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A similarity measure for complex numbers Authors: T. Birnbaum Publication Date: Jun. 2017
Abstract: Complex numbers are encountered in many applications due to integraltransformssuch as Fourier or Laplace or simply due to implicit wave phenomena. Butwhilecountless similarity measures exist for real valued data, its difficult tocompare two complex datasets. One has to resort to either comparing the datainparts, e.g. real/imaginary or absolute value/phase, or resort to errormeasuresbased on the mean squared error (MSE). Often applications provide a goodintuition of which mismatches are tolerable. We therefore propose a versatilesimilarity measure that allows an easy local and global interpretation. Itis specifically crafted for complex-valued data and guarantees scaleinvariance, invariance under constant phase shifts, symmetry with respect tothe arguments, but can be freely adjusted to match the desired notion ofsimilarity. We show that it globally matches the MSE and hope to provide auseful template for anybody required to compare complex-valued datasets.
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