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Simultaneously orthogonal expansion of two processes is one of the major mathematical tools for solving the problem of optimum detection of Gaussian signals in Gaussian noise. This expansion yields two integral equations: a homogeneous equation for the threshold and an inhomogeneous one for the test statistic of an optimum decision rule. After reviewing the optimum detection theory leading to the integral equations, four examples are presented to illustrate techniques of solving these equations and determination of the thresholds and test statistics. These techniques involve only elementary calculus and simple linear algebra. Finally, by way of example, an asymptotic interpretation of "white noise" in the context of optimum detection theory is given.