# Multibeam Wideband Transmit Beamforming Using 2D Sparse FIR Trapezoidal Filters
## 3 Proposed Low-Complexity Multibeam Wideband Transmit Beamformer
### Minimax Design of the 2D Sparse FIR Filter
#### 3.3.1 Step 1  
We obtain an intermediate sparse impulse response in the weighted minimax sense. The objective function to be minimized is expressed as  
\[\underset{\mathbf{h}}{\text{minimize}}\ \left[\max_{(\omega_{x}, \omega_{ct})\in\mathcal{F}} W(\omega_{x},\omega_{ct})|H(e^{i\omega_{x}},e^{i\omega_{ct}})\right.\] \[-H_{d}(e^{i\omega_{x}},e^{i\omega_{ct}})|+\mu\|\mathbf{h}\|_{1}\right] \tag{16}\]  
where \(H(e^{i\omega_{x}},e^{i\omega_{ct}})\) is given in (13), \(H_{d}(e^{i\omega_{x}},e^{i\omega_{ct}})\) is given in (10), and \(W(\omega_{x},\omega_{ct})\) is a weighting function used to control the passband ripple and stopband ripple, which affects the side-lobes of the transmit beamformer across the bandwidth. Parameter \(\mu\) is a small positive number (typically between 0.001 and 1 [54]), and \(\mathcal{F}\) is the region corresponding to the passband and stopband, i.e., without the transition band [52]. Due to the centrosymmetric impulse response of \(H(e^{i\omega_{x}},e^{i\omega_{ct}})\), only the region \([-\pi,\pi]\times[0,\pi]\) in the 2-D frequency domain \((\omega_{x},\omega_{ct})\) is considered to define \(\mathcal{F}\). By introducing an upper bound \(\beta\) for \(W(\omega_{x},\omega_{ct})|H(e^{i\omega_{x}},e^{i\omega_{ct}})-H_{d}(e^{i\omega _{x}},e^{i\omega_{ct}})|\) over a finite \(M\) set of frequency grids \(\Omega_{d}=\{\omega_{i}=(\omega_{1}^{(i)},\omega_{2}^{(i)}),1\leq i\leq M\} \subseteq\mathcal{F}\), (16) is reduced to a tractable constrained problem as  
\[\underset{\mathbf{h}}{\text{minimize}}\ \beta+\mu\|\mathbf{h}\|_{1}\]\[\text{subject to : }W(\omega_{i})|H(e^{i\omega_{1}^{(i)}},e^{i\omega_{2}^{(i)} })-\] \[H_{d}(e^{i\omega_{1}^{(i)}},e^{i\omega_{2}^{(i)}})|\leq\beta,1\leq i\leq M \tag{17}\]  
To deal with the nondifferentiability of \(\|\mathbf{h}\|_{1}\), taking an upper bound for each entry of \(\mathbf{h}\), i.e.,  
\[|h_{i}|\leq d_{i}\text{ for }1\leq i\leq n \tag{18}\]  
with \(n=1+\frac{N_{t}+N_{t}N_{t}+N_{c}}{2}\), leads (17) to  
\[\text{minimize }\beta+\mu\sum_{i=1}^{n}d_{i}\] \[\text{subject to : }W(\omega_{i})|\mathbf{c}(\omega_{i})^{\text{T}} \mathbf{h}-\] \[H_{d}(e^{i\omega_{1}^{(i)}},e^{i\omega_{2}^{(i)}})|\leq\beta \text{ for }1\leq i\leq M\] \[|h_{i}|\leq d_{i}\text{ for }1\leq i\leq n \tag{19}\]  
Treating the bounds \(\beta\) and \(d_{i}(1\leq i\leq n)\) as auxiliary design variables in (19), the optimization problem becomes a convex optimization problem, which can be converted as a linear programming (LP) problem [52] expressed as  
\[-H_{d}(e^{i\omega_{x}},e^{i\omega_{ct}})|+\mu\|\mathbf{h}\|_{1}\right] \tag{16}\]0 \[-H_{d}(e^{i\omega_{x}},e^{i\omega_{ct}})|+\mu\|\mathbf{h}\|_{1}\right] \tag{16}\]1  
where  
\[\mathbf{d} = \begin{bmatrix}d_{1}\\ d_{2}\\ \vdots\\ d_{n}\end{bmatrix} ,\mathbf{h}= \begin{bmatrix}h_{1}\\ h_{2}\\ \vdots\\ h_{n}\end{bmatrix} ,\mathbf{e}_{\mathbf{n}}= \begin{bmatrix}1\\ 1\\ \vdots\\ 1\end{bmatrix} _{n\times 1},\mathbf{x}= \begin{bmatrix}\beta\\ \mathbf{d}\\ \mathbf{h}\end{bmatrix} ,\] \[\mathbf{f} = \begin{bmatrix}1\\ \mu\mathbf{e}_{\mathbf{n}}\\ \mathbf{0}\end{bmatrix} ,\mathbf{A}= \begin{bmatrix}\mathbf{A}_{\mathbf{1}}\\ \mathbf{A}_{\mathbf{2}}\end{bmatrix} ,\mathbf{b}= \begin{bmatrix}\mathbf{b}_{ \mathbf{1}}\\ \mathbf{0}\end{bmatrix} , \tag{21}\]  
with  
\[\mathbf{A}_{\mathbf{1}}= \begin{bmatrix}1&\mathbf{0}&W(\omega_{1}) \mathbf{c}^{\text{T}}(\omega_{1})\\ \vdots&\vdots&\vdots\\ 1&\mathbf{0}&W(\omega_{M})\mathbf{c}^{\text{T}}(\omega_{M})\\ 1&\mathbf{0}&-W(\omega_{1})\mathbf{c}^{\text{T}}(\omega_{1})\\ \vdots&\vdots&\vdots\\ 1&\mathbf{0}&-W(\omega_{M})\mathbf{c}^{\text{T}}(\omega_{M})\end{bmatrix} , \mathbf{A}_{\mathbf{2}}= \begin{bmatrix}\mathbf{0}&\mathbf{I}_{\mathbf{n}}& \mathbf{I}_{\mathbf{n}}\\ \mathbf{0}&\mathbf{I}_{\mathbf{n}}&-\mathbf{I}_{\mathbf{n}}\end{bmatrix} ,\] \[\mathbf{b}_{\mathbf{1}}=