A SIPLIFIED DESIGN OF ULTIDIENSIONAL TRANSFER FUNCTION ODELS Stefan Petrausch, Rudof Rabenstein utimedia Communications and Signa Procesg, University of Erangen-Nuremberg, Cauerstr. 7, 958 Erangen, GERANY Te: +9 93 85 89; fax: +9 93 85 889 e-mai: stepe,rabe @LNT.de ABSTRACT utidimensiona transfer functions are an effective description of physica systems with time and space as independent variabes. They aow to construct discrete modes which are suitabe for computer impementation. The derivation of mutidimensiona transfer functions is straightforward for systems with sef-adjoint spatia operators. It is based on the appication of suitabe functiona transformations for the time and space variabes. For the more genera case of non sef-adjoint operators the derivation is more invoved. ere two different types of eigenvaue probems have to be considered, which ead to a suitabe spatia transformation and its inverse. In this contribution, the reations between these two eigenvaue probems are expored. By deriving some principa properties, a simpified design procedure for the resuting mutidimensiona transfer function modes is given. INTRODUCTION The description of mutidimensiona systems in the spectra domain is a topic that has been under investigation for quite some time. The current state of the art and appications for rea-time sound synthesis are summarized in []. A brief synopsis is given here. utidimensiona physica systems with time and space as independent variabes are frequenty described by partia differentia equations. Such a characterization is derived from the fundamenta aws of science and may be very detaied if required. any software packages exist for the numerica soution of partia differentia equations. Unfortunatey, their main purpose is restricted to numerica anaysis ug work stations or persona computers. They are of itte use for rea-time appications, hardware-in-the-oop, or for impementation on embedded systems. An aternative approach reies on a spectra description of mutidimensiona systems. Simiar to one-dimensiona systems, such a description can be derived ug functiona transformations for the time and space variabes. For the time variabe, the Lapace transformation turns the time derivatives into mutipications with the tempora frequency variabe and propery considers the initia conditions. For the space variabe and the boundary conditions, a spatia transformation can be derived ug resuts from the Sturm- Liouvie theory for boundary vaue probems. This method is straightforward for probems with sefadjoint spatia operators. For the more genera case of non sef-adjoint operators, a carefu anaysis of the eigenvaue probems associated with the spatia operator of the partia differentia equations and its adjoint operator is required. In genera terms, the eigenvaues and eigenfunctions of the adjoint operator determine the transformation from the space variabes to the spatia frequency variabes (aso caed wave numbers). The eigenvaues and eigenfunctions of the origina operator determine the inverse transformation. By virtue of this spatia transformation, a transfer function mode of a mutidimensiona system may be derived. It has been shown that such transfer function descriptions provide an exceent starting point for the derivation of discrete time modes suitabe for computer impementation. Detais of the theory, appications to various probems, and impementation exampes are given in []. This contribution presents new resuts on the eigenvaue probems for the spatia operator and its adjoint. These resuts are based on an in-depth investigation of the properties of genera operators in functiona spaces [3]. By estabishing reations between the soutions of both types of eigenvaue probems, a simpified representation of the spatia transformation is possibe. Some parts of the genera procedure which are aready we documented [, ] have been incuded in a rather concise fashion for the sake of competeness. Section introduces a generic form of a partia differentia equation. Section 3 briefy shows the appication of the Lapace transformation with respect to time. Some usefu properties of scaar products are compied in section. Section 5 investigates the eigenvaue probems of the spatia operators and introduces the spatia transformation. The resuting transfer function modes are presented in section 6. Finay the simpified design method is compied in section 7 and demonstrated by the exampe of a dispersive string in section 8. Section 9 concudes this paper. PROBLE DESCRIPTION This section introduces the notation for the cass of probems under consideration. The anaysis of physica systems which depend on time and space eads to a set of partia differentia equations (PDEs) aong with initia and boundary conditions.. Partia Differentia Equation The anaysis of a physica system according to the fundamenta aws of science resuts in a number of differentia equations for various dependent variabes. Arranging these equations in a unified notation eads to a vector PDE of the
form L + C t y(x, t) = f e(x, t). () ere, t and x are the independent variabes for time and space, respectivey. The space coordinate is a vector with one, two, or three components and is defined on a spatia region with boundary. y(x, t) denotes the vector of dependent variabes and f e is the vector of excitation signas. Depending on the type of probem, C is a mass or capacitance matrix. L is an operator of spatia derivatives of the form L = A + I. () A eements in () and () are matrices of size N N, respectivey vectors of size N. I denotes the identity matrix. The first order spatia derivative is defined according to the spatia dimensions of y(x, t).. Initia Conditions The initia conditions describe the initia state of the system at t = and have to be defined in the compete spatia region y(x, ) = y i(x), x. (3).3 Boundary Conditions The boundary conditions are given as a set of r N inear equations, acting on y(x, t) for x. By adding N r zero-ines one can aways achieve a N N system of equations f b y(x, t) = φ(x, t), x. () f b is a matrix of size N N with rank r, f b denotes the hermitian matrix, and φ(x, t) is a vector of boundary excitations. 3 LAPLACE TRANSFORATION A first step towards a mutidimensiona transfer function mode is the appication of the Lapace transformation. It removes the tempora derivative from () and turns the initiaboundary vaue probem from section into a boundary vaue probem (Capita etters for time-dependent functions denote Lapace transformation) [L + sc] Y(x, s) = F e(x, s) + Cy i(x), x fb Y(x, s) = Φ(x, s), x. (5) SCALAR PRODUCT AND ADJOINT OPERATOR The second step towards the mutidimensiona transfer function mode requires a transformation which removes the spatia derivatives and considers the boundary conditions. Unfortunatey, there is no unique transformation for a kinds of boundary conditions and a kinds of spatia domains. Instead, it is necessary to construct such a transformation for the specia probem at hand. Before presenting the design process of the spatia transformation, some properties of scaar products and adjoint operators are discussed [ 6].. Scaar Product A scaar product is a mapping of two eements u, v from a vector space over the fied (usuay or ) onto u, v. The foowing properties of scaar products are required in the seque (a denotes the conjugate compex vaue): a u, v = au, v = u, a v a, (6) Cu, v = u, C v, (7) u + u, v = u, v + u, v. (8) ere, the scaar product is defined by integration over the spatia domain u, v = v u dx. (9) Aso the resut of an integration over the boundary can be expressed by scaar products u, v + u, v = as can be shown through integration by parts. v u dx, (). Adjoint Operator An operator L is caed adjoint operator to L, if the so caed Green s identity hods Lu, v u, L v =, () where u and v satisfy the boundary conditions resp. adjoint boundary conditions 5 EIGENALUE PROBLES fb u =, () f b v =. (3) The spatia transformation shoud decompose any eement u of the vector space into a inear combination of the eigenfunctions K(x, β) of the operator L, so that the appication of L simpifies to a inear scaing of these eigenfunctions. To this end, the eigenvaue probems of the operator L and its adjoint have to be anayzed. 5. Eigenvaue Probem The eigenfunctions of the operator L (see ()) satisfy the eigenvaue probem LK(x, β) = βck(x, β), x fb K(x, β) =, x. () It is soved by N inear independent vectors, which can be arranged as an invertibe matrix K (x, β). The eigenvectors K(x, β) satisfy aso the boundary condition of the eigenvaue probem () and can be represented as the product of K (x, β) with the constant vector c K (x, β) = K (x, β) c. (5) 5. Adjoint Eigenvaue Probem The eigenfunctions of the adjoint operator have to satisfy L β) = βc K(x, β), x f b K(x, β) (6) =, x. For finite regions, the eigenvaues β and β can ony adopt discrete vaues, denoted by β µ and β ν, where µ, ν. Furthermore, ony simpe eigenvaues are assumed. For a treatment of mutipe eigenvaues, see [3].
5.3 Usefu Properties The foowing properties wi be shown now (dependency on x and β is omitted in the seque): L = A I (7) fb = I f b (8) β = β (9) K = K for a constant vector c with c () c c =. () 5.3. Adjoint operator and boundary condition Ug the scaar product properties (7), (8), () and the definition of L in (), LK, K can be expressed as LK, K = (A + I ) K, K = AK, K + K, K = K, A K K, K + = K, A I K + K K dx K K dx.() Since K and K satisfy the boundary conditions of the eigenvaue probem () resp. adjoint eigenvaue probem (6), the boundary term in equation () can be expressed as K K K dx = Then () becomes LK, K = K, K f b K = A I dx + K + I f b K dx. (I f b ) K K dx. Obviousy the condition for the adjoint operator () hods for ce K K dx = LK, K = K, L K (3) L = L = A I () fb = I f b (5) K f b K dx + f b K K dx =. (6) 5.3. Adjoint eigenvaues Inserting the eigenvaue probem () and the adjoint eigenvaue probem (6) into the condition for adjoint operators (3) gives From (6) and (7) foows βck, K = K, βc K. (7) β = β. (8) 5.3.3 Adjoint eigenvectors According to (5), the eigenvaues K can be represented as the product of an invertibe matrix K and a constant vector c, where K by itsef soves the differentia equation of the eigenvaue probem () (A + I ) K = βck. (9) Left and right mutipication of (9) with K K A + K ( K ) K = βk C, gives Ug the differentiation rue K = K ( K) K resuts in K A A K = βk C, I K = β C K. (3) Right mutipication of (3) with an arbitrary constant vector c eads to a differentia equation of the adjoint eigenvaue probem A I K = βc K with the adjoint eigenvector K x, β = K (x, β) c. (3) Furthermore, with (5) and (5) one can write the boundary condition of the eigenvaue probem () as I fb K c =, (3) Some rearrangements yied the property () for c: K K c = f b K c c c = K f b K c f b = c c = c K K c c c = (33) This competes the proof of the properties (7 ). Their appication does not ony simpify the foowing definitions of the spatia transformation and its inverse. It aso aows to give a concise design procedure of the resuting transfer function modes. 5. Biorthogonaity The biorthogonaity between different eigenfunctions can be easiy proven by the appication of the extended Green s identity. The concise notations K µ = K(x, β µ) and K ν = K(x, β ν) denote eigenfunctions of the operator L resp. L with different eigenvaues β µ and β ν. From equations (7), (8) foows with (7) (β µ β ν) CK µ, K ν =. (3) Equation (3) is satisfied either for µ = ν or by CK µ, K ν = for µ = ν. Thus the eigenfunctions K µ and K ν are biorthogona with respect to the weighting matrix C CK µ, K ν = K N CK dx = µ for µ = ν, for µ = ν
with N µ = CK µ, K µ = c K (x, β µ)ck (x, β µ) dx c. (35) 5.5 Sturm-Liouvie Transformation Ug the eigenfunctions of the operator L and its adjoint operator L, the spatia transformation and its inverse can be defined. The forward-transformation is given by Ȳ (µ, s) = Y (x, s) = CY(x, s), K(x, β µ) = K (x, β µ)cy(x, s) dx = c K (x, β µ)cy(x, s) dx. (36) Due to the biorthogonaity of the eigenfunctions, the inverse transformation simpifies to a sum over a possibe µ: Ȳ(µ, s) = µ N µ Ȳ (µ, s)k (x, β µ)c. (37) This type of transformation is aso caed a Sturm-Liouvie (SL) transformation, ce the eigenvaue probems are socaed Sturm-Liouvie probems (see []). 5.6 Differentiation Theorem As aready mentioned at the beginning of section 5, the Sturm-Liouvie transformation has beneficia properties concerning the differentia operator L. The scaar product of LY with the adjoint eigenvector can be reformuated anaogousy to the procedure in equation (6) and (7). ere the term fb Y does not vanish, but it can be repaced by the Lapace transform Φ(x, s) of the boundary vaue according to (5) LY, K = Y, L K + with = Y, βc K + = β CY, K + K Y dx f b K Y + K fb Y dx K Φ(x, s) dx = β Ȳ (µ, s) Φ(µ, s), (38) Φ(µ, s) = K (x, µ)φ(x, s) dx. (39) The resut is a simpe reation between the transforms of Y and LY, which ey resembes the form of the differentiation theorem of the Lapace transformation. 6 TRANSFER FUNCTION DESCRIPTION The fina step towards a mutidimensiona transfer function is now we prepared. Appication of the scaar product with the adjoint eigenfunctions K(x, µ) to both sides of the differentia equation (5) and Sturm-Liouvie transformation (36) with the differentiation theorem (38) resuts in LY, K + s CY, K = F e, K + Cy i, K β µ Ȳ (µ, s) + sȳ (µ, s) = F e(µ, s) + ȳ i(µ) + Φ(µ, s) () with the abbreviation F e(µ, s) = F e, K. Reordering of () gives the desired transfer function mode Ȳ (µ, s) = s + β µ Φ(µ, s) + F e(µ, s) + ȳ i(µ), () which can be discretized and impemented with methods described e.g. in []. 7 SUARY OF TE SIPLIFIED DESIGN PROCEDURE Expoiting the properties derived in section 5.3 aows a very concise description of the design process. Since the eigenvaues and eigenvectors of the adjoint probem are so ey reated to the origina PDE, the transfer function mode can be designed directy from the origina probem. Thus the spectra description of non sef-adjoint probems becomes as simpe as for sef-adjoint probems. The design process of the transfer function mode is as foows:. Set up the eigenvaue probem () for the operator L from () and sove it for the eigenvaues β µ and the matrix K from (5).. Invert K and choose c and c according to (). 3. Use (5) and (3) to obtain K and K from K, K c, and c.. Obtain Φ(µ, s), Fe(µ, s), ȳ i(µ) by SL transformation., 5. Compute Ȳ (µ, s) from a discrete impementation of the mutidimensiona transfer function. 6. Compute the soution by inverse SL transformation. The benefits of this simpified design procedure are iustrated in the next section by a rea word probem. 8 SIULATION OF DISPERSIE AND DAPED STRINGS Assume the foowing partia differentia equation (PDE) as a eaborated physica description of dispersive string vibration with homogeneous initia conditions (ICs) and fixed boundaries as boundary conditions (BCs) PDE : ρa ÿ T S y + y + d ẏ d 3ẏ = f e(x, t), IC : y(x, t) t= =, ẏ(x, t) t= =, BC : y() =, y() =, y () =, y () =. 8. ector PDE A probem description in the form of equation () can be easiy achieved by introducing the vector of output variabes y(x, t) = ẏ y y T Sy y A = T S d C =, f e = The system matrices resut in ρa d 3 T. (), B = I, f e(x, t).
The boundary conditions are aso represented in vector notation, where according to section.3 a square boundary operator f b is achieved by the insertion of zero-ines =f b y(x b, t) = 8. Lapace Transformation =φ(x b,t), x b ;. (3) The first step towards a transfer function mode is the Lapace-transformation (see section 3). It does not introduce additiona terms due to the homogeneous boundary conditions and resuts in a boundary vaue probem according to equation (5) which is not expicity shown here. 8.3 Eigenvaue Probem The further procedure to achieve a transfer function mode foows the summary from section 7. Thereby the greatest effort usuay has to be spent on the soution of the eigenvaue probem (). For this case it takes the form T S d = β + I x K = ρa d 3 K, () where K = K(x, β) has to fufi the homogeneous boundary conditions K(x b, β) = for x b,. (5) As in this specia case a system matrices are constants with respect to space, equation () is obviousy satisfied by the transition matrix = with K (x, β) = e (βc := := A)x β e x β e x e x e x e 3e x x e x 3e x β e x β e x e x e x e e x x e x e x T S βd 3 (T S βd 3) β (βρa d ), T S βd 3 + (T S βd 3) β (βρa d ), 3 := T S, := T S. The fufiment of the boundary conditions (5) determines the scaing vector c (see equation(5) and restricts the eigenvaues β to discrete vaues β µ, µ. As resut we achieve an anaytic expression for the eigenvectors β µ K(x, µ) = (6) T S + and the corresponding eigenfrequencies β µ = σ µ + jω µ, (7) d σ µ = ρa + d3, (8) ρa EI ω µ = sign(µ) b T S + σ µ. (9) ρa ρa These eigenfrequencies are the harmonics of the physica system. The constant σ µ (8) is the damping and the constant ω µ (9) is the anguar frequency of the µ-th harmonic. 8. Adjoint Eigenvaue Probem Thanks to the simpified design method proposed in this paper, it is no onger necessary to sove the adjoint eigenvaue probem expicity, i.e. to repeat the procedure in section 8.3 with the matrices from (6). Instead we appy step and 3 from the summary in section 7, which roughy havens the effort of the compete design procedure. In detai we achieve the adjoint eigenvaues β µ = βµ (see equation (8)) and the adjoint eigenvectors K(x, µ) = β µρa d x βµd 3. (5) 8.5 Transfer Function ode and Anaytic Resuts The ast three steps of the isting in section 7 are straightforward and ead to an easy to impement structure of a set of weighted second order recursive systems. Detais can be found in [], there this exampe is discussed with an expicit evauation of the adjoint eigenvaue probem. owever, for the sake of competeness and to demonstrate the capabiity of the FT, the transfer function mode and the anaytic soution of the probem are given here in ed form. The transfer function mode compies with equation (). Φ(µ, s) and ȳ i(µ) are zero due to the homogeneous boundary and initia conditions, so that ony the externa excitation term F e(µ, s) remains. In time-frequency domain the transfer function mode is written to Ȳ (µ, s) = s + β µ xe Fe(s), (5)
acements where the eigenfrequencies β µ are given in (7) - (9) and F e(s) is the Lapace transformed of the excitation function f e(t). Appication of the inverse Lapace transformation and the inverse Sturm-Liouvie transformation eads to an anaytic soution in the space and time domain y(x, t) = µ= j ω µ ρa β T S + xe e β µt f e(t). (5) Figure depicts some simuation resuts, which are achieved from a discrete impementation of equation (5). One can see severa snapshots of the defection y(x N, t) as a function of the normaized space coordinate. The snapshots are taken a few miiseconds after the strong dispersive string was excited at the center (x N =.5) by a band-imited impuse. y(xn,.ms) y(xn,.8ms) y(xn,.6ms) y(xn,.ms).........6.6.6.6.8.8.8.8 9 Concusions In this paper, the soution of initia boundary vaue probems with the hep of mutidimensiona transfer function modes is covered. Based on fundamenta theory a new simpified design method for the genera case of probems with non sef-adjoint operators is derived. First a description is given of how to formuate the initia boundary vaue probem in form of a partia differentia equation with appropriate initia and boundary conditions. Then the process to achieve a mutidimensiona transfer function mode is demonstrated in detai, motivated by severa proofs of fundamenta properties of eigenvaue and adjoint eigenvaue probems. As a novety to prior work [, ] it is shown how to construct the forward transformation without soving the adjoint eigenvaue probem expicity. A unknowns in the adjoint probem, the operator itsef, the eigenvaues, and even the eigenfunctions, are shown to be directy reated to the origina probem. As the soution of the eigenvaue probem consumes the major part of work during the design process, it is obvious that this new design method simpifies the appication of mutidimensiona transfer function modes for non sef-adjoint operators significanty. Finay, in the ast section of this paper, this simpification has been demonstrated by the rea word probem of dispersive and damped string vibration. References [] L. Trautmann and R.Rabenstein, Digita Sound Synthesis by Physica odeing Ug the Functiona Transformation ethod. New York: Kuwer Academic/Penum Pubishers, 3. [] R. Rabenstein and L. Trautmann, Spectra description of mutidimensiona systems with the Sturm-Liouvie transformation. In Proc. Int. Workshop on Spectra ethods and utirate Signa Procesg (SSP ), Pua, Croatia, June. EURASIP. [3]. Dymkou, R. Rabenstein and P. Steffen, Appication of Operator Theory to Discrete Simuation of Continuous Systems, 6th Internationa Symposium on athematica Theory of Networks and Systems (TNS), Leuven, Begium, Juy,. [] S. assani, athematica Physics. New York: Springer- erag, 999. [5] R.. Churchi, Operationa athematics. New York: cgraw-i, 3. edition, 999. [6] T.W. Körner, Fourier Anaysis. Cambridge: Cambridge University Press, 988. n y in mm normaized space coordinate x N = x Figure : Snapshots of the defection y(x N, t) at severa points in time after the string was excited by a band-imited impuse. The dispersion due to the fourth-order spatia derivative in the PDE causes the high frequencies to trave faster than the ow ones.