Omnidirectional plane-wave destruction

Appendix: Separability of the 2D maxflat filter

Following Thiran (1971), the maxflat condition of 1D fractional delay filter $B_1(1/Z_1)/B_1(Z_1)$ is expressed as

$$\sum_{k_1=-N}^N(\frac{p_1}{2}-k_1)^{2j_1+1}b_{k_1}(p_1)=0\;, \qquad j_1=0,\dots,2N-1.$$ (22)

With the additional constraint (Chen et al., 2013)

$$\sum_{k_1=-N}^Nb_{k_1}(p_1)=1,$$ (23)

we can obtain the unique $b_{k_1}(p_1)$ . Similarly, $b_{k_2}(p_2)$ satisfies the following linear system:

$$\left\{\begin{array}{l} \displaystyle{\sum_{k_2=-N}^N(\frac{p_2}{... ...,\dots,2N-1 \displaystyle{\sum_{k_2=-N}^Nb_{k_2}(p_2)=1} \end{array}\right..$$ (24)

Substituting $b_{k_1}(p_1)b_{k_2}(p_2)$ into the 2D maxflat equation 12, for all $j_1,j_2=0,\dots,2N-1$ , we obtain

$$\sum_{k_1=-N}^N(\frac{p_1}{2}-k_1)^{2j_1+1}b_{k_1}(p_1) \sum_{k_2=-N}^N(\frac{p_2}{2}-k_2)^{2j_2+1}b_{k_2}(p_2)=0.$$ (25)

Also, for $j_1=2N$ , and $j_2=0,\dots,2N-1$ or $j_2=2N$ , and $j_1=0,\dots,2N-1$ , equation 12 still holds true.

Substituting $b_{k_1}(p_1)b_{k_2}(p_2)$ into the additional constraint in Equation 13,

$$\sum_{k_1=-N}^Nb_{k_1}(p_1)\sum_{k_2=-N}^Nb_{k_2}(p_2)=1.$$ (26)

In other words, $b_{k_1}(p_1)b_{k_2}(p_2)$ is a solution of the combined linear system of equation 12 and 13. This linear system must have a unique solution, therefore

$$f_{k_1k_2}(p_1,p_2)=b_{k_1}(p_1)b_{k_2}(p_2).$$ (27)

Omnidirectional plane-wave destruction