Poisson approximation for excursions of adaptive algorithms

This paper analyzes excursions of adaptive algorithms. The distribution of the number of excursions in n units of time is approximated by a Poisson distribution. The mean and distribution of the time of the occurrence of the first excursion are approximated by those of an exponential distribution. Expressions for the error in the approximations are derived. The approximations are shown to hold asymptotically as the excursion defining set converges to the empty set and as the algorithm´s step size μ converges to zero. The validity of the approximations is tested on a variety of examples. The updates of the error between the estimated and optimal weights for many adaptive filters (for example the least mean square algorithm and its “signed” variants) are of the form of the equation given for the excursions of adaptive algorithms