Polyspectrum modeling using linear or quadratic filters

The polyspectrum modeling problem using linear or quadratic filters is investigated. In the linear case, it is shown that, if the output pth-order cumulant function of a filter, driven by a white noise, is of finite extent, then the filter necessarily has a finite-extent impulse response. It is proved that every factorable polyspectrum with a non-Gaussian white noise can also be modeled with a quadratic filter driven by a Gaussian white noise. It is shown that, if the quadratic filter has a finite-extent impulse response, then the output pth-order cumulant function is of finite extent; and if the output pth-order cumulant function of a quadratic filter is of finite extent, then the impulse response may or may not be of finite extent. It is shown that there exist finite and infinite extent pth-order cumulant functions that are not factorable but can be modeled with quadratic filters.<>