Fundamental limits on detection in low SNR under noise uncertainty

In this paper we consider the problem of detecting whether a frequency band is being used by a known primary user. We derive fundamental bounds on detection performance in low SNR in the presence of noise uncertainty - the noise is assumed to be white, but we know its distribution only to within a particular set. For clarity of analysis, we focus on primary transmissions that are BPSK-modulated random data without any pilot tones or training sequences. The results should all generalize to more general primary transmissions as long as no deterministic component is present. Specifically, we show that for every 'moment detector' there exists an SNR below which detection becomes impossible in the presence of noise uncertainty. In the neighborhood of that SNR wall, we show how the sample complexity of detection approaches infinity. We also show that if our radio has a finite dynamic range (upper and lower limits to the voltages we can quantize), then at low enough SNR, any detector can be rendered useless even under moderate noise uncertainty.