Estimation Problems with Singular Information Matrices: Cramer-Rao Bound and Its Application to Blind Channel Estimation

This thesis focuses on estimation problems with singular Fisher information matrices (FIM), especially on Cramer-Rao bounds (CRB) for such problems. CRB is a lower bound for variances of unbiased estimators. Conventional view about such problems is that CRB provides no useful information, because if the FIM is singular, there will be no unbiased estimators. Unbiased estimators, however, may exist if the possible values of unknown parameters are constrained to a specific set. This leads to the study of CRB for constrained parameters. In Chapter 2, we derive the minimum CRB among all minimum constraint functions. This bound, unlike existing CRB for constrained parameters, whose value depends on the constraint function, is the bound for all minimum constraints, where minimum constraints refer to constraints imposing least number of independent constraints on the unknown parameters. In Chapter 3, we extend the well-known CRBs for constrained and unconstrained parameters, originally for real parameters, to the case of complex parameters. We prove that the CRB for unconstrained complex parameters can always have the same form as that for real parameters, and if the constraint function is holomorphic, the CRB for constrained complex parameters can also have the same form as that for real parameters. With the preparation work in Chapter 2 and 3, we are able to derive the CRB for blind channel estimators in wireless block transmission systems in Chapter
4. The derived CRB formula is applicable to all block transmission systems with arbitrary linear redundant precoders. We also apply the results in Chapter 2 to derive a minimum CRB, and compare the derived CRBs with simulation results of the generalized subspace-based blind channel estimation algorithm. In chapter 5, we derive the performance of the algorithm based on first order perturbation analysis. Simulation results show that the performance analysis is close to simulation results especially under high signal-to-noise ratios (SNR).