Beamforming is a signal processing technique where an array of antenna elements can be steered to transmit and receive radio signals in a specific direction. The usage of millimeter wave (mmWave) frequencies and multiple input multiple output (MIMO) beamforming are considered as the key innovations of 5th Generation (5G) and beyond communication systems. The technique initially performs a beam alignment procedure, followed by data transfer in the aligned directions between the transmitter and the receiver. Traditionally, beam alignment involves periodical and exhaustive beam sweeping at both transmitter and the receiver, which is a slow process causing extra communication overhead with MIMO and massive MIMO radio units. In applications such as beam tracking, angular velocity, beam steering etc., the beam alignment procedure is optimized by estimating the beam directions using first order polynomial approximations. Recent learning-based SOTA strategies for fast mmWave beam alignment also require exploration over exhaustive beam pairs during the training procedure, causing overhead to learning strategies for higher antenna configurations. In this work, we first optimize the beam alignment cost functions e.g. the data rate, to reduce the beam sweeping overhead by applying polynomial approximations of its partial derivatives which can then be solved as a system of polynomial equations using well-known tools from algebraic geometry. At this point, a question arises: 'what is a good polynomial approximation?' In this work, we attempt to obtain a 'good polynomial approximation'. Preliminary experiments indicate that our estimated polynomial approximations attain a so-called sweet-spot in terms of the solver speed and accuracy, when evaluated on test beamforming problems.

翻译：波束成形是一种信号处理技术，通过控制天线阵列的相位使无线电信号在特定方向上进行发射和接收。毫米波频率与多输入多输出波束成形被公认为第五代及未来通信系统的关键创新。该技术首先执行波束校准过程，随后在发射机与接收机之间沿校准方向进行数据传输。传统波束校准需要在发射机和接收机两端进行周期性的穷举波束扫描，这一过程在MIMO和Massive MIMO无线电单元中会带来额外通信开销。在波束跟踪、角速度测量、波束转向等应用中，通过使用一阶多项式逼近估计波束方向可优化波束校准流程。当前基于学习的最优毫米波快速波束校准策略在训练过程中仍需对穷举波束对进行探索，导致更高天线配置下的学习策略存在额外开销。本研究首先优化数据速率等波束校准代价函数，通过应用其偏导数的多项式逼近来降低波束扫描开销，进而利用代数几何中的经典工具将该逼近转化为多项式方程组求解。在此关键问题在于：何为"良好的多项式逼近"？本研究尝试获取此类最优逼近。初步实验表明，在测试波束成形问题上，我们所估计的多项式逼近在求解器速度与精度之间达到了所谓的"最优平衡点"。