Qualitative behavior of nonlinear differential equations describing adaptive filters using nonideal multipliers

We analyze the behavior of a well-known class of analog adaptive filters in which all the multipliers are replaced by nonideal multipliers for which the linearity requirement with respect to one of the inputs is relaxed. Applications of these filters have been proposed in many different engineering contexts which have in common the following idealized problem: a system has a vector input x
t
and a scalar output
z_{t} = h' x_{t}
, where
h
is an unknown lime-invariant coefficient vector. From a knowledge of x
t
and z
t
it is required to estimate
h
. The estimate vector is
f(\hat{h_{t}})
where h
t
is generated in the filter as the solution of a differential equation and
f(\cdot)
is a nonlinear map defined by the characteristics of the nonideal multipliers. The effectiveness of the filter is determined by the convergence properties of the misalignment vector,
r=h-f(\hat{h})
. With a weak nondegeneracy requirement on x
t
, the "mixing condition," we prove the exponential convergence of
\parallel r_{t} \parallel
. Considerable emphasis is placed on the analysis of the effects of various important departures from the idealized problem as when noise is present, the coefficient vector
h
is time varying, etc. The results show that in every important aspect the qualitative behavior is similar to that of the conventional filter in which ideal multipliers are used. However, new techniques are required for the analysis of this inherently nonlinear system.