# Multibeam Wideband Transmit Beamforming Using 2D Sparse FIR Trapezoidal Filters
## 3 Proposed Low-Complexity Multibeam Wideband Transmit Beamformer
### Proposed Sparse 2D FIR Multibeam Beamformer  
The proposed transmit beamformer comprises a 2-D sparse FIR filter or a filter bank, DACs, RF front-end including power amplifiers (PAs), and a ULA of broadband antennas, as shown in Figure 2. The proposed transmit beamformer is fully digital and supports for \(K\leq N\) independently steerable wideband beams in two modes: 1) one wideband signal is transmitted in \(K\) independent directions (denoted as single-signal multi-beam (SSMB)) and 2) \(K\) independent wideband signals are transmitted in \(K\) independent directions (denoted as multi-signal multi-beam (MSMB)). For the SSMB case, we employ a 2-D sparse FIR filter having \(K\) trapezoidal passbands, and for the MSMB case, we employ a filter bank consisting of 2-D sparse FIR filters having singe trapezoidal passband. Trapezoidal passbands are chosen because of their constant half-angle \(\delta_{k}\) throughout the passbands, as shown in Figure 4. This enables the multibeam beamformer to generate constant beamwidth in the main lobes throughout the wideband frequency range. We note that, in the MSMB case, a dedicated 2-D sparse FIR filter is used for each independent far-field plane-wave corresponding to an independent information-carrying signal with a digital combination of plane-wave components after the filter bank, allowing a single set of DACs and RF chain to support \(K\) independent beams using the ULA. In order to explain the operation of the transmit beamformer, we consider the SSMB case, where the 1-D signal \(w(n_{ct})\), \(n_{ct}\in\mathbb{Z}\) is required to transmit in the \(K\) directions \(\theta_{1},\theta_{2},\ldots,\theta_{K}\). We call \(\theta_{i}\) (\(\in[-90^{\circ},90^{\circ}]\), \(i=1,2,\ldots,K\)) as the DODs. Here, broadband antennas are fed with the 2-D mixed-domain signal \(s_{m}(n_{x}\Delta_{x},ct)\), where \((n_{x},ct)\in\mathbb{Z}\times\mathbb{R}\), which consists of appropriately scaled and delayed continuous-time counterparts of \(w(n_{ct})\) in order to achieve transmit beamforming. Note that the electromagnetic waves emanating from broadband antennas constructively interfere in the far field to form a beampattern with \(K\) beams along the directions of \(\theta_{1},\theta_{2},\ldots,\theta_{K}\). The 2-D sparse FIR filter \(H(z_{x},z_{ct})\) with \(K\) trapezoidal passbands is employed to scale and delay \(w(n_{ct})\) appropriately to realize \(K\) beams, where the \(i\)th (\(=1,2,\ldots,K\)) trapezoidal passband provides the scaled and delayed \(w(n_{ct})\) for the \(i\)th beam. This is analogous to the use of a 2-D filter with \(K\) trapezoidal passbands to realize \(K\) beams in the receive mode beamforming, as reviewed in Section 2. The DACs and the PAs in the RF chains are used for the discrete-time to continuous-time conversion and to realize the required power gain, respectively. Now, let us consider the processing of the 2-D sparse FIR filter. The 2-D discrete-domain input signal \(s_{in}(n_{x},n_{ct})\), \((n_{x},n_{ct})\in\mathbb{Z}^{2}\) of \(H(z_{x},z_{ct})\) is defined as  
\[s_{in}(n_{x},n_{ct})= \begin{cases}w(n_{ct}),&n_{x}=0\\ 0,&\text{otherwise}.\end{cases}  \tag{6}\]  
\(H(z_{x},z_{ct})\) should have trapezoidal-shaped passbands in the principal Nyquist square \(\mathcal{N}\) of the 2-D discrete frequency domain \((\omega_{x},\omega_{ct})\in\mathbb{R}^{2}\), as shown in Figure 4, to achieve the required scaled and delayed versions of the wideband input signal \(w(n_{ct})\), in the 2-Ddiscrete-domain output signal \(s_{out}(n_{x},n_{ct})\) of the digital filter. Here, \(\mathcal{N}=\{(\omega_{x},\omega_{ct})\in\mathbb{R}^{2}\), \(-\pi\leq\omega_{x},\omega_{ct}<\pi\}\). The output signal \(s_{out}(n_{x},n_{ct})\) is given by  
\[s_{out}(n_{x},n_{ct})=\underset{(i_{x},i_{ct})\in\mathcal{I}}{ \sum}h_{s}(i_{x},i_{ct})s_{in}(n_{x}-i_{x},n_{ct}-i_{ct}) \tag{7}\]  
where \(\mathcal{I}\) is the set containing the indices of the nonzero coefficients of the impulse response \(h_{s}(i_{x},i_{ct})\) of \(H(z_{x},z_{ct})\).