New Results on the Performance of a Well-Known Class of Adaptive Filters

Abstmct-We derive a broad range of theoretical results concerning the performance and limit.tioas of a class of rnJoe adaptive fikxs. Applicrtions of theae filters have been proposed m many d i f f m t enginedng contexts which have in common the Idlowing idealized identificrtion problem: A system has a vector input x ( t) and a scrln output z ( t) = h’x(t), where h is an unknown timeinvariant coeffi-cient vector. From a knowledge of x ( t) and z ( t) it is reqM,to esti-mate h. The filta considered here 8djusts an &ate vector h(t) in 8 contrd loop, thus d h- h KF[z(t) -?(t)] ~ ( t) dt where?(t) = $x(t) , F is a sui-, m general nonlinear, function, and K is the loop gain. The effectiveness of the filta is detemim $ by the conveqence p m p t i e a of the m i d i g m e n t vector, r = h- k. With weak nondegenency requirements on x(r) we prove the exponential convergence to zero of the norm IIr(t)II. For ail values of K, we give uppet and lower bounds on the convergence rate which n e tight in that both bounds have simh qualitative dependence on K. The depen-dence of these bounds on K is unexpected and imporhnt since it reveils bade limitations of the filters which rte not predicted by the conventional approximate method of analysis, the “method of averag-ing. ” By analyzing the effects of added forcing term u(t) in the con-trol equation we obtain uppes bounds to the effects on the convergence process of various important departures from the idealized model as when noise is present as an dditiod component of z(t), the coeffi-cient vector h is time-vrrying, and the integrators in a hardware imple-mentation have fdte menory