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This letter proposes a new algorithm for efficiently finding an unbiased RLS estimate of FIR models with noisy inputs. The unbiased estimate is obtained without knowing any a priori information via a new cost. Furthermore, to reduce computational complexity, the estimate is updated along the current input-vector direction and the corresponding gain is efficiently computed. In addition, to increase the convergence rate, the algorithm is extended to update the estimate along not only current but also past input-vector directions. Simulation results show that the proposed algorithm exhibits a fast convergence rate and an enhanced tracking performance with noisy correlated inputs.