Talk
Abstract:
Initial Solution of Pseudo-Range Equations
Mike Elgersma
Honeywell Technology Center
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.
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