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Optimally direction-independent numerical dispersion

Although the choice of the free parameter $ a$ which gives a maximally direction-independent numerical dispersion profile has been made, in the past, through computerized optimization procedures [4], we note here that it is possible to make a theoretical choice as well, based on a Taylor series expansion of the spectrum.

The spectral amplification factors for the interpolated scheme can be written in terms of the function $ B_{\mbox{{\scriptsize\boldmath $\beta$}}}$, or, equivalently, in terms of the function $ F_{\mbox{{\scriptsize\boldmath $\beta$}}}$. It should be clear, then, that if $ F_{\mbox{{\scriptsize\boldmath $\beta$}}}$ is directionally independent, then so are the amplification factors, and thus the numerical phase velocity (see §2.4) as well. Ideally, we would like $ F_{\mbox{{\scriptsize\boldmath $\beta$}}}$ to be a function of the spectral radius $ \Vert$$ \beta$$ \Vert _{2} = (\beta_{x}^{2}+\beta_{y}^{2})^{1/2}$ alone. Now examine the Taylor expansion of $ F_{\mbox{{\scriptsize\boldmath $\beta$}}}$ about $ \beta$$ ={\bf0}$:

$\displaystyle F_{\mbox{{\scriptsize\boldmath$\beta$}}} = -\Delta^{2}\Vert\mbox{...

The directionally-independent $ O(\Delta^{2})$ term reflects the fact that the scheme is consistent with the wave equation; higher order terms in general show directional dependence. The choice of $ a=2/3$, however, gives

$\displaystyle F_{\mbox{{\scriptsize\boldmath$\beta$}}} = -\Delta^{2}\Vert\mbox{...
...Vert _{2}^{4}+O(\Delta^{6}) \hspace{1.0in} \mbox{{\rm for}} \hspace{0.2in}a=2/3$    

and the directional dependence is confined to higher-order powers of $ \Delta$. Thus for this choice of $ a$, the numerical scheme is maximally direction independent about spatial DC. Note that this value of $ a$ does fall within the required bounds for a passive waveguide mesh implementation. The value of $ 0.62$ (for which the numerical dispersion profile is plotted in Figure 2), which is very close to $ 2/3$, was chosen by visual inspection of dispersion profiles for various values of $ a$.

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Copyright © 2005-12-28 by Julius O. Smith III<jos_email.html>
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