Consider a nonlinear, time-invariant system of the following form:
$g(t) = \left[h_1(t) \ast f(t)^1\right] + \left[h_2(t) \ast f(t)^2\right] + \left[h_3(t) \ast f(t)^3\right] + ...$
where $\ast$ denotes convolution, and the $h_n(t)$'s are different impulse responses of the system for each power of the input signal.
This is clearly a sort of nonlinear system that's "in between" a Taylor series and a Volterra series:
- To get a Taylor series, force the system to be memoryless by requiring the $h_n(t)$'s to be scaled delta distributions.
- To get a Volterra series, add in additional terms where the input is being multiplied by time-shifted versions of itself.
My questions:
- Is there a name for this sort of series?
- Is there a well-known characterization of the sorts of nonlinear time-invariant systems that can be completely described by this series?
- Does this series turn out to be useful in practical applications?
- Does anyone have any general reference material on this series and where it fits into the nonlinear signal "landscape"?