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Here we show that by using second-order statistics (SOS) of a system output all the spectrally equivalent (SE) systems can be derived, and the correct phase can then be chosen by applying appropriately non-linear techniques such as cumulants. In this paper, a new method is proposed that first extracts higher-order statistics (HOS) information (i.e. phase) and then computes directly the true system using only SOS, thus avoiding searching the whole SE set. This new approach depends minimally on HOS, thus avoiding the well-understood limitations of HOS calculations. Finally, comparisons are made with similar approaches through both simulation and analysis, and the advantages of this new method are documented.