A Method for Computing Moments of Quadratic Forms Involving Wrapped Random Variables

Abstract

This paper provides an analytical method for computing moments of quadratic forms containing independent wrapped random variables and observed complex phase data. Higher-order moments are obtained using block matrix representations (wherein each moment is defined in terms of a structured kernel matrix) and by noting the relationship between the expectation of a wrapped variable and the characteristic function of the associated linear variable. Higher-order kernel matrices are found to exhibit a “level in which the matrix is invariant to index exchanges between Kronecker block levels. This symmetry, a block matrix relative of higher-order tensor symmetry, is developed and leveraged to reduce the number of terms that must be computed in the moment algorithm. Results for wrapped normal variables are validated using Monte Carlo integration, whose moment estimates are shown to converge (with increasing ensemble size) to values obtained by the method presented here.