Omnidirectional plane-wave destruction

2D maxflat condition

The frequency response of the objective phase shift operator is $Z_1^{p_1}Z_2^{p_2}=e^{\mathrm j(w_1p_1+w_2p_2)}$ , where, $w_1,w_2$ are frequencies in radius in vertical and horizontal directions. We must design the coefficients $f_{k_1k_2}$ so that the allpass system $H_2(Z_1,Z_2)$ can obtain a similar linear phase response. The frequency response of $H_2(Z_1,Z_2)$ is

$$H_2(e^{\mathrm jw_1},e^{\mathrm jw_2}) =\frac{ F^*(e^{\mathrm jw_... ...2}) }{ F(e^{\mathrm jw_1},e^{\mathrm jw_2}) }=e^{-\mathrm j2\theta_F(w_1,w_2)},$$ (8)

where $\theta_F(w_1,w_2)$ is the phase of $F(Z_1,Z_2)$ , which takes the following form:

$$\theta_F(w_1,w_2)=-\tan^{-1}\frac{ \displaystyle{\sum_{k_1=-N}^N\... ... \displaystyle{\sum_{k_1=-N}^N\sum_{k_2=-N}^N f_{k_1k_2}\cos(k_1w_1+k_2w_2)} }.$$ (9)

The phase approximating error is $w_1p_1+w_2p_2+2\theta_F(w_1,w_2)$ . In order to obtain an analytical $f_{k_1k_2}$ , we remove $\tan^{-1}$ and redefine the phase approximating error as

$$\epsilon(w_1,w_2) = \tan(\frac{w_1p_1+w_2p_2}{2}) -\frac{ \displaystyle{\sum_{k_1=-N}... ...{ \displaystyle{\sum_{k_1=-N}^N\sum_{k_2=-N}^N f_{k_1k_2}\cos(k_1w_1+k_2w_2)} }$$

$$= \frac{ \displaystyle{\sum_{k_1=-N}^N\sum_{k_2=-N}^N f_{k_1k_2}\si... ... \displaystyle{\sum_{k_1=-N}^N\sum_{k_2=-N}^N f_{k_1k_2}\cos(k_1w_1+k_2w_2)} }.$$ (10)

The sine function in the numerator can be expressed by 2D Taylor's expansion as

$$\sin(w_1(\frac{p_1}{2}-k_1)+w_2(\frac{p_2}{2}-k_2))$$

$$= \sum_{j_1=0}^\infty\sum_{j_2=0}^\infty (-1)^{j_1+j_2}\frac{ (\fra... ...+1}(\frac{p_2}{2}-k_2)^{2j_2+1} }{(2j_1+1)!(2j_2+1)!} w_1^{2j_1+1}w_2^{2j_2+1}.$$

We use the maxflat phase criterion (Thiran, 1971), which means that the filter has a phase response as flat as the desired linear response. In the 2D case, the criterion is equivalent to the mathematical expression that the partial derivatives of the error function should be as small as possible. We set them to be zero:

$$\frac{\partial^{j_1+j_2}\epsilon(w_1,w_2)} {\partial w_1^{j_1}\partial w_2^{j_2}}=0 \qquad j_1,j_2=0,1,\dots\;,$$ (11)

which is equivalent to the following 2D maxflat condition:

$$\sum_{k_1=-N}^N\sum_{k_2=-N}^N (\frac{p_1}{2}-k_1)^{2j_1+1}(\frac{p_2}{2}-k_2)^{2j_2+1}f_{k_1k_2}=0.$$ (12)

Omnidirectional plane-wave destruction