SNR wall for energy detection with noise power estimation

In this work we perform an asymptotic analysis of estimated noise power (ENP) energy detector (ED) to derive the condition for the existence of the signal-to-noise ratio (SNR) wall phenomenon. We prove that an ED with noise estimation does not exhibit the SNR wall if the variance of the estimate reduces when the observation time increases. In the absence of SNR wall, we show that the maximum slope of the design curves (SNR vs. observation time for an arbitrary target probability of false alarm (PFA) and probability of detection (PD)), is equal to −5 dB/decade for the ideal ED, and can be reached also by an ENP-ED. Finally, we derive analytical expressions for the design curves when maximum likelihood (ML) noise power estimation is adopted, and we prove that, asymptotically, the SNR penalty with respect to ideal ED is of 1.5 dB, when the number of noise-only samples is equal to the number of observed samples