Peak sidelobe level distribution computation for ad hoc arrays using extreme value theory

Extreme Value Theory (EVT) is used to analyze the peak sidelobe level distribution for array element positions with arbitrary probability distributions. Computations are discussed in the context of linear antenna arrays using electromagnetic energy. The results also apply to planar arrays of random elements that can be transformed into linear arrays. Before EVT is introduced, the number of times a beampattern crosses a certain level in an upward direction is considered. For this upward-crossing method, the evaluation of the probability of exceeding a given peak sidelobe is investigated as a function of the antenna array spatial position variance in the asymptotic limit of a large number of array elements. For sparse arrays with small number of elements, Gaussian approximations to the beampattern distribution at a particular angle introduce inaccuracies to the probability calculations. EVT is applied without making these Gaussian approximations. A bound is given for how close using a certain number of beampattern samples will get to the true peak sidelobe level of a random array. It is shown that the peak sidelobe level distribution converges to a Gumbel distribution in the limit of a large number of beampattern samples when the number of elements is larger than ten. It is also shown that being in the domain of attraction of the Gumbel distribution occurs under weak convergence as the number of elements increases. An expression for the beampattern distribution at a particular angle is given for any number of array elements, and simulations show that it is in the domain of attraction of the Weibull distribution.