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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]

Frames are the foundation of the linear operators used in the decomposition and reconstruction of signals, such as the discrete Fourier transform, Gabor, wavelets, and curvelet transforms. The emergence of sparse representation models has shifted the emphasis in frame theory toward sparse-norm minimization problems. In this paper, we apply frame theory to the sparse representation of signals, wherein a frame is used for synthesis dictionary and a dual frame is used for analysis dictionary. Our objective was to formulate a novel dual-frame design in which the sparse vector obtained through the decomposition of any signal is also the sparse solution representing signals based on a reconstruction frame. Our findings demonstrate that this type of dual frame cannot be constructed for overcomplete frames, thereby precluding the use of any linear analysis operator to derive the sparse synthesis coefficient for signal representation. We also developed a novel dictionary learning algorithm (called Parseval K-SVD) to learn a tight-frame dictionary with normalized atoms. We then dealt with the problem of signal representation using frames from the analysis and synthesis perspective to derive optimization formulations for problems pertaining to image recovery. Our preliminary results demonstrate that the images recovered using this approach are correlated with the frame bounds of dictionaries, thereby demonstrating the importance of using different dictionaries for different applications. [ABSTRACT FROM AUTHOR]