# Design of True Time Delay Millimeter Wave Beamformers for 5G Multibeam Phased Arrays
## 2 Beamforming Networks Design  
### Blass Matrix Architecture  
In this task, we pursue the development of a novel dual-layer Blass matrix. Typical Blass matrices are large in size and suffer from excessive losses. Aiming to compact and highly efficient mm-Wave beamforming networks in this work, the classical Blass matrix topology is first transformed into a circular design, and then implemented in a two-layer topology. To our knowledge, this is the first multi-layer Blass design. Compared to the typical single-layer Blass matrix, the proposed dual-layer network achieves 60% size reduction. An additional key point of our implementation are the couplers. Typical approaches use bondwires and bridges [48], two factors that decrease the efficiency of the beamformer and increase the design complexity in regular Blass matrices. Here, to minimize the losses and significantly reduce the design complexity, a dual-layer directional coupler is designed and used. The Blass matrix was first introduced in 1960 [28], and its network is shown in Figure 3. In its original form, the Blass matrix consists of \(N\) traveling wave feed lines (rows) cross-connected to a set of \(M\) transmission lines (columns), each one feeding an antenna element of the array. \(M\) and \(N\) are independent and they can be chosen arbitrarily, but always the number of beams \(M\) should be less (or equal) to the number of antenna elements. The other end of the line is terminated in a matched load (see Figure 3). This beamforming network provides \(M\) beams with an array of \(N\) elements. The interconnections are implemented with directional couplers of unequal power divisions as shown in Figure 3. For a uniform amplitude excitation of the array, the first (out of \(N\)) row of couplers divides the power to a \(1/N\) portion that flows towards the antenna element and a \((N-1)/N\) portion that flows along the column towards the rest of the network. Likewise, the second row of couplers divides the power into ratios of \(1/(N-1)\) and \((N-2)/(N-1)\) and so on. In addition, as long as we are moving farther away from the feeding ports and towards the radiating elements (element 1, element 2, \(\cdots\), element \(N\)) additional electrical length is introduced causing a true time delay (TTD) difference between the elements, which is responsible for the beam-steering of our array. By appropriately choosing the lengths of these paths, we can steer the beam at the desired direction. However, as long as we are moving towards higher port number (port 1, port 2, \(\cdots\), port \(M\)), the signal may travel to a specific radiating element along different paths (Figure 3). Take for example port 2 when it excites the radiating element 2. As shown in Figure 3, the desired signal is the one that can follow the path denoted with green line. However, there is also an additional path that the signal can take, which is the one denoted with red line. This second path is undesired and is referred to as spurious path, which is responsible for some performance degradation of the Blass matrix. Aiming to a compact mm-Wave beamforming network, a dual-layer semi-circular Blass matrix adopted herein is introduced [47]. Figure 4 shows a graphical representation of the circular topology. The semi-circular Blass matrix is designed so that beams 2 and 3 are identical to beam 1, but smartly rotated so that the transmission line between them introduces the desired time delays. Therefore, for the first beam, the time delay for the \(n\)th element is \(n\tau_{vertical}\), for the second beam is \(n(\tau_{vertical}+\tau_{horizontal})\), while for the third beam is \(n(\tau_{vertical}+2\tau_{horizontal})\). By determining \(\tau_{vertical}\) and \(\tau_{horizontal}\), each beam can be accurately steered in the desired direction. A critical component of the Blass matrix is, as expected, the coupler. The couplers used in this work are directional, but, unlike traditional couplers [48], they have their through port on the opposite side of the coupled port.