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Random noise places a fundamental physical limitation on the precision with which a signal may be observed or measured. In the case where signal energy arrives piecemeal in a repetitive manner, this energy may be integrated over a period of time to provide an increase in signal-to-noise ratio, thereby reducing this limitation. Several techniques have been suggested for performing this integration process; recent work with an electronic barrier-grid storage tube has shown that such a device exhibits many desirable features when used as an integrating device. An introduction to integrator theory in general and an analysis of the storage tube operating as an integrator are given. Calculations are made of the expected improvement in signal-to-noise ratio as a function of tube parameters and number of integrations. Evidence is presented to show that the improvement achieved experimentally is nearly equal to the calculated value.