Digital Waveguide Networks as Multidimensional Wave Digital Filters

1. INTRODUCTORY REMARKSMultidimensional wave digital ﬁlters(MDWDFs) [1, 2] have re-cently been applied toward the numerical simulation of distributedsystems. The basic procedure for deriving an algorithm is similarto that which was originally developed for deriving wave digitalﬁlter[3, 4] structures from lumped analog networks, though in thatcase, the application was to ﬁlterdesign, and not explicitly to sim-ulation. In the lumped case, one begins from a given analog net-work structure, composed typically of RLCelements (and possiblymore exotic devices such as transformers, gyrators, etc.). One thenappl;ies a continuous-to-discretespectral mapping to each reactiveelement, and, after the introduction of wave variables [3], ends upwith a recursible ﬁlterstructure. The spectralmapping (aparticulartype of bilinear transform, which corresponds, in the discrete-timedomain, to the use of the trapezoid rule of numerical integration)is chosen so that the energetic properties of the analog network aremirrored by the discrete-timestructure. It has the form(1)where is the continuous-timefrequency variable and is theunit delay in the frequency domain. is the sampling period. In-deed, the digital ﬁltertopology is the same as that of the originalanalog ﬁlter, and structures so derived possess many desirable nu-merical properties (including guaranteed stability even in a ﬁniteword-lengthcomputer implementation) [4]. In effect, such a ﬁl-ter is an explicit numerical integration scheme for the system ofordinary differential equations (ODEs) which describes the timeevolution of the state of the analog network.The procedure is very similar in multiple dimensions (MD). For simulation purposes, however, a suitablemultidimensionalKir-choff circuit (MDKC) representation [2] of a given system of par-tial differential equations (PDEs) must be ﬁrstfound; discretiza-tion is then performed in the same way as in the lumped case,namely through generalized MD spectral transformations whichpreserve passivity in a MD sense [5], and the introduction of wavevariables. A bare-bonessummary of the basics of MDWD ele-ments and connections is given in Section 2; for a thorough treat-ment, we refer the reader to [1, 2, 5]. The resulting algorithms are similar to those of digital waveg-uide networks (DWNs) [6], in that they are composed of scatteringjunctions (called adaptors in the WDF context, but functionallyidentical) and shifting operations; all operations are performed onwave variables in either case. Indeed, DWNs can be directly ap-plied to the same types of PDEs, namely those of hyperbolic type[7] (for which propagation speed is bounded). The two types ofstructures, are, however, not the same, though they both exhibit thesame good numerical properties (a direct result of their underlyingdiscrete passivity). In particular, DWNs can be shown to be equiv-alent to simple ﬁnitedifference schemes [8], but MDWD networkscan be identiﬁedwith multi-stepschemes of higher degree, andtheir respective numerical dispersion properties are distinct [9]. Inaddition, there has not as yet been a convenient MD representationfor a DWN as there is for a MDWDF (i.e. a MD circuit), and thusno simple way of arriving at a DWN from a given system of PDEs.It is shown in Section 3 that DWNs and MDWD networks canbe uniﬁedin a straightforward way. In fact, a DWN may be de-rived from an MD circuit representation in a way nearly identicalto the wave digital approach—the differencesare in the form of thecircuit representation and the type of spectral mapping used. Wewill examine, in particular, the (1+1)D transmission line system inthe presence of distributed spatially-varying material parameters(inductance and capacitance). The same result may be applied inorder to generate DWNs from the existing MD circuit representa-tions for various more complex systems in higher dimensions. Inparticular, it is possible to arrive at DWNs which simulate varioussystems of interest in musical physical modelling applications, in-cluding beam [10, 9] plate and shell dynamics problems [9], aswell as full (3+1)D elastic solid vibration [10, 9]. 2. REVIEW OF MULTIDIMENSIONAL W AVE DIGITALFILTERSWe brieﬂy outline here the MDWD discretization procedure, aspresented initially in [1, 2]; unfortunately, space limitations pre-clude a full review of lumped WDF principles (a prerequisite forthe material in this section). We refer the reader to [3, 4] for athorough introduction to WDFs.An D dynamic distributed problem is deﬁnedwith re-spect to the variables . Here, the ,are the spatial coordinates, and is physical time. (Forsimplicity, we will leave out any discussion of boundary condi-tions in this paper.) Generally speaking, the wave digital approachto numerical integration entails modelling a system of PDEs as anetwork of interconnected circuit elements which are themselvesdistributed objects. If the model system is passive, it is usuallypossible to represent it by a distributed network whose elementsare all individually passive; sources (driving functions), if they arepresent, are excepted in this respect. Passivity-preservingspectralmappings or integration rules can then be applied to discretize theelements individually; when reconnected, these elements form adiscrete passive network which simulates the model system.DAFX-1