The Wave Digital Filter Brass Mouthpiece Model. University of Edinburgh. Faculty of Music. Department of Physics and Astronomy. Figure 2. Input impedance of ...
The Wave Digital Filter Brass Mouthpiece Model Maarten van Walstijn and Murray Campbell
University of Edinburgh Faculty of Music Department of Physics and Astronomy
Introduction
TH EORETICAL
Wave digital filter (WDF) techniques are applied to develop an efficient discrete-time 2-port mouthpiece model that can be used in travelling-wave based modelling of brass instruments. WDF techniques were principally developed for simulation of analogue networks [3,4], and are used here to discretise the classical simple lumped element mouthpiece model [1,2]. In this model, the mouthpiece cup volume is assumed to behave as an acoustic compliance, and the backbore as an inductance with resistance. Dietz [2] introduced a slightly more complex version of this model, by adding a capacitor that represents the compliance of the backbore. (see figure 1).
L
(Z ) A
R
C1
Z in
W D F 3 -p o r t P a ra l l e l C a p a c i to r
(Z ) C
W D F 2 -p o r t S e ri e s In d u c to r
(Z ) D
W D F 2 -p o r t S e ri e s R e s i s to r
C2
(Z ) E
W D F 3 -p o r t P a ra l l e l C a p a c i to r
(Z )
(Z )
z -1
z -1
B
ZL
(Z ) G
A ir C o lu mn Model
F
C1 = 8.45pF L = 4350Kg/m
4
OPTIM IZED
V1 = 1.20ml
C1 = 10.16pF
M = 1.74mg
L = 4303Kg/m
R = 2.5M Ω
V1 = 1.44ml 4
M = 1.73mg
R = 1.19M Ω
C 2 = 14.16pF
V 2 = 2.01ml
C 2 = 12.37pF
V2 = 1.76ml
Table 1. Parameters of the trumpet mouthpiece model. For capacitance (C) and inductance (L), the corresponding volume (V) and mass (M) are given, respectively.
Experimental results by Dietz [2] indicate a resistance R ≈ 2.5 MΩ for a trumpet mouthpiece. To verify those parameter values for a Wick 3E trumpet mouthpiece, we compared a measured input impedance with equations (1), whereby the load impedance Z L of the trumpet was measured using pulse reflectometry techniques [5]. Deviations from the measured input impedance Z in* were found for both the simple model (in peak positions) and the complex model (in peak amplitude). In order to obtain a better fit, the parameters of the complex model were optimised using a simplex search method. The optimisation algorithm was set to minimise the error
∑ 2000
ε=
Figure 1. Lumped element model of a brass mouthpiece (top) and its wave digital filter implementation (bottom).
Z in (ω ) =
jω C1 1 + jω C1 ( R + j ω L + Z L )
100 Me a s u re d O p timis e d
(1a)
ω 2 LC2 Z L − jω ( L + RC2 Z L ) − R − Z L Z in (ω ) = jω 3 LC1C2Z L + ω 2 ( LC1 + RC1C2 Z L ) + jω ( RC1 + C2 Z L ) + 1
(3)
with weighting function W ( f ) that favours the musically important resonances. The optimised parameter values are displayed in table 1, and the resulting impedance curve is compared to the measured curve in figure 2.
Im p e d a n c e (M e g a O h m )
The main acoustical function of the brass mouthpiece is to enhance the input amplitude peaks of the load impedance Z L (that represents the instrument aircolumn) in the musically important frequency range. The input impedance of the network for the simple and complex model respectively, is:
W ( f ) ⋅ Z in* ( f ) − Z in ( f ) ,
f = 50
(1b)
80
60
40
20
An additional effect of the mouthpiece is that it shifts the impedance peaks somewhat. In other words, it has to have the proper effective length.
0 0
200
400
600
800
1000
1200
1400
1600
1800
2000
F r e q u e n c y (H z )
Figure 2. Input impedance of the trumpet.
Discretisation The model is segmented into 2-port units that are individually mapped into the digital domain using the 2-port and 3-port WDF approach [3,4] (see figure 1). The realisability of the filter structure is ensured by choosing the port resistances (in acoustic analogy corresponding to characteristic impedances) in between the units such that delay-free loops are avoided. At first sight this seems to pose some problems when connecting the mouthpiece model with the lips and the instrument. A full simulation of a brass instrument would exhibit an immediate reflection of backward-travelling waves at the lips. Therefore the local port resistance Z A is determined by the discretisation process and will therefore differ from the characteristic impedance of the mouthpiece entrance, which results in a different reflection function than the actual acoustical one. However, as can be seen from equations (1), the input impedance is independent of the choice of port resistances, thus the discrete model does still represent the proper acoustical functioning. The connection with the lead-pipe must also be designed such that no delay-free loop results. This requires that Z G is taken such that the implementation of the conical junction with the lead-pipe does not have an immediate reflection.
Time Domain Simulation Due to the fact that brass mouthpieces reflect much more main-bore wave energy than woodwind mouthpieces, brass instruments exhibit a relatively long reflection function. In time-domain simulations it is therefore efficient to represent the mouthpiece with the WDF model. The reflection function of the instrument aircolumn (without mouthpiece) does not exhibit the multiple reflections due to waves bouncing back and forth between the mouthpiece and the bell. Thus a much shorter reflection function can be used. Furthermore, the WDF mouthpiece model can be applied straightforward in a digital waveguide model.
Conclusions The WDF brass mouthpiece model is easy to ‘plug-in’ as a model unit and has low computational costs. Its parameters can be optimised to render a brass model in which the resonances of the instrument are well preserved. The model is relatively simple and its parameters are very intuitive, which makes it particularly useful for brass modelling with musical sound synthesis purposes.
Estimation of the Lumped Model Parameters References
According to the analogy between electrical and acoustical systems, the theoretical values for C1 , L and C2 , are: C1 =
V1 σ c2
L=
σl S
C2 =
V2 , σ c2
[1] (2)
where l is the length and S the cross-section of the backbore, and V1 and V2 = l ⋅ S represent the mouthcup and backbore volume, respectively. Since the backbore has a non-uniform cross-section, L and C2 are computed as an integral of the radius as a function of distance [2].
[2] [3] [4] [5]
Backus, J. Input Impedance Curves for the Brass Instruments, J. Acoust. Soc. Am., Vol. 60 (2), 1976. Dietz, P.H., Simulation of Trumpet Tones via Physical Modeling, Master Thesis, Dept. of Electrical Engineering, Bucknell Univ., Lewisburg (USA), 1988. Fettweis, A., Wave Digital Filters, Proc. Of the IEEE, Vol. 74 (2), 1986. Lawson, S. and Mirzai, A., Wave Digital Filters, Ellis Horwood, New York, 1990. Sharp, D.B., Acoustic Pulse Reflectometry for the Measurement of Musical Wind Instruments, Ph.D. Thesis, Univ. of Edinburgh, 1996.
