A Proof of the Spherically Symmetric Overbounding Theorem For Linear Systems

Graham W. Pulford

Peer Reviewed

Abstract: Rife and Gebre-Egziabher’s recent paper [1] considers linear mappings of random variables from spherically symmetric probability distributions in the context of overbounding correlated ranging errors. This theory has potential implications for ensuring the integrity of differential GPS (DGPS) navigation systems. We first give a succinct rederivation of their preliminary result. The extension to a cascade of two discrete-time, linear time-invariant filters, the main result in [1], is then formulated using a precise definition of spherical symmetry for scalar random processes. The relationship between the input and output probability densities when the input density is spherically symmetric is derived using linear systems theory. This yields an alternative statement of the spherically symmetric overbounding theorem in terms of the system matrices. A precise definition is given of the filter impulse response coefficients and the multi-dimensional convolution operation that is needed to ensure the validity of the overbounding theorem in [1]. The equivalence between the two representations is formally proved. Numerical simulations are provided that support the derived results.
Published in: NAVIGATION, Journal of the Institute of Navigation, Volume 55, Number 4
Pages: 283 - 292
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