Memoryless Nonlinearities

Memoryless or instantaneous nonlinearities form the simplest and most commonly implemented form of nonlinear element. Furthermore, many complex nonlinear systems can be broken down into a linear system containing a memoryless nonlinearity.

Given a sampled input signal $x(n)$, the output of any memoryless nonlinearity can be written as

$$y(n) = f(x(n))$$

where $f(\cdot)$ is ``some function'' which maps numbers to real numbers. We exclude the special case $f(x)=\alpha x$ which defines a simple linear gain of $\alpha$.

The fact that a function may be used to describe the nonlinearity implies that each input value is mapped to a unique output value. If it is also true that each output value is mapped to a unique input value, then the function is said to be one-to-one, and the mapping is invertible. If the function is instead ``many-to-one,'' then the inverse is ambiguous, with more than one input value corresponding to the same output value.