Abstract
Recovering M sources from N mixtures in underdetermined cases, i.e., M > N, is a great challenge, especially for insufficiently sparse sources in noisy cases. To solve this problem, an improved underdetermined blind source separation (UBSS) method is proposed based on single source points (SSPs) identification and l0-norm. Firstly, we present a mixing matrix estimation method based on SSPs that is identified by directly searching the identical normalized time–frequency (TF) vectors of mixed signals. This method considers the linear representation relations among these TF vectors and therefore could achieve more accurate SSPs identification even in noisy cases. Then, we prove that a non-SSP will be misjudged as SSP with probability zero under some assumptions, which guarantees the stability and effectiveness of the proposed method. Secondly, SSPs are only searched in some optimal frequency bins so that all SSPs in these frequency bins can be identified at one time. Then, the mixing matrix is estimated using hierarchical clustering technique. Thirdly, to recover source signals with real number of active sources, a l0-norm-based source recovery method is proposed which would be transformed to find the submatrix with the least column of the mixing matrix that can linearly represent TF vectors of mixed signals. Therefore, source signals can be recovered with the real number of active sources, which improves the estimation accuracy of source signals. Some experiments are carried out to show the effectiveness of the proposed method. The present research could improve the estimation accuracy of sources for insufficiently sparse sources with noise in underdetermined cases.
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References
F. Abrard, Y. Deville, A time–frequency blind signal separation method applicable to underdetermined mixtures of dependent sources. Signal Process. 85, 1389–1403 (2005). https://doi.org/10.1016/j.sigpro.2005.02.010
A. Aissa-El-Bey, N. Linh-Trung, K. Abed-Meraim et al., Underdetermined blind separation of nondisjoint sources in the time-frequency domain. IEEE Trans. Signal Process. 55, 897–907 (2007). https://doi.org/10.1109/TSP.2006.888877
P. Bofill, M. Zibulevsky, Underdetermined blind source separation using sparse representations. Signal Process. 81, 2353–2362 (2001). https://doi.org/10.1016/S0165-1684(01)00120-7
P. Bofill, Underdetermined blind separation of delayed sound sources in the frequency domain. Neurocomputing 55, 627–641 (2003). https://doi.org/10.1016/S0925-2312(02)00631-8
W. Cui, S. Guo, L. Ren et al., Underdetermined blind source Separation for linear instantaneous mixing system in the non-cooperative wireless communication. Phys. Commun. 45, 01255 (2021). https://doi.org/10.1016/j.phycom.2020.101255
Févotte, C., Ozerov, A.: Notes on nonnegative tensor factorization of the spectrogram for audio source separation: statistical insights and towards self-clustering of the spatial cues. In: 7th International Symposium on Computer Music Modeling and Retrieval, Málaga, Spain, June 21–24, Springer: Berlin, pp. 102–115 (2011). https://doi.org/10.1007/978-3-642-23126-1_8
W. Fu, X. Bai, F. Shi et al., Mixing matrix estimation algorithm for underdetermined blind source separation. IEEE Access 9, 136284–136291 (2021). https://doi.org/10.1109/ACCESS.2021.3114169
Z. Koldovský, P. Tichavský, A.H. Phan et al., A two-stage mmse beamformer for underdetermined signal separation. IEEE Signal Process. Lett. 20, 1227–1230 (2013). https://doi.org/10.1109/LSP.2013.2285932
H. Li, Y.H. Shen, J.G. Wang et al., Estimation of the complex-valued mixing matrix by single-source-points detection with less sensors than sources. Trans. Emerg. Telecommun. Technol. 23, 137–147 (2012). https://doi.org/10.1002/ett.1517
Y. Li, S.-I. Amari, A. Cichocki et al., Underdetermined blind source separation based on sparse representation. IEEE Trans. Signal Process. 54, 423–437 (2006). https://doi.org/10.1109/TSP.2005.861743
Y. Li, W. Nie, F. Ye et al., A mixing matrix estimation algorithm for underdetermined blind source separation. Circuits Syst. Signal Process. 35, 3367–3379 (2016). https://doi.org/10.1007/s00034-015-0198-y
B. Liu, V.G. Reju, A.W. Khong, A linear source recovery method for underdetermined mixtures of uncorrelated ar-model signals without sparseness. IEEE Trans. Signal Process. 62, 4947–4958 (2014). https://doi.org/10.1109/TSP.2014.2329646
C. Liu, C. Zhang, Remove artifacts from a single-channel EEG based on VMD and SOBI. Sensors 22, 6698 (2022). https://doi.org/10.3390/s22176698
J. Lu, W. Cheng, Y. Chu et al., Post-nonlinear blind source separation with kurtosis constraints using augmented lagrangian particle swarm optimization and its application to mechanical systems. J. Vib. Control 25, 2246–2260 (2019). https://doi.org/10.1177/1077546319852483
J. Lu, W. Cheng, D. He et al., A novel underdetermined blind source separation method with noise and unknown source number. J. Sound Vib. 457, 67–91 (2019). https://doi.org/10.1016/j.jsv.2019.05.037
J. Lu, W. Cheng, Y. Zi, A novel underdetermined blind source separation method and its application to source contribution quantitative estimation. Sensors 19, 1413 (2019). https://doi.org/10.3390/s19061413
B. Ma, T. Zhang, Underdetermined blind source separation based on source number estimation and improved sparse component analysis. Circuits Syst. Signal Process. 40(7), 3417–3436 (2021). https://doi.org/10.1007/s00034-020-01629-x
G. Mahé, G. Suzumura, L. Moisan et al., A nonintrusive audio clarity index (NIAC) and its application to blind source separation. Signal Process. 194, 108448 (2022). https://doi.org/10.1016/j.sigpro.2021.108448
J. Miettinen, E. Nitzan, S.A. Vorobyov et al., Graph signal processing meets blind source separation. IEEE Trans. Signal Process. 69, 2585–2599 (2021). https://doi.org/10.1109/TSP.2021.3073226
D. Peng, Y. Xiang, Underdetermined blind source separation based on relaxed sparsity condition of sources. IEEE Trans. Signal Process. 57, 809–814 (2009). https://doi.org/10.1109/TSP.2008.2007604
V.G. Reju, S.N. Koh, Y. Soon, An algorithm for mixing matrix estimation in instantaneous blind source separation. Signal Process. 89, 1762–1773 (2009). https://doi.org/10.1016/j.sigpro.2009.03.017
S. Rickard, The DUET Blind Source Separation (Springer, Dordrecht, 2007), pp.217–237
R. Saab, O. Yilmaz, M.J. McKeown et al., Underdetermined anechoic blind source separation via lq-basis-pursuit with q < 1. IEEE Trans. Signal Process. 55, 4004–4017 (2007). https://doi.org/10.1109/TSP.2007.895998
J. Sun, Y. Li, J. Wen et al., Novel mixing matrix estimation approach in underdetermined blind source separation. Neurocomputing 173, 623–632 (2016). https://doi.org/10.1016/j.neucom.2015.08.008
J.J. Thiagarajan, K.N. Ramamurthy, A. Spanias, Mixing matrix estimation using discriminative clustering for blind source separation. Digit. Signal Process. 23, 9–18 (2013). https://doi.org/10.1016/j.dsp.2012.08.002
P. Tichavsky, Z. Koldovsky, Weight adjusted tensor method for blind separation of underdetermined mixtures of nonstationary sources. IEEE Trans. Signal Process. 59, 1037–1047 (2011). https://doi.org/10.1109/TSP.2010.2096221
L. Wang, X. Yin, H. Yue et al., A regularized weighted smoothed L0 norm minimization method for underdetermined blind source separation. Sensors 18(12), 4260 (2018). https://doi.org/10.3390/s18124260
S. Xie, L. Yang, J.-M. Yang et al., Time-frequency approach to underdetermined blind source separation. IEEE Trans. Neural Netw. Learn. Syst. 23, 306–316 (2012). https://doi.org/10.1109/TNNLS.2011.2177475
Y. Xie, K. Xie, S. Xie, Underdetermined blind source separation of speech mixtures unifying dictionary learning and sparse representation. Int. J. Mach. Learn. Cybern. 12(12), 3573–3583 (2021). https://doi.org/10.1007/s13042-021-01406-5
O. Yilmaz, S. Rickard, Blind separation of speech mixtures via time-frequency masking. IEEE Trans. Signal Process. 52, 1830–1847 (2004). https://doi.org/10.1109/TSP.2004.828896
L. Zhen, D. Peng, Z. Yi et al., Underdetermined blind source separation using sparse coding. IEEE Trans. Neural Netw. Learn. Syst. 28, 3102–3108 (2017). https://doi.org/10.1109/TNNLS.2016.2610960
Acknowledgements
This research was funded in part by the National Key Research and Development Project (NO. 2020YFB1709801), in part by Jiangsu Provincial Double-Innovation Doctor Program (NO. 202030364), in part by Scientific Research Foundation for NUAA (NO. YAH20008), and in part by Natural Science Fund for Colleges and Universities in Jiangsu Province (NO. 22KJB470006). The authors would also like to thank Yingxian Zhang from Huazhong University of Science and Technology for her useful suggestions.
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JL contributed to conceptualization and writing—original draft preparation; JL and WQ helped in methodology and visualization; JL, QY, and KX performed formal analysis and investigation; JL and SL helped in writing—review and editing; JL and WQ done funding acquisition; JL and SL done validation.
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Appendices
Appendix A
Theorem 1
The nonzero normalized TF vector \({\tilde{\mathbf{x}}}\left( {u{\prime} ,v{\prime} } \right)\) is not identical with the nonzero normalized TF vector \({\tilde{\mathbf{x}}}\left( {\psi{\prime} ,\omega{\prime} } \right)\) with probability one if at least one of \({\tilde{\mathbf{x}}}\left( {u{\prime} ,v{\prime} } \right)\) and \({\tilde{\mathbf{x}}}\left( {\psi{\prime} ,\omega{\prime} } \right)\) is non-SSP.
Proof
Assume that the set of active sources at \(\left( {u{\prime} ,v{\prime} } \right)\) and \(\left( {\psi{\prime} ,\omega{\prime} } \right)\) are \(\mathcal{H}\) and \(\mathcal{T}\), respectively. The number of elements in \(\mathcal{H}\), \(\mathcal{T}\) and their union are \(m\), \(n\), and \(l\), respectively. The indices of the \(m\) active sources at \(\left( {u{\prime} ,v{\prime} } \right)\) are denoted as \(\phi_{1} ,\;\phi_{2} ,\; \ldots ,\;\phi_{m}\), i.e., \(\mathcal{H} = \left\{ {s_{{\phi_{1} }} \left( {u{\prime} ,v{\prime} } \right),} \right.\)\(\left. {s_{{\phi_{2} }} \left( {u{\prime} ,v{\prime} } \right),\; \ldots ,\;s_{{\phi_{m} }} \left( {u{\prime} ,v{\prime} } \right)} \right\}\). According to Assumption 3, \(m \le N - 1\) and \(n \le N - 1\). Since \({\tilde{\mathbf{x}}}\left( {u{\prime} ,v{\prime} } \right)\) and \({\tilde{\mathbf{x}}}\left( {\psi{\prime} ,\omega{\prime} } \right)\) are nonzero vectors, then \(m \ge 1\) and \(n \ge 1\). Thus, \(1 \le m \le N - 1\) and \(1 \le n \le N - 1\) can be obtained.
Four cases are discussed as follows.
Case 1: \(\mathcal{H} \ne \mathcal{T}\) and \(l \le N\).
Suppose that there are \(k\) elements (\(k \le m\) and \(k \le n\)) in the intersection of \(\mathcal{H}\) and \(\mathcal{T}\). Without loss of generality, the \(k\) elements can be set as \(s_{{\phi_{1} }} \left( {u^{\prime},v^{\prime}} \right),\;s_{{\phi_{2} }} \left( {u^{\prime},v^{\prime}} \right),\; \ldots ,\;s_{{\phi_{k} }} \left( {u^{\prime},v^{\prime}} \right)\). Obviously, the other \(n - k\) elements in \(\mathcal{T}\), denoted as \(s_{{\gamma_{1} }} \left( {u^{\prime},v^{\prime}} \right),\;s_{{\gamma_{2} }} \left( {u^{\prime},v^{\prime}} \right),\; \ldots ,\;s_{{\gamma_{n - k} }} \left( {u^{\prime},v^{\prime}} \right)\), come from the set \(\left\{ {s_{{\phi_{m + 1} }} \left( {u^{\prime},v^{\prime}} \right),\;s_{{\phi_{m + 2} }} \left( {u^{\prime},v^{\prime}} \right),\; \ldots ,\;s_{{\phi_{M} }} \left( {u^{\prime},v^{\prime}} \right)} \right\}\), then we obtain \(\mathcal{T} = \left\{ {s_{{\phi_{1} }} \left( {u^{\prime},v^{\prime}} \right),\; \ldots ,\;s_{{\phi_{k} }} \left( {u^{\prime},v^{\prime}} \right),} \right.\)\(\left. {s_{{\gamma_{1} }} \left( {u^{\prime},v^{\prime}} \right),\; \ldots ,\;s_{{\gamma_{n - k} }} \left( {u^{\prime},v^{\prime}} \right)} \right\}\).
Then, at \(\left( {u^{\prime},v^{\prime}} \right)\) and \(\left( {\psi^{\prime},\omega^{\prime}} \right)\), Eq. (3) can be rewritten as Eq. (A1) and Eq. (A2), respectively.
Now, we assume that \({\tilde{\mathbf{x}}}\left( {u^{\prime},v^{\prime}} \right) = {\tilde{\mathbf{x}}}\left( {\psi^{\prime},\omega^{\prime}} \right)\), then \({\mathbf{x}}\left( {u^{\prime},v^{\prime}} \right) = \alpha {\mathbf{x}}\left( {\psi^{\prime},\omega^{\prime}} \right)\) will hold, i.e.,
Eq. (A3) can be rewritten as
The matrix form of Eq. (A4) is
where \({\mathbf{A^{\prime}}}{ = }\left[ {{\mathbf{a}}_{{\phi_{1} }} ,\; \ldots ,\;{\mathbf{a}}_{{\phi_{k} }} ,{\mathbf{a}}_{{\phi_{k + 1} }} ,\; \ldots ,\;{\mathbf{a}}_{{\phi_{m} }} \;{\mathbf{a}}_{{\gamma_{1} }} ,\; \ldots ,\;{\mathbf{a}}_{{\gamma_{n - k} }} } \right]\) and \({\mathbf{s^{\prime}}} = \left[ {s_{{\phi_{1} }} \left( {u^{\prime},v^{\prime}} \right) - \alpha s_{{\phi_{1} }} \left( {\psi^{\prime},\omega^{\prime}} \right),\; \ldots ,\;} \right.s_{{\phi_{k} }} \left( {u^{\prime},v^{\prime}} \right) - \alpha s_{{\phi_{k} }} \left( {\psi^{\prime},\omega^{\prime}} \right),\)\(s_{{\phi_{k + 1} }} \left( {u^{\prime},v^{\prime}} \right),\;\left. { \ldots ,\;s_{{\phi_{m} }} \left( {u^{\prime},v^{\prime}} \right),\; - \alpha s_{{\gamma_{1} }} \left( {\psi^{\prime},\omega^{\prime}} \right),\; \ldots ,\; - \alpha s_{{\gamma_{n - k} }} \left( {\psi^{\prime},\omega^{\prime}} \right)} \right]^{{\text{T}}}\).
According to Assumption 2, \({\mathbf{A^{\prime}}}\) is of full column rank due to \(l \le N\). Therefore, Eq. (A5) has a unique zero solution, meaning \({\mathbf{s^{\prime}}} = {\mathbf{0}}\), i.e.,
As \(\mathcal{H} \ne \mathcal{T}\), there exists at least one element in the set \(\left\{ {s_{{\phi_{k + 1} }} \left( {u^{\prime},v^{\prime}} \right),} \right.\; \ldots ,\;s_{{\phi_{m} }} \left( {u^{\prime},v^{\prime}} \right),\)\(\left. {s_{{\gamma_{1} }} \left( {\psi^{\prime},\omega^{\prime}} \right),\; \ldots ,\;s_{{\gamma_{n - k} }} \left( {\psi^{\prime},\omega^{\prime}} \right)} \right\}\), and without loss of generality, we suppose \(s_{{\phi_{k + 1} }} \left( {u^{\prime},v^{\prime}} \right)\) exists.
However, we have \(s_{{\phi_{k + 1} }} \left( {u^{\prime},v^{\prime}} \right) = 0\) from Eq. (A6), which is a contradiction with the above result.
Therefore, \({\tilde{\mathbf{x}}}\left( {u^{\prime},v^{\prime}} \right) = {\tilde{\mathbf{x}}}\left( {\psi^{\prime},\omega^{\prime}} \right)\) is not hold in Case 1.
Case 2: \(\mathcal{H} \ne \mathcal{T}\) and \(l > N\).
Similar to Case 1, we also suppose that there are also \(k\) elements, i.e., \(s_{{\phi_{1} }} \left( {u^{\prime},v^{\prime}} \right),\;s_{{\phi_{2} }} \left( {u^{\prime},v^{\prime}} \right),\; \ldots ,\;s_{{\phi_{k} }} \left( {u^{\prime},v^{\prime}} \right)\), in the intersection of \(\mathcal{H}\) and \(\mathcal{T}\). Thus, Eq. (A1) and Eq. (A2) can be also obtained. It can be seen that \({\mathbf{x}}\left( {u^{\prime},v^{\prime}} \right)\) is in the m-dimensional subspace spanned by \({\mathbf{a}}_{{\phi_{1} }} ,\;{\mathbf{a}}_{{\phi_{2} }} ,\; \ldots ,\;{\mathbf{a}}_{{\phi_{m} }}\) and \({\tilde{\mathbf{x}}}\left( {\psi^{\prime},\omega^{\prime}} \right)\) is in the n-dimensional subspace spanned by \({\mathbf{a}}_{{\phi_{1} }} ,\;{\mathbf{a}}_{{\phi_{2} }} ,\; \ldots ,\;{\mathbf{a}}_{{\phi_{k} }} ,\;{\mathbf{a}}_{{\gamma_{1} }} ,\)\(\;{\mathbf{a}}_{{\gamma_{2} }} ,\; \ldots ,\;{\mathbf{a}}_{{\gamma_{n - k} }} \;\).If \({\tilde{\mathbf{x}}}\left( {u^{\prime},v^{\prime}} \right) = {\tilde{\mathbf{x}}}\left( {\psi^{\prime},\omega^{\prime}} \right)\), Eq. (A7) will hold.
where \(\beta\) is a real constant. Eq. (A7) shows that \({\mathbf{x}}\left( {u^{\prime},v^{\prime}} \right)\) is also in the n-dimensional subspace spanned by \({\mathbf{a}}_{{\phi_{1} }} ,\;{\mathbf{a}}_{{\phi_{2} }} ,\; \ldots ,\;{\mathbf{a}}_{{\phi_{k} }} ,\;{\mathbf{a}}_{{\gamma_{1} }} ,\;{\mathbf{a}}_{{\gamma_{2} }} ,\; \ldots ,\;{\mathbf{a}}_{{\gamma_{n - k} }}\).
Let \(\mathcal{P}\) be the union of the m-dimensional subspace and the n-dimensional subspace, and \(\mathcal{Q}\) be the intersections of these two subspaces. Clearly, \(\mathcal{Q}\) has a measure zero in \(\mathcal{P}\), indicating that any given TF vector in \(\mathcal{P}\) happens to be in \(\mathcal{Q}\) with probability zero. Therefore, Eq. (A7) will hold with probability zero, i.e., \({\tilde{\mathbf{x}}}\left( {u^{\prime},v^{\prime}} \right) = {\tilde{\mathbf{x}}}\left( {\psi^{\prime},\omega^{\prime}} \right)\) is not hold with probability one in Case 2.
Case 3: \(\mathcal{H} = \mathcal{T}\) and \(m \le N - 1\).
From Eq. (3), at \(\left( {u^{\prime},v^{\prime}} \right)\), we have
Since \(\mathcal{T} = \mathcal{H}\), then we have
If \({\tilde{\mathbf{x}}}\left( {u^{\prime},v^{\prime}} \right) = {\tilde{\mathbf{x}}}\left( {\psi^{\prime},\omega^{\prime}} \right)\), then \({\mathbf{x}}\left( {u^{\prime},v^{\prime}} \right) = \kappa {\mathbf{x}}\left( {\psi^{\prime},\omega^{\prime}} \right)\), i.e.,
Thus, we can have
Eq. (A11) can be rewritten in matrix form as
where \({\mathbf{A^{\prime}}}{ = }\left[ {{\mathbf{a}}_{{\phi_{1} }} ,\; \ldots ,\;{\mathbf{a}}_{{\phi_{m} }} } \right]\) and \({\mathbf{s^{\prime}}} = \left[ {s_{{\phi_{1} }} \left( {u^{\prime},v^{\prime}} \right) - \kappa s_{{\phi_{1} }} \left( {\psi^{\prime},\omega^{\prime}} \right),\;} \right. \ldots ,\;\left. {s_{{\phi_{m} }} \left( {u^{\prime},v^{\prime}} \right) - \kappa s_{{\phi_{m} }} \left( {\psi^{\prime},\omega^{\prime}} \right)} \right]^{{\text{T}}}\). As \(m < N\),\({\mathbf{A^{\prime}}}\) is of full column rank, indicating that Eq. (A12) has unique zero solution, that is,
As at least one of \({\tilde{\mathbf{x}}}\left( {u^{\prime},v^{\prime}} \right)\) and \({\tilde{\mathbf{x}}}\left( {\psi^{\prime},\omega^{\prime}} \right)\) is non-SSP, if \(\mathcal{H} = \mathcal{T}\), we get \(m \ge 2\). Thus, the following equation can be obtained.
From Assumption 4, source signals are mutually independent, which implies that Eq. (A14) will hold with probability zero. Therefore, \({\tilde{\mathbf{x}}}\left( {u^{\prime},v^{\prime}} \right) = {\tilde{\mathbf{x}}}\left( {\psi^{\prime},\omega^{\prime}} \right)\) is not hold with probability one in Case 3.
Case 4: \(\mathcal{H} = \mathcal{T}\) and \(m \ge N - 1\).
Case 4 is a contradiction with Assumption 3 due to \(m \ge N - 1\).
From the above analysis of Cases 1–4, we can conclude that \({\tilde{\mathbf{x}}}\left( {u^{\prime},v^{\prime}} \right) = {\tilde{\mathbf{x}}}\left( {\psi^{\prime},\omega^{\prime}} \right)\) is not hold with probability one if at least one of \({\tilde{\mathbf{x}}}\left( {u^{\prime},v^{\prime}} \right)\) and \({\tilde{\mathbf{x}}}\left( {\psi^{\prime},\omega^{\prime}} \right)\) is non-SSP.
This completes the proof of Theorem 1.
Appendix B
Theorem 2
For any given vector \({\mathbf{x}}\left( {t_{0} ,\;k_{0} } \right)\) with \(r\) active sources, \(r < N\), there does not exist a vector \({\mathbf{s^{\prime}}}\left( {t_{0} ,\;k_{0} } \right)\) that \(\left\| {{\mathbf{s^{\prime}}}\left( {t_{0} ,\;k_{0} } \right)} \right\|_{0} < r\) and \({\mathbf{x}}\left( {t_{0} ,\;k_{0} } \right) = {\mathbf{As^{\prime}}}\left( {t_{0} ,\;k_{0} } \right)\) with probability one.
Proof
Assume that the indices of \(r\) active sources are \(\chi_{1} ,\;\chi_{2} ,\; \ldots ,\;\chi_{r}\), then, Eq. (21) can be obtained with \(\left\| {{\mathbf{s}}_{{\chi_{1} ,\;\chi_{2} ,\; \ldots ,\;\chi_{r} }} \left( {t_{0} ,\;k_{0} } \right)} \right\|_{0} = r\), which indicates that \({\mathbf{x}}\left( {t_{0} ,\;k_{0} } \right)\) belongs to the r-dimensional subspace spanned by the column of \({\mathcal{A}}_{\;r}^{ * }\).
Assume that there exists a vector \({\mathbf{s^{\prime}}}\left( {t_{0} ,\;k_{0} } \right)\) that \({\mathbf{x}}\left( {t_{0} ,\;k_{0} } \right) = {\mathbf{As^{\prime}}}\left( {t_{0} ,\;k_{0} } \right)\) and \(\left\| {{\mathbf{s^{\prime}}}\left( {t_{0} ,\;k_{0} } \right)} \right\|_{0} = k\), \(k < r\). The indices of \(k\) nonzero elements of \({\mathbf{s^{\prime}}}\left( {t_{0} ,\;k_{0} } \right)\) are denoted as \(\sigma_{1} ,\;\sigma_{2} ,\; \ldots ,\;\sigma_{k}\), then
where \({\mathcal{A}^{\prime}}_{\;k} = \left[ {{\mathbf{a}}_{{\sigma_{1} }} ,\;{\mathbf{a}}_{{\sigma_{2} }} ,\; \ldots ,\;{\mathbf{a}}_{{\sigma_{k} }} } \right] \in {\mathcal{A}}_{\;k}\) and \({\mathbf{s^{\prime}}}_{{\sigma_{1} ,\;\sigma_{2} ,\; \ldots ,\;\sigma_{k} }} \left( {t_{0} ,\;k_{0} } \right) = \left[ {s^{\prime}_{{\sigma_{1} }} \left( {t_{0} ,\;k_{0} } \right),\;s^{\prime}_{{\sigma_{2} }} \left( {t_{0} ,\;k_{0} } \right),\; \ldots ,\;s^{\prime}_{{\sigma_{k} }} \left( {t_{0} ,\;k_{0} } \right)} \right]^{{\text{T}}}\).
Two cases are considered:
Case 1: \({\mathcal{A}^{\prime}}_{\;k}\) is a submatrix of \({\mathcal{A}}_{\;r}^{ * }\).
Without loss of generality, suppose that \({\mathcal{A}^{\prime}}_{\;k}\) is the first \(k\) columns of \({\mathcal{A}}_{\;r}^{ * }\), i.e., \({\mathcal{A}^{\prime}}_{\;i} = \left[ {{\mathbf{a}}_{{\chi_{1} }} ,\;{\mathbf{a}}_{{\chi_{2} }} ,\; \ldots ,\;{\mathbf{a}}_{{\chi_{k} }} } \right]\). Then we has \({\mathcal{A}}_{\;r}^{ * } {\mathbf{s}}_{{\chi_{1} ,\;\chi_{2} ,\; \ldots ,\;\chi_{r} }} \left( {t_{0} ,\;k_{0} } \right) = {\mathcal{A}^{\prime}}_{\;k} {\mathbf{s^{\prime}}}_{{\sigma_{1} ,\;\sigma_{2} ,\; \ldots ,\;\sigma_{k} }} \left( {t_{0} ,\;k_{0} } \right)\), that is,
Thus we can obtain
According to Assumption 2, \({\mathcal{A}}_{\;r}^{ * }\) is of full column rank, therefore \(\left[ {s_{{\chi_{1} }} \left( {t_{0} ,\;k_{0} } \right) - s^{\prime}_{{\sigma_{1} }} \left( {t_{0} ,\;k_{0} } \right),\;\; \ldots ,} \right.\)\(\left. {s_{{\chi_{k} }} \left( {t_{0} ,\;k_{0} } \right) - s^{\prime}_{{\sigma_{k} }} \left( {t_{0} ,\;k_{0} } \right),\;s_{{\chi_{k + 1} }} \left( {t_{0} ,\;k_{0} } \right),\;\; \ldots ,\;s_{{\chi_{r} }} \left( {t_{0} ,\;k_{0} } \right)} \right]^{{\text{T}}} = {\mathbf{0}}\), that is, \(s_{{\chi_{k + 1} }} \left( {t_{0} ,\;k_{0} } \right) = \;\; \ldots = \;s_{{\chi_{r} }} \left( {t_{0} ,\;k_{0} } \right) = 0\), which is a contradiction \(\left\| {{\mathbf{s}}_{{\chi_{1} ,\;\chi_{2} ,\; \ldots ,\;\chi_{r} }} \left( {t_{0} ,\;k_{0} } \right)} \right\|_{0} = r\).
Therefore, Eq (B1) does not hold in Case 1, i.e., the Theorem 2 is true in Case 1.
Case 2: \({\mathcal{A}^{\prime}}_{\;k}\) is not a submatrix of \({\mathcal{A}}_{\;r}^{ * }\).
If \({\mathcal{A}^{\prime}}_{\;k}\) is not a submatrix of \({\mathcal{A}}_{\;r}^{ * }\), then it must be a submatrix of one of other elements in set \({\mathcal{A}}_{\;r}\) because \(k < r\). Therefore, suppose that \({\mathcal{A}^{\prime}}_{\;k}\) is a submatrix of \({\mathcal{A}^{\prime}}_{\;r}\), where \({\mathcal{A}^{\prime}}_{\;r} \in {\mathcal{A}}_{\;r}\) and \({\mathcal{A}^{\prime}}_{\;r} \ne {\mathcal{A}}_{\;r}^{ * }\). From Eq. (B1), \({\mathbf{x}}\left( {t_{0} ,\;k_{0} } \right)\) belongs to the k-dimensional subspace spanned by the column of \({\mathcal{A}^{\prime}}_{\;k}\), therefore \({\mathbf{x}}\left( {t_{0} ,\;k_{0} } \right)\) must belong to the r-dimensional subspace spanned by the column of \({\mathcal{A}^{\prime}}_{\;r}\).
However, for any given TF vector with \(r\) active sources, it belongs to only one r-dimensional subspace spanned by the columns of the element in \({\mathcal{A}}_{\;r}\) with probability one, which indicates that \({\mathbf{x}}\left( {t_{0} ,\;k_{0} } \right)\) belongs to two different r-dimensional subspaces spanned by the column of \({\mathcal{A}}_{\;r}^{ * }\) and \({\mathcal{A}^{\prime}}_{\;r}\), respectively, at the same time with probability zero. As a result, \({\mathbf{x}}\left( {t_{0} ,\;k_{0} } \right)\) belongs to the k-dimensional subspace spanned by the column of \({\mathcal{A}^{\prime}}_{\;k}\) with probability zero, i.e., the Theorem 2 is true in Case 2.
This completes the proof of Theorem 2.
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Lu, J., Qian, W., Yin, Q. et al. An Improved Underdetermined Blind Source Separation Method for Insufficiently Sparse Sources. Circuits Syst Signal Process 42, 7615–7639 (2023). https://doi.org/10.1007/s00034-023-02470-8
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DOI: https://doi.org/10.1007/s00034-023-02470-8