Peak sidelobe level gumbel distribution for arrays of randomly placed antennas

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. 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. It is shown that the peak sidelobe level distribution converges weakly to a Gumbel distribution in the limit of a large number of beampattern samples. This result is for both sparse and dense arrays of randomly placed antennas over a large aperture. The definition of a large aperture in this context is ambiguous, but a possible rule-of-thumb is that it is at least a wavelength.