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Vector Wave Variables
It is straightforward to extend wave digital filtering principles to the vector case (this has been outlined in [131]; the same idea has apeared in the context of digital waveguide networks in [166,169]). For a
-component vector one-port element with voltage
and current
, it is posible to define wave variables
and
by
![$\displaystyle \begin{eqnarray}{\bf a} &=& {\bf v} + {\bf Ri}\\ {\bf b} &=& {\bf v} - {\bf Ri} \end{eqnarray}$](img433.png) |
(2.43a) |
for a
symmetric positive definite matrix
; power-normalized quantities may be defined by
![$\displaystyle \begin{eqnarray}\underline{{\bf a}} &=& \frac{1}{2}\left({\bf R}^...
...\frac{1}{2}\left({\bf R}^{-T/2}{\bf v} - {\bf R}^{1/2}{i}\right) \end{eqnarray}$](img436.png) |
(2.44a) |
where
is some right square root of
, and
is its transpose. The power absorbed by the vector one-port will be
![$\displaystyle w_{inst} = \left({\bf a}^{T}{\bf R}^{-1}{\bf a}-{\bf b}^{T}{\bf R...
...\bf a}}-\underline{{\bf b}}^{T}\underline{{\bf b}}\right) = 4{\bf v}^{T}{\bf i}$](img439.png) |
(2.45) |
Kirchoff's Laws, for a series or parallel connection of
-component vector elements with voltages
and
,
can be written as
and the resulting scattering equations will be
in terms of the wave variables
,
defined as per (2.43) and the port resistance matrices
,
. These are the defining equations of a vector adaptor; their schematics are essentially the same as those of Figure 2.12, except that they are drawn in bold--see Figure 2.14. As before, we use the same representation for power-normalized waves.
Figure 2.14:
Three-port vector adaptors-- (a) a vector series adaptor and (b) a vector parallel adaptor.
![\begin{figure}\begin{center}
\begin{picture}(400,120)
\par % graphpaper(0,0)(40...
...(a)}
\put(340,-20){(b)}
\end{picture} \end{center} \vspace{0.1in}
\end{figure}](img448.png) |
Subsections
Next: Coupled Inductances and Capacitances
Up: Wave Digital Elements and
Previous: Signal and Coefficient Quantization
Stefan Bilbao
2002-01-22