(t) and d-'(t) are respectively greater than -1 and lower than 1. Vt eR( d- (t + d (t)) V ' d+ (t - d- (t)) d+ (t) d- (t) (8) 3.1.2 Graphical Interpretation Provided that r- or r7 are inverse functions, their curves are symmetrical about the r = t axis, which also represents the identity time-scale (Fig. 2). In this graphical representation, d+ at time to measures the vertical distance between the axis of symmetry and curve r7, whereas d- measures the vertical distance between the axis and curve r-, which is also the
Page 00000003 show ill Eq. 9 that the output wave is deduced from d- by applyirtg the iliverse relatiort g1l: S =eo r- g- ogoeor g-1ld o 0 g- oct Therefore a complete corttirtuous-time rtortlirtear orteway propagatiort model of the waveguide irtcludes first a system mappirtg the irtput wave irtto a delay, thert a delay filter corttrolled by feedback cortvertirtg irtput-delay irtto output-delay, artd filially a third system mappirtg back the delay irtto the output wave. 4 Discrete-time System 4.1 Ideal Filter Sirtce the discrete versiort of a corttirtuous-time delay filter is art ideal fractiortal delay whose trartsfer furtctiort is z d, d beirtg a rtortirtteger delay2, we may cortsider the discrete-time system of Fig. 4 as art irttuitive courtterpart of the corttirtuous-time system of Fig. 3. But the irtstarttarteous feedback loop is Rot realizable , so we propose the implemerttatiort described irt Fig. 5. Thert the ideal complete discreteFigure 4: Nortrealisable ideal discretizatiort ++ Figure 5: Realisable ideal discretizatiort 4.2 Filter Approximations Ideal fractiortal delay filters may be approached by differertt digital filters artd differertt structures . Because of the feedback loop, we rteed a particularly stable implemerttatiort for the fractiortal time-varyirtg delay filter. That is why we have focused mairtly ort FIR implemerttatiorts, artd more particularly ort Lagrartge Irtterpolator Filters (LIFs). LIFs approximate the ideal fractiortal delay filters at low frequertcy, artd the valid frequertcy rartge irtcreases with the LIF order . However irt our case, it appears that orders greater thart 2 do rtot sigrtificarttly improve the frequertcy resportse. 4.3 Filter Aliasing Sirtce the origirtal corttirtuous-time system is rtortlirtear, aliasirtg effects may appear due to the highfrequertcy comportertt gerteratiort. This comportertt gerteratiort beirtg irttrirtsic to the origirtal system, it seems difficult to derive a theoretical dicrete-timne equivalertt which irtsures a bartd-limited output for arty bartd-limited irtput. Therefore, for pragmatical reasorts, whertever aliasirtg occurs, we suggest to oversample the filter, to limit the rtegative slope of the irtput sigrtal or to irttroduce a low-pass filter irt the feedback loop. 5 Acoustics 5.1 Basic Asumptions From measuremertts (see ) artd dimertsiortless artalysis (see ), we show that rtortlirtear propagatiort (at high amplitude of playirtg) as well as visco-thermal bourtdary-layer effects must be takert irtto accourtt to describe acoustic propagatiort irt the slide of a tromborte. Neglectirtg high order corttributiorts (such as rtortlirtear irtteractiorts irt the mairt flow or rtortlirtearities irt the bourtdary-layers), the solutiort irt the cylirtdrical pipe may be described as a lirtear combirtatiort of art irtcomirtg artd art outgoirtg plarte waves where both waves are submitted to the viscothermal losses artd the rtortlirtear distortiort durirtg the propagatiort. Neglectirtg the viscothermal losses, the rtortlirtear distortiort of a simple wave is described irt  equatiort (Eq. 1) with the propagatiort speed c beirtg a furtctiort of the fluid velocity a (co beirtg the lirtear sourtd speed, y the ratio of the specific heats for gases): Fig ure 6: Complete discrete-time rtortlirtear orte-way propagatiort model of the waveguide time model of Fig. 6 implemertts the model of a timevaryirtg propagatiort speed irt waveguide, artd replaces the classical digital delay-lirte z -d used for cortstartt propagatiort speed waveguide. The irtteger parameter p cart be set freely but must be greater thart orte. 2 The expression z-d is nothing more than a symbolic notation since the z-transform of the impulse response of the ideal filter does not actually exist. c(u) co + (y 1 2 (10)
Page 00000004 (o) linear p+ (incoming wave), (+) nonlinear p+, (*) nonlinear p+ (o) linear p- (outgoing wave), (+) nonlinear po, (*) nonlinear p7 Frequency (Hz) (o) linear slide output pressure (+) nonlinear slide output pressure Figure 8: Comparison of the amplitudes of the first ten harmonics: linear propagation vs. nonlinear propagation + + lips delay line b lips bell model bel o__4- delay line 4 Pi Po Figure 7: Simulation model 5.2 Simulations Fig. 7 displays the model we used to demonstrate some of the effects that nonlinear propagation introduces in a waveguide system simulating a windinstrument, such as a trombone. This model includes a one-mass lips model as nonlinear excitator, two linear/nonlinear delay-lines for propagating the incoming and outgoing waves, and a low-pass filter for the linear modelling of the bell (see ). Comparisons are made based on simulations of the two systems including linear or nonlinear propagation driven by the same parameters. It appears clearly (Fig. 8) that the nonlinear propagation produces high frequency components which makes the sound brassier. It seems also that the nonlinear propagation plays a rather subtle role for low frequencies (here lower than 800Hz) corresponding to the reflexion function bandwidth of the bell. But the simulations show that, even if the motion of the lips is perturbed, the frequencyspectrum of the input incoming wave p+ is not modified much. That would support  where it is supposed that the nonlinear propagation has a rather small effect on the self oscillation process at steadystate, compared to the effect on the radiated sound. 6 Conclusion We have proposed in this paper an efficient digital nonlinear system based on fractional delay filters. It replaces the traditional delay-line for simulating the nonlinear propagation which happens in some wind instruments. This digital system has been used for demonstrating some of the effect of the nonlinear propagation in a particular digital waveguide system. This simple system has already proved that it can be used for sound synthesis purposes. References  Ph. Depalle and S. Tassart. Fractional delay lines using Lagrange interpolators. In Proc. Int. Computer Music Conf. (ICMC'96), Hongkong, August 1996.  A. Hirschberg, J. Gilbert, R. Msallam, and A. Wijnands. Shock waves in trombones. J. Acoust. Soc. Am., 3(99):1754-1758, 1996.  M. Karjalainen, A. Harma, and U. K. Laine. Realizable warped IIR filters and their properties. In Proc. IEEE ICASSP'97, pages 2205-2208, Munich, April 1997.  T. I. Laakso, V. Vilimaki, M. Karjalainen, and U. K. Laine. Splitting the unit delay. IEEE Signal Processing magazine, 13(1), 1996.  R. Msallam, S. Dequidt, S. Tassart, and R. Causse. Physical model of the trombone including nonlinear propagation effects. In Proc. International Symposium on Musical acoustics (ISMA'97), Edinburgh (Scotland), August 1997.  G. B. Whitham. Linear and Nonlinear Waves. Wiley-Interscience Publication, 1974.