Peak sidelobe level gumbel distribution of antenna arrays with random phase centers

The maximum value of an antenna array's sidelobe beampattern, or radiation pattern in the power domain, is an important parameter determining its performance. In this paper, when array antenna elements have random phase centers, we approximate the maximum sidelobe value, or peak sidelobe level, as a Gumbel distribution using the extreme value theory (EVT). Before the EVT result, an expression for the beampattern distribution at each angle in the array field of view is found. From this expression, the pointwise convergence of the difference between the beampattern and exponential distributions in the limit of a large number of antennas is obtained. Using the exponential distribution approximation, EVT is applied with samples of the beampattern. A bound is given for the difference between the beampattern sample maximum and its true maximum in the sidelobe region.