This report considers the problem of the calculation of the distribution function of the output of a linear filter with a random square—wave input. The systems considered are the finite-time integrator, the RC low- pass filter, and certain restricted higher-order filters. The inputs are square-waves in which the lengths of axis-crossing intervals are random, but statistically independent. For the finite-time integrator with a coin-toss square-wave input, a difference equation for the characteristic function of the output is derived and solved. The continuity and differentiability properties of the distribution function of the output of an RC low—pass filter are discussed. Under specified conditions on an RC low-pass filter with a coin-toss square—wave input, the distribution function of the output is constructed. For the same problem, a functional equation is derived for the characteristic function of the output, and a recurrence relation is obtained for certain moments of the output. For a general square-wave input, an integral equation is derived for the characteristic function of the output of an RG low-pass filter at an axis—crossing of the input. From this equation a second recurrence relation for the moments of the output is obtained. For the coin-toss square-wave input, certain higher-order systems are also considered. In particular, when a second-order system is tuned to the clocking rate of the input, the problem is reduced to an equivalent first-order problem, and the distribution function for the output at an arbitrary instant is expressed in terms of a related function for the RC filter.
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