# Angle-of-Arrival Estimation Using Difference Beams in Localized Hybrid Arrays
## 1 Introduction  
Due to the superior balance between performance and cost, a hybrid antenna array is regarded as an excellent candidate for millimeter wave (mmWave) communication systems [1; 2]. Typically, the hybrid array is composed of multiple analog subarrays with phase controllable antenna elements. It includes two kinds of conventional configurations, i.e., localized and interleaved arrays in the light of the topology of subarrays. As the localized array is easier for schematic design and hardware implementation, it is more suitable for building a massive array. Angle-of-arrival (AoA) acquisition of the incoming mmWave signal is of considerable importance for signal reception, since the mmWave channels are dominated by the line-of-sight (LOS) propagation. A wide range of its applications including localization and tracking to mmWave communication systems, e.g., 5G mmWave cellular networks [2] and satellite-borne communications [1], have been increasingly studied in recent years. AoA estimation using a localized array suffers from the phase ambiguity problem, which has been progressively studied in [3; 4; 5; 6; 7]. Each of these solutions leverages the cross-correlations between neighbouring subarrays for an AoA estimate. Phase ambiguity stems from an undetermined integer multiple of \(2\pi\) between \(Nu\) and the argument of cross-correlations, where \(N\) is the number of antennas in a subarray, \(u=\frac{2\pi}{\lambda}d\sin\theta\), \(\theta\) the elevation AoA, \(\lambda\) the wavelength, and \(d\) the adjacent antenna spacing. With the identical phase shift deployment over all subarrays for constructive combination of cross-correlations, the work in [3] proposed a differential beam search algorithm to go through all possible beams and determine \(u\) with the largest output power. However, it incurs a long scanning period that linearly increases with the length of a subframe and \(N\). To avoid a long scanning period, the authors in [4] studied a phase shift configuration with different values in different subarrays to eliminate the ambiguity by directly estimating \(u\). Their ingenious idea is that (1) \(Nu\) is estimated by rectifying the signs of cross-correlations and then combining them coherently; (2) After calibrating subarray output signals with the estimated \(Nu\), one takes their inverse discrete Fourier transform (IDFT) and calculates the correlations of the Fourier coefficients to uniquely recover \(u\). The work in [5; 6] further generalized the phase shiftdesign in [4] to narrowband and wideband systems respectively, and revealed that the strongest cross-correlation takes the opposite sign from the remaining cross-correlations. Following this finding with an improved calibration accuracy of cross-correlations, the AoA can be speedily and reliably estimated even in low signal-to-noise ratio (SNR) regimes. In [7], an enhanced AoA estimation for a polarized mmWave signal was studied using a localized hybrid dual-polarized array, where polarization diversity combining was employed to improve the estimation of phase offset between adjacent subarrays. With the cross-correlation based algorithm, a multi-AoA estimation scheme with a combiner design was proposed in [8], where the paths for different users are identified by exploiting the low correlation property of the pseudo-random sequences. With a digital array, MUSIC and ESPRIT [9] are the classical methods used in high-resolution AoA estimation. The work in [10; 11] applied them to a localized array. Although accurate estimation can be achieved, the computational complexity incurred from singular value decomposition grows cubically with the total number of antennas [5], which makes the applications of these methods impractical in mmWave massive arrays. In [12], an auxiliary beam pair (ABP) design was proposed to provide high-resolution AoA estimation via amplitude comparison relating to each ABP. It, however, needs to scan all the directions of interest exhaustively, and the resolution is subject to the beamwidth and SNR.