Understanding the explosive divergence of the FTF algorithm

Along with its many desirable properties the fast transversal filter (FTF) algorithm suffers from explosive divergence. This type of divergence occurs when the algorithm is seemingly performing its operations normally, producing usable solutions, when the algorithm appears to suddenly produce extremely large errors and an obviously useless solution. Although it is known that a loss of backward consistency is the cause for the resultant perturbations, i.e., a violation to interrelationships between update parameters are not explicitly enforced by the update equations, it is not known why the algorithm suffers explosive divergence rather than a divergence that grows as a continuous function over time. Algorithms have been proposed to circumvent this problem but it remains to be shown through theoretical justification whether these algorithms have remedied the problem or only put it off to some later iteration. Here, we provide a rationale to explain the explosive character of divergence that is inherent to the manner in which the FTF algorithm is derived.