Amplitude Estimation and Source Separation for Harmonics Within Two Mixtures Based on a Min-Rank Criterion

Exponentially decaying sinusoidal signal model is generally applied in a wide range of engineering and research fields. In order to analyze the signal components quantitatively, we need to extract the important parameters, i.e., amplitudes, frequencies, and damping factors, from the signals. In some practical applications, we can obtain multiple signals from multiple sensors or with multiple procedures. In such a way, the acquired signals are composed of similar sources or components but with different intensities. How to separate different sources and estimate the intensities of each source within the signals is essential for further analysis. In this paper, we separated different sources within two mixtures by estimating the relative intensities of the sources within mixtures. First, to evaluate a “clean” separation result, we proposed a minimum rank criterion, i.e., the Hankel matrices of the isolated sources should have the minimum rank, which corresponds to a simplified signal composition. Second, in order to minimize the rank of the Hankel matrices, we reformulated the objective function as a generalized eigenvalue decomposition problem, which can be solved based on the minimum perturbation theory. By this method, we can detect the number and intensities of common harmonic components within two mixtures, and isolate two complicated source signals with highly overlapped spectra from the mixtures. The proposed method can be applied for signal simplification or component quantification in a wide range of areas, such as biomedical or electrical engineering, where the signals are usually modeled as a sum of sinusoidal components.