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Extracting multiple unknown sources from a single observation of a single-channel is an ill-posed problem encountered in a variety of applications. This paper characterizes the ambiguity of solutions to the source separation problem, and then proposes a novel adaptive-operator-based approach to deriving solutions based on a combination of separation operators and domain-specific knowledge related to sources. The proposed scheme involves transforming the original problem into a new problem, in which data-dependent operators and the unknown sources are variables to be optimized. We demonstrate that a solution to the proposed optimization problem must reside in the null spaces of the operators, and any such solution also provides an optimal value to the original problem. We then demonstrate the applicability of the proposed method to the separation of sparse sources as well as AM-FM sources. Note that the proposed scheme outperformed corresponding state-of-the-art methods in noiseless as well as noisy environments. Finally, we demonstrate the efficacy of the proposed scheme in separation tasks based on real-world ECG data (i.e., extracting fetal ECG signals from noisy observations in which maternal and fetal ECGs recordings are superimposed) and electrical data (i.e.,separating singularities from harmonic components in an observation of noisy data related to surges in electrical current). [ABSTRACT FROM AUTHOR]

We propose a novel operator-based model called the null space component analysis (NCA) to solve the noisy blind source separation (BSS) problem. Theoretically, we show that the NCA can resolve the rotation ambiguity in the BSS problem. In a set of m linearly independent source signals, the basic principle of the NCA is to associate each signal with a separating operator that includes the signal in its null space and repels other signals from the space. We show that the model can act as a constraint on the source signals in the noisy BSS problem. In contrast to the ICA-based and the sparsity-based approaches, NCA is a deterministic and data-adaptive algorithm that can solve both the under-determined and the over-determined BSS problems. To demonstrate the algorithm׳s efficiency, we process several types of signals, including real-life signals obtained from biomedical systems, and compare the results with those derived by other methods. [ABSTRACT FROM AUTHOR]