Modeling Using Linear or Quadratic Filters

The polyspectrum modeling problem using linear or quadratic filters is investigated. In the linear case, we show 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. Next, we establish expressions of the output polyspectrum of a quadratic filter, driven by a Gaussian white noise. We prove that every factorable polyspectrum with a non-Gaussian white noise, can also he modeled with a quadratic filter driven by a Gaussian white noise. We show that, if the quadratic filter has a finite extent impulse response, then the output p th-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, Some examples are given to illustrate the polyspectrum modeling using quadratic filters, in particular, we show that there exist finite and infinite extent p th-order cumulant functions that are not factorable hut can he modeled with quadratic filters.