


 Applications of planewave destruction filters  

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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 planewave filters to the response of the exact
phaseshift operator at low frequencies.
The Taylor series expansion of the phaseshift operator
around the zero frequency takes the form

(24) 
The Taylor expansion of the sixpoint implicit finitedifference operator
takes the form
Matching the corresponding terms of expansions (A1)
and (A2), we arrive at the system of nonlinear equations
System (A3A5) does not uniquely constrain the
filter coefficients , , and because
equation (A4) simply follows from (A3) andb
ecause all the coefficients can be multiplied simultaneously by an
arbitrary constant without affecting the ratios in
equation (A2). I chose an additional constraint in the
form

(29) 
which ensures that the filter does not alter the zero
frequency component. System (A3A5) with the
additional constraint (A6) resolves uniquely to
the coefficients of filter (9) in the main text:
The 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.)



 Applications of planewave destruction filters  

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