New methods of radar detection performances analysis

Original methods of radar detection performances analysis are derived for a fluctuating or non-fluctuating target embedded in additive and a priori unknown noise. This kind of noise can be, for example, the sea or ground clutter encountered in surface-sited radar for the detection of target illuminated at low grazing angles or in high resolution radar. For these cases, the spiky clutter tends to have a statistic which strongly differs from the gaussian assumption. Therefore, the detection theory becomes difficult to perform since the nature of statistics has to be known. The new methods proposed here are based on the parametric modelisation of the moment generating function of the noise envelope by Padé approximation and lead to a powerful estimation of its probability density function. They allow to evaluate the radar detection performances of target embedded in any noise without knowledge of the closed form of its statistic and allow in the same way to take into account any possible fluctuation of the target. These methods have been tested successfully on synthetic signals and have been performed on experimental signals such as ground clutter. 1. DESCRIPTION OF THE PROBLEM The radar detection of a target against a background of unwanted clutter due to echoes from the sea or land is a problem of interest in the radar field. For many years, the statistic of quadrature components of the radar clutter was supposed to be jointly gaussian because of the low radar resolution capabilities : in this case, the clutter was viewed as a sum of responses from a very large number of elementary scatterers (Central Limit Theorem). The current systems have now improved their resolution capabilities and hence their performances of detection. However, as resolution has increased, the statistic of the additive noise have no longer been observed to be gaussian. Recent experimentations conducted in ONERA indicate that large deviation from Rayleigh statistics are observed in situations such as low grazing angle illumination or with high resolution radars. In such cases, due to the spiky nature of the clutter, the empirical distribution exhibits both higher tails and larger standard deviation to mean than predicted by the Rayleigh distribution. Therefore, many works have been devoted to fit empirical models of distribution to experimental data. This is the case of the compound gaussian processes [1, 2], also called Spherically Invariant Random Processes (SIRP) which allow to modelise the multivariate probability density function (PDF) of the envelope of the clutter returns, taking into account the possible spatial or temporal correlation of the processes. The well known log-normal, Weibull and K-distribution densities [3] belong to this class of distributions but the main problems for this kind of parametrization are the quality of the estimation of the SIRP parameters and the complexity of the optimal detector implementation. We propose in this paper to analyze the performances of radar detection of a target embedded in any combination of clutter and thermal noise without the knowledge of the closed form of the densities of the noises. The estimation of the noise envelope density is only performed according to the modelisation by Padé approximation of the moment generating function (MGF) of the noise envelope. This method is based on the estimation of all the n-order moments of the noise envelope. The goal of this paper being not to derive a method of evaluating the best estimation of the moments, they will be supposed exactly estimated. This kind of modelisation allows to derive, for a constant false alarm rate, the simple form of the probability of detection of a target with constant or fluctuating envelope embedded in a complex noise fully characterized by the moments of its envelope. 2. GENERAL RELATIONS OF THE DETECTION THEORY We consider here the basic problem of detecting the presence or absence of a complex signal s(t) with envelope A in a set of measurements y(t) = yI(t) + i yJ(t) corrupted by a sum of independent additive complex noises corresponding to the clutter echoes c(t) and white gaussian thermal noise n(t). This problem can be described mathematically in terms of a hypothesis test between the following pair of statistical hypothesis : H0 : y(t) = n(t) + c(t) (1) H1 : y(t) = s(t) + n(t) + c(t) (2) If we note pH0(r) the probability density of the noise envelope |n(t) + c(t)|, the detection threshold θ is fixed by the value of the given probability Pfa of false alarm. Pfa = ∫ +∞ θ pH0(r) dr (3) while, denoting pH1(r) the PDF of the envelope of the complex signal embedded in noise |s(t) + n(t) + c(t)|, the detection probability Pd is classically given by : Pd = ∫ +∞