## Analysis of Nonlinear Filters

There is no general theory of nonlinear systems. A nonlinear system with memory can be quite surprising. In particular, it can emit any output signal in response to any input signal. For example, it could replace all music by Beethoven with something by Mozart, etc. That said, many subclasses of nonlinear filters can be successfully analyzed:

- A nonlinear, memoryless, time-invariant ``black box'' can be ``mapped out'' by measuring its response to a scaled impulse at each amplitude , where denotes the impulse signal ( ).
- A memoryless nonlinearity followed by an LTI filter can similarly be
characterized by a stack of impulse-responses indexed by amplitude (look up
*dynamic convolution*on the Web).

One often-used tool for nonlinear systems analysis is Volterra series [4]. A Volterra series expansion represents a nonlinear system as a sum of iterated convolutions:

*Volterra kernels*. The special notation indicates that the second-order kernel is fundamentally two-dimensional, meaning that the third term above (the first nonlinear term) is written out explicitly as

In the special case for which the Volterra expansion reduces to

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A Musical Time-Varying Filter Example