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In this paper, we consider the problem of detecting a random spatially distributed signal source by an array of sensors. We start with an approximate likelihood ratio (LR) detector and analyze its performance. Using the generalized likelihood ratio (GLR) approach, we then derive detectors under several assumptions on the available statistics. The performance of these detectors is evaluated, and the effect of the angular spread of the source is investigated. The detection performance behaves differently under different scenarios. We notice that the degrees of freedom (DOF) of the distributions of the detection statistics depend on both the signal angular spread and the number of data snapshots. Specifically, at a high SNR level and with small degrees of freedom, an increase of angular spread improves the detection performance. However, with large degrees of freedom, the increase of angular spread reduces detection performance. We provide a detailed discussion of the behavior of detection performance under various conditions. A comparison between the GLR detectors and conventional beamformer detectors is made by computer simulations. The results indicate that the GLR detectors perform better as the angular spread becomes large than that of the conventional beamformer detectors.