A tight upper bound of the average absolute error in a constant step-size sign algorithm

A direct performance index of the adaptive filtering sign algorithm (SA) is the average absolute error (AAE) at the output of the filter. Adopting this performance index, an easy analysis of SA is achieved under a weak assumption. It is proved, for both deterministic and random inputs to the filter, that the AAE has a tight upper bound that exceeds the minimum AAE by half the product of the step size and power of the filter input. The assumption used is existence of average squared and average absolute values of filter input signals. A practical interest of the result is that it provides a formula for the biggest step size as a function of tolerable adaptation-noise-to-desired-signal ratio