| We address the problem of estimating a distorted signal in the presence of noise in the case that very little prior knowledge is available regarding the nature of the signal, the distortion, or the noise. More specifically, we assume that the distorting system is linear but otherwise unknown, that the signal is drawn from a sequence of independent and identically distributed random variables of unknown distribution, and that the noise is independent of the signal but also has unknown distribution. We refer to this problem as “blind estimation without priors,” where the term “blind” captures the notion that signal estimates are obtained blindly with regard to knowledge of the distortion and interference.; Since its origins nearly half a century ago, there has evolved a large body of theoretical and practical knowledge regarding the blind estimation problem. Even so, very fundamental questions still remain. For example: (i) How good are blind estimates compared to their non-blind counterparts? (ii) When do we know that a blind estimation algorithm will return estimates of the desired signal versus a component of the interference? Though both of these questions have long histories within the research community, existing results have been either approximate and/or limited to special cases that leave out many problems of practical interest.; This dissertation presents answers to the questions above for the well-known Shalvi-Weinstein (SW) and constant modulus (CM) approaches to blind linear estimation. All results are derived in a general setting: vector-valued infinite impulse response channels, constrained vector-valued auto-regressive moving-average estimators, and near-arbitrary forms of signal and interference. First, we derive concise expressions tightly upper bounding the mean-squared error (MSE) of SW and CM-minimizing estimates which are principally a function of the optimum MSE achievable in the same setting. Second, we derive similar bounds for the average squared parameter error in CM-based blind channel identification. Third, we present sufficient conditions for gradient-descent (GD) initializations that guarantee convergence to CM-minimizing estimates of the desired user. These conditions principally involve the signal-to-interference ratio of the initial estimates. Finally, we propose a novel approach to CM-GD implementation that greatly reduces the implementation complexity of the standard CM-GD algorithm while still retaining its mean behavior. |
