Six-dimensional movable antenna (6DMA) is an innovative and transformative technology to improve wireless network capacity by adjusting the 3D positions and 3D rotations of antennas/surfaces (sub-arrays) based on the channel spatial distribution. For optimization of the antenna positions and rotations, the acquisition of statistical channel state information (CSI) is essential for 6DMA systems. In this paper, we unveil for the first time a new \textbf{\textit{directional sparsity}} property of the 6DMA channels between the base station (BS) and the distributed users, where each user has significant channel gains only with a (small) subset of 6DMA position-rotation pairs, which can receive direct/reflected signals from the user. By exploiting this property, a covariance-based algorithm is proposed for estimating the statistical CSI in terms of the average channel power at a small number of 6DMA positions and rotations. Based on such limited channel power estimation, the average channel powers for all possible 6DMA positions and rotations in the BS movement region are reconstructed by further estimating the multi-path average power and direction-of-arrival (DOA) vectors of all users. Simulation results show that the proposed directional sparsity-based algorithm can achieve higher channel power estimation accuracy than existing benchmark schemes, while requiring a lower pilot overhead.
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