#### Technical Report USC-SIPI-140

“System Identification Using Cumulants”

by Ananthram Swami

October 1988

Since second-order statistics are inadequate in describing non-Gaussian mixed phase linear process, certain higher-order statistics, namely cumulants, and their Fourier transforms, polyspectra, are used to estimate the parameters of linear processes. In the presence of additive (colored) Gaussian noise, cumulants are powerful analysis tools, because they effectively transform the data to a high signal-to-noise ratio domain.

Several algorithms to estimate the MA parameters of an ARMA model are developed; in particular, one of these algorithms permits the simultaneous estimation of the AR parameters and the impulse response coefficients. These algorithms are extended to multi-channel, multi-dimensional and non-causal models. It is shown that consistent AT estimate may be obtained from the 'normal' equations based on a finite set of 1-D slices of the cumulant; based on this, an AR order determination algorithm is proposed.

A Kronecker-product based approach is used to obtain a compact representation of the cumulants of vector processes; based on this, time-and lag recursive, as well as closed-form, expressions for the cumulants of the state and output processes of a time-varying, non-stationary, multiple output state-space model, are developed. A cumulant-based algorithm for the estimation of the matrices of the state-space model is developed.

By introducing an unconventional orthogonality condition, a set of four linear prediction problems, leading to the cumulant-based 'normal' equations is proposed; time-and order-recursive estimates of the multichannel AR parameters are obtained via a double lattice being excited by the observed process, and the other lattice being excited by an instrumental process. Some convergence results are presented, and the joint-process estimation problem is also addressed.

Cumulants of complex processes are defined, and a cumulant based approach to the harmonic retrieval and direction of arrival problems is proposed; techniques for detecting and quantifying quadratic and cubic phase coupling are developed. A viable definition for the cumulants of non-random signals is proposed, and the deterministic realization problem is addressed.

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