# Multibeam Wideband Transmit Beamforming Using 2D Sparse FIR Trapezoidal Filters
### Proposed 2-D FIR Transmit Beamformer  
We propose a fully digital transmit beamforming method based on 2-D sparse FIR trapezoidal filters and a uniform linear array (ULA) of wideband Vivaldi antennas [33], as shown in Figure 2. The proposed transmit beamformer is different from and improves on the standard FIR filter-and-sum transmit beamformers [34; 35; 36], where separate one-dimensional (1-D) FIR filters are employed for each antenna to implement the fractional time delays to achieve wideband operation. In particular, the proposed 2-D sparse FIR is optimal in the minimax sense and is designed using convex optimization techniques. Full-wave electromagnetic simulations using real antenna models confirm that the proposed wideband transmit beamformer can achieve multibeam transmission in the frequency range of 1.3-2.8 GHz, with more than 70% fractional bandwidth. Importantly, the proposed transmit beamformer provides considerable reduction in the computational complexity compared with nonsparse filter-and-sum transmit beamformers [34; 35; 36], 2-D nonsparse FIR filters [37; 38; 39; 40; 41; 42], and 2-D sparse FIR filter [32], without deteriorating beam directionality and causing increases in the side-lobe level. Furthermore, the computational efficiency of our sparse FIR filter design makes them suitable for real-time beamforming applications in wideband systems, while the linear phase response within the passband of the filter contributes to maintaining signal fidelity during beam steering.  
The proposed sparse 2-D FIR multibeam beamformer architecture is well-suited for both sub-6 GHz (legacy) band transmit beamforming, frequency range three (FR3) beamforming up to 28 GHz, as well as mmWave operation for emerging 5G/6G applications. Multibeam beamforming systems with sparse trapezoidal FIR filters offer several advantages, including flexibility in beam control, improved signal quality due to sharp transition bands, and the potential for real-time implementation with the sparse filter design. These advantages make them suitable for various applications. In the context of wireless communications, our work on multibeam beamforming could be beneficial for cellular networks to improve network capacity and user experience, especially in dense urban environments [43; 44]. Additionally, for mmWave communication systems, the ability to form narrow beams with low side-lobes using 2D trapezoidal FIR filters is crucial for establishing high-data-rate links [29; 30]. Multibeam beamforming allows radars to track multiple targets simultaneously or scan a wider area electronically [45; 46; 47]. Two-dimensional trapezoidal FIR filters can help shape the radar beam for specific applications like target tracking orclutter suppression. We note that the underlying concepts are equally applicable for receive mode as well, even though we present only the transmit mode in this paper. Therefore, in remote sensing, multibeam systems can be used to acquire data from different observation angles [48; 49]. Furthermore, 2D trapezoidal FIR filters can help achieve the desired beam patterns for specific features of interest in the scene.