Some mathematical results on the effects on digital adaptive filters of implementation errors and noise

Recently, we have shown that a necessary and sufficient condition for uniform convergence in a well-known class of digital adaptive filters is that the inputs satisfy a ´mixing condition´. In the case of mixing inputs, system noise, or errors incurred in the digital implementation, with bounded size, can produce only bounded behavior of the filter. We show here that, in many applications, inputs may be expected not to mix. We analyze the effects of two specific types of implementation errors in both the mixing and nonmixing cases. We show that arbitrarily small bounds on each type of error leads to unbounded behavior of the filter if the input is nonmixing and both types of errors are present. However, if only one type of error is present, then unbounded behavior does not occur if the error is small.