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A performance coefficient corresponding to an extended form of the signal-to-noise ratio at the output of an estimatorcorrelator detector is introduced and discussed. A suboptimum approach is then considered to the detection of signals transmitted through a purely random channel and corrupted by additive Gaussian noise, with arbitrary signal-to-noise ratios at the channel output. An estimator-correlator detector is postulated and the linear filter that maximizes the performance coefficient is determined. This performance coefficient is then once more maximized with respect to the transmitted waveform and the results of this analysis are applied to cases characterized by arbitrary large signal-to-noise ratios and Gaussian quasi-stationary undersea acoustic channels.