Ziv-Zakai Bound for Target Nonlinear Parameter Estimation

Nonlinear parameter estimation for targets is one of the fundamental problem in statistical signal processing, and has attracted a lot of research interest in the past few decades, where higher estimation accuracy is one of the key objectives. Since there is no general closed-form expression on minimum mean-squared error (MSE the lower bounds on MSE becomes the benchmark on performance evaluation for estimation algorithms. In the past half century, researches are devoted to find a lower bound with global tightness, strong physical interpretability, and ease of use in specific estimation scenarios. Therein, Ziv-Zakai bound (ZZB) has been proven as one of the globally tightest lower bounds. It establishes the intuitive relationship between the estimation error and the probability of error of a hypothesis testing problem, which provides better physical interpretability compared with other lower bounds. However, it still remains challenging for ZZB in specific estimation scenarios with respect to the ease of use. To this end, this dissertation provides a detailed literature review on the lower bounds on MSE for target nonlinear parameter estimation, focuses on the research on the ZZB for typical target nonlinear parameter estimation problems (e.g., distance and angle), and derives the explicit/closed-form and easy-to-use ZZB expressions for specific parameter estimation problems. The main contents of this dissertation are shown as follows: 1. An explicit ZZB for compressive time delay estimation is derived. This dissertation incorporates the ZZB derivation into the general compressive time delay estimation model, and breaks through the restriction of white Gaussian noise and specific random sensing kernel on the ZZB derivation, from which the derived ZZB is appropriate for arbitrary compressive sensing (CS) kernel and the arbitrary Gaussian colored noises. Thus, compared with the previous work, the derived ZZB shows better applicability to actual estimation scenario. According to the commonly used model such as deterministic model and Gaussian stochastic model, the derived ZZB is formulates as the function of CS kernel and the noise covariance matrix. The derived ZZB provides the global tight bound and accurately predicts the threshold signal-to-noise ratio (SNR) entering the asymptotic region for the minimum mean squared error estimator. 2. The closed-form ZZB for one-dimensional (1D) multi-source direction-of-arrival (DOA) estimation is derived, which reveals the relationship between the MSE convergence in the a priori performance region and the number of sources. According to the coherence/incoherence hybrid multi-source signal model, this dissertation formulates the ZZB as the function of the coherence coefficients among different sources, and introduces the order statistics to analyze the effect brought by the ordering process on the MSE convergence in the a priori performance region. The derived ZZB provides a unified expression for both the underdetermined and overdetermined multi-source DOA estimation. 3. The closed-form ZZB for two-dimensional (2D) multi-source DOA estimation is derived. This dissertation considers the partially correlated signal model and incorporates the correlated coefficient matrix into the ZZB derivation, such that the derived ZZB is formulated as the function of the correlated coefficient matrix. This dissertation discusses the substantial difference of the matching process between the estimated and the true DOAs in 2D multi-source DOA estimation and that in 1D multi-source DOA estimation, and demonstrate that, in 2D multi-source DOA estimation, the minimum Euclidean distance criterion is required to realize the matching process rather than an ordering process only along azimuth or elevation, which need the stochastic Euclidean bipartite matching problem to analyze the effect of the minimum Euclidean distance criterion on the MSE convergence in the a priori performance region. Meanwhile, this dissertation also provides the closed-form ZZB for azimuth/elevation estimation under 2D DOA signal model.