Nonlinear Systems with a Restricted Additivity Property

A special but nontrivial class of time-invariant systems is described, which is neither linear nor memoryless. This class of systems is characterized by the "semiadditive" property that the response to the sum of two nonoverlapping input time functions is equal to the sum of the responses to each input acting alone. These semiadditive systems are found to be amenable to characterization, analysis, and synthesis with a simplicity almost comparable to that of linear systems. In particular, it is shown that a semiadditive system is completely specified by an amplitude dependent step response function, a "needle pulse" response function of time and amplitude, or an amplitude dependent frequency response function. Explicit input-output relationships are presented, which include 1) an additivity integral that generalizes the superposition integral of linear systems, 2) a relation that yields the output spectrum directly from the input time function, and 3) a relation that yields the spectral density of the response to a stationary Gaussian input process directly from the autocorrelation function of the input. An analogy exists between semiadditive nonlinear systems and linear time-varying systems by interchanging the roles of amplitude and time.