Learned Spectrum Sensors Under Noise-Shape Uncertainty: An Invariant-Detection Analysis

Deep spectrum sensors are reported to work below the SNR wall of the energy detector, a gain usually credited to learned structure. We examine which statistic they compute. Seven trained detectors, including a DetectNet replica and a transformer, detect Gaussian surrogates with the signal’s spectrum as often as the signals (gap −0.1 to +0.8 points), and on identical windows they behave like the locally most powerful invariant (LMPI) test with the training spectrum as template. To first order in SNR, no detector can distinguish a signal from its surrogate, so near the detection limit training arrives at this test. Its price is an SNR wall set by noise-shape uncertainty. We derive the wall in closed form for any signal spectrum, show that it is attainable, and show that it is infinite when the spectrum varies less than the noise shape is uncertain. In 24 configurations, second-order detectors stop at the wall, while cyclostationary detectors pass it at the rate a noncentral chi-square law predicts. On recorded RTL-SDR noise, receiver roll-off raises false alarms to 17–30%, and on real signals the networks miss off-band transmissions as the LMPI template predicts. Calibration over measured noise
shapes restores false-alarm control. 

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