Omnidirectional plane-wave destruction

Directional separability

The combined linear system can be solved analytically. It is based on the property that $f_{k_1k_2}$ can be decoupled into the product of two terms, as shown in the appendix,

$$f_{k_1k_2}(p_1,p_2)=b_{k_1}(p_1)b_{k_2}(p_2),$$ (14)

where $b_{k_1}(p_1)$ and $b_{k_2}(p_2)$ denote coefficients of the 1D maxflat fractional delay filter used to approximate $Z_1^{p_1}$ and $Z_2^{p_2}$ respectively. Equation 14 implies that the 2D maxflat filter $H_2(Z_1,Z_2)$ is equivalent to a cascade of two 1D maxflat filters,

$$H_2(Z_1,Z_2)= \frac{B_1(1/Z_1)}{B_1(Z_1)} \frac{B_2(1/Z_2)}{B_2(Z_2)}.$$ (15)

$B_1(Z_1)$ and $B_2(Z_2)$ can be designed in the same way as for the line-interpolating PWD, whose coefficients are given by equation 5. $H_2(Z_1,Z_2)$ can be implemented by applying the 1D maxflat filter on each direction independently.

This separability of the maxflat linear phase filter extends to 3D and higher dimensions.

Omnidirectional plane-wave destruction