## 3 Methods
### Beamformer Design  
If the \(2N\)-th target broadside beampattern is given, (11) can also be formulated in the modal domain with a symmetric form  
\[B^{(2N)}(k,\theta)=\sum\limits_{n=-2N}^{2N}\gamma_{n}e^{in\theta} \tag{31}\]  
where  
\[\gamma_{n}=\left\{\begin{matrix}0,&n=\pm 1,\pm 3,\ldots,\pm(2N-1),\\ \sum\limits_{p=|n|/2}^{N}\frac{\alpha_{N,p}}{2^{2\mathbf{p}}}\binom{2p}{p-|n| /2},&n=0,\pm 2,\pm 4,\ldots,\pm 2N.\end{matrix}\right. \tag{32}\]  
where \(\left(\cdot\atop\cdot\right)\) is combinations, and \(|\cdot|\) represents absolute value. With the vector form  
\[B^{(2N)}(k,\theta)=\mathbf{\gamma}^{T}\mathbf{e}(\theta) \tag{33}\]  
where  
\[\mathbf{\gamma}=\left[\gamma_{-2N}\quad\cdots\quad\gamma_{2N}\right]^{T} \tag{34}\]  
\[\mathbf{e}(\theta)=\left[e^{-2N\theta}\quad\cdots\quad e^{2N\theta}\right]^{T} \tag{35}\]  
In the series expansion method, the Jacobi-Anger expansion is used to match the resulting beampattern to the target beampattern in (31). The Jacobi-Anger expansion is  
\[e^{-i\xi}\cos\theta=\sum\limits_{n=-\infty}^{+\infty}\beta_{n}(\xi)e^{in\theta} \tag{36}\]  
where  
\[\beta_{n}(\xi)=(-i)^{n}J_{n}(\xi). \tag{37}\]  
is the circular harmonic coefficient and \(J_{n}(\cdot)\) is the nth-order Bessel function of the first kind.  
Applying (36) into the exponential of (2) yields  
\[B(k,\theta)=\sum\limits_{l=-L_{0}}^{L_{0}}w_{l}^{*}(k)\sum\limits_{n=-\infty} ^{+\infty}\beta_{n}(kx_{l})e^{in\theta} \tag{38}\]  
In order to obtain a \(2N\)th-order broadside target radiation pattern, the infinite series is truncated to the order \(2N\):  
\[B(k,\theta)=\sum\limits_{n=-2N}^{2N}e^{in\theta}\sum\limits_{l=-L_{0}}^{L_{0} }\beta_{n}(kx_{l})w_{l}^{*}(k). \tag{39}\]  
If the radiation pattern in (39) is consistent with the target beampattern in (31), one can obtain  
\[\sum\limits_{l=-L_{0}}^{L_{0}}\beta_{n}(kx_{l})w_{l}^{*}(k)=\gamma_{n}\quad n =0,\pm 1,\ldots,\pm 2N \tag{40}\]  
For the design of a broadside differential beamformer, the distortionless constraint in the broadside direction is required:  
\[\mathbf{w}^{H}\mathbf{g}(k,\pi/2)=1 \tag{41}\]  
Combing (40)-(41),  
\[\mathbf{\Phi w}=\eta \tag{42}\]where  
\[\mathbf{\Phi}=\begin{bmatrix}\beta_{-2N}&\cdots&\beta_{2N}&\mathbf{g}(k,\pi/2)\end{bmatrix}^ {H} \tag{43}\]  
\[\beta_{n}=\begin{bmatrix}\beta_{n}(kx_{-L_{0}})&\cdots&\beta_{n}(kx_{L_{0}}) \end{bmatrix}^{T} \tag{44}\]  
\[\eta=\begin{bmatrix}\gamma^{T}&1\end{bmatrix}^{T} \tag{45}\]  
With the fact that \(\beta_{n}(\cdot)=\beta_{-n}(\cdot)\) and (32), one can obtain  
\[\widetilde{\mathbf{\Phi}}\mathbf{w}=\widetilde{\eta} \tag{46}\]  
where  
\[\widetilde{\mathbf{\Phi}}=\begin{bmatrix}\beta_{0}&\cdots&\beta_{2N}&\mathbf{g}(k,\pi/2)\end{bmatrix}^{H} \tag{47}\]  
is \((2N+2)\times L\) full-rank matrix  
\[\widetilde{\eta}=\begin{bmatrix}\widetilde{\gamma}^{T}&1\end{bmatrix}^{T} \tag{48}\]  
where \(\widetilde{\mathbf{\gamma}}=\begin{bmatrix}\gamma_{0}&\cdots&\gamma_{2N}\end{bmatrix}^ {T}\) is the coefficients of the desired beampattern. With the number loudspeakers \(L>2N+2\), the minimum-norm solution of (46) is  
\[\mathbf{w}=\widetilde{\mathbf{\Phi}}^{H}(\widetilde{\mathbf{\Phi}}\widetilde{\mathbf{ \Phi}}^{H})^{-1}\widetilde{\mathbf{\eta}} \tag{49}\]  
To deal with the ill-posed inverse problems existing in (49), Tikhonov regularization is used to calculate the minimum-norm solution,  
\[\mathbf{w}=\widetilde{\mathbf{\Phi}}^{H}(\widetilde{\mathbf{\Phi}}\widetilde{\mathbf{\Phi }}^{H}+\mu\mathbf{I}_{2N+2})^{-1}\widetilde{\mathbf{\eta}} \tag{50}\]  
where \(\mu\) is the regularization parameter, which is a preselected non-negative number to enhance robustness.