This paper addresses optimal periodic training signal design for frequency offset estimation in frequency selective multipath Rayleigh fading channels. For a fixed transmitted training signal energy and within a fixed length block, the optimal periodic training signal structure (the optimal location of identical training sub-blocks) and the optimal training sub-block signal are presented. The optimality is based on the minimum Cramer-Rao bound (CRB) criterion. Based on the snap-shot CRB, the

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... imal periodic training structure is derived. The optimal training sub-block signal is obtained by utilizing the average CRB and the received training signal statistics. The optimal training structure with optimal training signals achieves substantial performance improvement over non-optimal training structure with non-optimal training signals.