# Initial Solution of Pseudo-range Equations

Wednesday, August 16, 2000 - 2:50pm - 3:45pm

Keller 3-180

Michael Elgersma (Honeywell)

The pseudo-range equations are a system of quadratic equations. When the receiver may be outside the constellation of transmitters, standard iterative techniques may not converge to the correct solution. In this case, it may be neccessary to use the known solution to the system of quadratic equations.

Each quadratic equation has the same quadratic terms, so differences of the quadratic equations give linear equations. If there is at least one more equation than unknowns, then the differences of the quadratic equations give at least as many linear equations as unknowns. When there is no noise, these linear equations have a unique solution.

When the data is noisy, the system of quadratic equations is overdetermined. One approach would be to simply do a least-squares solution to the system of linear equations obtained from differences of the quadratic equations. The relationship between the linear least-squares solution and the best fit solution of the original overdetermined quadratic equations is explored. In order to obtain a region of possible solutions, the system of quadratic equations can be converted into a system of quadratic inequalities which capture the uncertainty in the measured data. Differences of these quadratic inequalities give systems of linear inequalities. It will be shown that the solution set of the system of linear inequalities is a convex polyhedra which contains the intersection of all the solution sets of the quadratic inequalities. Therefore the easily computable solution set of a system of linear inequalities can be used to bound the solution set of the original system of quadratic inequalities.

Open questions include the tightness of the above bound, and how the technique could be extended to deal with multipath issues.

Each quadratic equation has the same quadratic terms, so differences of the quadratic equations give linear equations. If there is at least one more equation than unknowns, then the differences of the quadratic equations give at least as many linear equations as unknowns. When there is no noise, these linear equations have a unique solution.

When the data is noisy, the system of quadratic equations is overdetermined. One approach would be to simply do a least-squares solution to the system of linear equations obtained from differences of the quadratic equations. The relationship between the linear least-squares solution and the best fit solution of the original overdetermined quadratic equations is explored. In order to obtain a region of possible solutions, the system of quadratic equations can be converted into a system of quadratic inequalities which capture the uncertainty in the measured data. Differences of these quadratic inequalities give systems of linear inequalities. It will be shown that the solution set of the system of linear inequalities is a convex polyhedra which contains the intersection of all the solution sets of the quadratic inequalities. Therefore the easily computable solution set of a system of linear inequalities can be used to bound the solution set of the original system of quadratic inequalities.

Open questions include the tightness of the above bound, and how the technique could be extended to deal with multipath issues.