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In millimeter wave (mmWave) communications, the beamforming gains of large-scale antenna arrays are used to combat severe path losses and improve communication link quality. This requires beam training (BT) to determine the optimal beam direction and achieve beam alignment. The hierarchical codebook with multi-resolution training beams is widely used in BT to reduce training overhead. Unfortunately, due to hardware restrictions, the transition area between two adjacent training beams often leads to error propagation in a traditional BT scheme, which limits the accuracy of beam alignment. In this paper, we propose an improved BT scheme under hierarchical codebook. First, we define an index to measure the power difference of the received signals. Once the index is lower than the predefined threshold, the optimal beam direction is considered to be in the transition area, and an extra interval that contains the transition area is proposed for further beam searching. Then, we analyze the optimal threshold and the probability of beam misalignment based on the ideal beam pattern, proving that the error rate exponentially decreases with the signal-to-noise ratio (SNR) under a single-path channel with constant gain. Finally, simulation results demonstrate that the proposed scheme can significantly improve the accuracy of beam alignment and is applicable to most existing beam patterns. INDEX TERMS Hierarchical codebook, millimeter wave, beam training, beam alignment, error propagation, beam alignment error rate.

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Channel estimation for millimeter wave (mmWave) systems is challenging due to their large antenna arrays. Owing to the sparse scattering nature of the mmWave channel, channel estimation can be performed by estimating the directions and the gains of paths. Most existing schemes assume that a wideband mmWave channel exhibits a common sparsity in the frequency domain. Unfortunately, they ignore the beam squint effect caused by the wide bandwidth, resulting in severe performance loss. In this paper, we investigate the wideband channel estimation problem with beam squint. Specifically, by utilizing measurement signals at all subcarriers, we propose a spatial spectrum-based scheme for a subarray architecture that requires only a single training slot. We first prove that the scheme can accurately obtain the spatial spectrum from a theoretical perspective. Then, we design the beamforming weights of the subarray to avoid pseudo peaks and analyze the inherent spectrum ambiguity phenomenon under the subarray architecture. Finally, to cope with beam squint, we divide the entire bandwidth into multiple subbands and design a combination criterion for the spatial spectrum of each subband. During this process, we prove that the spectrum ambiguity is eliminated, and the joint estimation of the path directions can be obtained. Simulation results demonstrate that the proposed scheme has better estimation accuracy than other methods and significantly reduces the required number of training slots.