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By the method of Darling and Siegert, a differential equation is obtained for the characteristic function of the output of a linear system when the input is a random telegraphic signal. For the ideal integrator with finite memory and for the RC low-pass filter, the solutions agree with previous results. For a truncated exponential weighting function, the characteristic function is obtained in terms of Bessel functions. For a particular ratio of the constants, the probability density of the output is rectangular except for delta functions. The applicability of this rectangular solution to other systems is investigated.