The optimal power law for the detection of a Gaussian burst in a background of Gaussian noise

The optimal power law is analyzed using characteristic function techniques. For a signal occupying more than about λ=0.5 of the integration time, the standard square-law energy detector is close to optimal (among the class of power-law detectors). The optimal power is ν*=3 for λ≈0.3, increasing rapidly as λ decreases below this value. The SNR advantage of the optimal processor is not significant for λ>0.5, but exceeds 1 dB for λ<0.1. It is further shown that a cube-law processor yields close-to-optimal results over a wide range of λ. For the representative values of probability of false alarm (P_fa) and probability of detection (P_d) chosen in the performance analysis, the results do not depend strongly on the number of points N that are integrated. Calculations performed for other values of P _d and P_fa also show that the results do not depend strongly on the value of these two parameters