Bisection is a mathematical and numerical method that involves dividing an interval into two equal parts and then selecting the subinterval where a particular property or condition is satisfied. This process is repeated iteratively to narrow down the range and converge towards a solution. Bisection is employed in various fields, including numerical analysis, optimization, and root-finding algorithms. Its simplicity and reliability make it a valuable tool for solving equations and locating critical points in mathematical and computational applications.


Beam Training Optimization in Millimeter-wave Systems under Beamwidth, Modulation and Coding Constraints

Millimeter-wave (mmWave) bands have the potential to enable significantly high data rates in wireless systems. In order to overcome intense path loss and severe shadowing in these bands, it is essential to employ directional beams for data transmission. Furthermore, it is known that the mmWave channel incorporates a few number of spatial clusters necessitating additional time to align the corresponding beams with the channel prior to data transmission. This procedure is known as beam training (BT). While a longer BT leads to more directional beams (equivalently higher beamforming gains), there is less time for data communication. In this paper, this trade-off is investigated for a time slotted system under practical constraints such as finite beamwidth resolution and discrete modulation and coding schemes. At each BT time slot, the access point (AP) scans a region of uncertainty by transmitting a probing packet and refines angle of arrival (AoA) estimate based on user equipment (UE) feedback. Given a total number time slots, the objective is to find the optimum allocation between BT and data transmission and a feasible beamwidth for the estimation of AoA at each BT time slot such that the expected throughput is maximized. It is shown that the problem satisfies the optimal substructure property enabling the use of a backward dynamic programming approach to find the optimal solution with polynomial computational complexity. Simulation results reveal that in practical scenarios, the proposed approach outperforms existing techniques such as exhaustive and bisection search.

Robust Beam Tracking and Data Communication in Millimeter Wave Mobile Networks

Millimeter-wave (mmWave) bands have shown the potential to enable high data rates for next generation mobile networks. In order to cope with high path loss and severe shadowing in mmWave frequencies, it is essential to employ massive antenna arrays and generate narrow transmission patterns (beams). When narrow beams are used, mobile user tracking is indispensable for reliable communication. In this paper, a joint beam tracking and data communication strategy is proposed in which, the base station (BS) increases the beamwidth during data transmission to compensate for location uncertainty caused by user mobility. In order to evade low beamforming gains due to widening the beam pattern, a probing scheme is proposed in which the BS transmits a number of probing packets to refine the estimation of angle of arrival based on the user feedback, which enables reliable data transmission through narrow beams again. In the proposed scheme, time is divided into similar frames each consisting of a probing phase followed by a data communication phase. A steady state analysis is provided based on which, the duration of data transmission and probing phases are optimized. Furthermore, the results are generalized to consider practical constraints such as minimum feasible beamwidth. Simulation results reveal that the proposed method outperforms well-known approaches such as optimized beam sweeping.