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Parameter Estimation Employing Parameter Estimation Employing Recursive Recursive EigenvalueEigenvalue DecompositionDecomposition
for MIMOfor MIMO--OFDM Using OFDM Using Maximum Likelihood Detector Maximum Likelihood Detector
with Arraywith Array CombiningCombining
Fan Lisheng, Kazuhiko Fukawa, Hiroshi Suzuki
Tokyo Institute of Technology
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Background
MIMO-OFDM
OFDM + MIMO
MIMO ・・・Improve the system capacity
OFDM ・・・Utilize the spectrum efficiently →high-speed data transmission
Cochannel interference→ Performance is degraded severely
!
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Co-channel interference in the uplink of MIMO-OFDM
Random accessRandom access
Incident angle Incident angle differencedifference
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MLD with Array CombingZF, MMSE, MLD cannot workTo suppress co-channel interference, MLD with the array combing has been proposed
Post-FFT AC PrePre--FFT ACFFT AC
FFT
FFT
Array
combiningMLD
FFTArray
combining MLD
FFT
Array combing: Suppress the co-channel interference
by prewhitening filtering
Pre-FFT AC is superior when using limited preamble symbols
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Motivation
The drawback of Pre-FFT ACLarge complexity in parameter update due to the non-recursive eigenvaluedecomposition (EVD)
A recursive EVD is applied to the parameter estimation of Pre-FFT AC
Reduce the complexity to about 25 percentsSuitable for real-time application
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Pre-FFT ACArray combining
Decision
feedback
Recursive EVD
From desired user
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Error signal in the time domain
: Error signal of the branch metric generatorcaused by the interference and noise
• : AC output- : Coefficient of AC- : Received signal
( , )r by l m ( )lb
H m≅w xlb
w( )mx
( , ) ( , ) ( , )b r b s bl m y l m y l mα = −( , )bl mα
• : Replica signal- : Channel impulse response- : Candidateof the transmitted signal
( , )s by l m ˆ( , ) ( )Hbl m m≅ c s
( , )bl mcˆ( )ms
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whenwhen
Parameter estimation
••To suppress the interference, To suppress the interference, prewhiteningprewhitening filtering is filtering is employed to the error signalemployed to the error signal::
0=
The parameter estimation should be based on EVD
Parameter vector Autocorrelation
matrix
0= b bl l≠
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Comparison between EVD methods
Newton’s iterative search
Low computational complexity
Cyclic Jacobi algorithmLarge computational
complexity
Recursive EVDNonrecursive EVD
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Recursive EVD
Auto-correlation matrix of is
: Forgetting factor
Solve the eigenvalues of the current matrixbased on the EVD result of previous matrix
: Eigenvalues of the previous matrix
: Eigenvectors of the previous matrix
: Eigenvalues of the current matrix: Eigenvectors of the current matrix
( )x mR( 1)x m−R
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Recursively solving the eigenvalues
ext ext( ) ( 1) ( ) ( )Hm m m mλ= − +R R X Xext ( )Hiy m= X p
ext ext{1 ( )[ ( 1) ] ( )} 0Him m m yλ β+ − − =X R I X
( ( ) ) 0i im β− =R I p
ext ext1 ( )[ ( 1) ] ( ) 0Him m mX R I Xλ β+ − − =
0y= ext ( )HjmX p ext ext( ) ( )H
j j m mβ λξ= +X X
0y ≠
Find a accurate root of
ext ext( ) 1 ( )[ ( 1) ] ( )Hf m m mX R I Xβ λ β= + − −
( ) 0f β = 12
( ) 0if β = 1'
( )( )
ii i
i
ff
ηη η
ηββ ββ
+ = −
NewtonNewton’’s iterative search algorithms iterative search algorithm::
: number of iterationsη
Newton’s iterative search
A0λ>
1( ) [ ( ), ( )]i i iβ β β −∈B A A
The interlacing property:Hλ= +B A cc
: Symmetric matrix
1[ , ]i i iβ λξ λξ −∈
Increasement
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Small increasement
'
( ) 0( )
i
i
ff
η
ηββ<
2
ext2
1
( )'( ) 0
( )
HLj
j j
mf β
λξ β=
= >−∑
q X
( )f β
'
( ) 0( )
i
i
ff
η
ηββ>
is a monotone increasing function
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Large increasement
The midpoint of the new intervalThe midpoint of the new interval is is used for the next iterationused for the next iteration
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Flow chart of iterative search
Select the midpoint of the interval
Yes
No
Obtain the next iteration value
Calculate the eigenvectors
Convergence threshold
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The convergence of the search algorithm
Guaranteed to convergenceQuadratic convergent
K. B. Yu, “Recursive updating the eigenvalue decomposition of a covariance matrix”, IEEE Trans. Acoust. Speech Sig. Proc. , vol. 39, pp. 1136-1145, May 1991.
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Simulation conditions
Incident angleDifference:60 DegDeviation: 4 Deg
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Computational complexity
(Dimension of the auto-correlation matrix)
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Average BER versus average CNR
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Average BER versus average CIR
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Average BER versus Doppler Spread
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Conclusion
A recursive EVD based algorithm was applied to the parameter estimation of Pre-FFT ACIt can reduce the total computational complexity to less than 25 percents (More suitable for real-time application)
Pre-FFT AC with recursive EVD can achieve the same performance as that of Pre-FFT AC with the conventional non-recursive EVD
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Thanks a lot for your attention