Volterra Systems With More Than One Input Port-Distortion in a Frequency Converter

This paper deals with nonlinear, time-invariant systems with memory which (i) have more than one input port, and (ii) are driven by inputs which are essentially sums of sine waves. The paper consists of two parts. Part I is concerned with a system which has two inputs, xu(t) and xv(t), and one output y(t). Two results for single-input systems are generalized: (i) an expression is given for an arbitrary frequency component of y(t) when xu(t) and x,,(i) are 1255 1256 T H E BELL SYSTEM TECHNICAL JOURNAL, OCTOBER 1973 finite sums of sine waves, and (ii) an expression is given for the power spectrum of y (t) when xu (t) is a stationary, zero-mean, Gaussian noise, and xv(t) is a single sine wave. Although most of the discussion in Part I deals with systems having two input ports, many of the results can be formally generalized to systems with more than two inputs. Part II is devoted to an example which shows how results given in Ref. 1 for a one-input Volterra system can be used to examine systems consisting of a single two-terminal nonlinear element imbedded in a linear network containing sources. The transformation from a multiinput to a single-input system is based upon Thevenin's theorem (see, for example, Anderson and Leon2). The example treated here is a frequency converter using a nonlinear capacitor. Particular attention is paid to computing the limiting form of the expression for the third-order distortion when the signal and pump amplitudes become small. The procedure we use in Part II is essentially a systemization of a procedure used by Gardiner and Ghobrial 3 to study the distortion performance of a varactor frequency converter.