Integer Parameter Estimation in Linear Models withApplications to GPS 1

We consider parameter estimation in linear models when some of the parameters are known to be integers. Such problems arise, for example, in positioning using phase measurements in the global positioning system (GPS.) Given a linear model, we address two problems: 1. The problem of estimating the parameters. 2. The problem of verifying the parameter estimates. Under Gaussian measurement noise: Maximum likelihood estimates of the parameters are given by solving an integer least-squares problem. Theoretically, this problem is very diicult to solve (N P-hard.) Verifying the parameter estimates (computing the probability of correct integer parameter estimation) is related to computing the integral of a Gaussian PDF over the Voronoi cell of a lattice. This problem is also very diicult computation-ally. However, by using a polynomial-time algorithm due to Lenstra, Lenstra, and Lovv asz (LLL algorithm): The integer least-squares problem associated with estimating the parameters can be solved ef-ciently in practice. Sharp upper and lower bounds can be found on the probability of correct integer parameter estimation. We conclude the paper with simulation results that are based on a GPS setup. Throughout this paper, we assume that the observation y 2 R n is related to the unknown vectors x 2 R p (real) and z 2 Z q (integer) through (1) where A 2 R np (full column rank) and B 2 R nq are known matrices. The measurement noise v 2 R n is assumed to be Gaussian with zero-mean and covari-assumed without loss of generality, as we can always rescale equation (1) by the square root of the covari-ance matrix of v. Given y, A, and B, our goal is to nd and verify estimates of the unknown parameters x and z. Some previous results are given in 6] and 7] and references therein. We consider maximum likelihood (ML) estimates x ML and z ML for x and z respectively that maximize the probability of observing y, i.