# Mixing Matrix Estimation of Underdetermined Blind Source Separation Based on Data Field and Improved FCM Clustering

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Model of MIMO Radar Signals

## 3. Detection Method Based on Single-Source Principal Value of Complex Angular

#### 3.1. Model of Underdetermined Blind Source Separation

#### 3.2. Single-Source Principal Value of Complex Angular Detection

## 4. Mixing Matrix Estimation Based on Two-Step Preprocessing Fuzzy Clustering

#### 4.1. Fuzzy C-Means Clustering Algorithm

**Step 1**: Set the number of cluster centers $c$, weighted index $m$, iteration stop threshold $epsm$, maximum iteration number $Maxiter$, initial cluster center $P0$, and initialization iteration number to 0;

**Step 2**: Calculate the initial distance matrix $D$;

**Step 3**: The membership matrix is updated by Equation (13);

**Step 4**: The cluster center is updated by Equation (14), and the number of iterations is added one;

**Step 5**: Then the distance matrix is calculated again, and the objective function is calculated by Equation (10). If the result is smaller than the iteration stop threshold or exceeds the maximum iteration number, the algorithm terminates and outputs the result, otherwise it jumps to step 4.

#### 4.2. Introduce Data Field to Select the Number of Cluster Centers

#### 4.3. Using the Particle Swarm Optimization Algorithm to Optimize the Clustering Center

## 5. Implementation Steps of the Improved Method

**Step 1**: The data field of the aliasing MIMO radar observed signals with single-source principal value of complex angular detection is obtained, and the number of the data center to be clustered is acquired by the equipotential graph, denoted by $c$;

**Step 2**: Initialize the parameters: $N$ represents the population size, the number of clusters is $c$, $m$ stands for the weighting index, the cognitive and social parameters are expressed as ${c}_{1}$ and ${c}_{2}$ respectively. The inertia factor is denoted as $\omega $, the maximum number of iterations is set as $Maxiter$, the velocity upper limit to ${v}_{\mathrm{max}}$ and the iteration stop threshold are indicated as $epsm$ and $e$;

**Step 3**: Initialize the population of particles and generate the initial population randomly;

**Step 4**: Initialize the membership degree matrix ${U}_{ik}$, and use single step FCM to calculate the set of initial clustering center;

**Step 5**: Initialize the value of fitness function by using Equation (18);

**Step 6**: The best judgment: The current fitness value of the particle is compared with its individual extremum $pbest$, and if it is greater than $pbest$, the current position is assigned to $pbest$. Furthermore, the individual extremum of the particle is compared with the global extremum $gbest$ of the population, and if it is greater than $gbest$, $gbest$ will be updated to the current position;

**Step 7**: Population evolution: According to Equations (16) and (17), we can acquire the velocity and position of the next generation of particles and produce a new generation of population to be new cluster centers;

**Step 8**: We can achieve FCM clustering by using the number and locations of clustering centers respectively obtained by the data field and PSO algorithm. The iteration times denoted as $iter$ and the value of the optimal clustering center in the iter-th iteration can be obtained;

**Step 9**: If the number of iterations reaches $Maxiter$ or the fitness reaches the threshold, the global optimal solution $gbest$ and the current membership matrix $U$ can be obtained, meanwhile, the algorithm terminates; otherwise, returns to step 6.

## 6. Simulation Results and Analysis

#### 6.1. Evaluation Criteria of Estimation Error

#### 6.2. Simulation Procedures

## 7. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Pedersen, M.S.; Wang, D. Two-microphone separation of speech mixtures. IEEE Trans. Neural Netw.
**2008**, 19, 475–492. [Google Scholar] [CrossRef] [PubMed][Green Version] - Abolghasemi, V.; Ferdowsi, S. Fast and incoherent dictionary learning algorithms with application to fMRI. Signal Image Video Process.
**2015**, 9, 147–158. [Google Scholar] [CrossRef] - Xiang, W.; Huang, Z.; Zhou, Y. Semi-Blind Signal Extraction for Communication Signals by Combining Independent Component Analysis and Spatial Constraints. Sensors
**2012**, 12, 9024–9045. [Google Scholar] - Aziz, M.A.E.; Khidr, W. Nonnegative matrix factorization based on projected hybrid conjugate gradient algorithm. Signal Image Video Process.
**2015**, 9, 1825–1831. [Google Scholar] [CrossRef] - Wang, H.; Li, R.; Tang, G. A Compound Fault Diagnosis for Rolling Bearings Method Based on Blind Source Separation and Ensemble Empirical Mode Decomposition. PLoS ONE
**2014**, 9, e109166. [Google Scholar] [CrossRef] [PubMed] - Hattay, J.; Belaid, S.; Lebrun, D. Digital in-line particle holography: Twin-image suppression using sparse blind source separation. Signal Image Video Process.
**2015**, 9, 1767–1774. [Google Scholar] [CrossRef] - Romano, J.M.T.; Attux, R.; Cavalcante, C.C. Unsupervised Signal Processing: Channel Equalization and Source Seperation; CRC Press: Boca Raton, FL, USA, 2016. [Google Scholar]
- Naik, G.R.; Wang, W. Blind Source Separation; Springer: Berlin/Heidelberg, Germany, 2014. [Google Scholar]
- 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] - Puigt, M.; Deville, Y. Time–frequency ratio-based blind separation methods for attenuated and time-delayed sources. Mech. Syst. Signal Process.
**2005**, 19, 1348–1379. [Google Scholar] [CrossRef][Green Version] - Arberet, S.; Gribonval, R.; Bimbot, F. A robust method to count and locate audio sources in a stereophonic linear anechoic mixture. In Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing, ICASSP 2007, Honolulu, HI, USA, 15–20 April 2007. [Google Scholar]
- Arberet, S.; Gribonval, R.; Bimbot, F. A robust method to count and locate audio sources in a multichannel underdetermined mixture. IEEE Trans. Signal Process.
**2010**, 58, 121–133. [Google Scholar] [CrossRef][Green Version] - Kim, S.G.; Yoo, C.D. Underdetermined blind source separation based on subspace representation. IEEE Trans. Signal Process.
**2009**, 57, 2604–2614. [Google Scholar] - Dong, T.; Lei, Y.; Yang, J. An algorithm for underdetermined mixing matrix estimation. Neurocomputing
**2013**, 104, 26–34. [Google Scholar] [CrossRef] - Xu, J.; Yu, X.; Hu, D. A fast mixing matrix estimation method in the wavelet domain. Signal Process.
**2014**, 95, 58–66. [Google Scholar] [CrossRef] - Ai, X.F.; Luo, Y.J.; Zhao, G.Q. Underdetermined blind separation of radar signals based on tensor decomposition. Syst. Eng. Electron.
**2016**, 38, 2505–2509. [Google Scholar] - Guo, Q.; Ruan, G.; Nan, P. Underdetermined Mixing Matrix Estimation Algorithm Based on Single Source Points. Circuits Syst. Signal Process.
**2017**, 36, 4453–4467. [Google Scholar] [CrossRef] - Giachetta, G.; Mangiarotti, L.; Sardanashvily, G. Advanced Classical Field Theory; World Scientific: Singapore, 2009. [Google Scholar]
- Kante, B.; Germain, D.; De Lustrac, A. Near field imaging of refraction via the magnetic field. Appl. Phys. Lett.
**2014**, 104, 021909. [Google Scholar] [CrossRef] - Li, D.; Wang, S.; Gan, W. Data Field for Hierarchical Clustering. Int. J. Data Warehous. Min.
**2011**, 7, 43–63. [Google Scholar] - Filho, T.M.S.; Pimentel, B.A.; Souza, R.M.C.R. Hybrid methods for fuzzy clustering based on fuzzy c-means and improved particle swarm optimization. Expert Syst. Appl. Int. J.
**2015**, 42, 6315–6328. [Google Scholar] [CrossRef] - Sun, J.; Li, Y.; Wen, J. Novel mixing matrix estimation approach in underdetermined blind source separation. Neurocomputing
**2016**, 173, 623–632. [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]

**Figure 3.**(

**a**) Distribution map on unit-positive hemispherical; (

**b**) Potentiometric map on the X1–X2 plane; (

**c**) Potentiometric map on the X1–X3 plane; (

**d**) Potentiometric map on the X2–X3 plane.

© 2018 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 (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

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.
https://doi.org/10.3390/sym10010021

**AMA Style**

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(1):21.
https://doi.org/10.3390/sym10010021

**Chicago/Turabian Style**

Guo, Qiang, Chen Li, and Guoqing Ruan. 2018. "Mixing Matrix Estimation of Underdetermined Blind Source Separation Based on Data Field and Improved FCM Clustering" *Symmetry* 10, no. 1: 21.
https://doi.org/10.3390/sym10010021