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This paper treats the detection of pulsed signals in the presence of receiver noise for the case of randomly fluctuating signal strength. The system considered consists of a predetection stage, a square law envelope detector, and a linear postdetection integrator. The main problem is the calculation of the probability density function of the output of the postdetection integrator. The analysis is carried out for a large family of probability density functions of the signal fluctuations and for very general types of correlation properties of the signal fluctuations. The effects of nonuniform beam shape and of nonuniform weighting of pulses by the postdetection integrator are also taken into account. The function which is actually evaluated is the Laplace transform of the probability density function of the integrator output. In many of the cases treated, the resulting Laplace transform has an inverse of known form. In such cases the evaluation of the probability density function would require the computation of a finite number of constants; in practice this would usually require the use of computing machinery, but would be perfectly feasible with presently available computing machinery.