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Physical Room Modeling with FDNs

A flat amplitude response at low frequencies, while desirable in several practical situations, is not found in actual rooms. Therefore, if the goal is to model the reverberation of a physical room, the way indicated by Theorem 2 is not appropriate. Somewhat happily, the FDN can be the kernel of a model of rectangular room, and its parameters can be interpreted in a physical and geometrical framework. In this section, we give only a sketch of this framework, since the details of the underlying metaphor are beyond the scope of this paper, and can be found in [23].

Consider a lossless shoe-box shaped room, having length $l_x$, depth $l_y$, and height $l_z$. For such a room, it is possible to compute analytically the frequencies of the normal modes [17] as

f_{n_x,n_y,n_z} = \frac{c}{2}
\end{displaymath} (35)

where $n_x,n_y,n_z=0,1,2,\dots$, and $c$ is the speed of sound in air. Each normal mode is associated with a direction in space, whose cosines, made by the wave propagation with respect to the $x$, $y$, and $z$ axes, are
\frac{v_x}{v}=\frac{n_x c}{2 f l_x}, \qquad
\frac{v_y}{v}=\frac{n_y c}{2 f l_y}, \qquad
\frac{v_z}{v}=\frac{n_z c}{2 f l_z},
\end{displaymath} (36)

where $v$ is the magnitude of the (vector) spatial frequency, and the subscripts in $f_{n_x,n_y,n_z}$ have been dropped for conciseness.

The triple ${\bf n}=(n_x, n_y, n_z)$ completely characterizes a normal mode. All the triples which are multiples are associated with a harmonic series of frequencies and with the same direction in space. This suggests that any harmonic series of normal mode frequencies can be obtained by means of a linear resonator (in other words, a comb filter) whose length in seconds is set to $d_0 = 1/f_0$, where $f_0$ is the fundamental frequency of the harmonic series. Therefore, we can decompose the modal distribution of the response of an actual room into harmonic subsets (a harmonic of $f_{n_x,n_y,n_z}$ is obtained by multiplying $n_x$, $n_y$, and $n_z$ by the same integer). Sorting these harmonic subsets according to their fundamental frequencies and taking the reciprocals of the N lowest fundamental frequencies yields a parallel comb filter representation of the room (i.e., an FDN with diagonal feedback matrix), so that the FDN reproduces the lowest eigenfrequencies exactly. This procedure was already outlined in [8] as a mean of identifying the parallel comb-filter parameters from a measured impulse response.

We can elaborate the representation further by interpreting the quantity $d_0$ as the time taken by a plane wavefront to travel a certain distance along the direction (36) in space. In fact, a normal mode and all its harmonically related multiples can be thought of as a plane wave bouncing back and forth in the closed environment [17]. For a finite medium, in order to support such an infinite plane wave, the planar fronts have to be bent at the walls such that they form a constant-area closed surface. It can be verified that the time $d_0$ is the time interval between two successive collision of plane wavefronts.

Once established that, in an idealized rectangular room, each harmonic subset of normal modes can be represented by a linear resonator oriented along a given direction in space, we can introduce other ``second-order'' effects into the basic model.

Let us consider an octant in space. Taking the first $N$ fundamental frequencies in the harmonic-subset decomposition of the normal modes corresponds to sampling in space along $N$ directions. An object in any point of the space will provide scattering among the $N$ directions. The walls themselves, when they are not ideally smooth, scatter the waves among different directions. We can think of lumping all these diffusion effects and representing them in the non-diagonal elements of the scattering matrix of a FDN. With some approximation, it was also shown in [23] that an isotropic object in a non-diffusive rectangular room can be represented by a circulant matrix, provided that the spatial sampling is almost uniform and the proper ordering of directions is chosen.

The geometric interpretation allows one to properly excite the modes according to the position of the sound source, by just replacing each coefficient in the vector $\bf b$ by a suitable cascade of FIR comb filters [23]. The position where we listen to the sound is related to the $\bf c$ coefficients in a similar way. It is also quite easy to take into account the radiation pattern of the source and the directivity of the pick-up. Perhaps more importantly, the absorption coefficients of the walls can be made direction-dependent, as they are found in reality, as they affect the different ``linear resonators'' differently.

In the model at hand, the matrix element $a_{i,j}$ scales the signal transmission from mode $j$ to mode $i$. The diagonal of the feedback matrix determines the strength of the ``standing waves'' set up along each pattern. Equivalently, we can think of a DWN modeling the parallel junction of $N$ acoustic tubes, where each tube gives rise to a harmonic subset of normal modes.

The physical modeling viewpoint is limited by the fact that only $N$ ``standing-wave paths'' in the room are being simulated, and all non-specular reflections are being forced to enter some subset of the supported ray paths.

In the model, the diffusivity of the whole reverb is lumped in the properties of the scattering matrix. This is a dramatic simplification, but it allows better control of diffusivity in isolation from other room parameters.

The geometrical interpretation is useful for computing the lengths of the delay lines according to the dimensions of a particular room, since each wavefront path corresponds to a normal mode. In previous work on artificial reverberation [25,16], the choice of the delay-line lengths in the all-pass and comb-filter sections is a primary issue. Typically, the choice is guided by heuristic rules or number-theoretic criteria, and a lot of trial and error is often necessary to obtain good values.

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``Circulant and Elliptic Feedback Delay Networks for Artificial Reverberation'', by Davide Rocchesso and Julius O. Smith III, preprint of version in IEEE Transactions on Speech and Audio, vol. 5, no. 1, pp. 51-60, Jan. 1996.

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Copyright © 2005-03-10 by Davide Rocchesso and Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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