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A large class of physical signals may be characterized by mode parameters (natural frequencies and damping coefficients) and initial conditions. The mode parameters, usually governed by well understood physical models, are often accurately known apriori. Conversely, very little is usually known about initial conditions. The problem of interest here is to detect signals with known modes, but unknown initial conditions, in additive Gaussian noise of unknown level. The signals may also be characterized as impulse responses of ARMA systems with unknown MA parts. We derive an F-statistic that is optimum (uniformly most powerful) in the class of all receivers that are invariant to certain tranformations of the data. We argue that the invariances are natural constraints. The statistic we derive provides constant false alarm rate performance.