Optimum-distributed signal detection system design is studied for cases with statistically dependent observations from sensor to sensor. The common parallel architecture is assumed. Here, each sensor sends a decision to a fusion center that determines a final binary decision using a nonrandomized fusion rule. General sensor cases are considered. A discretized iterative algorithm is suggested that can provide approximate solutions to the necessary conditions for optimum distributed sensor decision rules under a fixed fusion rule. The algorithm is shown to converge in a finite number of iterations, and the solutions obtained are shown to approach the solutions to the original problem, without discretization, as the variable step size shrinks to zero. In the formulation, both binary and multiple-bit sensor decisions cases are considered. Illustrative numerical examples are presented for two-L, three-L, and four-sensor cases, in which a common random Gaussian signal is to be detected in Gaussian noise. Some unexpected properties of distributed signal detection systems are also proven to be true. In an L-sensor-distributed detection system, which uses 1 bits in the decisions of the first 1 sensors, the last sensor should use no greater than 21 bits in its decision. Using more than this number of bits cannot improve performance. Further, in these cases, a particular fusion rule, which depends only on the number of bits used in the sensor decisions, can be used without acrificing any performance. This fusion rule can achieve optimum performance with the correct set of sensor decision rules.