Unbiased Adaptive Estimations of the Fourth-Order Cumulant for Real Random Zero-Mean Signal

In this paper, a consistent efficient estimator of the fourth-order cumulant for real discrete-time random i.i.d. (at least up to order 8) zero-mean signal is proposed, in both, batch and adaptive versions. In batch version, the proposed estimator is not only consistent, but also unbiased and efficient. The systematical theoretical and experimental studies with comparisons between the proposed estimator and three other estimators of the fourth-order cumulant (the natural or the traditional one, the trivial unbiased estimator for the known power case and the fourth
k
-statistics), are undertaken, for both, normal and uniform processes. Then, the adaptive versions of the estimators (all, except the fourth
k
-statistics), are given and studied in detail. The convergence in mean and the convergence in mean square analyses are performed for them, first theoretically, then empirically. Finally, the whole set of analyses carried out for both batch and adaptive versions shows that from many points of view the proposed estimator is interesting for use in versatile signal processing applications, especially in real-time and short-term ones.