Recursive Methods for Computation of One-Dimensional Cumulants

One-dimensional cumulants are useful for solving a wide variety of system-theoretic and signal processing problems. How can 1D cumulants be calculated? This paper shows how to compute 1D third- and fourth- order cumulants for a single-input, single-output linear time-invariant system that is described by a state-variable model. Results are given for both stationary and nonstationary systems. Emphasis is placed on computational procedures that are recursive. For stationary systems, our algorithms are recursive with respect to the cumulant lag. Because the cumulant is nonsymmetric with respect to lag zero, different algorithms are needed for positive and negative lags. For nonstationary systems, our algorithms are recursive either in time or lag, or in both time and lag. Our recursive algorithms are presented in two ways, namely matix form or Kronecker product form. The Kronecker product form is especially useful for fourth-order cumulant calculations.