# Design of True Time Delay Millimeter Wave Beamformers for 5G Multibeam Phased Arrays
## 1 Introduction
Specifically, such systems are comprised of the following components, as shown in Figure 2: (i) the antenna array; (ii) the \"RF precoding\", i.e., the analog RF beamformer that is driven by a number of transmitting RF chains; and (iii) a number of digital-to-analog converters (DACs) that combine the \"digital baseband precoding\" (which plays the role of the software defined radio (SDR) for the transmitter) into the analog RF beamforming networks. The receiver follows the same logic and appears at the bottom of Figure 2. Notably, hybrid beamforming combines the low cost, simplicity, and ease of implementation of analog beamformers with powerful abilities of digital beamformers, such as comprehensive interference rejection and beamsteering towards any desired position (at least theoretically) [25]. Therefore, by combining the analog and digital beamformers, novel communication systems can be developed that meet the needs of 5G [26]. Our paper is focused on analogue beamformers that represent an important component of hybrid beamforming systems. Figure 1: Illustration of different 5G communication cells. Many different analog RF beamforming implementations have been developed during the last 60 years following the topologies proposed by Butler [27], [PERSON] [28], and Nolen [29] matrices, as well as Rotman lenses [30]. The most common analog beamforming network is based on the Butler matrix, which is known for its high efficiency. However, it can only produce uniform amplitude distributions that are known for their high side-lobe level (\(SLL\)) values [31], and high number of crossovers that cause many undesired effects, such as increased insertion loss, mismatched junctions, etc. [32]. In the literature, limited works have been proposed for the development of Butler matrices with non-uniform amplitude distributions [33] and the reduction of crossovers [34]. Additionally, Butler matrices exhibit significantly narrower bandwidths than the most wideband antenna arrays currently available. A design with bandwidth of 3:1 (based on a \(10-dB\) return loss), \(1-dB\) insertion loss variation, and \(7^{\circ}\) phase deviation was recently reported by [PERSON] et al. [35]. Even though Butler matrices have been extensively used at the frequency range of 1-4 GHz, there are limited works at higher frequencies and mm-Waves. [PERSON] et al. [36] introduced a \(4\times 4\) two-layer Butler matrix at 5.8 GHz using coplanar wave-guide technology. [PERSON] et al. [37], introduced a compact substrate integrated waveguide (SIW) multi-folded \(4\times 8\) Butler matrix combined with a \(8\times 10\) SIW slot-coupled patch array at 38 GHz accomplishing a size reduction of 53.5% in its longitudinal direction. [PERSON] et al. [38] introduced a new class of Butler matrices with inherent bandpass filter transfer functions. Specifically, they presented the synthesis and design of a \(2\times 2\), 180 hybrid coupler at 10 GHz and a \(4\times 4\) Butler matrix with an equal-ripple four-pole Chebyshev bandpass characteristic centered at 12.5 GHz. More recently [PERSON] et al. [39] presented a dually-polarized Butler matrix for base stations with polarization diversity operating at 60 GHz. Finally, [PERSON] et al. [40] utilized additive manufacturing techniques to introduce a \(3D\)-printed modified butler matrix for a monopulse radar based on gap waveguides at W-Band. Nolen matrices outperform Butler matrices as they can produce an arbitrary number of beams. However, similar to Butler matrices, they are usually limited to uniform amplitude tapering and only  
Figure 2: Hybrid beamforming architecture. few reported attempts have attempted to fix this problem [41]. In addition, Nolen networks operate at relatively narrowband frequency range, compared to wideband antenna arrays. [PERSON] et al. [42], introduced a relatively broadband Nolen matrix using substrate integrated waveguide (SIW) technology, which provided constant phase over a 11.7% frequency bandwidth centered at 77 GHz.