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Starting with the canonical discrete-time detection formula for random signals in Gaussian noise, we demonstrate how different noise models lead to different limiting forms Of the detector. A Brownian motion model leads to the well-known estimator-correlator formula. With a band-limited white noise model, the limiting form contains an additional term: the posterior conditional variance. This second form, which is the Radon-Nikodym derivative of two finitely additive Gaussian measures, was derived by Balakrishnan using a different formulation . We thus provide an alternative development and interpretation of this second detector, and illustrate the effect of model assumptions on going from discrete-time to continuous-time problems.