Horn Reflectance Filter

A general characteristic of musically useful horns is that their internal bore profile is well approximated with a Bessel horn [2]. 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 [9].

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 $y(n)=a e^{c n}$, for $n=0,1,2,\ldots,N-1$, and zero afterwards. We can use this truncated exponential to efficiently implement the initial growing trend in the horn response ($c > 0$). We found empirically that improved accuracy is obtained by using the sum of an exponential and a constant, i.e.,

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

The truncated constant $b$ can also be generated using a one-pole TIIR filter, with its pole set to $z=1$. In this case, no multiplies are needed, except for the single scale factor $b$. The transfer function of the TIIR filter for modeling a single segment of the horn impulse response as an offset exponential can be written as

$$H(z) = h_{0} \frac{1-p^{N+1}z^{-(N+1)}}{1 - pz^{-1}} + b \frac{1-z^{-(N+1)}}{1 - z^{-1}}.$$ (1)

The remaining reflection impulse response has a decaying trend, and can therefore be modeled accurately with diverse conventional filter design techniques. Here, the Steiglitz-McBride IIR filter design algorithm was applied [3].

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.

Figure 2: Bessel horn response (solid) compared with digital filter approximation (dashed) in terms of impulse response (a), magnitude (b) (up to Nyquist) and phase delay (c) (up to bell cut-off). The vertical line in (a) indicates the segmentation into an growing exponential and a decaying tail.