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We find a new expression for the solving statistics of fast fluctuating Gaussian pulses with arbitrary-function envelope, based on which we can receive much simpler processing algorithms of random signals with unknown parameters in comparison with available analogues. For example, the synthesis and analysis of maximum likelihood detector and measurer of a high-frequency random pulse with unknown time parameter is carried out. The asymptotically exact expressions for detection and estimation characteristics including anomalous effects are presented. By methods of statistical computer modeling the adequacy of the considered analytical approach of the statistical analysis of random pulsed signals is corroborated, the working capacity of offered detector and measurer is established, and applicability borders of asymptotically exact formulas for their characteristics are also defined.