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Physical Audio Signal Processing
    Digital Waveguide Theory
       Loaded Waveguide Junctions

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Loaded Waveguide Junctions

In this section, scattering relations will be derived for the general case of N waveguides meeting at a load. When a load is present, the scattering is no longer lossless, unless the load itself is lossless. (i.e., its impedance has a zero real part). For $ N>2$, $ v^{+}_i$ will denote a velocity wave traveling into the junction, and will be called an ``incoming'' velocity wave as opposed to ``right-going.''H.7

Figure H.26: Four ideal strings intersecting at a point to which a lumped impedance is attached. This is a series junction for transverse waves.
\includegraphics[width=\twidth]{eps/fNstrings}

Consider first the series junction of $ N$ waveguides containing transverse force and velocity waves. At a series junction, there is a common velocity while the forces sum. For definiteness, we may think of $ N$ ideal strings intersecting at a single point, and the intersection point can be attached to a lumped load impedance $ R_J(s)$, as depicted in Fig.H.26 for $ N=4$. The presence of the lumped load means we need to look at the wave variables in the frequency domain, i.e., $ V(s) = {\cal L}\{v\}$ for velocity waves and $ F(s) = {\cal L}\{f\}$ for force waves, where $ {\cal L}\{\cdot\}$ denotes the Laplace transform. In the discrete-time case, we use the $ z$ transform instead, but otherwise the story is identical. The physical constraints at the junction are

$\displaystyle V_1(s) = V_2(s) = \cdots = V_N(s)$ $\displaystyle \isdef$ $\displaystyle V_J(s)$ (H.75)
$\displaystyle F_1(s) + F_2(s) + \cdots + F_N(s)$ $\displaystyle =$ $\displaystyle V_J(s) R_J(s) \isdef F_J(s)$ (H.76)

where the reference direction for the load force $ F_J$ is taken to be opposite that for the $ F_i$. (It can be considered the ``equal and opposite reaction'' force at the junction.) For a wave traveling into the junction, force is positive pulling up, acting toward the junction. When the load impedance $ R_J(s)$ is zero, giving a free intersection point, the junction reduces to the unloaded case, and signal scattering will be energy preserving. In general, the loaded junction is lossless if and only if re$ \left\{R_J(j\omega)\right\}\equiv0$, and it is memoryless if and only if im$ \left\{R_J(j\omega)\right\}\equiv0$.

The parallel junction is characterized by

$\displaystyle F_1(s) = F_2(s) = \cdots = F_N(s)$ $\displaystyle \isdef$ $\displaystyle F_J(s)$ (H.77)
$\displaystyle V_1(s) + V_2(s) + \cdots + V_N(s)$ $\displaystyle =$ $\displaystyle F_J(s)/R_J(s) \isdef V_J(s)$ (H.78)

For example, $ F_i(s)$ could be pressure in an acoustic tube and $ V_i(s)$ the corresponding volume velocity. In the parallel case, the junction reduces to the unloaded case when the load impedance $ R_J(s)$ goes to infinity.

The scattering relations for the series junction are derived as follows, dropping the common argument `$ (s)$' for simplicity:

$\displaystyle R_J V_J$ $\displaystyle =$ $\displaystyle \sum_{i=1}^N F_i = \sum_{i=1}^N (F^+_i + F^-_i)$ (H.79)
  $\displaystyle =$ $\displaystyle \sum_{i=1}^N (R_i V^+_i - R_i \underbrace{V^-_i}_{V_J-V^+_i})$ (H.80)
  $\displaystyle =$ $\displaystyle \sum_{i=1}^N (2 R_i V^+_i - R_i V_J)$ (H.81)

where $ R_i$ is the wave impedance in the $ i$th waveguide, a real, positive constant. Bringing all terms containing $ V_J$ to the left-hand side, and solving for the junction velocity gives
$\displaystyle V_J$ $\displaystyle =$ $\displaystyle 2\left(R_J + \sum_{i=1}^N R_i\right)^{-1} \sum_{i=1}^N R_i V^+_i$ (H.82)
  $\displaystyle \isdef$ $\displaystyle \sum_{i=1}^N{\cal A}_i(s) V^+_i(s)$ (H.83)

(written to be valid also in the multivariable case involving square impedance matrices $ R_i$ [442]), where

$\displaystyle {\cal A}_i(s) \isdef \frac{2R_i}{R_J(s) + R_1 + \cdots + R_N}$ (H.84)

Finally, from the basic relation $ V_J = V_i = V^+_i + V^-_i$, the outgoing velocity waves can be computed from the junction velocity and incoming velocity waves as

$\displaystyle V^-_i(s) = V_J(s) - V^+_i(s)$ (H.85)

Similarly, the scattering relations for the loaded parallel junction are given by

$\displaystyle F_J(s)$ $\displaystyle =$ $\displaystyle \sum_{i=1}^N{\cal A}_i(s) F^+_i(s), \quad {\cal A}_i(s) \isdef \frac{2\Gamma _i}{\Gamma _J(s) + \Gamma _1 + \cdots + \Gamma _N}$ (H.86)
$\displaystyle F^-_i(s)$ $\displaystyle =$ $\displaystyle F_J(s) - F^+_i(s)$ (H.87)

where $ F_J(s)$ is the Laplace transform of the force across all elements at the junction, $ \Gamma _J(s)$ is the load admittance, and $ \Gamma _i=1/R_i$ are the branch admittances.

It is interesting to note that the junction load is equivalent to an $ N+1$st waveguide having a (generalized) wave impedance given by the load impedance. This makes sense when one recalls that a transmission line can be ``perfectly terminated'' (i.e., suppressing all reflections from the termination) using a lumped resistor equal in value to the wave impedance of the transmission line. Thus, as far as a traveling wave is concerned, there is no difference between a wave impedance and a lumped impedance of the same value.

In the unloaded case, $ R_J(s)=0$, and we can return to the time domain and define (for the series junction)

$\displaystyle \alpha_i = \frac{2R_i}{R_1 + \cdots + R_N}$ (H.88)

These we call the alpha parameters, and they are analogous to those used to characterize ``adaptors'' in wave digital filters [136]. For unloaded junctions, the alpha parameters obey

$\displaystyle 0\leq\alpha_i \leq 2$ (H.89)

and

$\displaystyle \sum_{i=1}^N\alpha_i = 2$ (H.90)

In the unloaded case, the series junction scattering relations are given (in the time domain) by

$\displaystyle v_J(t)$ $\displaystyle =$ $\displaystyle \sum_{i=1}^N \alpha_i v^+_i(t) \protect$ (H.91)
$\displaystyle v^-_i(t)$ $\displaystyle =$ $\displaystyle v_J(t) - v^+_i(t)$ (H.92)

The alpha parameters provide an interesting and useful parametrization of waveguide junctions. They are explicitly the coefficients of the incoming traveling waves needed to compute junction velocity for a series junction (or junction force or pressure at a parallel junction), and losslessness is assured provided only that the alpha parameters be nonnegative and sum to $ 2$. Having them sum to something less than $ 2$ simulates a ``resistive load'' at the junction.

Note that in the lossless, equal-impedance case, in which all waveguide impedances have the same value $ R_i=R$, (H.88) reduces to

$\displaystyle \alpha_i = \frac{2}{N}$ (H.93)

When, furthermore, $ N$ is a power of two, we have that there are no multiplies in the scattering relations (H.91). This fact has been used to build multiply-free reverberators and other structures using digital waveguide meshes [439,526,401,529].

An elaborated discussion of $ N=2$ strings intersection at a load is given in in §M.2. Further discussion of the digital waveguide mesh appears in §H.12.

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Next: Two Coupled Strings
written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.


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