Abstract
The model of inherent connection between underdetermined blind signal separation and compressed sensing (CS) is analyzed first; then, the mathematical model of underdetermined blind signal reconstruction is built using CS. More specifically, the mixing matrix is estimated by exploiting the wavelet packet transform and k-means clustering methods up to permutation and scaling indeterminacy, and then, the measurement matrix and the measurement equation are obtained. To reconstruct the underdetermined sparse source signals, the proposed semi-blind compressed reconstruction algorithm is derived based on the blind signal reconstruction model and compressive sampling matching pursuit (CoSaMP) method. Our simulation results demonstrate that the proposed scheme is effective, irrespective of artificial data or real data. Moreover, the proposed scheme can be adjusted for different applications by modifying the mixing matrix estimation method and CoSaMP method with respect to the correspondence conditions.
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Notes
According to the number of source signals and mixed signals, there are three basic types of BSS models: overdetermined, determined and underdetermined. When the number of the source signals is less than or equal to that of the mixed signals, the BSS model is called overdetermined or determined; otherwise, it is called underdetermined.
Here, \(\tilde{\beta }\) is a book keeping shift index selected from the original shift index \(\beta \). \(\tilde{\beta }\) with nonzero elements represent the corresponding rows of the decomposition coefficients of mixed signals, which are used for source signals reconstruction.
To use complex-valued data with k-means clustering, one method is to split the real and complex parts to obtain real data in twice as many dimensions. Alternatively, one could modify the calculation of the squared Euclidean distance from something like sum(\((\cdot )\cdot {}^{\wedge }2\)) to something like \(\hbox {sum}(\hbox {abs}(\cdot )\cdot {}^{\wedge }2\)).
In the conventional BSS algorithms, a meaningful result is obtained when \(\hbox {S/R}>10\,\hbox {dB}\).
The complex-valued noiseless mixed signals are the same as the first experiment in this subsection.
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Acknowledgments
The authors would like to thank the editor and anonymous referees for their insightful comments and suggestions, which have greatly improved the paper. This research is financially supported by the National Natural Science Foundation of China (Nos. 61401401, 61571401, 61301150, 61172086, U1204607), the China Postdoctoral Science Foundation (Nos. 2014M561998, 2015T80779) and the open research fund of the National Mobile Communications Research Laboratory, Southeast University (No. 2016D02).
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Wang, F., Li, R., Wang, Z. et al. Compressed Blind Signal Reconstruction Model and Algorithm. Circuits Syst Signal Process 35, 3192–3219 (2016). https://doi.org/10.1007/s00034-015-0189-z
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DOI: https://doi.org/10.1007/s00034-015-0189-z