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“Finite-Sample Covariances of Second-, Third-, and Fourth-Order Sample Cumulants in Narrowband Array Processing”
by Thomas Kaiser and Jerry M. Mendel
September 1996
Recently, fourth-order cumulants were successfully applied in the area of narrowband array signal processing. For example, the virtual-ESPRIT-Algorithm (VESPA) [8] for direction-finding and recovery of independent sources can also calibrate an array of unknown configuration. Furthermore with extended VESPA [16] direction-finding of highly correlated or coherent sources is possible. In addition, fourth-order cumulants can also be used for estimation of the range as well as the angle in the near-field case [2].
This large number of new algorithms motivates a performance analysis to compare the higher-order statistics based algorithms with the conventional second-order statistics based algorithms. Up to now, for higher-order statistics based algorithms only asymptotic results are available for direction-finding. These results are restricted to a certain class of communication signals [1].
In this technical report we avoid these restrictions by deriving the finite-sample covariance of
- the second-order sample cumulant (moment), - the third-order sample cumulant (moment), and - the fourth-order sample cumulant
for: finite data length, any kind of random signals, any kind of noises, any array shapes, and, arbitrary sensors. We do this, because most performance analyses are based on one of these covariances. Consequently, this report provides the fundamentals for very general performance analyses of second- and higher-order statistics based array-processing algorithms.