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We have recently proposed a cross-spectrum based method for the two-channel blind identification problem. Two observations were assumed available, corresponding to the output of two channels driven by the same input. The channels were estimated based on the phase of the cross-correlation of the observations, filtered through their minimum-phase equivalent filters. Through simulations that method was found to be robust to additive noise, and significantly outperformed existing methods at low signal-to-noise ratios. In this paper we provide an analytic justification of the robustness of that method. We study analytically the propagation of noise in the phase estimate and in the channel estimates. Validation of the analytical results is provided by means of Monte Carlo simulations.