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.Thus, a unit delay z 1 may bereplaced by(for frequency-dependent wave velocity)That is, each delay element becomes an allpass filter which approximates the requireddelay versus frequency.A diagram appears in Fig.10.14, where Ha (z) denotes theallpass filter which provides a rational approximation to z c 0 / c ( É ).For computability of the string simulation in the presence of scattering junctions,there must be at least one sample of pure delay along each uniform section of string.This means for at least one allpass filter in Fig.10.14, we must have Ha ( ") = 0which implies Ha (z) can be factored as z 1H' (z), where H' (z) is a causal, stablea aallpass.In a systolic VLSI implementation, it is desirable to have at least one realdelay from the input to the output of every allpass filter, in order to be able to pipelinethe computation of all of the allpass filters in parallel.Computability can be arrangedin practice by deciding on a minimum delay, (e.g., corresponding to the wave velocityat a maximum frequency), and using an allpass filter to provide excess delay beyondthe minimum.Because allpass filters are linear and time invariant, they commute like gain factorswith other linear, time-invariant components.Fig.10.15 shows a diagram equivalentto Fig.10.14 in which the allpass filters have been commuted and consolidated attwo points.For computability in all possible contexts (e.g., when looped on itself), a PRINCIPLES OF DIGITAL WAVEGUIDE MODELS OF MUSICAL INSTRUMENTS453Figure 10.14 Section of a stiff string where allpass filters play the role of unit delayelements.single sample of delay is pulled out along each rail.The remaining transfer function,3Hc (z) = zH (z) in the example of Fig.10.15, can be approximated using any allpassafilter design technique [Laakso et al., 1996, Lang and Laakso, 1994, Yegnanarayana,1982].Alternatively, both gain and dispersion for a stretch of waveguide can beprovided by a single filter which can be designed using any general-purpose filterdesign method which is sensitive to frequency-response phase as well as magnitude;examples include equation error methods (such as used in the Matlab invfreqz ()function [Smith, 1983, pp.48 50]), and Hankel norm methods [Gutknecht et al.,1983, Beliczynski et al., 1992].In the case of a lossless, stiff string, if H (Z) denotes the consolidated allpass transfercfunction, it can be argued that the filter design technique used should minimize thephase-delay error, where phase delay is defined by(Phase Delay)Pc(É) 0 ïøïø "cMinimizing the Chebyshev norm of the phase-delay error, c / (É) ,approximates minimization of the error in mode tuning for the freely vibrating string[Smith, 1983, pp.182 184].Since the stretching of the overtone series is typicallywhat we hear most in a stiff, vibrating string, the worst-case phase-delay error seemsa good choice in such a case.However, psychoacoustic experiments are necessaryto determine the error tolerance and the relative audibility of different kinds of errorbehaviors. 454 APPLICATIONS OF DSP TO AUDIO AND ACOUSTICSFigure 10.15 Section of a stiff string where the allpass delay elements are consolidatedat two points, and a sample of pure delay is extracted from each allpass chain.Alternatively, a lumped allpass filter can be designed by minimizing group delay,(Group Delay)The group delay of a filter gives the delay experienced by the amplitude envelope of anarrow frequency band centered at É, while the phase delay applies to the carrier atÉ, or a sinusoidal component at frequency É [Papoulis, 1977].As a result, for propertuning of overtones, phase delay is what matters, while for precisely estimating (orcontrolling) the decay time in a lossy waveguide, group delay gives the effective filterdelay seen by the exponential decay envelope.To model stiff strings, the allpass filter must supply a phase delay which decreasesas frequency increases.A good approximation may require a fairly high-order filter,adding significantly to the cost of simulation.To a large extent, the allpass orderrequired for a given error tolerance increases as the number of lumped frequency-dependent delays is increased.Therefore, increased dispersion consolidation is ac-companied by larger required allpass filters, unlike the case of resistive losses. PRINCIPLES OF DIGITAL WAVEGUIDE MODELS OF MUSICAL INSTRUMENTS455Part II ApplicationsWe will now review selected applications in digital waveguide modeling, specificallysingle-reed woodwinds (such as the clarinet), and bowed strings (such as the violin).In these applications, a sustained sound is synthesized by the interaction of the digitalwaveguide with a nonlinear junction causing spontaneous, self-sustaining oscillationin response to an applied mouth pressure or bow velocity, respectively.This type ofnonlinear oscillation forms the basis of the Yamaha VL series of synthesizers ( VL standing for virtual lead ).10.9 SINGLE-REED INSTRUMENTSA simplified model for a single-reed woodwind instrument is shown in Fig.10.16.Figure 10.16 A schematic model for woodwind instruments.If the bore is cylindrical, as in the clarinet, it can be modeled quite simply using abidirectional delay line [Smith, 1986a, Hirschman, 1991].If the bore is conical, suchas in a saxophone, it can still be modeled as a bidirectional delay line, but interfacingto it is slightly more complex, especially at the mouthpiece [Benade, 1988, Gilbertet al., 1990, Smith, 1991, Välimäki and Karjalainen, 1994a, Scavone, 1997] Becausethe main control variable for the instrument is air pressure in the mouth at the reed, itis convenient to choose pressure wave variables
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