Loaded Junctions

In discrete-time modeling of acoustic systems, it is often useful to attach waveguide junctions to external dynamic systems which act as a load. We speak in this case of a loaded junction. The load is expressed in general by its complex admittance and can be considered a lumped circuit attached to the distributed waveguide network.

Examples of loaded junctions can be found in certain representations of finger holes in wind-instrument modeling [93,80], or in bridge terminations in stringed-instrument modeling [98] (see the example in the next section). They are also a natural way to load a waveguide mesh to simulate air above a membrane [31]. Sometimes the load can be time-varying and even nonlinear, as in the case of a piano hammer hitting a string [127,7]; in this case, the hammer is modeled as a mass-spring system in which the spring stiffness nonlinearly depends on its compression, and the impedance ``seen'' by the string changes continuously during contact. Finally, the loaded junction equations can be used to interface a DWN with other types of physical modeling simulations (e.g., a WDF): In such a case, the attached load is simply the driving-point impedance of the external simulation.

To derive the scattering matrix for the loaded parallel junction of $N$ lossless acoustic tubes, the Kirchhoff's node equation is reformulated so that the sum of velocities meeting at the junction equals the exit velocity (instead of zero). For the series junction of transversely vibrating strings, the sum of forces exerted by the strings on the junction is set equal to the force acting on the load (instead of zero).

The load admittance $\Gamma_L$ is regarded as a lumped driving-point admittance [128], and the equation

$$U_L(z) = \Gamma_L(z) p_J(z)$$ (48)

expresses the relation at the load. In acoustic modeling applications, $\Gamma_L$ is normally first obtained in the continuous-time case as a function of the Laplace transform variable $s$. Denote the continuous-time Laplace transform by $\Gamma_L^c(s)$, and the inverse Laplace transform by $\gamma_L^c(t)$. To preserve the tuning of audio resonances of acoustic systems, the load admittance is usually converted to discrete time by the impulse-invariant method, i.e., simple sampling of $\gamma_L^c(t)$ in the time domain. In simpler cases in which there is only one resonance to be transformed from the $s$ plane to the $z$ plane, such as a piano-hammer model (a mass-spring system) or the individual terms of a partial fraction expansion, the bilinear transform $s = \alpha(z-1)/(z+1)$ may be used [65].

In the impulse-invariant case, to avoid aliasing, we require $\Gamma_L^c \simeq 0$ for frequencies near and above half the sampling rate. Since the driving-point admittances of acoustic systems are not generally bandlimited (consider the driving-point impedance of an elementary mass $\Gamma_L^c(s)=1/ms$ or spring $\Gamma_L^c(s)=Ks$), anti-aliasing filtering should be applied to $\gamma_L^c(t)$. The bilinear transform is inherently free of aliasing since the entire $j\omega$ axis in the $s$ plane is mapped only once to the unit circle in the $z$ plane, but as a result of this mapping, the frequency axis is warped, and the behavior at continuous-time frequencies can be mapped to the same discrete-time frequencies at only three points (dc, half the sampling rate, and one more frequency which is arbitrary). For this reason, the impulse-invariant method is normally preferred when the precise tuning of more than one spectral feature is important.

In case of a time-varying or nonlinear load, as in the piano-hammer model, to avoid aliasing, the load impedance should vary sufficiently slowly and/or be sufficiently weakly nonlinear so that the low-pass filtering in the DWN will attenuate modulation and/or distortion products to insignificance before aliasing occurs in the recursive network.

The scattering matrix for a loaded parallel junction of $N$ lossless, scalar waveguides with real wave impedances can be rewritten as [93]

$${\mbox{\boldmath$A$}}_L = \left[ \begin{array}{llll} \frac{2... ... \dots & \frac{2 \Gamma_{N}}{\Gamma_J} -1 \end{array} \right]$$

where

$$\Gamma_J = \Gamma_L + \sum_{i=1}^{N}{\Gamma_{i}}$$ (49)

For the more general case of $N$ $m$-variable physical waveguides, the expression of the scattering matrix is that of (48), with

$$S=\left[{\mbox{\boldmath$1$}}^T\left({\sum_{i=1}^{N}{{\mbox{... ...Gamma$}}_i}}\right){\mbox{\boldmath$1$}}+ \Gamma_L\right]^{-1}$$ (50)