Recovering the Poles from Fourth-Order Cumulants of System Output

The problem of identifying the pole 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 non-zero fourth-order cumulant function at zero lag. There is no other restriction on the probability distribution of the driving noise. The system is assumed to be causal and exponentially stable, but is not required to be minimum phase. The system output is observed in additive, possibly non-Gaussian, noise. We show 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 fourth-order cumulants of the noisy output sequence. This fundamental result then leads to a linear solution to the problem of the pole recovery. We conclude by briefly discussing implications of these results for signal processing and system identification with higher-order statistics including: model fitting with colored noise, estimation of cumulant spectra of linear processes, and linear identification of noncausal autoregressive model parameters.