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An analytical expression is derived for the average output signal-to-interference-plus-noise ratio of optimum combining for a spatially correlated array of antennas in the presence of a single interferer and Rayleigh fading. Using the derived expression and based on an asymptotic analysis of the eigenvalues of dense correlation matrices, the asymptotic performance of optimum combining is evaluated as the number of the antennas increases while the total physical size of the array is fixed. Two different scenarios are considered, namely, fixed average received power per antenna and fixed total average received power. It is shown that in the former scenario, the average output signal-to-interference-plus-noise ratio is asymptotically a linear function of the number of the antennas while in the latter scenario it eventually saturates at a certain value. The slope of the asymptotic line in the former scenario as well as the value of the saturation limit in the latter scenario are derived in terms of the point spectrum of the underlying array correlation function. The case of multiple interferers is examined by simulation and is shown to exhibit similar asymptotic behaviour to the case of one interferer.