A bound for the magnitude frequency domain characteristics associated with the outputs of a wide class of nonlinear systems is derived as a relatively simple function of the generalized frequency response functions and properties of the system inputs. It is shown how the practical computation of the new bound can be easily performed for nonlinear systems with finite but arbitrary order nonlinearities and worked examples are included. The paper is divided into two parts. In Part 1, an expression for the output magnitude bound is derived, properties of the result are discussed and general procedures for the practical computation of the bound are developed. In Part 2 the practical computations associated with applying the bound to the polynomial nonlinear autoregressive model with exogenous input are discussed.