Department of Mathematics Colloquium

University of Idaho

Spring 2013

Thursday,  April 25, 3:30-4:20 pm, room TLC 047

Refreshments in Brink 305 at 3:00 pm

An Operator-Theoretic View of Radar

 

Douglas Cochran


School of Electrical, Computer and Energy Engineering

Arizona State University


Abstract

A monostatic active radar consists of a co-located transmitter and receiver.  The transmitter emits a radio signal that propagates into the environment and is scattered by objects in the "scene." Some of the scattered energy returns to the transmitter as echoes of the transmitted signal. The time between the signal transmission and the echo return is proportional to the distance between the transmitter and the object that produced the echo.  If the scattering object is moving, the echo carries information about the radial component of this motion because of the Doppler effect, which is well approximated in most radars by a shift in the frequency of the transmitted signal.  So a standard radar model represents the received signal as a linear superposition of time-delayed and frequency-shifted replicates of the transmitted signal. This presentation gives an overview of the Hilbert-Schmidt class of linear operators on L^2(R) and their use in modeling radar scenes, particularly in view of the Hilbert space of such operators having a basis of time-frequency shift operators. This kind of phase-space harmonic analysis, which extends to L^2(G) for arbitrary locally compact abelian groups G, is underpinned by a rich algebraic structure involving an action of the Weyl-Heisenberg group on L^2 and maximal subgroups that are isotropic with respect to a symplectic form. Some aspects of this more general setting will be introduced if time permits.