Integer parameter estimation in linear models with applications to GPS

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 carrier phase measurements in the global positioning system (GPS), where the unknown integers enter the equations as the number of carrier signal cycles between the receiver and the satellites when the carrier signal is initially phase locked. Given a linear model, we address two problems: (1) the problem of estimating the parameters and (2) the problem of verifying the parameter estimates. We show that with additive Gaussian measurement noise the maximum likelihood estimates of the parameters are given by solving an integer least-squares problem. Theoretically, this problem is very difficult computationally (NP-hard); verifying the parameter estimates (computing the probability of estimating the integer parameters correctly) requires computing the integral of a Gaussian probability density function 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 (1982), the LLL algorithm, the integer least-squares problem associated with estimating the parameters can be solved efficiently 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 synthetic GPS setup.