This paper investigates the signal activity detection problem for spread spectrum signals in low probability of detection communication systems under Gaussian disturbance with unknown covariance matrix. We assume that the receiver is equipped with an adaptive antenna array with general array configuration, comprising of a primary array with high gain beam and a reference array with low gain beam. It has been shown that this reference array structure can benefit interference suppression. Since it is difficult to derive the uniformly most powerful detector for the detection problem, we resort to using the Wald test and generalized likelihood ratio test (GLRT) schemes to design the detectors. Analytical performance of the proposed Wald test and GLRT is derived, indicating that the proposed two detectors bear constant false alarm rate property. Finally, numerical simulations are carried out, which verify the correctness of the theoretical analysis. Besides, it is revealed that the use of reference channel can facilitate to enhance the detection performance, but an increasing in the antenna number does not necessarily lead to an improvement in detection performance.