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The adaptor equations for a connection of
ports, in either the series (2.31) or parallel (2.32) case, may be written as
![$\displaystyle {\bf b} = {\bf Sa}$](img387.png) |
(2.37) |
where
and
, and where we have
![$\displaystyle {\bf S}$](img390.png) |
![$\displaystyle =$](img140.png) |
|
![$\displaystyle \mbox{\boldmath$\alpha$}$](img392.png) ![$\displaystyle _{s}{\bf 1}^{T}\hspace{1.0in}$](img393.png) |
|
Series adaptor (voltage waves) |
(2.38) |
![$\displaystyle {\bf S}$](img390.png) |
![$\displaystyle = -$](img394.png) |
|
![$\displaystyle {\bf I}_{M} + {\bf 1}$](img395.png) ![$\displaystyle \mbox{\boldmath$\alpha$}$](img392.png) ![$\displaystyle _{p}^{T}\hspace{1.0in}$](img396.png) |
|
Parallel adaptor (voltage waves) |
(2.39) |
Here
is an
vector containing all ones,
is the
identity matrix, and ![$ \alpha$](img400.png)
and ![$ \alpha$](img400.png)
are defined by
The sum of the elements of either ![$ \alpha$](img400.png)
or ![$ \alpha$](img400.png)
is 2.
For power wave variables, we have a similar relationship,
![$\displaystyle \underline{{\bf b}} = \underline{{\bf S}}\,\underline{{\bf a}}$](img405.png) |
(2.40) |
where
![$\displaystyle \underline{{\bf S}}$](img406.png) |
![$\displaystyle =$](img140.png) |
|
![$\displaystyle {\bf I}_{M} - \sqrt{\mbox{\boldmath$\alpha$}_{s}}\sqrt{\mbox{\boldmath$\alpha$}_{s}}^{T}\hspace{1.0in}$](img407.png) |
|
Series adaptor (power-normalized waves) |
|
![$\displaystyle \underline{{\bf S}}$](img406.png) |
![$\displaystyle = -$](img394.png) |
|
![$\displaystyle {\bf I}_{M} + \sqrt{\mbox{\boldmath$\alpha$}_{p}}\sqrt{\mbox{\boldmath$\alpha$}_{p}}^{T}\hspace{1.0in}$](img408.png) |
|
Parallel adaptor (power-normalized waves) |
|
Here the square root sign indicates an entry-by-entry square root of a vector (all entries of ![$ \alpha$](img400.png)
and ![$ \alpha$](img400.png)
are non-negative).
Defining the Euclidean norm of a column vector
as
, it is easy to show that a power normalized scattering matrix
is norm-preserving in either the series or parallel case, i.e., we have
![$\displaystyle \Vert\underline{{\bf b}}\Vert _{2} = \Vert\underline{{\bf a}}\Vert _{2}$](img411.png) |
(2.41) |
For voltage waves, we have the preservation of a weighted
norm, i.e.,
![$\displaystyle \Vert{\bf b}\Vert _{{\bf P},2} = \Vert{\bf a}\Vert _{{\bf P},2}$](img412.png) |
(2.42) |
where
; in this case,
is an
positive definite diagonal matrix simply given by diag
. It should be clear that (2.41) and (2.42) are merely re-statements of power conservation at a memoryless, lossless
-port.
Note that multiplying
or
by a vector requires, in either the series of parallel case,
adds and multiplies; in particular, it is cheaper than a full
matrix multiply.
Next: Signal and Coefficient Quantization
Up: Adaptors
Previous: Adaptors
Stefan Bilbao
2002-01-22