Swift and accurate alignment of transmitter (Tx) and receiver (Rx) beams is a fundamental design challenge to enable reliable outdoor millimeter-wave communications. In this paper, we propose a new Optimized Two-Stage Search (OTSS) algorithm for Tx-Rx beam alignment via spatial scanning. In contrast to one-shot exhaustive search, OTSS judiciously divides the training energy budget into two stages. In the first stage, OTSS explores and trains all candidate beam pairs and then eliminates a set of less favorable pairs learned from the received signal profile. In the second stage, OTSS takes an extra measurement for each of the survived pairs and combines with the previous measurement to determine the best one. For OTSS, we derive an upper bound on its misalignment probability, under a single-path channel model with training codebooks having an ideal beam pattern. We also characterize the decay rate function of the upper bound with respect to the training budget and further derive the optimal design parameters of OTSS that maximize the decay rate. OTSS is proved to asymptotically outperform state-of-the-art beam alignment algorithms, and is numerically shown to achieve better performance with limited training budget and practically synthesized beams.
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