  This paper considers a distributed wave-based sensing system that probes a
scene consisting of multiple interacting idealized targets. Each sensor is a
collocated transmit-receive pair that is capable of transmitting arbitrary
wideband waveforms. We address the problem of finding the space-time transmit
waveform that provides the best target detection performance in the sense of
maximizing the energy scattered back into the receivers. Our approach is based
on earlier work that constructed the solution by an iterative time-reversal
(TR) process. In particular, for the case of idealized point-like scatterers in
free space, we examine the frequency dependence of the eigenvalues of the TR
operator, and we show that their behavior depends on constructive and
destructive interference of the waves traveling along different paths. In
addition, we show how these eigenvalues are connected to the poles of the
Singularity Expansion Method. Our study of the frequency behavior distinguishes
this work from most previous TR work, which focused on single-frequency
waveforms and noninteracting targets. The main result of the present paper is
that the TR process provides an automated, guaranteed way to find the resonant
frequencies of the scattering system.
