Chong-Yung Chi ( 祁忠勇 )
description
Transcript of Chong-Yung Chi ( 祁忠勇 )
BLIND SOURCE SEPARATION BY KURTOSIS MAXIMIZATION WITH APPLICATIONS IN WIRELESS
COMMUNICATIONS
Chong-Yung Chi (祁忠勇 )
Institute of Communications Engineering &Department of Electrical Engineering
National Tsing Hua UniversityHsinchu, Taiwan 30013, R.O.C.
Tel: +886-3-5731156, Fax: +886-3-5751787E-mail: [email protected]
http://www.ee.nthu.edu.tw/cychi/
Acknowledgments: The viewgraphs were prepared through Chun-Hsien Peng’s helps.
Invited talk at I2R, Singapore, July 18, 2006.
OUTLINE
1. Introduction to Blind Source Separation (BSS)
2. FKMA and MSC Procedure
3. Turbo Source Extraction Algorithm (TSEA)
4. Non-cancellation Multistage Source (NCMS) Separation Algorithms
NCMS-FKMA
NCMS-TSEA
5. Simulation Results --- Part 1
6. Turbo Space-time Receiver for CCI/ISI Reduction
7. Simulation Results --- Part 2
8. Conclusions
1
FKMA: Fast Kurtosis Maximization AlgorithmMSC: Multistage Successive Cancellation
1. Blind Source Separation (BSS)
Instantaneous Mixture of Sources
][1 nx
][nxP
][1 ns
][nsK
Noise P Output Measurements
Unknown
mixing matrix KP
][1 nw
][nwP
][ns ][nx
(Mutually Indep. but Colored)
A
2
Applications: array signal processing, wireless communications and biomedical signal processing, etc [1-3].
1
[ ] [ ] [ ] [ ] [ ] K
i ii
n n n s n n
x A s w a w
GOAL Extract all the source signals with only measurements .
][nx][nsi
(Memoryless channel)
is the basis vector that spans the subspace of
][nsi (ith column of i A)a
Algorithm
Whitening
Multistage
AMUSE 1. Statistically mutually uncorrelated2. Zero mean3. Temporal colored with distinct power spectra
1. Zero mean
2. Gaussian
SOBI
FOBI 1. Statistically mutually independent2. Zero mean3. Distinct fourth-order moments
EFOBI
FastICA1. Statistically mutually independent2. Zero mean 3. Non-Gaussian (Non-zero fourth-order cumulants, e.g., kurtosis)
MSC-FKMA
MSC-TSEA
NCMS-FKMA
NCMS-TSEA
P KA
P K
P K
],[nsk ][nw
Statistically independent
Kk ,...,1
SOS
HOS
Existing BSS Algorithms
SOS: Second-order Statistics
HOS: Higher-order Statistics
3
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
AMUSE and SOBI Algorithm Using SOS:
Step 1: Prewhitening by Eigenvalue Decomposition (EVD)
]}[][{ H nnE xxRx
K ..., , , 21
Kfff ..., , , 21
: largest K eigenvalues of xR
: associated K eignevectors of
2ˆw : average of the other (smallest) P-K eigenvalues of (assuming that )
xRIwwRw
2H ]}[][{ wnnE
(PxP matrix)
EVDxR
H1
2 21
ˆ [ , ..., ]ˆ ˆ
K
w K w
f fD
][ˆ][][ˆ][ nnnn wDUsxDz
U : KxK unitary matrix
(dimension-reduced whitening spatial processing)
(whitening matrix)
AMUSE: Algorithm for Multiple Unknown Signals Extraction (Tong et al., 1990 [1])SOBI: Second-order Blind Identification (Belouchrani et al., 1997 [2])
5
{
]}[][{][ knnEk HzzRz
Step 2: Estimation of the Unitary Matrix fromU
Prewhitening by EVD
][nx ][nz EVD of
U2/])[][( kk H
zz RR
U Joint Diagonalization of }1|][{ Jiki ..., ,Rz
(AMUSE)
(SOBI)
(KxK matrix)
][ˆ][ˆ H nn zUs
(demixing matrix)
(spatial processing for simultaneous extraction of all the K sources)
Step 3: Source Separation and Channel Estimation
# #ˆ ˆ ˆ A W D U ( : pseudo-inverse)#
6
Hˆ ˆW U D
(mixing matrix estimate)
(A2) are modeled as
: zero-mean non-Gaussian independent identically distributed (i.i.d.) process with ; is statistically independent of for all .
Assumptions:(A1) The unknown mixing matrix is of full column rank with . P K
][nw(A3) is zero-mean Gaussian and statistically independent of .
}z ,z ,z ,z{cum 4321 : fourth-order joint cumulant of random variables
KP A
][ns
][][][ nbnuns iii ][nui
Stable LTI System
][nbi
} ..., ,2 ,1{ ],[ Kinsi
][nui
0]}[{4 nuC i][nui ][nu j
ji
2. FKMA and MSC Procedure
(referred to as kurtosis of )
FKMA: Fast Kurtosis Maximization Algorithm (Chi and Chen, 2001 [4,5])
MSC: Multistage Successive Cancellation 7
4321 z ,z ,z ,z
}z ,z ,z ,z{cum}z{C4 z
where
is the characteristic function of random variables
01
11
1
),,(ln
)(},,{cum
M
M
MM
MM jxx
}{ )(1
11),,( MM xxjM eE
.,,1 Mxx
Definition of Definition of HOS (i.e., (i.e., Cumulants):): (Bartlett, 1955, Brillinger, 1975, etc)
Assume that are zero-mean random variables. Then
}{},{cum 2121 xxExx
}{}{}{}{
}{}{}{},,,{cum
}{},,{cum
32414231
432143214321
321321
xxExxExxExxE
xxExxExxxxExxxx
xxxExxx
4321 , , , xxxx
22224**4 }{}){(2}{},,,{cum}{ xExExExxxxxC
(referred to as kurtosis of )x
8
}][{
]}[{])[()( 22
4
neE
neCneJJ v
Maximization
][][][ T nsnne kk xv
( is an unknown complex scale
factor and )
k},,1{ Kk
(noise-free case(noise-free case))
Fast Kurtosis Maximization Algorithm (FKMA) (Chi et al., 2001 [4, 5])
Optimum
9
v
Closed-form solution for : Not existent
Gradient-type iterative algorithms for finding a local optimum : Not very computationally efficient
v
)( )1()1()1()( iiii J vQvv
v
where Q is a positive-definite matrix depending on the algorithm used, and μ is the step size such that
)()( )1()( ii JJ vv
magnutude of normalized kutorsis of [ ]e n
Criterion [7]:
Criterion [7]:
Fast Kurtosis Maximization Algorithm (FKMA) (Chi et al., 2001 [4, 5])
9
)1(1
)1(1)(
i
ii
dR
dRv
Compute
At the th iterationi
][nx
][)1( ne i
YesSuper-expoAlgorithm(SEA)
nential
No
( ) ( 1)( ) ( )i iJ J v v?
][)( ne i
][)( ne i
To the thiteration
)1( i
Update through a gradient type optimization algorithm such that
)(iv
)()( )1()( ii JJ vv
T* *{ [ ] [ ]}E n n XR R x x
( 1) ( 1) ( 1) ( 1) * *cum{ [ ], [ ], ( [ ]) , [ ]}i i i ie n e n e n n d x
(PxP matrix)
Algorithm:
}][{
]}[{])[()( 22
4
neE
neCneJJ v
Maximization
][][][ T nsnne kk xv
( is an unknown complex scale
factor and )
k},,1{ Kk
(noise-free case(noise-free case))
Optimumv
magnutude of normalized kutorsis of [ ]e n
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 .
( [ ])is n ][nsi
( [ ])is n][nsi
10
Observations: The FKMA itself is an exclusive spatial processing algorithm.
The smaller the value of , the worse the performance of the FKMA for finite SNR and finite data length .
( [ ])is nN
By (A2)
( [ ]) ( [ ]) ( [ ])i i iJ s n s n J u n
where
(absolute normalized kurtosis of )][nsi
(entropy measure of the stable sequence ) ][nbi
(equality holds only as , i.e., minimum entropy of ) ][][ nnbi ][nbi
4
22
[ ]( [ ])
0 ( [ ]) 1( [ ])
[ ]
ii k
ii
ik
b kJ s n
s nJ u n
b k
MSC Procedure
][nx
][ˆ nsk
Estimate One Source Signal Using FKMA
Obtain
}{}{
2|][ˆ|
][ˆ][ˆ
nsE
nsnE
k
kk
x
a
Update by][nx
][ˆˆ][ nsn kkax Next Stage
Each Stage of the Multistage Successive Cancellation (MSC)
Procedure
( : th column of )ka k A
11
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].
][ˆ nsk ka A
NOTE
ka
][nx
][nx
][][TSEA nvn vv
: vector for extracting a colored source signal , i.e.,
removing spatial interference due to the mixing matrix . ((spatial spatial
filterfilter))
][nskv 1 PA
: single-input single-output (SISO) deconvolution (or higher-order whitening) filter of order to restore from . ((temporal temporal
filterfilter))
L][nv
][nuk ][nsk
where
T TTSEA TSEA[ ] [ ] [ ] [ ] [ ]
k
n n n k n k
v x v x
3. Turbo Source Extraction Algorithm
Source Separation Filter:
12
( [ ])J nMaximization
Design Criterion:
T TTSEA
T
[ ] [ ] [ ] [ ] [ ] [ ]
ˆ[ ] [ ] [ ] (spatial processing)
[ ] [ ] [ ] (temporal processing)k
n n n n e n v n
e n s n n
n v n n
v x v y
v x
y x
(A bank of same temporal filters)
( [ ]) ( [ ])
( [ ]) ( [ ])kn J u n
J e n e n
(Extracted Source)(Extracted Source)
Turbo Source Extraction Algorithm (TSEA) (Chi et al., 2003 [3])
Signal processing procedure at the th cyclei
Step 1
FKMA(s)][ˆ][ )1( nvnv i
)(ˆ iv
Temporal ProcessingTemporal Processing Spatial ProcessingSpatial Processing
][)ˆ(
][)1(T)( n
nii yv
][][
][)1(
nvn
ni
x
y
][nx
Step 1
(a)(a) (b)(b)
13
][ˆ][
][)( nvne
ni
FKMA(t) )(ˆ ivv
][ˆ )( nv i
][ˆ
][][ T
ns
nne
k
xv
Step 2
][nxStep 2
(b)(b) (a)(a)
(Extracted Source)(Extracted Source)
T [ ] e n
T[ [0], [1], , [ ]]v v v L T]][ , ],1[ ],[[][ Lnenenen e
][][][])[][(
][][ˆ][][][)(
)()(
ngnunvnbnu
nvnsnvnen
kkki
kkk
ik
i
Interpretations:
14
Why? Performance of TSEA is superior to FKMA.
][][][ )( nvnbng ikk
Increasing is equivalent to increasing
( [ ]) ( [ ]) ( [ ])kJ n n J u n
1) 1) TemporaTemporal l ProcessiProcessing:ng:
T)(1 ])[~ , ],[][][][~ , ],[~(][~ nsnnvnsnsnsn K
ikk s
][~][~][][][ )( nwnnvnn i sAxy
( [ ]) ( [ ]) ( [ ]) and ( [ ]), k k ln s n s n s n l k
2) Spatial 2) Spatial Processing:Processing:
4
22
[ ]( [ ]) ( [ ]),
[ ]
kmk
km
g mn s n
g m
(b)(b)
(b)(b)
Remarks:
15
The performance gain of the TSEATSEA reaches the maximum as long as the order LL (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 MSC procedure. The resultant BSS algorithm that uses the TSEA, is referred to as MSC-TSEA,MSC-TSEA, also outperforms the MSC-FKMAMSC-FKMA, at the extra expense of the temporal processing at each stage.
TSEA is computationally efficient with super-exponential convergence rate and P parameters for spatial processingP parameters for spatial processing and L+1 parameters for temporal processing,L+1 parameters for temporal processing, respectively.
N SNR
Constrained
Criterion: ˆ ˆ ˆ, , , , 2, 3, , K a a a1 2C -1
4. Non-Cancellation Multistage Source Separation Algorithms NCMS-FKMA
16
T T Targ max{ ( ) ( [ ]) : [ ] [ ], }J J e n e n n v
v v v x v C 0 -1
where
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 .
T T[ ] [ ] [ ] [ ]k ke n n n s n x v x 1
S
k[ ]ks n S
Constraint
( ) v C (unconstrained optimization (unconstrained optimization problem)problem) Targ max{ ( ) ( [ ]) : [ ] [ ]}J J e n e n n x
vv
[ ] [ ]n nx C x
C: projection matrixP P
Unconstrained Criterion:
v
CObtain by
SVD of and][nx
17
C][][ nn xCx ˆ
-1a
[ ]nx
][ˆ nsk
(F-a)(F-a)Estimate One Source Signal
Using FKMA
Obtain
}{}{
2|][ˆ|
][ˆ][
nsE
nsnE
k
k
x
][nx
][nx
P/)1, ,1 ,1( T)0( v(Initial (Initial Condition)Condition) (0) v v
Good Good Initial Initial
ConditionCondition
Good Good Initial Initial
ConditionCondition
ˆaˆa
[ ]e n
Signal Processing Procedure of NCMS-FKMANCMS-FKMA
v(F-b)(F-b)
Estimate One Source Signal
Using FKMA
Remarks:
18
The constrained source extraction filter obtained in (F-a)(F-a) provides a suitable initial conditionsuitable initial condition for the unconstrained source extraction filter in (F-b),(F-b), which accordingly leads to one distinct source estimatedistinct source estimate obtained at each stage neither involving cancellation nor imposing any constraints on the source extraction filter, as well as faster convergence than (F-a).(F-a). Therefore, unlike the MSC-FKMA, the NCMS-FKMA is free fromfree from the error propagation effects the error propagation effects at each stage.
v
v[ ]e n
As the MSC-TSEAMSC-TSEA performs better than the MSC-FKMAMSC-FKMA, the NCMS-TSEANCMS-TSEA also performs better than the NCMS-FKMANCMS-FKMA at the moderate expense of extra computational load for the temporal processing of the TSEA.
CObtain by
SVD of and][nx
19
C][][ nn xCx ˆ
-1a
][][ˆ nensk
(T-a)(T-a)Estimate One Source Signal
Using TSEA
Obtain
}{}{
2|][ˆ|
][ˆ][
nsE
nsnE
k
k
x
][nx
][nx
P/)1, ,1 ,1( T)0( v
(Initial (Initial Condition)Condition) (0) v v
Good Good Initial Initial
ConditionCondition
Good Good Initial Initial
ConditionCondition ˆa
[ ]e n
Signal Processing Procedure of NCMS-TSEA NCMS-TSEA
v
(T-b)(T-b)Estimate One Source Signal
Using TSEA
(0)[ ] [ ]v n v n
[ ]v n[ ]nx
ˆa
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
}][{
}][][{SNR
2
2
nE
nnE
w
wx
][nui
][nw
}1{
][nsi ][nui ][nbi
20
Output (extracted) signal to interference-plus-noise ratio (Output SINR)
K
iiK 1
SINR1
SINROutput
Four cases are considered as follows:
Part A: Performance of NCMS-FKMA and NCMS-TSEA mixing matrix (taken from Chang et al., 1998 [9]) (P=5, K=4)
4 5
5 ..., 1, ,0 )10
1(exp][
nn
nbi
i
4593.04807.01983.05731.0
6640.04216.02644.03558.0
2504.02661.04959.06107.0
2097.01157.07494.03397.0
4914.07120.02887.02380.0
A
A
21
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.
N
( [ ])is n i i
N
K
kk nJK
1
])[()1(
(or ) for all , 5.0i ( [ ]) 0.2368i is n i 5L
Figure 1. Simulation results (Output SINR versus SNR) of Case 1.
22
5 10 15 20 25 300
5
10
15
20
25
30
SNR (dB)
OU
TP
UT
SIN
R (
dB
)NCMS-TSEA, N=1500NCMS-TSEA, N=1000NCMS-TSEA, N=500 NCMS-FKMA, N=1500NCMS-FKMA, N=1000NCMS-FKMA, N=500
(or ) for all , , and SNR=30 dB1i ( [ ]) 0.1856i is n
Figure 2. Simulation results (Output SINR versus data length ) of Case 2.
N
i 5L
23
103
104
105
106
14
16
18
20
22
24
26
28
30
32
N
OU
TP
UT
SIN
R (
dB
)
NCMS-TSEANCMS-FKMA
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 110
12
14
16
18
20
22
24
26
28
30
32
OU
TP
UT
SIN
R (
dB
)
NCMS-TSEANCMS-FKMA
0.745 0.345 0.240 0.183 0.144 0.115 0.091 0.068 0.01
SNR=30 dB, , and2000N
Figure 3. Simulation results (Output SINR versus ) of Case 3.
5L
24
0 1 2 3 4 5 6 7 8 910
15
20
25
30
35
L
OU
TP
UT
SIN
R (
dB
)
NCMS-TSEA, =0.5 (or =0.2368)NCMS-TSEA, =1 (or =0.1856)
and (i.e., and ) for all
SNR=30 dB, 1i 5.0 ( [ ]) 0.1856is n 2368.0 i
Figure 4a. Simulation results (Output SINR versus the order of
the temporal filter ) of Case 4 (a).L
2000N
25
0 1 2 3 4 5 6 7 8 90.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
L
NCMS-TSEA, =0.5 (or =0.2368)NCMS-TSEA, =1 (or =0.1856)
J(
[n
]) k=
1
K
(1/ K
)k
Figure 4b. Simulation results (Output SINR versus the order of
the temporal filter ) of Case 4 (b).L
26
and (i.e., and ) for all
SNR=30 dB, 1i 5.0 ( [ ]) 0.1856is n 2368.0 i
2000N
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]) and SOBI algorithm (Belouchrani et al. 1997 [2])
4 5
5 ..., 1, ,0 ),10
1(exp][
nn
nbi
i
A
N L
)1(
)2.0 ,3.0 ,4.0 , 1( 4321
4644.0 ])[( ,3335.0 ])[( ,2706.0 ])[( , 1856.0])[( 4321 nsnsnsns
)3,2,1 , ( iii
Three cases are considered as follows: Case A:Case A: Output SINR1 versus SNR for and .
Case B:Case B: Output SINR versus SNR for and .
Case C:Case C: Output SINR versus for SNR = 20 dB and .
2000N 5L
2000N 5L
5LN
27
5 10 15 20 25 300
5
10
15
20
25
30
35
SNR (dB)
OU
TP
UT
SIN
R 1 (d
B)
NCMS-TSEAMSC-TSEANCMS-FKMAMSC-FKMAFastICA SOBI ALGORITHMAMUSE
Figure 5. Simulation results (Output SINR1 versus SNR) of Case ACase A.
28
5 10 15 20 25 300
5
10
15
20
25
30
SNR (dB)
OU
TP
UT
SIN
R (
dB
)
NCMS-TSEAMSC-TSEANCMS-FKMAMSC-FKMAFastICA SOBI ALGORITHMAMUSE
Figure 6. Simulation results (Output SINR versus SNR) of Case BCase B.
29
500 1000 1500 2000 2500 3000 3500 4000 4500 50006
8
10
12
14
16
18
20
22
N
OU
TP
UT
SIN
R (
dB
)
NCMS-TSEAMSC-TSEANCMS-FKMAMSC-FKMAFastICA SOBI ALGORITHMAMUSE
Figure 7. Simulation results (Output SINR versus data length ) of Case CCase C.
N
30
: a 3x2 mixing 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 the MSC-FKMA, AMUSE and SOBI algorithm
Case D:Case D: Output SINR versus
1 1 11(z) (1 0.5z )(1 0.8z )(1 4z )B
1 1 12 (z) [1 (0.5 )z ][1 (0.8 )z ][1 (4 )z ]B
A
4959.06107.0
7494.03397.0
2887.02380.0
A
N L
)1( )3 ,2 ,1 ,( iii
40.005.0
31
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.45
10
15
20
25
30
OU
TP
UT
SIN
R (
dB
)
NCMS-TSEAMSC-TSEANCMS-FKMAMSC-FKMAFastICA SOBI ALGORITHMAMUSE
Figure 8. Simulation results (Output SINR versus ) of Case Case D.D.
32
f1
f3
f5
f4f6
f2
f1
f7
CCICCI
GOALGOAL Enhance data rate, link quality, capacity, and coverage.
CCI:CCI: Co-channel Interference
ISI:ISI: Intersymbol Interference (due to multipath)
Space-time processingSpace-time processing using an antenna array has been used for
combating CCI and ISIcombating CCI and ISI in the receiver design [15-16].
][nh][nu
][nx
(Nois(Noise)e)
(Multipath (Multipath channel)channel)
][nw
Problem Statement: CCICCI and ISIISI Suppression in TDMA Cellular Wireless Communications
6. Turbo Space-time Receiver for CCI/ISI Reduction
33
][1 nx
][2 nx
][nxP
][1 nu
)],[( 1111 nh
)],[( 1212 nh
)],[( 2121 nh
)],[( 2222 nh
][2 nu
Consider the scenario where the base station is equipped with Consider the scenario where the base station is equipped with multiple antennas,multiple antennas, and the signal of interest and CCI are received and the signal of interest and CCI are received fromfrom multiple distinct directions of arrival (DOA), with a frequency- multiple distinct directions of arrival (DOA), with a frequency-selective fading channel for each DOAselective fading channel for each DOA. . ((a general scenarioa general scenario))
Signal Model:
34
][][][ nnn wAsx
““ISI-distorted’’ signal ISI-distorted’’ signal (colored signal)(colored signal) from jth DOA of user from jth DOA of user kk
( is no. of DOAsDOAs of user k)
th-order channel impulse response of jthjth DOADOA of user k
where
1 2{ } ( , ,..., )KP A A A A 1 2{ } ( ( ), ( ),..., ( ))k kk P p k k kp A a a a
T1 2[ ] ( [ ], [ ],..., [ ])s
kk k k kpn s n s n s n
[ ] [ ] [ ] 1, 2,..., kj kj k ks n h n u n j p
( ) : kja steering vector of jthjth DOADOA of user k
[ ] :kjh n kjL
The received signal from the desired user and CCIs (users) can be The received signal from the desired user and CCIs (users) can be expressed as expressed as anan instantaneous mixture of multiple sourcesinstantaneous mixture of multiple sources
pk
1K
1
K
kk
p (total no. of DOAs or ’’sources(total no. of DOAs or ’’sources”)”)
35
T T T T1 2[ ] ( [ ], [ ],..., [ ])s s s sKn n n n
][][1
nnK
kkk wsA
(A1) The unknown DOA matrix is of full column rank and
P
36
(A3) is zero-mean Gaussian, and statistically independent of for all .
Assumptions: A
(A2) The data sequence of user 1 (the desired signal)(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).(of CCI). k
][nuk
][1 nu0]}[{C 14 nu
][nw ][nuk
1K
1
K
kk
p (total no. of DOAs or ’’sources(total no. of DOAs or ’’sources”)”)
P
block mutually independent colored sourcesK
T1 2[ ] ( [ ], [ ],..., [ ])s
kk k k kpn s n s n s n
correlated colored non-Gaussian sourcespk
T T T T1 2[ ] ( [ ], [ ],..., [ ])s s s sKn n n n
block mutually independent colored sources; mutually independent random variables for each n
Mutually independentcolored sources
Case II: Each user has multiple DOAs with disjoint domains of support of multipath channel impulse responses, i.e.,
Case I: Each user has a single DOA with multiple paths (Venkataraman et al., 2003), i.e.,
1, 1, 2,...,kp k K
11 1( ( ), , ( ))K A a a T11 1[ ] ( [ ], , [ ])Kn s n s ns
1 1[ ] [ ] [ ]k k ks n u n h n
2* *[ ] [ ] [ ] [ ] [ ] 0, ki kj ki kj kl
E s n s n h l h l E u n i j
T T T T1 2[ ] ( [ ], [ ],..., [ ])s s s sKn n n n
[ ] [ ] 0 and 1ki kjh n h n i j k K
37
K
1[ ] [ ]je n s n ][][ 1 nun Temporal Filter
Spatial Filter
Conventional Cascade Space-Time Receiver (CSTR) (Jelitto and Fettweis, 2002)
][nv][nx
v
Space-time Processor
For Cases I and II, the conventional CSTR has been reported for CCI and ISI suppression
In CAMSAP-06, we proposed two space-time receivers based on kurtosis maximization for these two cases and a discussion of the proposed space-time receivers for the general scenario.general scenario.
Other existing structures: full-dimension (joint) ST processing, reduced dimension ST processing (prewhitening followed by joint ST processing).
38
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
}][{
]}[{])[()( 22
4
neE
neCneJJ v
MaximizationT[ ] [ ] [ ]
[ ] [ ]
kj kj
kj k kj
e n n s n
u n h n
v x
(noise-free case)
Kurtosis Maximization (Ding ad Nguyen, 2000):
v
v
is an unknown complex scale factorkj
} , ,2 ,1{ Kk {1, 2, , }kj p and
magnutude of normalized kutorsis of [ ]e n
)1(1
)1(1)(
i
ii
dR
dRv
Compute
At the th iterationi
][nx
][)1( ne i
YesSuper-expoAlgorithm(SEA)
nential
No
)( )( )1()( ii JJ vv
?
][)( ne i
][)( ne i
To the thiteration
)1( i
Update through a gradient type optimization algorithm such that
)(iv
)( )( )1()( ii JJ vv
]}[][{ T* nnE xxR
]}[ ,])[( ],[],[{cum **)1()1()1()1( nnenene iiii xd
(PxP matrix)
Fast Kurtosis Maximization Algorithm (FKMA) (Chi and Chen, 2001):
40
Blind CSTR Using FKMA
T1 1 1[ ] [ ] [ ] [ ] [ ] j je n n s n u n h n v x
Spatial processing using FKMA for CCI suppression
With a suitable initial conditionsuitable initial condition for , FKMA will converge at a super-exponential rate with for high SNR.
v
41
T1[ ] [ ] [ ] [ ] [ ]n v n e n n u n ev
T]][ , ],1[ ],0[[ Lvvv vT]][ , ],1[ ],[[][ Lnenenen e
where
Temporal processing using FKMA for ISI removal
(L: order of the temporal filter)
1[ ] [ ]je n s n1[ ] [ ]n u n Temporal
FilterSpatial Filter ][nv
][nxv
Space-time Processor
11 , ,2 ,1 ,][][ pjnsne j
the CCI, (i.e., ). So
The performance of the spatial filter (to suppress CCI) using FKMA is The performance of the spatial filter (to suppress CCI) using FKMA is
worse for smaller and worse for larger , worse for smaller and worse for larger , leading to limited performance of the temporal filter of the blind CSTR.
It can be easily shown that (Chi et al., 2003)
1 1 1( [ ]) ( [ ]) ( [ ]), j jJ s n s n J u n
where 1
1
4
10
1 22
10
[ ]0 ( [ ]) 1
[ ]
j
j
L
jm
jL
jm
h ms n
h m
42
1( [ ]),js n 1 jL
Usually, the ISI-distorted (desired) signalISI-distorted (desired) signal , has higher power than all
1 [ ]js n
implying that can be used as the initial conditioninitial condition for the spatial filter needed by the FKMA.
2 2
1 { [ ] } { [ ] }j k iE s n E s n 1k
1( )a jv
2H1 arg max { ( ) [ ] },
a xj E n (DOA estimate by delay-and-(DOA estimate by delay-and-
sum)sum)
][nv
v
TSTR[ ] [ ]n v nv v
TTSTR[ ] [ ] [ ]n n n v x
Space-Time Filter for Source Extraction (Chi et al. 2003, 2006):
])[( nJ Maximization
Optimum
Design Criterion:
T TTSTR
1 1 1 1
[ ] [ ] [ ] [ ]
[ ] [ ] [ ]j j
n n n n
s n v n u n
v x v y
T1 1 1[ ] [ ] [ ] [ ]j j je n s n n s n v x
Blind Turbo Space-Time Receiver (TSTR)
43
(noise-free case)
[ ] [ ] [ ]n n v n y x
( ) T
1
[ ] [ ]
[ ]
i
j
e n n
s n
v x
Spatial FilterSpatial Filter
FKMAFKMA ][][ )1( nvnv i
][)( nv i)(iv
)(ivv Temporal Temporal
FilterFilterFKMAFKMA
( )2 1[ ] [ ]i n u n
][][
][)(
nvn
ni
x
y
(S2)(S2)
(S1)(S1)][)(1 ni
][nx
][nxTemporal Temporal FilterFilter
Spatial FilterSpatial Filter
Signal processing procedure at the th cycle: i
Proposed Blind TSTR Using FKMA
])[( )(1 nJ i ])[( )(
2 nJ i
44
i=i+1
CSTR
CSTR
( ) ( ) ( ) ( )2 1
( )1 1 1 1 1 1
[ ] [ ] [ ] [ ] [ ]
( [ ] [ ]) [ ] [ ] [ ]
i i i ij
ij j
n e n v n s n v n
u n h n v n u n g n
Interpretations:
Why? Performance of blind TSTR is superior to blind CSTR.
1 [ ]jg n
Increasing is equivalent to increasing
])[( )(2 nJ i
1) 1) TemporaTemporal l ProcessiProcessing:ng:
( 1) ( ) T11 1 1 2 1[ ] ( [ ],..., [ ] [ ] [ ] [ ], , [ ], , [ ] )
K
i ij j k Kpn s n s n s n v n n s n s n s
][~][][][][ )1( nwnnvnn i sAxy 2) Spatial 2) Spatial
Processing:Processing:
4
1( )2 122
1
[ ]( [ ]) ( [ ])
[ ]
jmi
j
jm
g mn s n
g m
( )2 1 1( [ ]) ( [ ]) ( [ ]) and ( [ ]), 1, l i
j j kln s n s n s n k j
45
Compared with the blind CSTR, the proposed blind TSTR is the proposed blind TSTR is insensitive to insensitive to the value of the value of (i.e., robust against channel with multiple paths or severe ISI).
Remarks:
It can be proven that
for all , implying the guaranteed convergencethe guaranteed convergence of the proposed blind TSTR. Typically, the number of cyclesthe number of cycles spent by the TSTR before convergence, is equal to 2 or 32 or 3. The computational load of the blind TSTR is approximately 2 or 32 or 3 times that of the blind CSTR.
i
Because the design of and that of are coupled in a coupled in a constructiveconstructive and boosting manner and boosting manner, the proposed blind TSTR outperforms the blind CSTR for all , and meanwhile their performance difference is their performance difference is largerlarger for larger . for larger .
v
])[( ])[( ])[( ])[( 1)(
2)(
1)1(
2 nuJnJnJnJ iii
][nv
LL
1( [ ])js n
46
1 [ ]jh n
CASE I: CCI suppression by CCI suppression by
Performance of the blind TSTR:
TSTR[ ] [ ]n v nv vTTSTR[ ] [ ] [ ], n n n v x where
T1 11 1
1 1 1
[ ] ( ) [ ] [ ] [ ] residual CCI and noise
[ ]
n h n v n u n
u n
v a
CASE II: CCI suppression byCCI suppression by
1
T T1 1 1 1 1 1
1 1 1
[ ] ( ) [ ] [ ] [ ] ( ) [ ] [ ]* [ ]
residual CCI and noise [ ]
p
i i j jj i
i i
n h n v n u n h n v n u n
u n
v a v a
Multiple DOAs suppressed also byMultiple DOAs suppressed also by
1T
1 1 11
[ ] ( ) [ ] [ ] [ ] residual CCI and noise p
j jj
n h n v n u n
v a
GENERAL CASE: CCI suppression CCI suppression byby
the spatial filterthe spatial filter and the temporal filterand the temporal filter combine the signalscombine the signalsfrom all the DOAs in a constructive and boosting fashionfrom all the DOAs in a constructive and boosting fashion
v
1 1 1[ ]g u n 47
v
[ ]v n
v
v
v
: zero-mean, independent binary sequence of with equal probability
, ,
: white Gaussian noise vector SNR:
50 independent runs
01 402
Scenario of Case I
][nui
][nw
}1{
10P (array size)
}])[]([{]0[ HnnE ssRS : Diagonal matrix: Diagonal matrix
543211 z7073.0z4712.0z3089.0z7073.00.4325z 6178.0)z( H
543212 0.1622z0.1217z0.2839z0.3650z0.2839z0.4056)z( H
543213 0.1195z0.1992z0.2390z0.3586z0.3187z0.3984)z( H
603
22
2
12
2
11 1}][{
}][{
}][)({SNR
ww
nsE
nE
nsE
w
a
48
7. Simulation Results --- Part 2
Order of the temporal filter =20
SNR=20 dB
Data length =2000
L
N
402 01 603
49
Blind CSTRBlind CSTR Proposed Blind TSTRProposed Blind TSTR
50
Data length =2000
Order of the temporal filter =20
N
L
51
SNR=30 dB
Order of the temporal filter =20
L
52
SNR=30 dB
Data length =2000
N
53
: zero-mean, independent binary sequence of with equal probability
: real white Gaussian noise vector
SNR:
50 independent runs
][nui
][nw
}1{
10P (array size)
1 211(z) 0.5199 0.3639z 0.3119zH 1 2
21(z) 0.3562 0.3206z 0.1425zH
3 4 512 (z) 0.5754z 0.2466z 0.3288zH 3 4 5
22 (z) 0.3776z 0.2098z 0.2518zH
011 2012 4021 6022
54
: Diagonal matrix
}][{
}][)(][)({SNR
2
212121111
nE
nsnsE
w
aa
}])[]([{]0[ HnnE ssRS
Scenario of Case II
Order of the temporal filter =20
SNR=20 dB
Data length =2000
L
N55
011 2012 4021 6022
56
Blind CSTRBlind CSTR Proposed Blind TSTRProposed Blind TSTR
Data length =2000
Order of the temporal filter =20
N
L
57
SNR=30 dB
Order of the temporal filter =20
L
58
SNR=30 dB
Data length =2000
N
59
: zero-mean, independent binary sequence of with equal probability
: white Gaussian noise vector
SNR:
50 independent runs
Scenario of the general case
][nui
][nw
}1{
10P (array size)
011 2012 4021 6022
1 2 311(z) 1 0.7z 0.6z 0.5zH 1 2 3
21(z) 1 0.9z 0.4z 0.3zH 2 3 4 5
22 (z) z 0.9z 0.5z 0.6zH
: Block-diagonal matrix
}][{
}][)(][)({SNR 2
2
12121111
nE
nsnsE
w
aa
}])[]([{]0[ HnnE ssRS
60
543212 0.4z0.3z0.7z0.8z )z( H
Order of the temporal filter =20
SNR=30 dB
Data length =2000
011 2012 4021 6022
61
L
N
62
Blind CSTRBlind CSTR Proposed Blind TSTRProposed Blind TSTR
Data length =2000
Order of the temporal filter =20
N
L
63
SNR=30 dB
Order of the temporal filter =20
L
64
SNR=30 dB
Data length =2000
N
65
We have introduced a novel blind source extraction algorithm, TSEATSEA, which operates cyclically using the FKMAFKMA for both of the temporal processing and spatial processing. The proposed TSEATSEA outperforms the FKMA for in addition to sharing convergence speed and computational efficiency of the later at each cycle.
FKMAFKMA only involves spatial processing for extraction of one non-Gaussian (i.i.d. or colored) source from source mixtures. It performs well with super-exponential convergence rate, but its performance depends on the parameter .
8. Conclusions
0 ( [ ]) 1is n
( [ ]) 1is n
Because of performance degradation resultant from the error propagation in the MSC procedure, we further introduced two non-cancellation BSS algorithms, namely, NCMS-FKMA and NCMS-TSEA, that can extract a distinct source at each stage without error propagation.
66
The two BSS algorithms, NCMS-FKMA and NCMS-TSEA perform better than the existing MSC-FKMA and the MSC-TSEA, respectively, with moderately higher computational complexities. FKMA and TSEA are under investigation for CCI and ISI in MIMO wireless communications (e.g., OFDM and multi-rate CDMA) and other applications such as 2-D MIMO systems in biomedical signal processing (with certain constraints or partial correlation between source signals).
67
Some works of Part 1/Part 2 will be published in C.-Y. Chi and C.-H. Peng, “Turbo source extraction algorithm and non- cancellation source separation algorithms by kurtosis maximization,” IEEE Trans. Signal Processing, vol. 54, no. 8, pp. 2929-2942, Aug. 2006.
C.-H. Peng, C.-Y. Chi and C.-W. Chang, “Blind multiuser detection by kurtosis maximization for asynchronous multi-rate DS/CDMA systems,” EURASIP Journal on Applied Signal Processing, vol. 2006, Article ID 84930, 17 pages, 2006. doi:10.1155/ASP/2006/84930. (special issue: Multisenor Processing for Signal Extraction and Applications)
Thank you very Thank you very muchmuch
68
Background materials of the talk can be found in the following book: C.-Y. Chi, C.-C.Feng, C.-H. Chen and C.-Y. Chen, Blind Equalization Blind Equalization and System Identification and System Identification, London: Springer-Verlag, 2006.
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[5] C.-Y. Chi and C.-Y. Chen , C.-H. Chen and C.-C. Feng, ``Batch processing algorithms for blind equalization using higher-order statistics,'' IEEE Signal Processing Magazine, vol. 20, pp. 25-49, Jan. 2003.
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69
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