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Pilot-based estimation of the squared Euclidean norm of the channel vector of a Rayleigh fading system is considered. Unlike most previous work in the area of estimation of multiple antenna channels, the authors consider Bayesian estimation where the long-term channel statistics are known a priori. Closed-form expressions of the Minimum Mean Square Error (MMSE) estimator and its Mean Squared Error (MSE) are derived for the cases of either an unweighted or a weighted unitary pilot matrix. The problem of finding the optimal pilot weighting, in the sense of minimizing the average MSE, is considered and a simple algorithm is proposed to achieve an approximately optimal power allocation numerically.
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