Although the choice of the free parameter which gives a maximally direction-independent numerical dispersion profile has been made, in the past, through computerized optimization procedures [157], 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
, or, equivalently, in terms of the function
. It should be clear, then, that if
is directionally-independent, then so are the amplification factors, and thus the numerical phase velocity (see §A.1.4) as well. Ideally, we would like
to be a function of the spectral radius
** alone. Now examine the Taylor expansion of
about :
**

The directionally-independent term reflects the fact that the scheme is consistent with the wave equation; higher order terms in general show directional dependence. The choice of , however, gives

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