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

本論文的主題為消息矩陣不可逆的估測問題，特別是在這類問題中的Cramer-Rao下界限。Cramer-Rao下界限是不偏估計式變異數的一個下界限。現有文獻的推導結果指出，如果一個估測問題的消息矩陣為不可逆，則不存在不偏估計式，因此Cramer-Rao下界限無法提供任何有意義的資訊。然而，若我們將被估測參數的可能範圍以限制函數限制在某個集合之中，即使消息矩陣不可逆，不偏估計式依然可能存在。在本論文的第二章中，我們推導出對於任意最小限制的最小Cramer-Rao下界限。此新提出的下界限不同於現有文獻結果的地方在於，現有結果的Cramer-Rao下界限值是被限制函數所決定的；而本論文提出的新下界限則是所有可能最小限制的下界限，因此與限制函數的選擇無關。在第三章，我們將已有的Cramer-Rao下界限公式推廣至被估測參數為複數的情形。我們證明出，若被估測參數未被限制，則推廣的複數Cramer-Rao下界限之形式與已有的實數Cramer-Rao下界限可以完全相同；若限制函數為全純函數，則推廣的複數Cramer-Rao下界限之形式亦可與實數Cramer-Rao下界限相同。根據第二、三章的理論結果，我們在第四章推導出在無線區塊通訊系統中，盲信道估測的Cramer-Rao下界限。第四章的結果可適用於任意的區塊通訊系統參數設定，因此推廣了原本只適用於補零冗餘符碼的現有文獻結果。我們並利用第二章的結果推導出盲信道估測的最小Cramer-Rao下界限，並將推導出的Cramer-Rao下界限與一般化的基於子空間的盲信道估測演算法作效能比較。在第五章中，我們利用一階攝動分析的方法，推導出一般化的基於子空間的盲信道估測演算法之效能分析。分析結果在信噪比足夠高時，與模擬結果十分接近。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)