  This paper considers the joint transceiver design in a wireless sensor
network where multiple sensors observe the same physical event and transmit
their contaminated observations to a fusion center, with all nodes equipped
with multiple antennae and linear filters. Under the mean square error (MSE)
criterion, the joint beamforming design problem can be formulated as a
nonconvex optimization problem. To attack this problem, various block
coordinate descent (BCD) algorithms are proposed with convergence being
carefully examined. First we propose a two block coordinate descent (2-BCD)
algorithm that iteratively designs all the beamformers and the linear receiver,
where both subproblems are convex and the convergence of limit points to
stationary points is guaranteed. Besides, the thorough solution to optimizing
one single beamformer is given, which, although discussed several times, is
usually incomplete in existing literature. Based on that, multiple block
coordinate descent algorithms are proposed. Solving the joint beamformers'
design by cyclically updating each separate beamformer under the 2-BCD
framework gives birth to a layered BCD algorithm, which guarantees convergence
to stationary points. Besides that, a wide class of multiple BCD algorithms
using the general essentially cyclic updating rule has been studied. As will be
seen, by appropriately adjusting the update of single beamformer, fast
converging, highly efficient and stationary point achieving algorithms can be
obtained. Extensive numerical results are presented to verify our findings.
