Further Results on Energy Detection of Random Signals in Gaussian Noise

This article shows that the non-coherent detector for random signals which maximizes the generalized likelihood function is the same as the detector that maximizes the probability of correct detection at any specified false alarm probability by deriving the exact statistics for \Y\(p) (where Y is a Gaussian random variable and p is a positive real number) for the following two cases: (i) sample size L = 1 but for arbitrary real p > 0; (ii) arbitrary L but for p = 1, 2 or 4. This observation is in stark contrast to all earlier studies on the "improved" energy detector (that replaces the squaring operation of the signal amplitude in the classical energy detector with an arbitrary positive power p operation). Our analytical results are in excellent agreement with those obtained via Monte-Carlo simulations, and also highlight the inaccuracies with the Gamma density approximation for \Y\(p) employed by Chen [2]. Although different choices of p and L will yield distinct receiver operating characteristics (ROC) curves, the optimum p remains 2 (i.e., no gain over the classical energy detector) regardless of the values of the signal-to-noise ratio and/or the sample size L.