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Motivated by the developments on iterate averaging of recursive stochastic approximation algorithms and asymptotic analysis of sign-error algorithms for adaptive filtering, this work develops two-stage sign algorithms for adaptive filtering. The proposed algorithms are based on constructions of a sequence of estimates using large step sizes followed by iterate averaging. Our main effort is devoted to improving the performance of the algorithms by establishing asymptotic normality of a suitably scaled sequence of the estimation errors. The asymptotic covariance is calculated and shown to be the smallest possible. Hence, the asymptotic efficiency or asymptotic optimality is obtained. Then variants of the algorithm including sign-regressor procedures and constant-step algorithms are studied. The minimal window width of averaging is also dealt with. Finally, iterate-averaging algorithms for blind multiuser detection in direct sequence/code-division multiple-access (DS/CDMA) systems are proposed and developed, and numerical examples are examined.