Further results on Crame´r-rao bounds for parameter estimation in long-code DS/CDMA systems

In a recent paper, closed-form formulas for the Cramer-Rao bounds (CRBs) on the error variance of any unbiased estimator of the amplitudes, timing offsets, phase offsets, and directions-of-arrival (DoAs) of multiplexed asynchronous direct-sequence code division multiple access (DS/CDMA) signals adopting aperiodic spreading codes have been derived. These formulas depend on the active users´ spreading codes and pilot information symbols. In this paper, we obtain formulas for the modified CRB under the assumption that the spreading codes can be modeled as random nuisance parameters. Moreover, we also consider the case that the timing offsets are modeled as random variates and provide formulas for the hybrid CRB, i.e., the CRB corresponding to the situation that some of the parameters to be estimated are random variates rather than unknown deterministic quantities. Numerical examples showing the impact of the training length, signal-to-noise ratio (SNR), and the number of active users on the CRB are also reported and commented upon.