Bell Models

The flaring bell of a horn cannot be accurately modeled as a sparse digital waveguide, because traveling pressure waves only propagate without reflection in conical bores (which include cylindrical bores as a special case) [360].^10.24 Digital waveguides are ``sparse'' (free of internal scattering) only when there are long sections at a constant wave impedance.

The most cost-effective bell filters (and, more generally, ``flare filters'') to date appears to be the use of truncated IIR (TIIR) digital filters [544]. These filters use an unstable pole to produce exponentially rising components in the impulse response, but the response is cut off after a finite time, as is needed in the case of a bell impulse response. By fitting a piecewise polynomial/exponential approximation to the reflection impulse response of the trumpet bell, very good approximations can be had for the computational equivalent of approximately a 10th order IIR filter (but using more memory in the form of a delay line, which costs very little computation).

In more detail, the most efficient computational model for flaring bells in brass instruments seems to be one that consists of one or more sections having an impulse response given by the sum of a growing exponential and a constant, i.e.,

$$y(n) = \left\{\begin{array}{ll} a e^{c n} + b, & n=0,1,2,\ldots,N-1 \\ [5pt] 0, & \mbox{otherwise}. \\ \end{array} \right.$$

The truncated constant $b$ can also be generated using a one-pole TIIR filter, with its pole set to $z=1$ . The remaining reflection impulse response has a decaying trend, and can therefore be modeled accurately using one of many conventional filter design techniques. In [532,533], the Steiglitz-McBride IIR filter design algorithm [289] yielded good results from pulse-reflectometry data [429], as shown in Fig.9.60.

Figure 9.59: Example of a TIIR filter for generating a growing exponential or constant segment (from [532]).

Figure 9.60: Impulse-response and driving-point-impedance fit for the trumpet bell using two offset exponentials and two biquads designed as a 4th-order IIR filter using the Steiglitz-McBride algorithm (from [532]). The dot-dashed lines show the model segment boundaries.

The C++ class in the Synthesis Tool Kit (STK) implementing a basic brass synthesis model is called Brass.cpp.