Applications of Generalized Scattering

The generalized scattering-matrix transfer function can be used to describe the intersection of conical acoustic tubes having different taper angles [96,117,130,76]. Conical waveguides have a complex ``one-pole'' wave impedance in the continuous-time case [2,5]. As a result, the elements of the scattering matrix for intersecting cones become first-order filters in place of real scattering coefficients. It turns out that the entire scattering junction for adjoined conical sections can be implemented using a single first-order IIR filter, analogous to the one-multiply lattice-filter junction [76], and ``one-filter'' forms exist also for the normalized case, in the presence of losses, and when there is a discontinuity in cross-sectional area (as in piecewise cylindrical acoustic tubes). In the discrete-time case, higher order filters and a guard band are necessary for high accuracy at all audio frequencies.

An interesting property of conical waveguide junctions is that the reflection coefficient corresponding to reflection from a decreasing taper angle (e.g., from a cylinder to a converging cone), is unstable. That is, the impulse response of such a junction contains a growing exponential component. The growing exponentials are always canceled by reflections (e.g., from the conical tip), but special care must be taken when using unstable one-pole filters in a practical model [131], or else FIR filters may be used to implement the truncated growing exponential impulse response including the canceling reflection from the other end of the converging conical segment [85]. Another promising approach (so far not implemented to our knowledge), is to choose normalized waves instead of pressure waves for the simulation.

Results similar to the conical junction are obtained for models of toneholes in woodwind bores [119,115,85]. In this application, the tonehole can be regarded as a three-port junction in which two of the ports connect to the bore on either side of the tonehole, and the third port connects to the tonehole itself. While the wave impedance of the bore is real for a cylindrical tube such as the clarinet, the tonehole presents a complex lumped impedance at the junction. (Alternatively, the tonehole can be modeled as an extremely short waveguide terminated by a simple reflection when closed and a complex radiation reflectance when open.) As in the case of intersecting cones, the reflection and transmission coefficients at the tonehole junction become one-pole filters in the continuous-time case, and third-order filters perform very well in the discrete-time case [85,87]. In addition, it is possible to obtain low-order one-filter forms which handle all filtering in the tonehole junction [107]. However, unlike the conical junction case, the junction filter is always stable.

A third application similar to piecewise conical tube modeling and tonehole modeling is the modeling of coupled strings intersecting at a point driving a lumped impedance (e.g., a ``bridge'' model for a piano). Again, a ``one-filter junction'' is available [98], analogous to the one-multiply lattice-filter section. However, in this case, the one filter, while shared by all the strings, must contain all important body resonances (e.g., of a piano soundboard) ``seen'' through the bridge by the strings. A bridge filter is therefore not naturally a low-order filter. However, low-order approximations have been found to sound quite good, and it can be argued that strong coupling at isolated frequencies is musically undesirable. In other words, a smoother bridge reflectance, having a frequency response which follows only the upper envelope of the measured magnitude response, may be preferable.

The generalized scattering-matrix transfer function can also be used to simulate wave scattering by an object among discrete directions of arrival. If we consider $N$ waveguides to simulate wave propagation along a set of $N$ directions, the element $a_{i,j}$ of the scattering matrix can be interpreted as the coefficient applied to the signal coming in from direction $j$ and being transmitted out along direction $i$. The scattering coefficients are highly dependent on frequency, and this can be handled using frequency-dependent elements of the scattering matrix. For example, wavelengths which are small compared with the object tend to reflect, or scatter according to the texture of the object surface, while wavelengths which are larger than the object tend to diffract around it and send scattering components in all directions. Consider the simpler case of incident wave energy scattering from a textured wall. In this case, wavelengths larger than the ``grain size'' of the texture tend to reflect specularly, while smaller wavelengths are scattered into a range of new directions according to well known formulas [52]. As a rough approximation for this case, we might use a single high-pass filter for all of the $N-1$ transmitted waves, and the complementary low-pass filter for the reflected wave, as in the ``cross-over'' scheme proposed by Regalia et al. [71]. In such a scheme, low frequencies are reflected specularly while high frequencies are diffusely reflected along $N-1$ directions.