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We address the matched detector problem in the case the signal to be detected is imperfectly known. While in the standard detector the signal is known to lie along a particular direction, we consider the case where this direction is known up to additive white Gaussian noise. This somehow amounts to assuming that the signal lies in a cone the aperture of which depends upon the level of uncertainty. We build the associated generalized likelihood ratio (GLR), analyze its statistical properties, indicate how to set the threshold to achieve a given false alarm rate, and how to predict the associated probability of detection. The so-obtained detector reduces to the conventional one when the uncertainty vanishes and we analyze its behavior when the level of uncertainty, which has to be known a priori, is mis-evaluated. It appears that the sensitivity of the detector is quite low with respect to this kind of errors. More importantly several realistic examples are presented that indicate that the proposed detector remains quite efficient when the true signals are far from being of the assumed model and whatever the model of the uncertainty actually is. It is this robustness that makes the detector valuable.