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In most previously reported studies on locally optimum detection of signals, independent observations have been assumed in various noise environments. The use of an independent observation model may cause a considerable performance degradation in detection applications of modern high data-rate communication systems exhibiting dependence among interference components. In this paper, the detection of weak known and random signals is addressed in observations corrupted by multiplicative and first-order Markov additive noises. The asymptotic and finite sample-size performance of several detectors are obtained and compared: it is confirmed that the dependence of noise components need to be taken into account to maintain detection performance appropriately.