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Appendix A

Determining filter coefficients by Taylor expansion

This appendix details the derivation of equations (9) and (10). The main idea to match the frequency responses of the approximate plane-wave filters to the response of the exact phase-shift operator at low frequencies.

The Taylor series expansion of the phase-shift operator $e^{i \omega \sigma}$ around the zero frequency $\omega=0$ takes the form

e^{i \omega \sigma} \approx 1 + i\, \,\sigma\,\omega -
...frac{\sigma^3\,\omega^3}{6} +
\end{displaymath} (24)

The Taylor expansion of the six-point implicit finite-difference operator takes the form
$\displaystyle \frac{B_3(Z_t)}{B_3(1/Z_t)}$ $\textstyle =$ $\displaystyle \frac
{b_{-1}\,Z_t^{-1} + b_0 + b_1\,Z_t}
{b_1\,Z_t + b_0 + b_{-1...
...a} + b_0 + b_1\,e^{i \omega}}
{b_1\,e^{-i \omega} + b_0 + b_{-1}\,e^{i \omega}}$  
  $\textstyle \approx$ $\displaystyle 1 - \frac{2\,i \,\left( b_{-1} - b_1 \right) \,\omega}{b_0 + b_{-...
...\left( b_{-1} - b_1 \right)^2\,\omega^2}
{\left( b_0 + b_{-1} + b_1 \right)^2}$  
    $\displaystyle +
\left( b_{-1} - b_1 \right) \,
\left[ b_0^2 - b_0...
...^2 \right) \right] \,\omega^3}
{3\,\left(b_0 + b_{-1} + b_1\right)^3} + \ldots$ (25)

Matching the corresponding terms of expansions (A-1) and (A-2), we arrive at the system of nonlinear equations
$\displaystyle \sigma$ $\textstyle =$ $\displaystyle \frac{2\,\left( b_1 - b_{-1} \right) }
{b_0 + b_{-1} +b_1}$ (26)
$\displaystyle \sigma^2$ $\textstyle =$ $\displaystyle \frac{4\,\left( b_1 - b_{-1} \right)^2}
{\left(b_0 + b_{-1} +b_1 \right)^2}$ (27)
$\displaystyle \sigma^3$ $\textstyle =$ $\displaystyle \frac{2\,\left( b_1 - b_{-1} \right) \,
\left[ b_0^2 - b_0\,\lef...
...\,b_{-1}\,b_1 +
b_1^2 \right) \right] }
{\left(b_0 + b_{-1} + b_1 \right)^3}$ (28)

System (A-3-A-5) does not uniquely constrain the filter coefficients $b_{-1}$, $b_0$, and $b_1$ because equation (A-4) simply follows from (A-3) andb ecause all the coefficients can be multiplied simultaneously by an arbitrary constant without affecting the ratios in equation (A-2). I chose an additional constraint in the form
B_3(1) = b_{-1} + b_0 + b_1 = 1\;,
\end{displaymath} (29)

which ensures that the filter $B_3(Z_t)$ does not alter the zero frequency component. System (A-3-A-5) with the additional constraint (A-6) resolves uniquely to the coefficients of filter (9) in the main text:
$\displaystyle b_{-1}$ $\textstyle =$ $\displaystyle \frac{(1-\sigma)(2-\sigma)}{12}\;;$ (30)
$\displaystyle b_0$ $\textstyle =$ $\displaystyle \frac{(2+\sigma)(2-\sigma)}{6}\;;$ (31)
$\displaystyle b_1$ $\textstyle =$ $\displaystyle \frac{(1+\sigma)(2+\sigma)}{12}\;.$ (32)

The $B_5$ filter of equation (10) is constructed in a completely analogous way, using longer Taylor expansions to constrain the additional coefficients. Generalization to longer filters is straightforward.

The technique of this appendix aims at matching the filter responses at low frequencies. One might construct different filter families by employing other criteria for filter design (least squares fit, equiripple, etc.)

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