Acquisition of spread spectrum signals in Rayleigh fading

We analyze the acquisition of direct-sequence-spread-spectrum signals by an antenna array of M elements in a flat Rayleigh fading environment subject to additive stationary temporally-uncorrelated zero-mean Gaussian interference. The observations are complex baseband sampled signal vectors received at the array elements at discrete time instants. The pseudorandom code is periodic with period K and orthogonal to all its shifted versions. The acquisition process requires estimation of the received code lag from K consecutive M × 1 received signal vectors. We find the maximum likelihood estimate of the code lag when the data, which is binary, is constant during the acquisition process. Using the sufficient statistic of the estimate, we analyze the performance of the acquisition system. In the case of independent and identically distributed Rayleigh fading and spatially-white Gaussian interference, we derive a closed-form expression for the probability of correct decision. The bias of the code lag estimate and the mean-square estimation error are also presented.