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In this paper, signals in , a subspace of the space of square integrable signals defined on , are approximated by signals in , the one-dimensional subspace of spanned by the first function from the set of reversed time Laguerre functions. A system mapping into itself is associated with a system mapping into itself; the latter system is characterized by a gain-exponential describing function. This type of describing function is developed as an analysis tool for studying the transient response of a large class of nonlinear feedback systems. The contraction-mapping fixed-point theorem is used to develop conditions for the existence of a solution prior to the use of the exponential describing function to obtain an approximate solution.