Recovering the poles from third-order cumulants of system output

The problem of identifying the poles of single-input/single-output (SISO) linear stochastic systems from the higher-order statistics of noisy observations is considered. It is assumed that the system is driven by an independent and identically distributed non-Gaussian process with nonzero third-order cumulant function at zero lag. There is no other restriction of the probability distribution of the driving noise. The system is assumed to be of known order, causal, and exponentially stable, but is not required to be minimum phase. The system output is observed in additive, possibly non-Gaussian, noise. It is shown that if there are no pole-zero cancellations in the transfer function of the given system, then it is necessary and sufficient for a block Hankel matrix to have rank equal to the system order where the matrix is constructed from a partial set of third-order cumulants of the noisy output sequence. This fundamental result then leads to a linear solution to the problem of estimating the coefficients of the system characteristic polynomial from which the system poles can be found via root-finding