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Nonparametric estimators of frequency response functions are usually biased when the system to be identified has unknown, possibly correlated, additive noise on both the inputs and outputs. In this paper, it is shown how the use of a specific type of input force (random with periodic second-order statistics) leads to a simple, but unbiased, estimator, even in the presence of high levels of additive stationary noise at the inputs and the outputs. No other a priori information on the noise sources is required and these are even allowed to be mutually correlated. The statistical performance of the proposed Hα estimator is analyzed and a nontrivial formula is established for its variance. In turn, this analytical result is used for designing a suboptimal input signal. Finally, the predicted performance of the proposed estimator is validated with the aid of experimental results.