ML frequency estimation in systems with transmit diversity: pilot signals, complexity and accuracy

We investigate how pilot signals affect complexity and accuracy of maximum likelihood (ML) frequency estimation in frequency-flat channels with transmit diversity. We show that for arbitrary pilot signals, the complexity can be as low as O(N_t), where N_t is the number of transmit antennas. For linearly dependent pilot signals this can be further reduced down to O(1) with even better estimation accuracy. Lower and upper bounds of the Cramer-Rao lower bound (CRLB) over possible channel gains are derived. We define locally optimal pilot signals as those minimising the upper bound and show how to find such signals. The CRLB characterises local properties of ML estimates, while the ambiguity function allows analysis of global properties of estimates in the whole acquisition range. We analyse the ambiguity function for binary pilot signals, in particular Hadamard sequences and show how it relates to behaviour of frequency estimates over the signal-to-noise ratio and frequency acquisition range.