# Multibeam Wideband Transmit Beamforming Using 2D Sparse FIR Trapezoidal Filters
## 2 Review of Spectra of 2D Plane Waves Received by Uniform Linear Arrays  
We briefly review the spatiotemporal modeling of 2D plane waves received by a ULA and their spectra in this section. We consider the receive mode for the review because most of the related works on wideband beamforming using 2-D and three-dimensional filters consider receive-mode beamforming. A plane wave received by a ULA having \(N+1\) wideband antennas placed along the \(x\) axis in the three-dimensional space \((x,y,z)\in\mathbb{R}^{3}\) is shown in Figure 3a. Such a plane wave can be modeled as a 2-D discrete-domain spatiotemporal signal, after synchronous analog to digital conversion, which can be expressed as [50; 51]  
\[pw(n_{x},n_{ct})=w(\sin(\theta)n_{x}\Delta_{x}+n_{ct}\Delta_{ct}), \tag{1}\]  
where \((n_{x},n_{ct})\in\mathbb{Z}^{2}\), \(\theta\) (the zenith angle) denotes the direction of arrival (DOA) of the 2-D plane wave, \(\Delta_{x}\) and \(\Delta_{ct}\) are the sampling intervals in the \(x\) and \(ct\) domains, respectively, \(c\) is the constant speed of propagation of an electromagnetic wave, and \(w(n_{ct})\) specifies the 1-D temporal signal carried by the 2-D plane wave. The _ideal_ discrete-domain spectrum \(PW(\omega_{x},\omega_{ct})\) of \(pw(n_{x},n_{ct})\) can be obtained as [50; 51]  
\[PW(\omega_{x},\omega_{ct})=W(\omega_{ct})\delta(\omega_{x}-\sin(\theta)\omega_ {ct}), \tag{2}\]  
where \((\omega_{x},\omega_{ct})\in\mathbb{R}^{2}\), \(\omega_{ct}=\omega_{t}/c\), \(W(\omega_{ct})\) is the spectrum of \(w(n_{ct})\), and \(\delta(\cdot)\) is the 1-D continuous-domain impulse function. Note that \(\omega_{x}\) is the discrete-domain spatial frequency, and \(\omega_{ct}\) is the discrete-domain temporal frequency scaled by \(c\). The region of support (ROS) \(\mathcal{R}\) of \(PW(\omega_{x},\omega_{ct})\) can be obtained as [50; 51]  
\[\mathcal{R}=\{(\Omega_{x},\Omega_{ct})\in\mathbb{R}^{2}|(\Omega_{x}-\sin( \theta)\Omega_{ct})=0,W(\Omega_{ct})\
eq 0\}, \tag{3}\]  
which lies on a straight line going through the origin of the 2-D discrete frequency domain, as shown in Figure 3b. The angle \(\alpha\) between the ROS of the plane wave and the \(\omega_{ct}\) axis is given by [39; 50]  
\[\alpha=\tan^{-1}(\sin(\theta)). \tag{4}\]  
Figure 2: Architecture of the proposed transmit beamformer for the MSMB case, consisting of a filter bank of 2D sparse FIR trapezoidal filters, DACs, and RF front end, as well as a uniform linear array of broadband antennas. In the case of \(K\)\((\in\mathbb{Z}^{+})\) plane waves having different DOAs \(\theta_{k}\) and \(w_{k}(ct)\), \(k=1,2,\ldots,K\), received by a ULA, the _ideal_ discrete-domain spectrum \(PW_{K}(\omega_{x},\omega_{ct})\) is given by  
\[PW_{K}(\omega_{x},\omega_{ct})=\sum_{k=1}^{K}W_{k}(\omega_{ct})\delta(\omega_{ x}-\sin(\theta_{k})\omega_{ct}). \tag{5}\]  
In this case, the spectral ROS \(\mathcal{R}_{K}\) lies on \(K\) straight lines going through the origin inside the principal Nyquist square of the 2-D discrete frequency domain with \(\alpha_{k}=\tan^{-1}(\sin(\theta_{k}))\). In practice, the true ROS of the generated discrete-domain 2-D plane wave is a narrow parallelogram when the number of antennas is finite. In the case of a 2-D _broadband bandpass_ plane wave having a temporal bandwidth \(B_{ct}\) with center frequency \(\Omega_{ct}^{C}\), as shown in the Figure 4, the desired 2-D plane wave (PW) (signal of interest) can be extracted by using a 2D filter having _narrow-angle double-trapezoidal-shaped passbands_[39; 40]. This will attenuate noise and radio frequency interference (RFI) having DOAs that are different from the DOA of the signal of interest. The same analysis can be applied to the transmit beamforming, where 2-D wideband plane waves are emitted by a ULA in \(K\) different directions, i.e, with direction of departures  
Figure 4: Spectral ROS of broadband bandpass 2D plane wave having different DOAs or DODs, as well as multiple trapezoidal passbands of the 2D FIR filter. (DODs) \(\theta_{k}\); \(k=1,2,\ldots,K\).