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Summary: The Optimum Error Nonlinearity in LMS Adaptation with an
Independent and Identically Distributed Input
Tareq Y. Al-Na ouri Azzedine Zerguine & Maamar Bettayeb
Electrical Engineering Department Electrical Engineering Department
Stanford University King Fahd University of Petroleum & Minerals
Stanford, CA 94305 Dhahran 31261
USA Saudi Arabia
Abstract
The class of LMS algorithms employing a gen-
eral error nonlinearity is considered. The calculus
of variations is employed to obtain the optimum er-
ror nonlinearity for an independent and identically
distributed input. The nonlinearity represents a uni-
fying view of error nonlinearities in LMS adaptation.
In particular, it subsumes two recently developed
optimum nonlinearities for arbitrary and Gaussian
inputs. Moreover, several more familiar algorithms
such as the LMS algorithm, the least-mean fourth
LMF algorithm and its family, and the mixed norm
algorithm employ nonlinearities that are actually
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