# Sparse Component Analysis (SCA) Based on Adaptive Time—Frequency Thresholding for Underdetermined Blind Source Separation (UBSS)

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The UBSS–SCA-Based Method

## 3. Mixed Signal Generation

## 4. Mixing Matrix Estimation

#### 4.1. The Proposed Adaptive Time–Frequency Thresholding (ATFT) Method

- Let ${index}_{j}=[1,2,3,\dots ,D]$.
- The threshold of ATFT, $\alpha $, is obtained from the norm of the result of all mixtures and was computed by$$\alpha ={\left({\sum}_{d=0}^{D-1}{\left|{TF}_{{norm}_{j}}\right|}^{2}\right)}^{1/2}/D$$

- 3.
- According to Equation (5), the thresholding operation is defined as$${index}_{j}=\left\{\begin{array}{l}1,{TF}_{{norm}_{j}}>\frac{\mathrm{E}\left({TF}_{{norm}_{j}}\right)}{\alpha}\\ 0,otherwise\end{array}\right.$$

#### 4.2. Single Source Point (SSP) Detection

Algorithm 1: Mixing matrix estimation procedure using the proposed ATFT method |

Input: The mixed signal, $\widehat{X}\left(t\right)$.
- $\widehat{X}\left(t\right)$ is transformed from the time domain into the TF domain using STFT to produce $\stackrel{~}{X}\left(t,f\right)$.
- Each element in the mixture vectors, $\stackrel{~}{X}\left(t,f\right)$, is normalised to have a unit norm, ${TF}_{{norm}_{j}}=\sum _{i=1}^{j}{\left|{\stackrel{~}{X}}_{Mj}(t,f)\right|}^{2}$ where $j=\mathrm{1,2},3\dots ,D$. $D$ is the dimension.
- The mean of the unit norm, E$({TF}_{{norm}_{j}})=\frac{1}{D}\sum _{j=1}^{D}{TF}_{{norm}_{j}},\mathrm{i}\mathrm{s}\text{}\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{u}\mathrm{t}\mathrm{e}\mathrm{d}$.
- The threshold $\alpha $ is used to select the significant coefficient.
- Let ${index}_{j}=[1,2,3,\dots ,D]$$${index}_{j}=\left\{\begin{array}{l}1,{TF}_{{norm}_{j}}>\frac{\mathrm{E}\left({TF}_{{norm}_{j}}\right)}{\alpha}\\ 0,otherwise\end{array}\right.$$
- ${\stackrel{~}{X}}_{yz}\left(t,f\right)=\left\{\begin{array}{l}{\stackrel{~}{X}}_{ij}(t,f){index}_{j}=1\\ not\text{}select,otherwise\end{array}\right.$where $u=i$, $j=\mathrm{1,2},3\dots ,D,v=\mathrm{1,2},3\dots ,\widehat{D}$. $\widehat{D}$= new dimension
- For each ${\stackrel{~}{X}}_{yz}\left(t,f\right)$, the sparse coding coefficients are computed by using ${l}_{1}$-norm optimisation.
- The mixture of TF vectors with sparse coding coefficient vectors containing only one nonzero element is added into Ω.
- The elements are normalised on Ω.
Output: Normalised Ω. |

#### 4.3. Clustering

Algorithm 2: The procedure of mixing matrix estimation using hierarchical clustering |

Input: The extracted SSP vectors, $\mathrm{\Omega}=\{{y}_{1},{y}_{2},\dots ,{y}_{N}\}$.
- The clustering method is applied to the extracted SSPs to group its elements into $n$ clusters.
- The centres of these n clusters are calculated as the estimations for the columns of the mixing matrix to produce the estimated mixing matrix.
- The outliers are removed and ${C}_{new}$ is produced.
- The centres of the clusters are calculated as the estimated mixing matrix, $\stackrel{~}{A}$.
Output: The estimated mixing matrix, $\stackrel{~}{A}$ is formed. |

## 5. Source Recovery Estimation

Algorithm 3: Source recovery estimation |

Input: The mixed signal, $\widehat{X}\left(t\right)$, and the estimated mixing matrix, $\stackrel{~}{A}$.
- $\widehat{X}\left(t\right)\text{}\mathrm{i}\mathrm{s}\text{}\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{s}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{e}\mathrm{d}$ into the TF domain using STFT.
- All $M\times \left(M-1\right)$ submatrices of $\stackrel{~}{A}$ are placed into $A$.
- The source of TF representation is estimated for each TF point.
- Using inverse STFT, the estimated source signals are transformed back into the time domain.
Output: The estimated source signals, $\stackrel{~}{S}$. |

## 6. Numerical Simulations

#### 6.1. The Performance of Mixing Matrix Estimation

#### 6.2. The Performance of Source Recovery Estimation

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Comon, P. Independent component analysis, A new concept? Signal Process.
**1994**, 36, 287–314. [Google Scholar] [CrossRef] - Babatas, E.; Erdogan, A.T. Time and frequency based sparse bounded component analysis algorithms for convolutive mixtures. Signal Process.
**2020**, 173, 107590. [Google Scholar] [CrossRef] - Sawada, H.; Ono, N.; Kameoka, H.; Kitamura, D.; Saruwatari, H. A review of blind source separation methods: Two converging routes to ILRMA originating from ICA and NMF. APSIPA Trans. Signal Inf. Process.
**2019**, 8, e12. [Google Scholar] [CrossRef] - Wang, L. A Study on Multi-Subspace Representation of Nonlinear Mixture with Application in Blind Source Separation: Modeling and Performance Analysis. Ph.D. Thesis, Keio University, Tokyo, Japan, 2019. [Google Scholar]
- Guan, W.; Dong, L.; Cai, Y.; Yan, J.; Han, Y. Sparse component analysis with optimized clustering for underdetermined blind modal identification. Meas. Sci. Technol.
**2019**, 30, 125011. [Google Scholar] [CrossRef] - Guo, Q.; Ruan, G.; Liao, Y. A time-frequency domain underdetermined blind source separation algorithm for MIMO radar signals. Symmetry
**2017**, 9, 104. [Google Scholar] [CrossRef] - Guo, Q.; Li, C.; Ruan, G. Mixing matrix estimation of underdetermined blind source separation based on data field and improved FCM clustering. Symmetry
**2018**, 10, 21. [Google Scholar] [CrossRef] - Lu, J.; Cheng, W.; Zi, Y. A novel underdetermined blind source separation method and its application to source contribution quantitative estimation. Sensors
**2019**, 19, 1413. [Google Scholar] [CrossRef] - Yu, G. An underdetermined blind source separation method with application to modal identification. Shock Vib.
**2019**, 2019, 1–15. [Google Scholar] [CrossRef] - Qiu, P.; Zhang, Y.; Wang, Y.; Yin, Q.; Wang, Q. Underdetermined Speech Source Separation Based on Hybrid Clustering. In Proceedings of the 2021 40th Chinese Control Conference (CCC), Shanghai, China, 26–28 July 2021; Volume 430074, pp. 3077–3081. [Google Scholar]
- Su, Q.; Shen, Y.; Wei, Y.; Deng, C. Underdetermined blind source separation by a novel time–frequency method. AEU -Int. J. Electron. Commun.
**2017**, 77, 43–49. [Google Scholar] [CrossRef] - Li, Y.; Wang, Y.; Dong, Q. A novel mixing matrix estimation algorithm in instantaneous underdetermined blind source separation. Signal Image Video Process.
**2020**, 14, 1001–1008. [Google Scholar] [CrossRef] - He, X.; He, F. Underdetermined mixing matrix estimation based on artificial bee colony optimization and single-source-point detection. Multimed. Tools Appl.
**2020**, 79, 13061–13087. [Google Scholar] [CrossRef] - Li, Y.; Amari, S.I.; Cichocki, A.; Ho, D.W.C.; Xie, S. Underdetermined blind source separation based on sparse representation. IEEE Trans. Signal Process.
**2006**, 54, 423–437. [Google Scholar] - Liu, C.; Li, Y.; Nie, W. A new underdetermined blind source separation algorithm under the anechoic mixing model. Int. Conf. Signal Process. Proc. ICSP
**2016**, 2019, 1799–1803. [Google Scholar] - Zhen, L.; Peng, D.; Yi, Z.; Xiang, Y.; Chen, P. Underdetermined Blind Source Separation Using Sparse Coding. IEEE Trans. Neural Networks Learn. Syst.
**2017**, 28, 3102–3108. [Google Scholar] [CrossRef] - Eqlimi, E.; Makkiabadi, B.; Samadzadehaghdam, N.; Khajehpour, H.; Mohagheghian, F.; Sanei, S. A Novel Underdetermined Source Recovery Algorithm Based on k-Sparse Component Analysis. Circuits Syst. Signal Process.
**2019**, 38, 1264–1286. [Google Scholar] [CrossRef] - Cheng, W.; Jia, Z.; Chen, X.; Han, L.; Zhou, G.; Gao, L. Underdetermined convolutive blind source separation in time-frequency domain based on single source points and experimental validation. Meas. Sci. Technol.
**2020**, 31, 095001. [Google Scholar] [CrossRef] - Wang, T.; Yang, F.; Yang, J. Convolutive Transfer Function-Based Multichannel Nonnegative Matrix Factorization for Overdetermined Blind Source Separation. IEEE/ACM Trans. Audio Speech Lang. Process.
**2022**, 30, 802–815. [Google Scholar] [CrossRef] - Li, Y.; Cichocki, A.; Amari, S. Sparse component analysis for blind source separation with less sensors than sources. Ica
**2003**, 2003, 89–94. [Google Scholar] - Linh-Trung, N.; Aïssa-El-Bey, A.; Abed-Meraim, K.; Belouchrani, A. Underdetermined blind source separation of non-disjoint nonstationary sources in the time-frequency domain. In Proceedings of the 8th International Symposium on Signal Processing and its Applications, ISSPA, Sydney, NSW, Australia, 28–31 August 2005; IEEE: Piscatevi, NJ, USA, 2005; Volume 1, pp. 46–49. [Google Scholar]
- He, X.S.; He, F.; Cai, W.H. Underdetermined BSS Based on K-means and AP Clustering. Circuits Syst. Signal Process.
**2016**, 35, 2881–2913. [Google Scholar] [CrossRef] - Shi, F.; Liu, C. Mixing matrix estimation algorithm for underdetermined instantaneous mixing model. Int. J. Perform. Eng.
**2019**, 15, 337–345. [Google Scholar] [CrossRef] - Reju, V.G.; Koh, S.N.; Soon, I.Y. An algorithm for mixing matrix estimation in instantaneous blind source separation. Signal Process.
**2009**, 89, 1762–1773. [Google Scholar] [CrossRef] - Abrard, F.; Deville, Y. A time-frequency blind signal separation method applicable to underdetermined mixtures of dependent sources. Signal Process.
**2005**, 85, 1389–1403. [Google Scholar] [CrossRef] - Kim, S.G.; Yoo, C.D. Underdetermined blind source separation based on subspace representation. IEEE Trans. Signal Process.
**2009**, 57, 2604–2614. [Google Scholar] - Zhang, L.; Yang, J.; Guo, Z.; Zhou, Y. Underdetermined blind source separation from time-delayed mixtures based on prior information exploitation. J. Electr. Eng. Technol.
**2015**, 10, 2179–2188. [Google Scholar] [CrossRef] [Green Version] - Ye, F.; Chen, J.; Gao, L.; Nie, W.; Sun, Q. A Mixing Matrix Estimation Algorithm for the Time-Delayed Mixing Model of the Underdetermined Blind Source Separation Problem. Circuits Syst. Signal Process.
**2019**, 38, 1889–1906. [Google Scholar] [CrossRef] - Kumar, M.; Jayanthi, V.E. Underdetermined blind source separation using CapsNet. Soft Comput.
**2020**, 24, 9011–9019. [Google Scholar] [CrossRef] - Wang, J.; Chen, X.; Zhao, H.; Li, Y.; Yu, D. An Effective Two-Stage Clustering Method for Mixing Matrix Estimation in Instantaneous Underdetermined Blind Source Separation and Its Application in Fault Diagnosis. IEEE Access
**2021**, 9, 115256–115269. [Google Scholar] [CrossRef] - Wang, X.; Wang, S.; Huang, Z.; Du, Y. Structure regularized sparse coding for data representation. Knowl. -Based Syst.
**2019**, 174, 87–102. [Google Scholar] [CrossRef] - Bermant, P.C. BioCPPNet: Automatic bioacoustic source separation with deep neural networks. Sci. Rep.
**2021**, 11, 1–13. [Google Scholar] [CrossRef] [PubMed] - Frogs of Australia > Taxonomy. Available online: https://frogs.org.au/frogs/search.php (accessed on 10 December 2022).
- Yilmaz, Ö.; Rickard, S. Blind separation of speech mixtures via time-frequency masking. IEEE Trans. Signal Process.
**2004**, 52, 1830–1846. [Google Scholar] [CrossRef] - Reju, V.G.; Koh, S.N.; Soon, I.Y. Underdetermined convolutive blind source separation via time-frequency masking. IEEE Trans. Audio Speech Lang. Process.
**2010**, 18, 101–116. [Google Scholar] [CrossRef]

**Figure 4.**Scatterplot of mixed signals for a scenario with three sensors (M = 3) and four sources (N = 4) of bioacoustic signals. (

**a**) Mixtures in the time domain, (

**b**) Mixtures in the time–frequency domain.

**Figure 7.**Comparison of the performance of mixing matrix estimation using the ATFT method and Zhen’s method, tested on bioacoustic signals.

**Figure 9.**Comparison of the performance of mixing matrix estimation for three mixtures with different numbers of sources using the ATFT method.

**Figure 10.**Comparison of the performance of mixing matrix estimation for four mixtures with different numbers of sources using the ATFT method.

**Figure 11.**Comparison of the performance of source recovery estimation by the ATFT method and Zhen’s method on bioacoustic signals.

**Figure 12.**Simulation results of source recovery. First row: the original source signals of four bioacoustic signals. Second row: the estimated sources using the ATFT method. Third row: the estimated sources using Zhen’s method.

References | Mixing System | Single-Source Point Detection | Limitations |
---|---|---|---|

[25] | Two mixtures of three sources from two guitars and one voice | Time–frequency ratio of the mixtures (TIFROM) | Low estimation accuracy for noisy and insufficiently sparse sources |

[24] | Three, four and five mixtures each for four to seven speech utterances | Compared the absolute directions of the real and imaginary parts of the TF points in the mixing signals | Limited application, requires real-valued entries in the mixing matrix |

[26] | Three mixtures of four sets of sources consisting of the genres of music, speech, instruments and various sounds | An SSD algorithm that recognises the TF points occupied by a single source for each source. | Loses efficiency when the mixing matrix is complex and not real |

[27] | Two mixtures of three speech signals | Extracts prior information from the complex-valued mixing matrix at the receiver’s end | Too much computation or poor robustness |

[16] | Three mixtures of four speech sources | Sparse coding | Has a fixed parameter to select the STFT coefficients before SSP detection |

[28] | Two mixtures of four speech signals | An SSP detection technique based on the transformation matrix | The selection of the peak value used to determine the number of source signals is greatly affected by noise |

[29] | Two mixtures of two sets of sources consisting of three male and female speech signals | Calculates the mixing ratio | Sensitive to noise in real-world systems |

[13] | Three mixtures of six flutes | Calculating the mixing ratio | Sensitive to noise |

Database | Zhen’s Method | ATFT Method |
---|---|---|

Bioacoustic signals | 2.3321 | 1.9218 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Hassan, N.; Ramli, D.A.
Sparse Component Analysis (SCA) Based on Adaptive Time—Frequency Thresholding for Underdetermined Blind Source Separation (UBSS). *Sensors* **2023**, *23*, 2060.
https://doi.org/10.3390/s23042060

**AMA Style**

Hassan N, Ramli DA.
Sparse Component Analysis (SCA) Based on Adaptive Time—Frequency Thresholding for Underdetermined Blind Source Separation (UBSS). *Sensors*. 2023; 23(4):2060.
https://doi.org/10.3390/s23042060

**Chicago/Turabian Style**

Hassan, Norsalina, and Dzati Athiar Ramli.
2023. "Sparse Component Analysis (SCA) Based on Adaptive Time—Frequency Thresholding for Underdetermined Blind Source Separation (UBSS)" *Sensors* 23, no. 4: 2060.
https://doi.org/10.3390/s23042060