A geometrical derivation of the excess mean square error for Bussgang algorithms in a noiseless environment

Abstract

The steady-state excess mean square error (EMSE) is a useful performance criterion to measure how noisy Bussgang algorithms are. Thanks to a simple geometrical interpretation of LMS-like algorithms, the Pythagoras theorem gives us a general equation similar to the “fundamental energy conservation” of Mai and Sayed (IEEE Trans. Signal Process. 48 (1) (2000) 80). Thereafter, a simple, but general, closed form of the EMSE is derived for Bussgang algorithms when they have converged, i.e, when the optimal solution of cost function criterion is obtained. As an example of this closed form, the EMSE computation of the constant modulus algorithm is done and compared with the ones given in the literature.