Signal Distortion in Nonlinear Feedback Systems

This paper reports on some properties of the solutions to the functional equation $s_{2}(t) = \varphi[{\bf C}s_{2}(t) + s_{1}(t)],$ where ϕ is a nonlinear function, the operator C is a convolution, and s 1 is a known function belonging to a prescribed Banach space. The equation plays a central role in the theory of signal transmission through a general physical system containing linear time-invariant elements and a single time-variable nonlinear element. In particular we establish conditions under which s 2 (t) is the fixed point of a contraction mapping of the Banach space into itself and we discuss some consequences of this result. As a direct application, we consider the range of validity of two simple cascade flow graphs (i.e., flow graphs without feedback loops) for approximately determining the signal distortion in nonlinear feedback systems when the distortion is small. Our discussion is not restricted to specific types of nonlinear characteristics