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.It is interesting to define ¸i sin 1(ki), always possible for passive junctions since 1 d" ki d" 1, and note that the normalized scattering junction is equivalent to a 2Drotation:(10.50)APPLICATIONS OF DSP TO AUDIO AND ACOUSTICS442where, for conciseness of notation, the time-invariant case is written.While it appears that scattering of normalized waves at a two-port junction requiresfour multiplies and two additions, it is possible to convert this to three multiplies andthree additions using a two-multiply  transformer to power-normalize an ordinaryone-multiply junction.Transformer Normalization.The transformer is a lossless two-port defined by[Fettweis, 1986](10.51)The transformer can be thought of as a device which steps the wave impedance to a newvalue without scattering; instead, the traveling signal power is redistributed among the± ±force and velocity wave variables to satisfy the fundamental relations f = ± Rv(10.25) at the new impedance.An impedance change from Ri  1 on the left to R onithe right is accomplished using(10.52)as can be quickly derived by requiring.The parameter gican be interpreted as the  turns ratio since it is the factor by which force is stepped(and the inverse of the velocity step factor).Figure 10.10 illustrates aThe Three-Multiply Normalized Scattering Junction.three-multiply normalized scattering junction [Smith, 1986b].The one-multiply junc-tion of Fig.10.8 is normalized by a transformer.Since the impedance discontinuityis created locally by the transformer, all wave variables in the delay elements to theleft and right of the overall junction are at the same wave impedance.Thus, usinga" 1.transformers, all waveguides can be normalized to the same impedance, e.g., RiIt is important to notice that gi and 1/gi may have a large dynamic range in practice.For example, if k " [ 1 + , 1  ], the transformer coefficients may become as largei (n 1)as If is the  machine epsilon, i.e., = 2 for typical n-bit two scomplement arithmetic normalized to lie in [ 1, 1), then the dynamic range of thetransformer coefficients is bounded by Thus, while transformer-normalized junctions trade a multiply for an add, they require up to 50% more bits ofdynamic range within the junction adders.PRINCIPLES OF DIGITAL WAVEGUIDE MODELS MUSICAL INSTRUMENTS443Figure 10.10 A three-multiply normalized scattering junction.10.5.3 Junction PassivityIn fixed-point implementations, the round-off error and other nonlinear operationsshould be confined when possible to physically meaningful wave variables.Whenthis is done, it is easy to ensure that signal power is not increased by the nonlinearoperations.In other words, nonlinear operations such as rounding can be made passive.Since signal power is proportional to the square of the wave variables, all we needto do is make sure amplitude is never increased by the nonlinearity.In the case ofrounding, magnitude truncation, sometime called  rounding toward zero, is one wayto achieve passive rounding.However, magnitude truncation can attenuate the signalexcessively in low-precision implementations and in scattering-intensive applicationssuch as the digital waveguide mesh [Van Duyne and Smith, 1993].Another option iserror power feedback in which case the cumulative round-off error power averages tozero over time.A valuable byproduct of passive arithmetic is the suppression of limit cycles andoverflow oscillations.Formally, the signal power of a conceptually infinite-precisionimplementation can be viewed as a Lyapunov function bounding the squared amplitudeof the finite-precision implementation.To formally show that magnitude truncation is sufficient to suppress overflow os-cillations and limit cycles in waveguide networks built using structurally losslessscattering junctions, we can look at the signal power entering and leaving the junction.A junction is passive if the power flowing away from it does not exceed the powerflowing into it.The total power flowing away from the ith junction is bounded by theAPPLICATIONS OF DSP TO AUDIO AND ACOUSTICS444incoming power if(10.53)incoming poweroutgoing powerLet denote the finite-precision version of ’
.Then a sufficient condition for junctionpassivity is(10.54)(10.55)Thus, if the junction computations do not increase either of the output force amplitudes,no signal power is created.An analogous conclusion is reached for velocity scatteringjunctions.Passive Kelly-Lochbaum and One-Multiply Junctions.The Kelly-Lochbaum andone-multiply scattering junctions are structurally lossless [Vaidyanathan, 1993] be-cause they can be computed exactly in terms of only one parameter k (or ± i), and alliquantizations of the parameter within the allowed interval [ 1, 1] (or [0, 2]) correspondto lossless scattering.4 The structural losslessness of the one-multiply junction has beenused to construct a numerically stable, one-multiply, sinusoidal digital oscillator [Smithand Cook, 1992].In the Kelly-Lochbaum and one-multiply scattering junctions, because they arestructurally lossless, we need only double the number of bits at the output of eachmultiplier, and add one bit of extended dynamic range at the output of each two-input adder.The final outgoing waves are thereby exactly computed before they arefinally rounded to the working precision and/or clipped to the maximum representablemagnitude.For the Kelly-Lochbaum scattering junction, given n-bit signal samples and m-bitreflection coefficients, the reflection and transmission multipliers produce n + m andn + m + 1 bits, respectively, and each of the two additions adds one more bit.Thus, theintermediate word length required is n + m + 2 bits, and this must be rounded withoutamplification down to n bits for the final outgoing samples.A similar analysis givesalso that the one-multiply scattering junction needs n + m + 2 bits for the extendedprecision intermediate results before final rounding and/or clipping.Passive Four-Multiply Normalized Junctions.Unlike the structurally lossless cases,the (four-multiply) normalized scattering junction has two parameters, si ki and, and these can  get out of synch in the presence of quantization.PRINCIPLES OF DIGITAL WAVEGUIDE MODELS OF MUSICAL INSTRUMENTS 445c  de-Specifically, let si  denote the quantized value of si, and leti s i cinote the quantized value of ci.Then it is no longer the case in general that + = 1.As a result, the normalized scattering junction is not structurally lossless in the pres-ence of coefficient quantization.A few lines of algebra shows that a passive roundingrule for the normalized junction must depend on the sign of the wave variable beingcomputed, the sign of the coefficient quantization error, and the sign of at least one ofthe two incoming traveling waves.We can assume one of the coefficients is exact forpassivity purposes, so assume = 0 and define , where [x] denotesslargest quantized value less than or equal to x.In this case we have e" 0.Therefore,cand a passive rounding rule which guarantees need only look at the signbits of andPassive Three-Multiply Normalized Junctions.The three-multiply normalizedscattering junction is easier to  passify [ Pobierz caÅ‚ość w formacie PDF ]