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Higher order spectra analysis can be employed to reconstruct a corrupted signal via the third-order spectrum, i.e. the bispectrum. However, this technique is plagued by an unknown misplacement of the origin of the restored signal, this being true both for 1D signals such as audio and 2D signals such as images. In this contribution we tackle and solve the misplacement in the 1D case. We first present the details of the problem and the solution adopted, then we test our method on both synthetic and real 1D signals. Finally we simulate 1D turbulent signals and restore them, showing that our algorithm, once extended to higher dimensions, could be well suited for the restoration of images affected by air turbulence distortion.