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This paper is concerned with certain aspects of the problem of discriminating between two Gaussian processes. The emphasis is on determining approximate optimum detector structures which avoid some of the mathematical difficulties inherent in the evaluation of the exact optimum detector structure. To this end, an approach termed the "inverse operator" approach is presented which leads to approximate detector structures via the Neumann series expansion of linear operator theory. These approximate detectors are found by using a finite number of terms in an "optimun detector" expansion which results from the use of the above Neumann series expansion. A sufficient condition for the rapid convergence of the optimum detector expansion is found to be that the eigenvalues of a certain operator have magnitudes much less than unity. An upper bound is derived for the error incurred at the detector output by the use of a finite number of terms in the optimum detector expansion. Error probabilities are calculated for the case in which the detector outputs may be assumed approximately normally distributed. From an output "signal-to-noise" ratio point of view, it is shown that the performance of the optimum detector and approximate detectors will differ negligibly if the above eigenvalues have squared magnitudes much less than unity. Some upper bounds are derived for the largest eigenvalue (magnitude).