New results on the performance of a well-known class of adaptive filters

Abstract

We derive a broad range of theoretical results concerning the performance and limitations of a class of analog adaptive filters. Applications of these filters have been proposed in many different engineering contexts which have in common the following idealized identification problem: A system has a vector input x(t) and a scalar output z(t)=h'x(t), where h is an unknown time-invariant coefficient vector. From a knowledge of x(t) and z(t) it is required to estimate h. The filter considered here adjusts an estimate vector h <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">^</sup> (t) in a control loop, thus d/dt h <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">^</sup> = KF[z(t)-z <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">^</sup> (t)]x(t) where z <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">^</sup> (t) = h <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">^</sup> 'x(t), F is a suitable, in general nonlinear, function, and K is the loop gain. The effectiveness of the filter is determined by the convergence properties of the misalignment vector, r = h - h <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">^</sup> . With weak nondegeneracy requirements on x(t) we prove the exponential convergence to zero of the norm ||r(t)||. For all values of K, we give upper and lower bounds on the convergence rate which are tight in that both bounds have similar qualitative dependence on K. The dependence of these bounds on K is unexpected and important since it reveals basic limitations of the filters which are not predicted by the conventional approximate method of analysis, the "method of averaging." By analyzing the effects of an added forcing term u(t) in the control equation we obtain upper bounds to the effects on the convergence process of various important departures from the idealized model as when noise is present as an additional component of z(t), the coefficient vector h is time-varying, and the integrators in a hardware implementation have finite memory.