  In this letter, we investigate the channel estimation problem for MIMO
wireless communication systems with movable antennas (MAs) at both the
transmitter (Tx) and receiver (Rx). To achieve high channel estimation accuracy
with low pilot training overhead, we propose a tensor decomposition-based
method for estimating the parameters of multi-path channel components,
including their azimuth and elevation angles, as well as complex gain
coefficients, thereby reconstructing the wireless channel between any pair of
Tx and Rx MA positions in the Tx and Rx regions. First, we introduce a
two-stage Tx-Rx successive antenna movement pattern for pilot training, such
that the received pilot signals in both stages can be expressed as a
third-order tensor. Then, we obtain the factor matrices of the tensor via the
canonical polyadic decomposition, and thereby estimate the angle/gain
parameters for enabling the channel reconstruction between arbitrary Tx/Rx MA
positions. In addition, we analyze the uniqueness condition of the tensor
decomposition, which ensures the complete channel reconstruction between the
whole Tx and Rx regions based on the channel measurements at only a finite
number of Tx/Rx MA positions. Finally, simulation results are presented to
evaluate the proposed tensor decomposition-based method as compared to existing
methods, in terms of channel estimation accuracy and pilot overhead.
