Efficient time-domain models of wind instruments generally involve at least
three components: the mouthpiece and (lip-)reed interface, the main bore
plus any valve segments or tone holes, and the bell. Mouthpiece, lip and
reed models pose challenging problems which continue to be addressed.
A uniform cylindrical or conical bore is easily modeled in the digital
waveguide formulation [7] using a single delay line to
represent the associated round-trip propagation delay. (Associated losses
may be lumped elsewhere, such as at the mouthpiece.) The bell of the
instrument, assuming linearity, may be characterized by its reflection and
transmission impulse responses, and is therefore efficiently modeled as a
``lumped'' horn reflectance and transmittance. Other non-uniform tubular
segments are potentially well modeled using the well known piecewise
conical bore formulation, but its time-domain application has so far been
strongly limited due to numerical problems. In this paper we present
practical and efficient methods for modeling and implementing piecewise
conical bores and horn reflectances.^{2}

A straightforward but computationally expensive discrete-time model of the bell is to use its impulse response as a convolution filter. This yields a trivially designed finite-impulse-response (FIR) filter model for the reflectance. Alternatively, an infinite-impulse-response (IIR) digital filter can be designed to approximate the bell reflection response. In general, IIR filters can approximate a given impulse response with much less computation because they are recursive. However, prevalent phase-sensitive IIR filter-design methods perform poorly when applied to a measured bell reflectance. This is due mainly to the long, slowly rising, quasi-exponential portion of the time-domain response, arising from the smoothly flaring bore profile that is characteristic of musical horns. As a result, there is a need for more effective digital filter design techniques in this context.

Inspections of horn reflectances in the time domain suggest that a natural
modeling approach might consist of dividing the response into at least two
sections: an initial growing exponential, followed by a more oscillatory
``tail.'' The tail can be faithfully modeled using more conventional
filter-design methods. The most efficient way to model a growing
exponential is by means of an *unstable* one-pole filter, just as we
encounter in piecewise conical acoustic tubes [4].
Thus, the problems of modeling flared horns and piecewise conical bores
give rise to the problem of how to utilize unstable digital filters as
modeling elements without running into numerical problems.

It turns out that growing exponential impulse-response segments can be efficiently and practically devised using ``Truncated Infinite Impulse Response'' (TIIR) digital filtering techniques [10]. The basic idea of a TIIR filter is to synthesize an FIR filter as an IIR filter minus a delayed ``tail canceling'' IIR filter (which has the same poles as the first). That is, the second IIR filter generates a copy of the ``tail'' of the first so that it can be subtracted off, thus creating an FIR filter. When all IIR poles are stable, TIIR filters are straightforward. In the unstable case, the straightforward implementation fails numerically: While the filter tails always cancel in principle, the exponential growth of the roundoff-error eventually dominates. Thus, in the unstable case, TIIR filters must switch between two alternate instances of the desired TIIR filter (i.e., two pairs of tail-canceling IIR filters). The state of the ``off-duty'' filter is cleared in order to zero out the accumulating round-off noise. The key observation is that, because the desired TIIR filter functions as an FIR filter, it reaches exact ``steady state'' after only samples, where is the length of the synthesized FIR filter. As a result, a ``fresh instance'' of the TIIR filter, when ``ramped up'' from the zero state, is ready to be switched in exactly after only samples, even though the component IIR filters have not yet reached the same internal state as those of the TIIR filter being switched out.

An empirically derived trumpet bell reflectance was found to have an impulse response duration on the order of 10 ms (which is on the order of 400 samples at a 44.1 kHz sampling rate). While a length 400 FIR filter can faithfully model the trumpet-bell reflectance, use of TIIR methods reduces the complexity by well over an order of magnitude with good matching of the principal time-domain and frequency-domain features (accurately preserving the horn resonances in particular).

In the remainder of this paper, we will discuss (1) the needed class of TIIR filters, (2) TIIR modeling of flared horns, using a theoretically derived Bessel horn reflectance as the desired response, (3) construction and calibration of a TIIR-based digital waveguide trumpet model, based on experimental data, and finally (4) TIIR modeling of piecewise conical acoustic bores.

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