Integer parameter estimation in linear models with applications to GPS

Abstract

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 difficult to solve (NP-hard)); and 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 difficult computationally). However, by using a polynomial-time algorithm due to Lenstra, Lenstra, and Lovasz (LLL algorithm), the integer least-squares problem associated with estimating the parameters can be solved efficiently in practice; and 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.