Time delay in adaptive filtering

Adaptive filtering with delayed data is examined in detail. The recursive algorithm developed has two features: delayed signals are allowed, and parallel implementation via pipelining structure can be incorporated into the framework of the algorithm. The algorithm considered is a natural generalization of the classical adaptive filter procedures. It is shown that convergence with probability one is preserved when delayed signals appear in the recursive algorithm. A simple example is given to demonstrate the convergence properties. It is demonstrated that the delays do not harm the computation procedure as far as the convergence properties are concerned