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Blind Source Separation (BSS), Culling, Non-stationary, Variational Mode Decomposition (VMD).

The Fourier Transform (FT) is the single best-known technique for viewing and reconstructing signals. It has many uses in all realms of signal processing, communications, image processing, radar, optics, etc. The premise of the FT is to decompose a signal into its frequency components, where a coefficient is determined to represent the amplitude of each frequency component. It is rarely ever emphasized, however, that this coefficient is a constant. The implication of that fact is that Fourier Analysis (FA) is limited in its accuracy at representing signals that are time-varying, e.g. non-stationary. Another novel technique called empirical mode decomposition (EMD) was introduced in the late 1990s to overcome the limits of FA, but the EMD was shown to have stability issues in reconstructing non-stationary signals in the presence of noise or sampling errors. More recently, a technique called variational mode decomposition (VMD) was introduced that overcomes the limitations of both aforementioned methods. This is a powerful technique that can reconstruct non-stationary signals blindly. It is only limited in the choice of the number of modes, K, in the decomposition. In this paper, we discuss how K may be determined a priori, using several examples. We also present a new approach that applies VMD to the problem of blind source separation (BSS) of two signals, estimating the strong powered signal, termed the interferer, first and then extracting the weaker one, termed the signal-of-interest (SOI). The baseline approach is to use all the predetermined K modes to reconstruct the interferer and then subtract its estimate from the received signal to estimate the SOI. We then devise an approach to choose a subset of the K modes to better estimate the interferer, termed culling, based on a very rough a priori frequency estimate of the weak SOI. We show that the VMD method with culling results in improvement in the mean-square error (MSE) of the estimates over the baseline approach by nearly an order of magnitude.