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Adaptors for Wave Digital Elements

An adaptor is an $ N$-port memoryless interface which interconnects wave digital elements. Since each element's ``port'' is a connection to an infinitesimal waveguide section at some real wave impedance $ R_i$, and since the input/output signals are wave variables (traveling-waves within the waveguide), the adaptor must implement signal scattering appropriate for the connection of such waveguides. In other words, an $ N$-port adaptor in a wave digital filter performs exactly the same computation as an $ N$-port scattering junction in a digital waveguide network.F.2

This section first addresses the simpler two-port case, followed by a derivation of the general $ N$-port adaptor, for both parallel and series connections of wave digital elements.

As discussed in §7.2, a physical connection of two or more ports can either be in parallel (forces are equal and the velocities sum to zero) or in series (velocities equal and forces sum to zero). Combinations of parallel and series connections are also of course possible.

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 $ R_1$ and $ R_2$. b) Corresponding parallel force scattering junction (adaptor wave flow diagram) in Kelly-Lochbaum form. Compare with Fig.F.7.
\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:

\begin{eqnarray*} && f_1(n) = f_2(n) \isdef f_J(n)\\ && v_1(n) + v_2(n) = 0 \end{eqnarray*}

Following the same derivation leading to Eq.$ \,$(F.2), 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... ...amma _1 f^{{+}}_1 + \Gamma _2 f^{{+}}_2 }{\Gamma _1+\Gamma _2} . \end{eqnarray*}

The outgoing wave variables are given by

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

Defining the reflection coefficient as

$\displaystyle \rho \isdef \frac{R_2-R_1}{R_2+R_1} $

we have that the scattering relations for the two-port parallel adaptor are

\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 $ R_1$ to the port impedance of element 1, and $ R_2$ to the port impedance of element 2. An example is shown in Fig.F.35.

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.

Compatible Port Connections

Note carefully that to connect a wave digital element to port $ i$ of the adaptor, we route the signal $ f^{{-}}(n)$ coming out of the element to become $ f^{{+}}_i(n)$ on the adaptor port, and the signal $ f^{{-}}_i(n)$ coming out of port $ i$ of the adaptor goes into the element as $ f^{{+}}(n)$. Such a connection is said to be a compatible port connection. In other words, the connections must be made such that the arrows go in the same direction in the wave flow diagram.

General Parallel Adaptor for Force Waves

In the more general case of $ N$ wave digital element ports being connected in parallel, we have the physical constraints

    $\displaystyle f_1(n) = f_2(n) = \cdots = f_N(n) \isdef f_J(n)$ (F.14)
    $\displaystyle v_1(n) + v_2(n) + \cdots + v_N(n) = 0$ (F.15)

The derivation for the two-port case extends to the $ N$-port case without modification:
0 $\displaystyle =$ $\displaystyle \sum_{i=1}^N v_i$  
  $\displaystyle =$ $\displaystyle \sum_{i=1}^N\frac{f^{{+}}_i-f^{{-}}_i}{R_i}$  
  $\displaystyle =$ $\displaystyle \sum_{i=1}^N\frac{2f^{{+}}_i-f_J}{R_i}$  
  $\displaystyle \isdef$ $\displaystyle \sum_{i=1}^N \left(2\Gamma _if^{{+}}_i-\Gamma _i f_J \right)$  
$\displaystyle \,\,\Rightarrow\,\, \sum_{j=1}^N \Gamma _j f_J$ $\displaystyle =$ $\displaystyle \sum_{i=1}^N 2\Gamma _i f^{{+}}_i$  
$\displaystyle \,\,\Rightarrow\,\, f_J$ $\displaystyle =$ $\displaystyle \frac{\sum_{i=1}^N 2\Gamma _i f^{{+}}_i}{\sum_{j=1}^N \Gamma _j} . \protect$ (F.16)

The outgoing wave variables are given by

$\displaystyle f^{{-}}_i(n) = f_J(n) - f^{{+}}_i(n) $

Alpha Parameters

It is customary in the wave digital filter literature to define the alpha parameters as

$\displaystyle \fbox{$\displaystyle \alpha_i \isdef \frac{2\Gamma _i}{\sum_{j=1}^N \Gamma _j}$} \protect$ (F.17)

where $ \Gamma _i \isdef 1/R_i$ are the admittances of the wave digital element interfaces (or ``reference admittances,'' in WDF terminology). In terms of the alpha parameters, the force-wave parallel adaptor performs the following computations:
$\displaystyle f_J(n)$ $\displaystyle =$ $\displaystyle \sum_{i=1}^N \alpha_i f^{{+}}_i(n)\protect$ (F.18)
$\displaystyle f^{{-}}_i(n)$ $\displaystyle =$ $\displaystyle f_J(n) - f^{{+}}_i(n)\protect$ (F.19)

We see that $ N$ multiplies and $ 2N-1$ additions are required. However, by observing from Eq.$ \,$(F.17) that

$\displaystyle \sum_{i=1}^N \alpha_i = 2, $

we may implement $ \alpha_1$, say, as $ 2-\sum_{i=2}^N\alpha_i$ in order to eliminate one multiply. In WDF terminology, port 1 is then a dependent port.

Reflection Coefficient, Parallel Case

The reflection coefficient seen at port $ i$ is defined as

$\displaystyle \rho_i \isdef \left. \frac{f^{{-}}_i(n)}{f^{{+}}_i(n)} \right\vert _{f^{{+}}_j(n)=0, \forall j\neq i} \protect$ (F.20)

In other words, the reflection coefficient specifies what portion of the incoming wave $ f^{{+}}_i(n)$ is reflected back to port $ i$ as part of the outgoing wave $ f^{{-}}_i(n)$. The total outgoing wave on port $ i$ is the superposition of the reflected wave and the $ N-1$ transmitted waves from the other ports:

$\displaystyle f^{{-}}_i(n) = \rho_i f^{{+}}_i + \sum_{j\neq i} \tau_{ji} f^{{+}}_j \protect$ (F.21)

where $ \tau_{ji}$ denotes the transmission coefficient from port $ j$ to port $ i$. Starting with Eq.$ \,$(F.19) and substituting Eq.$ \,$(F.18) gives

\begin{eqnarray*} f^{{-}}_i(n) &=& f_J(n) - f^{{+}}_i(n)\\ &=& \left(\sum_{j=1... ...\alpha_i - 1)f^{{+}}_i(n) + \sum_{j\neq i} \alpha_j f^{{+}}_j(n) \end{eqnarray*}

Equating like terms with Eq.$ \,$(F.21), we obtain

$\displaystyle \rho_i$ $\displaystyle =$ $\displaystyle \alpha_i - 1 \protect$ (F.22)
$\displaystyle \tau_{ji}$ $\displaystyle =$ $\displaystyle \alpha_j, \quad (i\neq j) \protect$ (F.23)

Thus, the $ j$th alpha parameter is the force transmission coefficient from $ j$th port to any other port (besides the $ i$th). To convert the transmission coefficient from the $ i$th port to the reflection coefficient for that port, we simply subtract 1. This general relationship is specific to force waves at a parallel junction, as we will soon see.

Physical Derivation of Reflection Coefficient

Physically, the reflection coefficient seen at port $ i$ is due to an impedance step from $ R_i$, that of the port interface, to a new impedance consisting of the parallel combination of all other port impedances meeting at the junction. Let

$\displaystyle \Gamma _J(i) \isdef \sum_{i\neq j} \Gamma _i \protect$ (F.24)

denote this parallel combination, in admittance form. Then we must have

$\displaystyle \rho_i = \frac{R_J(i)-R_i}{R_J(i)+R_i} = \frac{\Gamma _i-\Gamma _J(i)}{\Gamma _i+\Gamma _J(i)} \protect$ (F.25)

Let's check this ``physical'' derivation against the formal definition Eq.$ \,$(F.20) leading to $ \rho_i = \alpha_i - 1$ in Eq.$ \,$(F.22). Toward this goal, let

$\displaystyle \Gamma _J \isdef \sum_{j=1}^N \Gamma _j $

denote the parallel combination of all admittances connected to the junction. Then by Eq.$ \,$(F.24), we have $ \Gamma _J = \Gamma _i + \Gamma _J(i)$ for all $ i$. Now, from Eq.$ \,$(F.17),

\begin{eqnarray*} \rho_i &\isdef & \alpha_i - 1 \;\isdef \; \frac{2\Gamma _i}{\... ..._i + \Gamma _J(i)} \;=\; \frac{R_J(i) - R_i}{\Gamma _J(i)-R_i} \end{eqnarray*}

and the result is verified.

Reflection Free Port

It is useful in practice, such as when connecting two adaptors together, to make one port reflection free. A reflection-free port is defined to have a zero reflection coefficient. For port $ i$ of a parallel adaptor to be reflection free, we must have, from Eq.$ \,$(F.25),

$\displaystyle R_i = R_J(i) \isdef \frac{1}{\sum_{i\neq j} \Gamma _i} $

Thus, the port's impedance must equal the parallel combination of the other port impedances at the junction. In this case, the junction as a whole ``perfectly terminates'' the reflection free port, so no reflections come back from it.

Connecting two adaptors at a reflection-free port prevents the formation of a delay-free loop which would otherwise occur [136]. As a result, multi-port junctions can be joined without having to insert unit elements (see §F.1.7) to avoid creating delay-free loops. Only one of the two ports participating in the connection needs to be reflection free.

We can always make a reflection-free port at the connection of two adaptors because the ports used for this connection (one on each adaptor) were created only for purposes of this connection. They can be set to any impedance, and only one of them needs to be reflection free.

To interconnect three adaptors, labeled $ A$, $ B$, and $ C$, we may proceed as follows: Let $ A$ be augmented with two unconstrained ports, having impedances $ R_1$ and $ R_2$. Add a reflection-free port to $ B$, and suppose its impedance has to be $ R_B$. Add a reflection-free port to $ C$, and suppose its impedance has to be $ R_C$. Now set $ R_1=R_B$ and connect $ B$ to $ A$ via the corresponding ports. Similarly, set $ R_2=R_C$ and connect $ C$ to $ A$ accordingly. This adaptor-connection protocol clearly extends to any number of adaptors.

Two-Port Series Adaptor for Force Waves

Figure F.6a illustrates a generic two-port description of the series adaptor.

Figure F.6: a) Two-port description of the adaptor implementing a series connection between reference impedances $ R_1$ and $ R_2$. b) Corresponding series force scattering junction (adaptor wave flow diagram) in Kelly-Lochbaum form.
\includegraphics[width=\twidth]{eps/lAdaptorSeries}

As discussed in §7.2, a series connection is characterized by a common velocity and forces which sum to zero at the junction:

\begin{eqnarray*} && f_1(n) + f_2(n) = 0\\ && v_1(n) = v_2(n) \isdef v_J(n) \end{eqnarray*}

The derivation can proceed exactly as for the parallel junction in §F.2.1, but with force and velocity interchanged, i.e., $ f\leftrightarrow v$, and with impedance and admittance interchanged, i.e., $ R\leftrightarrow \Gamma $. In this way, we may take the dual of Eq.$ \,$(F.14) to get

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

diagrammed in Fig.F.7. Converting back to force wave variables via $ f^{{+}}_i=R_iv^{+}_i$ and $ f^{{-}}_i=-R_iv^{-}_i$, and noting that $ (1+\rho)R_1/R_2 = 1-\rho$, we obtain, finally,

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

as diagrammed in Fig.F.6b. The one-multiply form is now

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

Figure F.7: Series velocity scattering junction in Kelly-Lochbaum form.
\includegraphics[scale=0.9]{eps/lscat_vel_series_renum}

General Series Adaptor for Force Waves

In the more general case of $ N$ ports being connected in series, we have the physical constraints

\begin{eqnarray*} && v_1(n) = v_2(n) = \cdots = v_N(n) \isdef v_J(n)\\ && f_1(n) + f_2(n) + \cdots + f_N(n) = 0 \end{eqnarray*}

The derivation is the dual of that in the parallel case (cf. Eq.$ \,$(F.16)), i.e., force and velocity are interchanged, and impedance and admittance are interchanged:

\begin{eqnarray*} 0 &=& \sum_{i=1}^N f_i \\ &=& \sum_{i=1}^NR_i\left(v^{+}_i-v... ...uad v_J &=& \frac{\sum_{i=1}^N 2R_i v^{+}_i}{\sum_{j=1}^N R_j} . \end{eqnarray*}

The outgoing wave variables are given by

$\displaystyle v^{-}_i(n) = v_J(n) - v^{+}_i(n) $

Beta Parameters

It is customary in the wave digital filter literature to define the beta parameters as

$\displaystyle \fbox{$\displaystyle \beta_i \isdef \frac{2R_i}{\sum_{j=1}^N R_j}$} \protect$ (F.26)

where $ R_i$ are the port impedances (attached element reference impedances). In terms of the beta parameters, the force-wave series adaptor performs the following computations:
$\displaystyle v_J(n)$ $\displaystyle =$ $\displaystyle \sum_{i=1}^N \beta_i v^{+}_i(n)\protect$ (F.27)
$\displaystyle v^{-}_i(n)$ $\displaystyle =$ $\displaystyle v_J(n) - v^{+}_i(n)\protect$ (F.28)

However, we normally employ a mixture of parallel and series adaptors, while keeping a force-wave simulation. Since $ f^{{+}}_i(n) = R_i v^{+}_i(n)$, we obtain, after a small amount of algebra, the following recipe for the series force-wave adaptor:

$\displaystyle f^{{+}}_J(n)$ $\displaystyle =$ $\displaystyle \sum_{i=1}^N f^{{+}}_i(n)\protect$ (F.29)
$\displaystyle f^{{-}}_i(n)$ $\displaystyle =$ $\displaystyle f^{{+}}_i(n) - \beta_if^{{+}}_J(n)\protect$ (F.30)

We see that we have $ N$ multiplies and $ 2N-1$ additions as in the parallel-adaptor case. However, we again have from Eq.$ \,$(F.26) that

$\displaystyle \sum_{i=1}^N \beta_i = 2, $

so that we may implement one beta parameter as 2 minus the sum of the rest, thus eliminating a multiplication by creating a dependent port.

Reflection Coefficient, Series Case

The velocity reflection coefficient seen at port $ i$ is defined as

$\displaystyle \rho^v_i \isdef \left. \frac{v^{-}_i(n)}{v^{+}_i(n)} \right\vert _{v^{+}_j(n)=0, \forall j\neq i} \protect$ (F.31)

Representing the outgoing velocity wave $ v^{-}_i(n)$ as the superposition of the reflected wave $ \rho^v_iv^{+}_i(n)$ plus the $ N-1$ transmitted waves from the other ports, we have

$\displaystyle v^{-}_i(n) = \rho^v_i v^{+}_i + \sum_{j\neq i} \tau^v_{ji} v^{+}_j \protect$ (F.32)

where $ \tau^v_{ji}$ denotes the velocity transmission coefficientvelocity!transmission coefficient from port $ j$ to port $ i$. Substituting Eq.$ \,$(F.29) into Eq.$ \,$(F.30) yields

\begin{eqnarray*} v^{-}_i(n) &=& v_J(n) - v^{+}_i(n)\\ &=& \left(\sum_{j=1}^N ... ... &=& (\beta_i - 1)v^{+}_i(n) + \sum_{j\neq i} \beta_j v^{+}_j(n) \end{eqnarray*}

Equating like terms with Eq.$ \,$(F.32) gives

$\displaystyle \rho^v_i$ $\displaystyle =$ $\displaystyle \beta_i - 1 \protect$ (F.33)
$\displaystyle \tau^v_{ji}$ $\displaystyle =$ $\displaystyle \beta_j, \quad (i\neq j)$ (F.34)

Thus, the $ j$th beta parameter is the velocity transmission coefficient from $ j$th port to any other port (besides the $ i$th). To convert the transmission coefficient from the $ i$th port to the reflection coefficient for that port, we simply subtract 1. These relationships are specific to velocity waves at a series junction (cf. Eq.$ \,$(F.22)). They are exactly the dual of Equations (F.22-F.23) for force waves at a parallel junction.

Physical Derivation of Series Reflection Coefficient

Physically, the force-wave reflection coefficient seen at port $ i$ of a series adaptor is due to an impedance step from $ R_i$, that of the port interface, to a new impedance consisting of the series combination of all other port impedances meeting at the junction. Let

$\displaystyle R_J(i) \isdef \sum_{i\neq j} R_i \protect$ (F.35)

denote this series combination. Then we must have, as in Eq.$ \,$(F.25),

$\displaystyle \rho_i = \frac{R_J(i)-R_i}{R_J(i)+R_i}$ (F.36)

Let's check this ``physical'' derivation against the formal definition Eq.$ \,$(F.31) leading to $ \rho^v_i = \beta_i - 1$ in Eq.$ \,$(F.33). Define the total junction impedance as

$\displaystyle R_J \isdef \sum_{j=1}^N R_j $

This is the series combination of all impedances connected to the junction. Then by Eq.$ \,$(F.35), $ R_J = R_i + R_J(i)$ for all $ i$. From Eq.$ \,$(F.26), the velocity reflection coefficient is given by

\begin{eqnarray*} \rho^v_i &\isdef & \beta_i - 1 \;\isdef \; \frac{2R_i}{R_J} -... ..._J(i)}\\ &=& \frac{R_i - R_J(i)}{R_i + R_J(i)}\\ &=& -\rho_i \end{eqnarray*}

Since

$\displaystyle \rho^v_i\isdef \frac{v^{-}_i(n)}{v^{+}_i(n)} = \frac{-f^{{-}}_i(n)/R_i}{f^{{+}}_i(n)/R_i} = - \frac{f^{{-}}_i(n)}{f^{{+}}_i(n)} \isdef -\rho_i $

the result follows.

Series Reflection Free Port

For port $ i$ to be reflection free in a series adaptor, we require

$\displaystyle R_i = R_J(i) \isdef \sum_{i\neq j} R_i \protect$ (F.37)

That is, the port's impedance must equal the series combination of the other port impedances at the junction. This result can be compared with that for the parallel junction in §F.2.2.

The series adaptor has now been derived in a way which emphasizes its duality with respect to the parallel adaptor.

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