# Multibeam Wideband Transmit Beamforming Using 2D Sparse FIR Trapezoidal Filters
## 4 Full-Wave Electromagnetic Simulation Results
### Single-Beam Beamforming  
Here, beamforming is achieved in a single direction, where the DOD is \(\theta\in[-90^{\circ},90^{\circ}]\). As an example of single-direction beamforming, consider the case where the DOD is \(30^{\circ}\), where \(\alpha=26.57^{\circ}\). We design 2-D FIR trapezoidal filters using the windowing method and optimization techniques with nonsparse and sparse impulse responses. Here, the design of the filter with the windowing technique is adapted from [39; 40] for the nonsparse impulse response whereas from [32] for the sparse impulse response. Furthermore, nonsparse filter designed using the optimization technique is adapted from [41] (without considering the mutual-coupling compensation for fair evaluation). For the windowing technique, the Dolph-Chebyshev window is used as the window function, because a Dolph-Chebyshev \begin{table}
\begin{tabular}{l c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Case**} & \multirow{2}{*}{\(\begin{matrix}(-28^{\circ},26^{\circ})\) DOD} \\ (**Nonsparse FIR**) \\ \end{tabular} & \multirow{2}{*}{\(\begin{matrix}(-28^{\circ},26^{\circ})\) DOD} \\ (**Sparse FIR**) \\ \end{tabular} & \multirow{2}{*}{\(\begin{matrix}(-28^{\circ},26^{\circ})\) DOD} \\ (**Sparse FIR**) \\ \end{tabular} & \multirow{2}{*}{\(\begin{matrix}(-48^{\circ},0^{\circ})\) DOD} \\ (**Nonsparse FIR**) \\ \end{tabular} & \multirow{2}{*}{\(\begin{matrix}(-48^{\circ},0^{\circ})\) DOD} \\ (**Sparse FIR**) \\ \end{tabular} & \multirow{2}{*}{\(\begin{matrix}(-48^{\circ},0^{\circ})\) DOD} \\ (**Sparse FIR**) \\ \end{tabular} \\ \hline DOD (degrees) & & & & & & & & & & & \\ Average & & & & & & & & & & & \\ Standard deviation & 1.17 & 1.03 & 1.23 & 1.46 & 2.28 & 0.00 & 2.24 & 2.22 & 0.00 & 2.21 \\ Gain (dB) & & & & & & & & & & \\ Average & & & & & & & & & & & \\ Side-lobe level \({}^{\dagger}\) (dB) & & & & & & & & & & & \\ Peak & & & & & & & & & & & \\ Average & & & & & & & & & & & \\ Filter passband ripple (dB) & & & & & & & & & & & \\ Along main beam directions & 1.84 & 2.98 & 2.64 & 2.61 & 1.23 & 2.21 & 1.23 & 1.23 & 1.90 & 2.81 & 1.90 \\ In the defined passband & & & & & & & & & & & \\ Filter stopband attenuation (dB) & & & & & & & & & & & \\ Average in the defined stopband & & & & & & & & & & & \\ DSP complexity & & & & & & & & & & \\ ADDs & & & & & & & & & & \\ MULs & & & & & & & & & & & \\ \hline \hline \end{tabular}  
* with respect to the average gain.  
\end{table} 
Table 2: Accuracy of the beam pattern and the DSP complexity of the proposed sparse transmit beamformer and a nonsparse counterpart for the SSMB mode. window provides an approximately equiripple frequency magnitude response ([55], Chapter 9.4), and therefore an approximately minimax design. Furthermore, for the sparse impulse response [32], hard thresholding with a threshold of \(2.544\times 10^{-4}\) is employed. The order of all of the filters is selected as \(14\times 64\), and the half-angle of the trapezoidal filter \(\delta\) is selected as \(3^{\circ}\). Furthermore, the maximum transition band is \(0.1\pi\) rad/sample. The proposed filter is designed with the parameters \(\mu=0.05\) and \(\epsilon_{th}=0.007144\). Furthermore, the passband and stopband weights are taken as \(1\), and the region \([-\pi,\pi]\times[0,\pi]\) in the 2-D frequency domain \((\omega_{x},\omega_{ct})\) is uniformly discretized into \(201\times 101\) points. The 2-D sparse FIR filters, i.e., those proposed and designed adapting the method in [41], have only 143 nonzero coefficients, whereas nonsparse filters have 975 coefficients. The magnitude frequency responses of the designed filters are shown in Figure 9a-d. Furthermore, Table 3 provides a comparison of the accuracy of the wideband beam patterns and DSP complexity for the proposed transmit beamformer and previously proposed filter designs.