Reference-Signal Sufficiency Criterion and Residual-Mismatch Sensitivity Analysis for CAF-Based Passive Detection

Abstract

Reference-signal quality is an important determinant of detection performance in passive radar based on the cross-ambiguity function (CAF). A reference signal may be obtained from the received direct-path signal, reconstructed from the illuminator waveform, or generated locally from available prior information. Existing assessment methods commonly focus on reference-channel quality, reconstruction error, or the fraction of known resources, but these quantities do not directly determine whether a selected reference signal is sufficient for a specified detection task. This paper defines a normalized cross-ambiguity efficiency between a deterministic candidate reference signal and the actual illumination waveform. The prescribed per-cell false-alarm probability, target detection probability, and full-waveform benchmark output signal-to-noise ratio (SNR) are converted into a required efficiency threshold. Sufficiency is then assessed using the worst-case detection performance over a specified residual delay-Doppler mismatch set. Among the candidates that satisfy the task, a partial order on information requirements is used to identify the Pareto-minimal set. Under a point-target model with white noise, the criterion applies to general deterministic finite-energy candidate reference signals. As an application, a 5G New Radio (NR) orthogonal frequency-division multiplexing (OFDM) waveform is considered. The known-energy fraction at perfect alignment is separated from the time-frequency support response under residual mismatch, yielding a computationally efficient diagonal approximation. Selected points are checked by direct CAF calculation using full time-domain in-phase/quadrature (IQ) waveforms with cyclic prefixes, and three residual-mismatch regions are examined. The results show that a candidate may meet the detection threshold at perfect alignment yet fail over a nonzero residual-mismatch region. Reference-signal applicability is therefore jointly determined by known energy, time-frequency support, residual mismatch, and the detection threshold; no candidate is universally optimal independently of the specified task.