A general characteristic of musically useful horns is that their internal bore profile is well approximated with a Bessel horn . Although any real instrument bell will show significant deviations from this approximation in its bore shape and acoustic reflectance, a theoretically derived Bessel horn reflection function may serve as a suitable generalized target-response for developing effective digital filter design techniques. In order to obtain such a target-response, the pressure reflectance of a Bessel horn that approximates the shape of a trumpet bell was computed as in .
As shown in Fig. 2, the Bessel horn
reflection impulse response has a slow, quasi-exponentionally growing
portion at the beginning, corresponding to the smoothly increasing taper
angle of the horn. A one-pole TIIR filter gives a
truncated exponential impulse response
, and zero afterwards. We can use this truncated
exponential to efficiently implement the initial growing trend in the horn
response (). We found empirically that improved accuracy is
obtained by using the sum of an exponential and a constant, i.e.,
In Fig. 2, the TIIR horn filter structure (using a 3rd-order IIR tail filter approximation) is compared with the theoretical response. The phase delay (directly proportional to the ``effective length'' of the bell for standing waves), has a particularly good fit, which is important for accurate musical resonance frequencies of a brass instrument.