Two-Port Parallel Adaptor for Force Waves

International Standard Book Number — Click for https://mathworld.wolfram.com/ISBN.html

The reflection coefficient in wave propagation is a constant which gives the ratio of the strength of a reflected wave to that of an incident wave at an impedance mismatch. — Click for http://en.wikipedia.org/wiki/Reflection_coefficient

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Wave Digital Filters (WDF) digitize analog circuits element-by-element using the bilinear transform and virtual traveling waves. — Click for https://ccrma.stanford.edu/~jos/pasp/Wave_Digital_Filters.html

The wave impedance in a propagation medium relates the force-related component of the wave motion to the displacement-related component of the motion. — Click for http://ccrma.stanford.edu/~jos/OnePorts/Impedance.html

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In physics, a wave is an oscillation that propagates through a medium (space-time, gas, fluid, or solid) — Click for https://en.wikipedia.org/wiki/Wave

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The wave impedance in a propagation medium is the force variable (such as pressure for acoustic waves) divided by the velocity variable. In ideal plane waves and traveling waves, the wave impedance is a real, positive number. For spherical waves in air, the wave impedance is different at each frequency. — Click for https://ccrma.stanford.edu/~jos/OnePorts/Impedance.html

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A force is required to change the momentum of an object. In the absence of external forces, momentum is conserved. For a mass m in flight, the momentum is m v, where v denotes the velocity of the mass. Newton's second law, F = m a, says that force equals mass times acceleration, i.e., force equals the time derivative of momentum. — Click for https://scienceworld.wolfram.com/physics/Force.html

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Two-Port Parallel Adaptor for Force Waves

Figure F.5a illustrates a generic parallel two-port connection in terms of forces and velocities.

**Figure:** a) Two-port description of the adaptor implementing a parallel connection between reference impedances and . b) Corresponding parallel force scattering junction (adaptor wave flow diagram) in Kelly-Lochbaum form. Compare with Fig.F.8.
$\includegraphics[width=\twidth]{eps/lAdaptorParallel}$

As discussed in §7.2, a parallel connection is characterized by a common force and velocities which sum to zero:

Following the same derivation as in §C.8.1, and defining $\Gamma _i=1/R_i$ for notational convenience, we obtain

$\begin{eqnarray*} 0 &=& v_1+v_2 \\ &=& \frac{f^{{+}}_1-f^{{-}}_1}{R_1} + \frac{f^{{+}}_2-f^{{-}}_2}{R_2} \\ &=& \frac{2f^{{+}}_1-f_J}{R_1} + \frac{2f^{{+}}_2-f_J}{R_2} \\ &\isdef & 2\Gamma _1f^{{+}}_1-\Gamma _1 f_J + 2\Gamma _2f^{{+}}_2-\Gamma _2 f_J \\ \,\,\Rightarrow\,\,\quad (\Gamma _1+\Gamma _2) f_J &=& 2\left[\Gamma _1 f^{{+}}_1 + \Gamma _2 f^{{+}}_2 \right] \\ \,\,\Rightarrow\,\,\quad f_J &=& 2 \frac{\Gamma _1 f^{{+}}_1 + \Gamma _2 f^{{+}}_2 }{\Gamma _1+\Gamma _2} . \end{eqnarray*}$

$\begin{eqnarray*} f^{{-}}_1(n) &=& f_J(n) - f^{{+}}_1(n) \\ f^{{-}}_2(n) &=& f_J(n) - f^{{+}}_2(n) \end{eqnarray*}$

$\begin{eqnarray*} f^{{-}}_1 &=& \rho f^{{+}}_1 + (1-\rho) f^{{+}}_2 \protect \\ f^{{-}}_2 &=& (1+\rho)f^{{+}}_1 - \rho f^{{+}}_2 \protect \end{eqnarray*}$

as diagrammed in Fig.F.5b. This can be called the Kelly-Lochbaum implementation of the two-port force-wave adaptor.

Now that we have a proper scattering interface between two reference impedances, we may connect two wave digital elements together, setting

to the port impedance of element 1, and

to the port impedance of element 2. An example is shown in Fig.F.37.

The Kelly-Lochbaum adaptor in Fig.F.5b evidently requires four multiplies and two additions. Note that we can factor out the reflection coefficient in each equation to obtain

$\begin{eqnarray*} f^{{-}}_1 &=& f^{{+}}_2 + \rho(f^{{+}}_1 - f^{{+}}_2)\\ f^{{-}}_2 &=& f^{{+}}_1 + \rho(f^{{+}}_1 - f^{{+}}_2) \end{eqnarray*}$

which requires only one multiplication and three additions. This can be called the one-multiply form. The one-multiply form is most efficient in custom VLSI. The Kelly-Lochbaum form, on the other hand, may be more efficient in software, and slightly faster (by one addition) in parallel hardware.

Two-Port Parallel Adaptor for Force Waves

``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4 Copyright © 2024-06-28 by Julius O. Smith III Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4
Copyright © 2024-06-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA), Stanford University