Department of Mathematics Colloquium
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Abstract |
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| 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.
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