Over-the-Scene Computation: Illumination-Programmed Functional Sensing of Unresolvable Populations

Abstract

Decisions about dense populations depend on how much will cross a boundary by a deadline and when the first member arrives. Conventional radar forms a range-Doppler map, whose cells add noise and whose detections merge unresolved members. We introduce over-the-scene computation, where illumination makes the scene compute such functionals without resolving members. With random echo phases, a correlator's mean output energy sums member reflectivities weighted by a kernel set by the waveform and template. The waveform's ambiguity function is this computation's transfer function, determining computable functionals and their noise cost. One readout aggregates members over a small delay-Doppler area but along a line segment growing with duration, longest for the chirp of matching slope among constant-envelope waveforms. Members predicted to share a range lie on such a line, so a chirp's mean beat spectrum is the predicted range profile at a look-ahead horizon set by its rate. For unbiased estimation, a Cramer-Rao bound over all equal-energy waveforms favors resolving members above unit signal-to-noise ratio (SNR) and grouping weaker ones into unit-SNR sets. At 0.65 members per cell, -10 dB member SNR, and one sweep, the chirp's bin sum inside the boundary comes within 16% of this bound in root-mean-square error and, with 77 times fewer samples per horizon, outperforms every unbiased estimator from a coherent processing interval of equal dwell, bandwidth, and energy. No finite set of waveform-template probes determines an unconstrained Rayleigh-fluctuating population's first arrival, but bisection localizes it if first arrivals outweigh later arrivals' sidelobes.