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A common problem in radar, navigation systems or pulse-time modulation systems is that of determination of the delay or time of arrival of a pulse or other distinctive modulation form. while considerable effort has been directed toward the optimization in various senses of the signal-to-noise ratios for such signals, the problem of the most accurate determination of time delay of signals in noise bas, with a few noteworthy exceptions, been taken for granted. It is of interest to examine this problem for its own sake, not only as indicative of optimum circuit practice, but, more important, in order to make clear the fundamental accuracy limitations and the manner in which these limitations depend upon signal and noise parameters. The consideration of this paper is limited to the case where there is no doubt as to the existence nor approximate position of a pulse but where the accuracy of determination of pulse position is limited by noise. This case is therefore more representative of the problems of high accuracy systems such as Loran or shoran than of radar.