Using Dual Approximation for Best Linear Unbiased Estimators in Continuous Time, with Application to Continuous-Time Phase Estimation

Best linear unbiased estimator (BLUE) theory is well established for discrete, finite-dimensional vectors, where methods of vector gradients can be used on a constrained optimization problem. However, when the observation is infinite-dimensional (e.g., continuous-time functions), the gradient-based approach can be problematic. We pose the BLUE problem as an instance of a dual approximation problem, which recasts the problem into finite dimensional space employing the principle of orthogonality, requiring no gradients for solution. To demonstrate the ideas, they are first developed on a finite-dimensional problem, then extended to infinite dimensional problems. We present an example application of phase estimation from continuous-time observations.