DW Displacement Inputs
We define general DW inputs as follows:
![]() |
![]() |
![]() |
(E.33) |
![]() |
![]() |
![]() |
(E.34) |
The



![$ [y^{+}_{n+2,m},y^{-}_{n+2,m}]^T$](http://www.dsprelated.com/josimages_new/pasp/img4651.png)
![$ [\underline{\upsilon}(n+2)^T,\underline{\upsilon}(n+1)^T]^T$](http://www.dsprelated.com/josimages_new/pasp/img4652.png)
Typically, input signals are injected equally to the left and right along the string, in which case

![\begin{displaymath}
\underbrace{\left[\!
\begin{array}{c}
\vdots\\
y^{+}_{n+2,m...
...ine{\upsilon}(n+1)
\end{array}\!\right]}_{\underline{u}(n+2)}.
\end{displaymath}](http://www.dsprelated.com/josimages_new/pasp/img4655.png)

![\begin{displaymath}
\underline{x}_W(n+2) = \mathbf{A}_W\underline{x}_W(n) +
\un...
...d{array}\!\right]}_{{\mathbf{B}_W}}
\underline{\upsilon}(n+2).
\end{displaymath}](http://www.dsprelated.com/josimages_new/pasp/img4657.png)
To show that the directly obtained FDTD and DW state-space models
correspond to the same dynamic system, it remains to verify that
. It is somewhat easier to show that
![\begin{eqnarray*}
\mathbf{T}\,\mathbf{A}_W&=& \mathbf{A}_K\,\mathbf{T}\\
&=&
\l...
...dots & \vdots & \vdots & \vdots & \vdots
\end{array}\!\right].
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/pasp/img4659.png)
A straightforward calculation verifies that the above identity holds,
as expected. One can similarly verify
, as expected.
The relation
provides a recipe for translating any
choice of input signals for the FDTD model to equivalent inputs for
the DW model, or vice versa.
For example, in the scalar input case (
), the DW input-weights
become FDTD input-weights
according to
![\begin{displaymath}
\left[\!
\begin{array}{l}
\qquad\vdots\\
y_{n+1,m-1}\\
y_{...
...psilon}(n+2)\\
\underline{\upsilon}(n+1)
\end{array}\!\right]
\end{displaymath}](http://www.dsprelated.com/josimages_new/pasp/img4662.png)

Finally, when




![\begin{displaymath}
\mathbf{B}_K=
\left[\!
\begin{array}{cc}
\vdots & \vdots\\
...
...0 \\
2 & 0 \\
1 & 0 \\
\vdots & \vdots
\end{array}\!\right]
\end{displaymath}](http://www.dsprelated.com/josimages_new/pasp/img4668.png)








Next Section:
DW Non-Displacement Inputs
Previous Section:
Proof that the Third-Order Time Derivative is Ill Posed