A Kronecker product formulation of the cumulant based realization of stochastic systems

We derive new recursive equations ad closed form expressions relating the parameters of an ARMA model (which may be non-minimum phase and/or non-causal) with the cumulants of its output, in response to excitation by a non-Gaussian i.i.d. process. Based on these relationships, cumulant-based stochastic realization algorithms are developed. The output noise may be colored Gaussian or i.i.d. non-Gaussian with unknown variance. When a state-space representation is adopted, the stochastic realization problem reduces to the realization of an appropriate Hankel matrix formed by cumulant statistics. Using a Kronecker product formulation, exact expressions are presented for identifying state-space quantities when output cumulants are provided, or for computing output cumulants when the state-space triple is known. Conditions for stationarity of a linear process, with respect to its cumulants, are also presented. If a transfer function approach is employed, alternative formulations are given to cover the case of non-causal models.