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Chong-Yung Chi ( 祁忠勇 )

B LIND S OURCE S EPARATION B Y K URTOSIS M AXIMIZATION W ITH A PPLICATIONS I N W IRELESS C OMMUNICATIONS. Chong-Yung Chi ( 祁忠勇 ). Institute of Communications Engineering & Department of Electrical Engineering National Tsing Hua University Hsinchu, Taiwan 30013, R.O.C.

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Chong-Yung Chi ( 祁忠勇 )

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  1. BLINDSOURCESEPARATIONBYKURTOSISMAXIMIZATION WITH APPLICATIONS IN WIRELESS COMMUNICATIONS Chong-Yung Chi (祁忠勇) Institute of Communications Engineering & Department of Electrical Engineering National Tsing Hua University Hsinchu, Taiwan 30013, R.O.C. Tel: +886-3-5731156, Fax: +886-3-5751787 E-mail: cychi@ee.nthu.edu.tw http://www.ee.nthu.edu.tw/cychi/ Invited talk at I2R, Singapore, July 18, 2006. Acknowledgments: The viewgraphs were prepared through Chun-Hsien Peng’s helps.

  2. OUTLINE • Introduction to Blind Source Separation (BSS) • FKMA and MSC Procedure • Turbo Source Extraction Algorithm (TSEA) • Non-cancellation Multistage Source (NCMS) Separation Algorithms • NCMS-FKMA • NCMS-TSEA • SimulationResults --- Part 1 • Turbo Space-time Receiver for CCI/ISI Reduction • SimulationResults --- Part 2 • Conclusions FKMA: Fast Kurtosis Maximization AlgorithmMSC: Multistage Successive Cancellation 1

  3. 1. Blind Source Separation (BSS) • Instantaneous Mixture of Sources Noise POutput Measurements (Mutually Indep. but Colored) Unknownmixing matrix (Memoryless channel) is the basis vector that spans the subspace of GOAL Extract all the source signals with only measurements . Applications:array signal processing, wireless communications and biomedicalsignal processing, etc [1-3]. 2

  4. Statistically independent • Existing BSS Algorithms SOS HOS SOS:Second-order Statistics HOS:Higher-order Statistics 3

  5. AMUSE:Algorithm for Multiple Unknown Signals Extraction (Tong et al., 1990 [1]) SOBI:Second-order Blind Identification (Belouchrani et al., 1997 [2]) FOBI:Fourth-order Blind Identification (Cardoso, 1989 [12]) EFOBI:Extended Fourth-order Blind Identification (Tong et al., 1991 [1]) FastICA:Fast Independent Component Analysis (Hyvarinen et al., 1997 [13, 14]) MSC:Multistage Successive Cancellation NCMS:Non-cancellation Multistage FKMA: Fast Kurtosis Maximization Algorithm TSEA:Turbo Source Extraction Algorithm 4

  6. { Þ • AMUSE and SOBI Algorithm Using SOS: Step 1: Prewhitening by Eigenvalue Decomposition (EVD) (PxP matrix) : largest K eigenvalues of EVD : associated K eignevectors of : average of the other (smallest)P-Keigenvalues of (assuming that) (whitening matrix) (dimension-reduced whitening spatial processing) : KxK unitary matrix AMUSE:Algorithm for Multiple Unknown Signals Extraction (Tong et al., 1990 [1]) SOBI:Second-order Blind Identification (Belouchrani et al., 1997 [2]) 5

  7. Step 2: Estimation of the Unitary Matrix from (KxK matrix) EVD of Prewhitening by EVD (AMUSE) Joint Diagonalization of (SOBI) Step 3: Source Separation and Channel Estimation (spatial processing for simultaneous extraction of all the K sources) (demixing matrix) ( : pseudo-inverse) (mixing matrix estimate) 6

  8. 2. FKMA and MSC Procedure • Assumptions: (A1) The unknown mixing matrix is of full column rank with . (A2) aremodeled as : zero-meannon-Gaussianindependent identically distributed (i.i.d.) process with ; is statistically independent of for all . Stable LTI System (A3) is zero-mean Gaussian and statistically independent of . : fourth-order joint cumulant of random variables (referred to as kurtosis of ) FKMA:Fast Kurtosis Maximization Algorithm (Chi and Chen, 2001 [4,5]) MSC:Multistage Successive Cancellation 7

  9. Assume that are zero-meanrandom variables. Then (referred to as kurtosis of ) where is the characteristic function of random variables • Definition of HOS (i.e., Cumulants): (Bartlett, 1955, Brillinger, 1975, etc) 8

  10. Closed-form solution for : Not existent • Gradient-type iterative algorithms for finding a local optimum : Not very computationally efficient where Q is a positive-definite matrix depending on the algorithm used, and μ is the step size such that • Fast Kurtosis Maximization Algorithm (FKMA) (Chi et al., 2001 [4, 5]) Criterion [7]: Optimum Maximization (noise-free case) magnutude of normalized kutorsis of ( is an unknown complex scale factor and ) 9

  11. (SEA) Algorithm nential -expo Super • Fast Kurtosis Maximization Algorithm (FKMA) (Chi et al., 2001 [4, 5]) Criterion [7]: Optimum Maximization (noise-free case) magnutude of normalized kutorsis of ( is an unknown complex scale factor and ) Algorithm: At the th iteration Yes Compute To the thiteration ? No Update through a gradient type optimization algorithm such that (PxP matrix) 9

  12. Observations: • The FKMA itself is an exclusive spatialprocessing algorithm. • The smaller the value of , the worse the performance of the FKMA for finite SNR and finite data length . By(A2) (absolute normalized kurtosis of ) where (entropy measure of the stable sequence ) (equality holds only as , i.e., minimum entropy of ) can be thought of as a measure of distance of from a Gaussian process, implying that the performance of the FKMA (which requires to be non-Gaussian [6]), depends on . 10

  13. NOTE The estimated sources and columns of obtained at later stages in the MSC procedure may become less accurate due to error propagation effects from stage to stage [6]. • MSC Procedure Each Stage of the Multistage Successive Cancellation (MSC)Procedure Estimate One Source Signal Using FKMA ( : th column of ) Obtain Update by Next Stage 11

  14. 3. Turbo Source Extraction Algorithm • Source Separation Filter: (A bank of same temporal filters) where : vector for extracting a colored source signal , i.e., removing spatial interference due to the mixing matrix .(spatial filter) : single-input single-output (SISO) deconvolution (or higher-order whitening) filter of order to restore from . (temporal filter) • Design Criterion: (Extracted Source) Maximization 12

  15. Step 2 Step 2 FKMA(t) (b) (a) (Extracted Source) Turbo Source Extraction Algorithm (TSEA) (Chi et al., 2003 [3]) Signal processing procedure at the th cycle Temporal Processing Spatial Processing Step 1 FKMA(s) (b) (a) Step 1 13

  16. Performance of TSEA is superior to FKMA. Why? Interpretations: 1) Temporal Processing: (b) Increasing is equivalent to increasing 2) Spatial Processing: (b) 14

  17. Remarks: • TSEA is computationally efficient with super-exponential convergence rate and P parameters for spatial processing and L+1 parameters for temporal processing, respectively. • The performance gain of the TSEA reaches the maximum as long as the order L (a parameter under our choice) of the temporal filter is sufficiently large. On the other hand, the asymptotic performance of FKMA approaches that of the TSEA as and . • All the sources can be extracted through the MSC procedure.The resultant BSS algorithm that uses the TSEA, is referred to as MSC-TSEA,also outperforms the MSC-FKMA, at the extra expense of the temporal processing at each stage. 15

  18. 4. Non-Cancellation Multistage Source Separation Algorithms • NCMS-FKMA Constraint Constrained Criterion: where (unconstrained optimization problem) Unconstrained Criterion: : projection matrix Theorem 1:Let be the set of all the extracted source signals up to stage . With (A1), (A2),and the noise-free assumption, the optimum where is an unknown non-zero constant and . 16

  19. Good Initial Condition • Signal Processing Procedure of NCMS-FKMA (Initial Condition) (F-b)Estimate One Source Signal Using FKMA (F-a)Estimate One Source Signal Using FKMA Obtain by SVD of and Obtain 17

  20. Remarks: • The constrained source extraction filter obtained in(F-a)provides asuitable initial conditionfor the unconstrained sourceextraction filter in(F-b),which accordingly leads to onedistinct source estimateobtained at each stage neither involving cancellation nor imposing any constraints on the source extraction filter, as well as faster convergence than(F-a).Therefore, unlike the MSC-FKMA, the NCMS-FKMA isfree from the error propagation effectsat each stage. • As the MSC-TSEA performs better than the MSC-FKMA, theNCMS-TSEA also performs better than the NCMS-FKMA at the moderate expense of extra computational load for the temporal processing of the TSEA. 18

  21. Good Initial Condition • Signal Processing Procedure of NCMS-TSEA (Initial Condition) (T-b)Estimate One Source Signal Using TSEA (T-a)Estimate One Source Signal Using TSEA Obtain by SVD of and Obtain 19

  22. 5. Simulation Results --- Part 1 • Parameters Used: • : zero-mean, independent binary sequence of with equal probability • : generated by filtering through the chosen FIR filters • : real white Gaussian noise vector • SNR: • 50 independent runs • Output (extracted) signal to interference-plus-noise ratio (Output SINR) 20

  23. Part A: Performance of NCMS-FKMA and NCMS-TSEA • mixing matrix (taken from Chang et al., 1998 [9]) (P=5, K=4) • Four cases are considered as follows: Case 1:Output SINR versus SNR for different data length . Case 2:Output SINR versus different data length . Case 3:Output SINR versus (or ) for all . Case 4:(a)Output SINR versus L. (b) versus L. 21

  24. (or) for all , Figure 1. Simulation results (Output SINR versus SNR) of Case 1. 22

  25. (or) for all , , and SNR=30 dB Figure 2. Simulation results (Output SINR versus data length ) of Case 2. 23

  26. SNR=30 dB, , and Figure 3. Simulation results (Output SINR versus ) of Case 3. 24

  27. SNR=30 dB, and(i.e., and) for all Figure 4a. Simulation results (Output SINR versus the order of the temporal filter ) of Case 4 (a). 25

  28. SNR=30 dB, and(i.e., and) for all Figure 4b. Simulation results (Output SINR versus the order of the temporal filter ) of Case 4 (b). 26

  29. Part B: Performance Comparison • The same mixing matrix in Part A and • Data length = 2000 and = 5 • Comparison with the MSC-FKMA, AMUSE (Tong et al. 1990 [1]) andSOBI algorithm (Belouchrani et al. 1997 [2]) • Three cases are considered as follows: Case A:Output SINR1 versus SNR for and . Case B:Output SINR versus SNR for and . Case C:Output SINR versus for SNR = 20 dB and . 27

  30. Figure 5. Simulation results (Output SINR1 versus SNR) of Case A. 28

  31. Figure 6. Simulation results (Output SINR versus SNR) of Case B. 29

  32. Figure 7. Simulation results (Output SINR versus data length ) of Case C. 30

  33. Case D:Output SINR versus • : a3x2mixing matrix by removing the last two rows and columns of the mixing matrix in Part A. (P=3, K=2) • Data length =1000, SNR=30 dB and =3. • Comparison with theMSC-FKMA, AMUSE and SOBI algorithm 31

  34. Figure 8. Simulation results (Output SINR versus ) of Case D. 32

  35. GOAL Enhance data rate, link quality, capacity, and coverage. 6. Turbo Space-time Receiver for CCI/ISI Reduction • Problem Statement: CCI and ISI Suppression in TDMA Cellular Wireless Communications (Noise) f5 f6 f4 f1 (Multipath channel) f7 f3 f2 f1 CCI:Co-channel Interference CCI ISI:Intersymbol Interference (due to multipath) Space-time processing using an antenna array has been used for combating CCI and ISI in the receiver design [15-16]. 33

  36. Signal Model: Consider the scenario where the base station is equipped with multiple antennas, and the signal of interest and CCI are received from multiple distinct directions of arrival (DOA), with a frequency-selective fading channel for each DOA. (a general scenario) 34

  37. The received signal from the desired user and CCIs (users) can be expressed as aninstantaneous mixture of multiple sources where ( is no.of DOAs of user k) “ISI-distorted’’ signal (colored signal) from jth DOA of user k steering vector of jthDOA of user k th-order channel impulse response of jthDOA of user k (total no. of DOAs or ’’sources”) 35

  38. Assumptions: (A1)The unknown DOA matrix is of full column rank and (A2)The data sequence of user 1 (the desired signal) is i. i. d. zero-mean non-Gaussian with , and meanwhile statistically independent of the other () zero-mean i. i. d. data sequences (of CCI). (A3)is zero-mean Gaussian, and statistically independent of for all . (total no. of DOAs or ’’sources”) correlated colored non-Gaussian sources block mutually independent colored sources 36

  39. Case I: Each user has a single DOA with multiple paths (Venkataraman et al., 2003), i.e., Mutually independentcolored sources • Case II: Each user has multiple DOAs with disjoint domains ofsupport of multipath channel impulse responses, i.e., block mutually independent colored sources; mutually independent random variables for each n 37

  40. Conventional Cascade Space-Time Receiver (CSTR)(Jelitto and Fettweis, 2002) For Cases I and II, the conventional CSTR has been reported for CCI and ISI suppression Space-time Processor Temporal Filter Spatial Filter In CAMSAP-06, we proposed two space-time receivers based on kurtosis maximizationfor these two cases and a discussion of the proposed space-time receiversfor the general scenario. Other existing structures: full-dimension (joint) ST processing, reduced dimension ST processing (prewhitening followed by joint ST processing). 38

  41. Kurtosis Maximization(Ding ad Nguyen, 2000): (noise-free case) Maximization magnutude of normalized kutorsis of is an unknown complex scale factor and • Closed-form solution for : Not existent • Gradient-type iterative algorithms for finding a local optimum : Not very computationally efficient • Applicable not only for Case I but also for Case II(Peng et al., ICICS 2005) 39

  42. (SEA) Algorithm nential -expo Super • Fast Kurtosis Maximization Algorithm (FKMA)(Chi and Chen, 2001): At the th iteration Yes Compute To the thiteration ? No Update through a gradient type optimization algorithm such that (PxP matrix) 40

  43. Blind CSTR Using FKMA Space-time Processor Temporal Filter Spatial Filter • Spatial processing using FKMA for CCI suppression With a suitable initial condition for , FKMA will converge at a super-exponential rate with for high SNR. • Temporal processing using FKMA for ISI removal where (L: order of the temporal filter) 41

  44. Usually, the ISI-distorted (desired) signal , has higher power than all the CCI, (i.e., ). So (DOA estimate by delay-and-sum) implying that can be used as the initial condition for the spatial filter needed by the FKMA. • It can be easily shown that (Chi et al., 2003) where The performance of the spatial filter (to suppress CCI) using FKMA is worse for smaller and worse for larger , leading to limited performance of the temporal filter of the blind CSTR. 42

  45. Blind Turbo Space-Time Receiver (TSTR) • Space-Time Filter for Source Extraction (Chi et al. 2003, 2006): • Design Criterion: (noise-free case) Optimum Maximization 43

  46. Proposed Blind TSTR Using FKMA Signal processing procedure at the th cycle: CSTR Spatial Filter Temporal Filter FKMA (S2) i=i+1 (S1) Spatial Filter Temporal Filter FKMA CSTR 44

  47. Why? Performance of blind TSTR is superior to blind CSTR. Interpretations: 1) Temporal Processing: Increasing is equivalent to increasing 2) Spatial Processing: 45

  48. Remarks: • It can be proven that for all , implying the guaranteed convergence of the proposed blind TSTR. Typically, the number of cycles spent by the TSTR before convergence, is equal to 2 or 3. The computational load of the blind TSTR is approximately 2 or 3 times that of the blind CSTR. • Because the design of and that of are coupled in a constructive and boosting manner, the proposed blind TSTR outperforms the blind CSTR for all , and meanwhile their performance difference is larger for larger . • Compared with the blind CSTR, the proposed blind TSTR is insensitive to the value of(i.e., robust against channel with multiple paths or severe ISI). 46

  49. Performance of the blind TSTR: where CASE I: CCI suppression by Multiple DOAs suppressed also by CASE II: CCI suppression by GENERAL CASE:CCI suppression by the spatial filterand the temporal filtercombine the signalsfrom all the DOAs in a constructive and boosting fashion 47

  50. 7. Simulation Results --- Part 2 • Scenario of Case I • : zero-mean, independent binary sequence of with equal probability • ,, • : white Gaussian noise vector • SNR: • 50 independent runs : Diagonal matrix (array size) 48

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