A theoretical result for processing signals that have unknown distributions and priors in white Gaussian noise

In many applications, observations result from the random presence or absence of random signals in independent and additive white Gaussian noise. When the observations are independent and the probabilities of presence of the signals are upper-bounded by some value in [0,1), a theoretical result is established for the noise standard deviation. The latter is the only positive real number satisfying a specific convergence criterion when the number of observations and the minimum amplitude of the signals tend to infinity. This convergence involves neither the probability distributions nor the probabilities of presence of the signals. An estimate of the noise standard deviation is derived from this theoretical result. A binary hypothesis test based upon this estimate is also proposed. This test performs the detection of signals whose norms are lower-bounded by some known real value and whose probabilities of presence are less than or equal to one half. Neither the estimate nor the test requires prior knowledge of the probability distributions of the signals. Experimental results are given for the case of practical importance where the signals are independent two-dimensional random vectors modelling modulated sinusoidal carriers. These experimental results suggest that the asymptotic conditions of the limit theorem are not so constraining and can certainly be significantly relaxed in practice. Typical applications concern radar, sonar, speech processing but also proximity sensing and Electronic (Warfare) Support Measures.