# Multibeam Wideband Transmit Beamforming Using 2D Sparse FIR Trapezoidal Filters
## 4 Full-Wave Electromagnetic Simulation Results
### Multibeam Beamforming
#### 4.1.2 Three Independent Signals Transmission in Three DODs
\begin{table}
\begin{tabular}{l c c c c c c c c c} \hline \hline \multirow{2}{*}{**Case**} & \multirow{2}{*}{\begin{tabular}{c} \((-28^{\circ},26^{\circ})\) DOD \\ (Nonsparse FIR) \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} \((-28^{\circ},26^{\circ})\) DOD \\ (Sparse FIR) \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} \((-45^{\circ},0^{\circ},45^{\circ})\) DOD \\ (Nonsparse FIR) \\ \end{tabular} } & \multirow{2}{*}{\
\begin{tabular}{c} \((-45^{\circ},0^{\circ},45^{\circ})\) DOD \\ (Sparse FIR) \\ \end{tabular} } & \multirow{2}{*}{
\begin{tabular}{c} \((-45^{\circ},0^{\circ},45^{\circ})\) DOD \\ (Sparse FIR) \\ \end{tabular} } \\ \hline \hline DOD (degrees) & \multirow{3}{*}{\(-28.2\)} & \multirow{3}{*}{\(26.0\)} & \multirow{3}{*}{\(-27.3\)} & \multirow{3}{*}{\(25.9\)} & \multirow{3}{*}{\(-45.0\)} & \multirow{3}{*}{\(0.0\)} & \multirow{3}{*}{\(44.9\)} & \multirow{3}{*}{\(-45.6\)} & \multirow{3}{*}{\(0.0\)} & \multirow{3}{*}{\(45.6\)} \\ Average & & & & & & & & \\ Standard deviation & \multirow{3}{*}{\(1.32\)} & \multirow{3}{*}{\(1.01\)} & \multirow{3}{*}{\(0.99\)} & \multirow{3}{*}{\(0.94\)} & \multirow{3}{*}{\(3.06\)} & \multirow{3}{*}{\(0.00\)} & \multirow{3}{*}{\(3.04\)} & \multirow{3}{*}{\(2.52\)} & \multirow{3}{*}{\(0.12\)} & \multirow{3}{*}{\(2.52\)} \\ Gain (dB) & & & & & & & & \\ Average & \multirow{3}{*}{\(10.44\)} & \multirow{3}{*}{\(10.56\)} & \multirow{3}{*}{\(10.28\)} & \multirow{3}{*}{\(10.47\)} & \multirow{3}{*}{\(8.25\)} & \multirow{3}{*}{\(8.52\)} & \multirow{3}{*}{\(8.29\)} & \multirow{3}{*}{\(7.77\)} & \multirow{3}{*}{\(8.37\)} & \multirow{3}{*}{\(7.99\)} \\ Side-to-beve \(\dagger\) (dB) & & & & & & & & & \\ Peak & \multirow{3}{*}{\(-17.18\)} & & \multirow{3}{*}{\(-13.83\)} & & \multirow{3}{*}{\(-17.76\)} & & \multirow{3}{*}{\(-16.54\)} & \multirow{3}{*}{\(\\\)Average & \multirow{3}{*}{\(-26.11\)}} & \multirow{3}{*}{\(-23.86\)} & & \multirow{3}{*}{\(-22.28\)} & & \multirow{3}{*}{\(-23.11\)} & \\ Filter passband ripple (dB) & & & & & & & & \\ Along main beam directions & \multirow{3}{*}{\(2.25\)} & \multirow{3}{*}{\(2.07\)} & \multirow{3}{*}{\(2.34\)} & \multirow{3}{*}{\(2.84\)} & \multirow{3}{*}{\(3.32\)} & \multirow{3}{*}{\(3.10\)} & \multirow{3}{*}{\(3.32\)} & \multirow{3}{*}{\(4.00\)} & \multirow{3}{*}{\(3.61\)} & \multirow{3}{*}{\(4.54\)} \\ In the defined passband & & & & & & & & & \\ Filter stopband attenuation (dB) & & & & & & & & \\ Average in the defined stopbands & \multirow{3}{*}{\(-28.86\)} & \multirow{3}{*}{\(-24.66\)} & & \multirow{3}{*}{\(-25.80\)} & \multirow{3}{*}{\(-23.36\)} & \multirow{3}{*}{\(120\)} & \multirow{3}{*}{\(164\)} & \multirow{3}{*}{\(120\)} & \multirow{3}{*}{\(164\)} & \multirow{3}{*}{\(120\)} \\ DSP complexity & & & & & & & & \\ ADDs & \multirow{3}{*}{\(974+974=1948\)} & \multirow{3}{*}{\(148+138=286\)} & \multirow{3}{*}{\(974+974=9222\)} & \multirow{3}{*}{\(120+164=120=404\)} & \multirow{3}{*}{\(161+83=61=205\)} \\ MULs & & & & & & & & & \\ \hline \hline \end{tabular} \({}^{\dagger}\) with respect to the average gain.  
\end{table} 
Table 1: Accuracy of the beam pattern and the DSP complexity of the proposed sparse transmit beamformer and a nonsparse counterpart for the MSMB mode.