Convergence of the sign algorithm for adaptive filtering with correlated data

Convergence of a decreasing gain sign algorithm (SA) for adaptive filtering is analyzed. The presence of the hard limiter in the algorithm makes a rigorous analysis difficult. Therefore, there are few results available. Such results normally include restrictive assumptions such as the assumptions that successive observation vectors are independent and the new error signal of the adaptive filter has a time invariant probability density function. The former assumption is not valid in the context of adaptive filtering since two successive observation vectors share most of their components, while the latter assumption is a restriction on the adaptive weights whose evolution is a priori unknown. In lieu of using these assumptions, an almost-sure convergence of the SA is proved under the assumption that the sequence of observation vectors is M-dependent. This assumption allows strong correlation between successive observations