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Page 376 ï~~Connections between Feedback Delay Networks and Waveguide Networks for Digital Reverberation Julius 0. Smith Center for Computer Research in Music and Acoustics (CCRMA) Stanford University Email: jos~ccrma.stanford.edu Davide Rocchesso Centro di Sonologia Computazionale Dipartimento di Elettronica e Informatica University degli Studi di Padova Email: email@example.com Abstract The order N feedback delay network (FDN) has been proposed for digital reverberation. Also proposed with similar advantages is the digital waveguide network (DWN). This paper notes that the FDN is isomorphic to a (normalized) waveguide network consisting of one (parallel) scattering junction joining N reflectively terminated branches. This correspondence gives rise to new generalizations in both cases. The feedback delay network (FDN), depicted in Fig. 1, has been proposed for digital reverberation applications (see the companion paper [Rocchesso and Smith, 1994] for references). These structures are characterized by a set of delay lines connected in a feedback loop through a "feedback matrix." a2,1 a2,2 a2,3 a]3,1 a3,2 a3,3 D t eg t s(n) h DW ma i ey t b _ms s$ n [c - Z d t Figure 1: Order 3 Feedback delay network. Digital waveguide networks (DWN) have also been proposed as a starting point for digital reverberator development [Smith, 1985]. Like FDNs, DWNs make it easy to construct high-order lossless systems. Ordinarily, the lossless prototype reverb erat or is judged for the quality of white noise it generates in response to an impulse signal. For smooth reverberation, the white noise should sound uniform in every respect. Subsequent introduction of lowpass filters into the prototype network (e.g., applied to junction pressure) serves to set the desired reverberation time vs. frequency. Since FDNs and DWNs appear to present very different approaches for constructing lossless prototypes, it is natural to ask what connections may exist between them, and whether there may be unique advantages of one over the other. Figure 2 illustrates an N-branch DWN which is structurally equivalent to an N-th order FDN. The waves traveling into the junction are associated with the FDN delay line outputs si(n), and the length of each waveguide is half the length of the corresponding FDN delay line mi (since a traveling wave must traverse the branch twice to complete a round trip from the junction to the termination and back). When mi is odd, we may replace the reflecting termination by a unit-sample delay, or we may define the branch medium such that the speed of propagation is slightly faster in one direction. s1(, 7 S) AN(n+mN) a,(n) SN(n) Figure 2: Waveguide network consisting of a single scattering junction to which N branches are connected. The far end of each branch is terminated with a perfect, non-inverting reflection, indicated by a black dot. As discussed in greater detail in the companion paper, the delay-line inputs (outgoing traveling waves) are computed by multiplying the delayline outputs (incoming traveling waves) by the Nby-N feedback matrix A = [ai,J: N si(n + mi) = E ai,jsj(n) j=1 The above notation coincides with that used in the companion paper. By defining p+ = st(n), Sound SynthesisTechniques 376 ICMC Proceedings 1994
Page 377 ï~~p= = si(n + mi), and A = [a.,j], we obtain the more usual DWN notation p_- =Ap+ where p+ is the vector of incoming traveling-wave samples arriving at the junction at time n, pis the vector of outgoing traveling-wave samples leaving the junction at time n, and A is the scattering matrix associated with the junction. To obtain lossless FDNs, prior work has focused on unitary feedback matrices. A matrix A is said to be unitary if A*A = I, where '.' denotes transposition and complex conjugation. Since every lossless scattering junction provides a lossless FDN matrix, are all of these matrices unitary? The answer is immediately no: Unitary scattering matrices arise only in the case of normalized waves, e.g., pressure waves which are multiplied by the square-root of the wave admittance of the waveguide in which they travel [Smith, 1987]. In such cases, the scattering matrix can be expressed as a Householder reflection A = 2T/1III2-I, where _T = [ /'T,..., v'r, and 1i is the wave admittance in the ith waveguide branch. Unnormalized scattering junctions can be expressed in the form of an "oblique" Householder reflection A = 21pT/ (_T, I_) - I, where 1_T [1,...,1] and rT = [r1,...,FN]. Thus, p+ is reflected about 1 and scaled based on its "shadow" along [. From these forms, we see that all junctions of N physical waveguides require only O(N) computations and thus do not span all lossless scattering matrices without further generalization. What are all lossless scattering matrices? From basic physical principles, a scattering matrix is lossless if and only if the total active complex power is scattering-invariant, i.e., if p+*rp+ = p-*rpA*rA = r (1) where I' is a Hermitian, positive-definite matrix which can be interpreted as a generalized junction admittance. For unitary A, we have I' = I. In the case of N traveling pressure waves scattering at a "parallel" junction, we obtain r = diag(I1,..., PN). Unless all branch admittances are identical, the scattering matrix is never unitary. In general, the Cholesky factorization r' = U*U gives an upper triangular matrix U which converts A to a unitary matrix via similarity transformation: A*rA = r =~ A*U*UA = U*U =. A = I, where A = UAU-1. Hence, the eigenvalues of every lossless scattering matrix lie on the unit circle. When U is diagonal, a physical waveguide interpretation always exists with U = diag([). A generalized waveguide interpretation exists for all U via "power equivalent junctions" [Smith, 19871 in which U acts as an ideal transformer (in the classical network theory sense) on the the vector of all N waveguide variables. It readily follows from similarity to A that A admits N linearly independent eigenvectors. Conversely, assume IAI = 1 for each eigenvalue of A, and that there exists a matrix T of linearly independent eigenvectors of A. Then the matrix T diagonalizes A to give T-'AT = D =Â~, T*A*T-* = D*, where D = diag(A1,...,AN). Multiplying, we obtain T*A*T-*T-lAT = D*D = I = A*T-*T-'A = T-*T-1. Thus, Eq. (1) is satisfied for r = T-*T-l which is Hermitian and positive definite. We may summarize as follows: Theorem: A scattering matrix (FDN feedback matrix) A is lossless if and only if its eigenvalues lie on the unit circle and its eigenvectors are linearly independent. Thus, lossless scattering matrices may be fully parametrized as A = T-1DT, where D is any unit-modulus diagonal matrix, and T is any invertible matrix. It can be quickly verified that all scattering matrices arising from the intersection of N physical waveguides possess one eigenvalue equal to 1 (corresponding to all incoming waves being equal) and N - 1 eigenvalues equal to -1 (corresponding to equal incoming waves on N - I branches, and a large opposite wave on the remaining branch which pulls the junction pressure to zero). Since only a subset of all N-by-N unitary matrices is given by a physical junction of N digital waveguides, (e.g., consider permutation matrices), the FDN point of view yields lossless systems outside the scope of single-junction waveguide networks. On the other hand, since only normalized waveguide junctions exhibit unitary scattering matrices, the DWN approach gives rise to new classes of lossless FDNs. Moreover, by considering more than one scattering junction, the DWN approach suggests a far larger class of lossless network topologies. References [Smith, 1985] J.O. Smith. A New Approach to Digital Reverberation using Closed Waveguide Networks, ICMC-85, Vancouver. CORMA STAN-M-31. [Smith, 1987] J.O. Smith. Music Applications of Digital Waveguides, Report STAN-M-39, CORMA, 1987. (Contains [Smith, ICMC-85J.) [Rocchesso and Smith, 1994] D. Rocchesso and J.O. Smith (companion paper). Circulant Feedback Delay Networks for Sound Synthesis and Processing, ICMC94, Aarhus, Denmark. ICMC Proceedings 1994 377 Sound Synthesis Techniques