Times-Of-Arrival, Maximum-Likelihood, Errors-in-Variables, Total-Least-Squares, Quadratic Eigen-decompositions: Hey Presto, a Solution!

Dr. Garry Newsam (DSTO and NICTA)

MSI Advanced Computation

DATE: 2006-08-28
TIME: 10:30:00 - 11:30:00
LOCATION: G 35 (John Dedman building)
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ABSTRACT:
Events such as explosions or radar pulses generate acoustic or electromagnetic signals, and measurements of the times that the associated signals arrive at known locations can be used to locate the original event. Applications range from locating artillery by sound ranging in World War I to GPS today. The equations relating /N/ measurements /*x*_n , t_n / to the unknown location /*x*, t/ are non-linear, but it has long been known that this system can be reduced to /N-1/ linear equations and one quadratic. On the face of it, however, this reduction does not appear to be much use in the over-determined case as the standard least-squares solution of the linear system has no particular relation to the desired maximum-likelihood estimate. The talk will show that nevertheless it is possible to take advantage of the existence of the equivalent, nearly linear system to simplify optimisation of the likelihood function. To do so, likelihood maximisation is recast as a constrained least squares problem, which then translates into an errors-in-variables problem for the equivalent, nearly linear system. Errors-in-variables problems are in turn solvable by total-least-squares techniques, with the present case being an apparently hitherto unstudied variant in that the same errors appear in two different columns: its solution turns out to be characterised in terms of a quadratic eigenvalue decomposition instead of the standard SVD. Finally the approach can be generalised: the talk will conclude by presenting an application in HF communications where locations and propagation paths are restricted to the surface of a sphere.

BIO:
Garry Newsam is Head of Signals Analysis Group in Defence Science and Technology Organisation (DSTO) and based in South Australia, but is currently visiting the VISTA group in NICTA until October. Prior to his present position he was head of DSTO's Image Analysis and Exploitation Group and has had wide experience in signal, image and radar processing. His first love, however, was numerical analysis; the talk is basically an exercise in numerical linear algebra and elementary statistics.