Lattice parameter estimation from sparse, noisy measurements

We consider a problem in which noisy measurements are made of the positions of points in a lattice. Some parameters of the lattice are known but others need to be estimated. In particular, it is not known a priori from which lattice point each measurement arises. In previous work [1-5], the authors have considered estimating the parameters of a one-dimensional lattice from measurements on the real Line. The application is period estimation from sparse, noisy measurements of a periodic event, e.g., estimation of baud in telecommunications signal processing. Here, we take a first step in generalising the results to higher-dimensional lattices, starting with two dimensions. We propose a model in which the lattice is square but the unknown parameters are a translation, rotation and scaling. An application is again in telecommunications, to blind detection of QAM. We propose an estimator based on the Bartlett point-process periodogram [6]. We show that, under certain conditions, the estimator is strongly consistent and obeys a central limit theorem. We demonstrate convergence to the limit with numerical simulations.