Abstract
Blind extraction of fetal electrocardiogram from the composite maternal electrocardiogram without noise and with additive Gaussian white noise (AGWN) is achieved by Barros and Cicochi (Neural Computat 13:1995–2003) and by Li and Zhang (Neurocomputing 71:1538–1542), respectively. Theoretical analysis shows that these two algorithms are essentially the same. In this paper, an intelligible method is proposed, which is based on the eigenvalue decomposition of the cross-correlation of whitened source signals at a given time tag. Simulation experiments show that the proposed method has better extraction performance. Moreover, a great deal of experiments with AGWN show that Li and Zhang’s method is slightly inferior to the Barros and Cicochi’s, which is opposite to the claim by Li and Zhang.
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\( {\mathbf{p}} = [p_{1} ,p_{2} , \ldots ]^{T} \) is called a canonical basis vector if \( p_{i} = 1 \) and \( p_{l} = 0,\forall l \ne i \).
Here, the foundation is
\( \left\{ \begin{gathered} E\left\{ {s_{i} \left( t \right)s_{i} \left( {t - \tau_{i} } \right)} \right\} > E\left\{ {s_{j} \left( t \right)s_{j} \left( {t - \tau_{i} } \right)} \right\} \hfill \\ E\left\{ {s_{i} \left( t \right)s_{j} \left( {t - \tau_{i} } \right)} \right\} = 0 \hfill \\ \end{gathered} \right.,\forall j \ne i \), under which \( R_{{{\mathbf{ss}}}} \left( {\tau_{i} } \right) \) is diagonal and its ith diagonal element is the maximum.
Proof: If \( {\mathbf{C}} \) is orthogonal, the characteristic polynomials of \( {\mathbf{D}} \) and \( {\mathbf{CDC}}^{T} \) are the same because:
\( \left| {{\mathbf{CDC}}^{T} - \lambda {\mathbf{I}}} \right| = \left| {{\mathbf{C}}\left( {{\mathbf{D}} - \lambda {\mathbf{I}}} \right){\mathbf{C}}^{T} } \right| = \left| {\left( {{\mathbf{D}} - \lambda {\mathbf{I}}} \right){\mathbf{CC}}^{T} } \right| = \left| {{\mathbf{D}} - \lambda {\mathbf{I}}} \right| \).
References
Barros AK, Cichocki A (2001) Extraction of specific signals with temporal structure. Neural Comput 13:1995–2003
Li YX, Zhang Y (2008) An algorithm for extracting fetal electrocardiogram. Neurocomputing 71:1538–1542
Li C, Liao G, Shen Y (2010) A reference-based blind source extraction algorithm. Neural Comput Appl 19:299–303
De Moor D (ed) Daisy: database for the identification of systems, http://www.esat.kuleuven.ac.be/sista/daisy
Acknowledgments
The authors are grateful to the anonymous reviewers for their valuable and constructive comments, which greatly improved the manuscript. This work was supported by NSFC under Grant Nos. 61101211 and 60736009.
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Li, C., Liao, G. Notes on two temporal structure–based methods for blind extraction of fetal electrocardiogram. Neural Comput & Applic 21, 1647–1650 (2012). https://doi.org/10.1007/s00521-012-0877-6
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DOI: https://doi.org/10.1007/s00521-012-0877-6