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SC-FDMA Oversampling MMSE Equalizer
YunXiang LIU, LeoINFINITUS, Nanyang Technological University
Yingcai Experimental School, University of Electronic Science and Technologyof China
May 14, 2013
YunXiang LIU, Leo ([email protected]) May 14, 2013 1 / 34
.. Outline
• Background
• Introduction to Oversampling MMSE Equalizer SC-FDMA System
• Simulation Results
• Conclusions
• Future Work
YunXiang LIU, Leo ([email protected]) May 14, 2013 2 / 34
.. Outline
• Background
• Introduction to Oversampling MMSE Equalizer SC-FDMA System
• Simulation Results
• Conclusions
• Future Work
YunXiang LIU, Leo ([email protected]) May 14, 2013 2 / 34
.. Outline
• Background
• Introduction to Oversampling MMSE Equalizer SC-FDMA System
• Simulation Results
• Conclusions
• Future Work
YunXiang LIU, Leo ([email protected]) May 14, 2013 2 / 34
.. Outline
• Background
• Introduction to Oversampling MMSE Equalizer SC-FDMA System
• Simulation Results
• Conclusions
• Future Work
YunXiang LIU, Leo ([email protected]) May 14, 2013 2 / 34
.. Outline
• Background
• Introduction to Oversampling MMSE Equalizer SC-FDMA System
• Simulation Results
• Conclusions
• Future Work
YunXiang LIU, Leo ([email protected]) May 14, 2013 2 / 34
.. Outline
• Background
• Introduction to Oversampling MMSE Equalizer SC-FDMA System
• Simulation Results
• Conclusions
• Future Work
YunXiang LIU, Leo ([email protected]) May 14, 2013 3 / 34
.. Background
• SC-FDMA system is a Pre-DFT OFDM system, which is similar to OFDM
• SC-FDMA (Single Carrier FDMA) is an attractive alternative to OFDMAespecially in the uplink communications where lower PAPR (peak to averagepower ratio) greatly benefits the mobile terminal, in terms of transmit powerefficiency and terminal costs.
• SC-FDMA has been adopted as the uplink multiple access scheme in 3GPPLong Term Evolution (LTE), or Evolved UTRA
• SC-FDMA effectively copes with frequency-selective fading channels by usingsimple frequency-domain equalization
YunXiang LIU, Leo ([email protected]) May 14, 2013 4 / 34
.. Background
• SC-FDMA system is a Pre-DFT OFDM system, which is similar to OFDM
• SC-FDMA (Single Carrier FDMA) is an attractive alternative to OFDMAespecially in the uplink communications where lower PAPR (peak to averagepower ratio) greatly benefits the mobile terminal, in terms of transmit powerefficiency and terminal costs.
• SC-FDMA has been adopted as the uplink multiple access scheme in 3GPPLong Term Evolution (LTE), or Evolved UTRA
• SC-FDMA effectively copes with frequency-selective fading channels by usingsimple frequency-domain equalization
YunXiang LIU, Leo ([email protected]) May 14, 2013 4 / 34
.. Background
• SC-FDMA system is a Pre-DFT OFDM system, which is similar to OFDM
• SC-FDMA (Single Carrier FDMA) is an attractive alternative to OFDMAespecially in the uplink communications where lower PAPR (peak to averagepower ratio) greatly benefits the mobile terminal, in terms of transmit powerefficiency and terminal costs.
• SC-FDMA has been adopted as the uplink multiple access scheme in 3GPPLong Term Evolution (LTE), or Evolved UTRA
• SC-FDMA effectively copes with frequency-selective fading channels by usingsimple frequency-domain equalization
YunXiang LIU, Leo ([email protected]) May 14, 2013 4 / 34
.. Background
• SC-FDMA system is a Pre-DFT OFDM system, which is similar to OFDM
• SC-FDMA (Single Carrier FDMA) is an attractive alternative to OFDMAespecially in the uplink communications where lower PAPR (peak to averagepower ratio) greatly benefits the mobile terminal, in terms of transmit powerefficiency and terminal costs.
• SC-FDMA has been adopted as the uplink multiple access scheme in 3GPPLong Term Evolution (LTE), or Evolved UTRA
• SC-FDMA effectively copes with frequency-selective fading channels by usingsimple frequency-domain equalization
YunXiang LIU, Leo ([email protected]) May 14, 2013 4 / 34
.. Background
• However, SC-FDMA also suffers from carrier frequency offset, which iscaused by oscillator instability, Doppler effect
• CFO destoys the orthogonality among subcarriers
• Inter-carrier Interference is introduced
• Frequency offset estimation and compensation is critical
YunXiang LIU, Leo ([email protected]) May 14, 2013 5 / 34
.. Background
• However, SC-FDMA also suffers from carrier frequency offset, which iscaused by oscillator instability, Doppler effect
• CFO destoys the orthogonality among subcarriers
• Inter-carrier Interference is introduced
• Frequency offset estimation and compensation is critical
YunXiang LIU, Leo ([email protected]) May 14, 2013 5 / 34
.. Background
• However, SC-FDMA also suffers from carrier frequency offset, which iscaused by oscillator instability, Doppler effect
• CFO destoys the orthogonality among subcarriers
• Inter-carrier Interference is introduced
• Frequency offset estimation and compensation is critical
YunXiang LIU, Leo ([email protected]) May 14, 2013 5 / 34
.. Background
• However, SC-FDMA also suffers from carrier frequency offset, which iscaused by oscillator instability, Doppler effect
• CFO destoys the orthogonality among subcarriers
• Inter-carrier Interference is introduced
• Frequency offset estimation and compensation is critical
YunXiang LIU, Leo ([email protected]) May 14, 2013 5 / 34
.. Background
• Oversampling MMSE Equalization may be also an effective way to combatthe frequency offset
• It is clear by noting that a dual problem of frequency offset for multicarriersystems is timing jitter for signal-carrier systems, which is traditionally solvedvia time-domain-oversampling
• Frequency oversampling should be expected to be robust against frequencyoffset
YunXiang LIU, Leo ([email protected]) May 14, 2013 6 / 34
.. Background
• Oversampling MMSE Equalization may be also an effective way to combatthe frequency offset
• It is clear by noting that a dual problem of frequency offset for multicarriersystems is timing jitter for signal-carrier systems, which is traditionally solvedvia time-domain-oversampling
• Frequency oversampling should be expected to be robust against frequencyoffset
YunXiang LIU, Leo ([email protected]) May 14, 2013 6 / 34
.. Background
• Oversampling MMSE Equalization may be also an effective way to combatthe frequency offset
• It is clear by noting that a dual problem of frequency offset for multicarriersystems is timing jitter for signal-carrier systems, which is traditionally solvedvia time-domain-oversampling
• Frequency oversampling should be expected to be robust against frequencyoffset
YunXiang LIU, Leo ([email protected]) May 14, 2013 6 / 34
.. Outline
• Background
• Introduction to Oversampling MMSE Equalizer SC-FDMA System
• Simulation Results
• Conclusions
• Future Work
YunXiang LIU, Leo ([email protected]) May 14, 2013 7 / 34
..Comparison between Conventional MMSE andOversampling MMSE
YunXiang LIU, Leo ([email protected]) May 14, 2013 8 / 34
.. Mathmatical Model-Transmitter
• DFT operation
Xi = FMdi
wheredi = [di
1, di2, ......., di
M]T
[FM]p,q = (1/√
M)e−j2π(pq/M)
i = 1, 2, ...,Q,Q = N/M
• Subcarrier Mapping
XiN×1 = Mi
TXiM×1
where MiT is subcarrier mapping matrix
YunXiang LIU, Leo ([email protected]) May 14, 2013 9 / 34
.. Mathmatical Model-Transmitter
• DFT operation
Xi = FMdi
wheredi = [di
1, di2, ......., di
M]T
[FM]p,q = (1/√
M)e−j2π(pq/M)
i = 1, 2, ...,Q,Q = N/M
• Subcarrier Mapping
XiN×1 = Mi
TXiM×1
where MiT is subcarrier mapping matrix
YunXiang LIU, Leo ([email protected]) May 14, 2013 9 / 34
.. Mathmatical Model-Transmitter
• DFT operation
Xi = FMdi
wheredi = [di
1, di2, ......., di
M]T
[FM]p,q = (1/√
M)e−j2π(pq/M)
i = 1, 2, ...,Q,Q = N/M
• Subcarrier Mapping
XiN×1 = Mi
TXiM×1
where MiT is subcarrier mapping matrix
YunXiang LIU, Leo ([email protected]) May 14, 2013 9 / 34
.. Mathmatical Model-Transmitter
• IDFT operation
x = F−1N
Q∑i=1
MiTXi
• Add Guard Interval
x = PGIF−1N
Q∑i=1
MiTXi
wherePGI = [IT
N, 0TNg×N]
T
YunXiang LIU, Leo ([email protected]) May 14, 2013 10 / 34
.. Mathmatical Model-Transmitter
• IDFT operation
x = F−1N
Q∑i=1
MiTXi
• Add Guard Interval
x = PGIF−1N
Q∑i=1
MiTXi
wherePGI = [IT
N, 0TNg×N]
T
YunXiang LIU, Leo ([email protected]) May 14, 2013 10 / 34
.. Mathmatical Model-Channel Model
• Channel Response
h =K−1∑k=0
hkδ(τ − τk)
whereK−1∑k=0
E[|hk|2
]= 1
YunXiang LIU, Leo ([email protected]) May 14, 2013 11 / 34
.. Mathmatical Model-Channel and Receiver
• Receiver Signal after Zeros Padding
r = D (ε)PzpHx + n
wherePzp =
[ITP0
T(L−P)×P
]T
D (ε) = diag1, e−j 2π
N ε, ..., e−j 2πN (L−1)ε
and H is a P × P lower triangular Toeplitz matrix with the first column[h0, h1, ..., hK−1, 0, ..., 0]
T and ε is normalized by subcarrier spacing 1/T
YunXiang LIU, Leo ([email protected]) May 14, 2013 12 / 34
.. Mathmatical Model-Receiver
• Channel Matrix Transformation
Hzp = PzpHPGI
By definingFL×L
∆=
1√L
exp(−j2πL np
)L×L
FL×N∆=
1√L
exp(−j2πL lp
)L×N
Hzp = FHL×LHFL×N
where H = diag H0,H1, ...,HL−1 and S = L/N, which is the oversamplingfactor
Hl∆= H
(l
ST
)=
K−1∑k=0
hk exp(−j2π[
lS
]τqT ), l = 0, 1, ...,L − 1.
YunXiang LIU, Leo ([email protected]) May 14, 2013 13 / 34
.. Mathmatical Model-Receiver
• Channel Matrix Transformation
Hzp = PzpHPGI
By definingFL×L
∆=
1√L
exp(−j2πL np
)L×L
FL×N∆=
1√L
exp(−j2πL lp
)L×N
Hzp = FHL×LHFL×N
where H = diag H0,H1, ...,HL−1 and S = L/N, which is the oversamplingfactor
Hl∆= H
(l
ST
)=
K−1∑k=0
hk exp(−j2π[
lS
]τqT ), l = 0, 1, ...,L − 1.
YunXiang LIU, Leo ([email protected]) May 14, 2013 13 / 34
.. Mathmatical Model-Receiver
• Receive Signal after L-point DFT
y = FL×LD (ε)PzpHPGIFHN×N
Q∑i=1
MiTXi + FL×LPzpn
= FL×LD (ε)FHL×LHFL×NFH
N×N
Q∑i=1
MiTXi + FL×Pn
= FL×LD (ε)FHL×LHΩΨ+ η
where
Ω = FL×NFHN×N,Ψ =
Q∑i=1
MiTXi, η = FL×Pn
Cη = E[ηηH] = σ2
nΓ, Γ = FL×PFHL×P
YunXiang LIU, Leo ([email protected]) May 14, 2013 14 / 34
.. Mathmatical Model - Receiver
• Oversampling MMSE Receiver
wH = E[dyH] · E
[yyH]†
=(
HΩ)H
HΩΩHHH +σ2
nσ2
sÆ
Based on Bayesian Gauss-Markov theorem• Estimation
Ψ = wHy
• De-mappingXi = MiΨ
• N-point IDFT operationdi = FH
N×NXi
YunXiang LIU, Leo ([email protected]) May 14, 2013 15 / 34
.. Mathmatical Model - Receiver
• Oversampling MMSE Receiver
wH = E[dyH] · E
[yyH]†
=(
HΩ)H
HΩΩHHH +σ2
nσ2
sÆ
Based on Bayesian Gauss-Markov theorem• Estimation
Ψ = wHy
• De-mappingXi = MiΨ
• N-point IDFT operationdi = FH
N×NXi
YunXiang LIU, Leo ([email protected]) May 14, 2013 15 / 34
.. Mathmatical Model - Receiver
• Oversampling MMSE Receiver
wH = E[dyH] · E
[yyH]†
=(
HΩ)H
HΩΩHHH +σ2
nσ2
sÆ
Based on Bayesian Gauss-Markov theorem• Estimation
Ψ = wHy
• De-mappingXi = MiΨ
• N-point IDFT operationdi = FH
N×NXi
YunXiang LIU, Leo ([email protected]) May 14, 2013 15 / 34
.. Outline
• Backgroud
• Introduction to Oversampling MMSE Equalizer SC-FDMA System
• Simulation Results
• Conclusions
• Future Work
YunXiang LIU, Leo ([email protected]) May 14, 2013 16 / 34
.. Simulation outline
• Simulation Setup
• Reconsctrction of OFDM Oversampling Equalizer based on [1]
• BER Performance Comparison between SC-FDMA and OFDM without CFO
• BER Performance Comparison between SC-FDMA and OFDM with CFO
• BER Performance of SC-FDMA under different CFO
• Effect of Oversampling Factor
• Oversampling Gain Discussion
• One Path Rayleigh Channel with Different CFO
YunXiang LIU, Leo ([email protected]) May 14, 2013 17 / 34
.. Simulation Setup
• Simulation System Parameters
Total Subcarriers 64
Pre-DFT subcarriers number 16
Modulation QPSK
Equalization Conventional and Oversampling
MMSE
Mapping Scheme Interleaved
CP number 16
Iteration Number 1000000
Channel 16path Rayleigh channel
Carrier Frequency Offset(CFO) [0,0.02,0.05,0.1]
Attention: Here CFO is normalized by subcarrier spacing
YunXiang LIU, Leo ([email protected]) May 14, 2013 18 / 34
..Reconsctruction of OFDM Oversampling Equalizer basedon [1]
0 2 4 6 8 10 12 14 16 18 2010
−4
10−3
10−2
10−1
100
Eb/N0(dB)
Ave
rage
BE
R
Performance Comparison of OFDM
Conv. MMSE with εg = 0
Conv. MMSE with εg = 0.02
Conv. MMSE with εg = 0.05Oversamping MMSE with εg = 0
Oversamping MMSE with εg = 0.02
Oversamping MMSE with εg = 0.05
• 1. Shi, Q., Liu, L., Member, S., and Guan, Y. L. (2010). Fractionally SpacedFrequency-Domain MMSE Receiver for OFDM Systems, 59(9), 4400-4407.
YunXiang LIU, Leo ([email protected]) May 14, 2013 19 / 34
..BER Performance Comparison between SC-FDMA andOFDM without CFO
0 2 4 6 8 10 12 14 16 18 2010
−7
10−6
10−5
10−4
10−3
10−2
10−1
100
EbN0
BE
R
Performance Comparison of OFDM and SC−FDMA
OFDM Conventional MMSEOFDM OverSampling MMSESC−FDMA Conventional MMSESC−FDMA Oversampling MMSE
YunXiang LIU, Leo ([email protected]) May 14, 2013 20 / 34
..BER Performance Comparison between SC-FDMA andOFDM with CFO
0 2 4 6 8 10 12 14 16 18 2010
−7
10−6
10−5
10−4
10−3
10−2
10−1
100
EbN0
BE
R
Performance Comparison of OFDM and SC−FDMA with CFO = 0.02
OFDM Conventional MMSEOFDM OverSampling MMSESC−FDMA Conventional MMSESC−FDMA Oversampling MMSE
YunXiang LIU, Leo ([email protected]) May 14, 2013 21 / 34
..BER Performance Comparison between SC-FDMA andOFDM with CFO
0 5 10 15 20 2510
−8
10−7
10−6
10−5
10−4
10−3
10−2
10−1
100
EbN0
BE
R
Performance Comparison of OFDM and SC−FDMA with CFO = 0.05
OFDM Conventional MMSEOFDM OverSampling MMSESC−FDMA Conventional MMSESC−FDMA Oversampling MMSE
YunXiang LIU, Leo ([email protected]) May 14, 2013 22 / 34
..BER Performance Comparison between SC-FDMA andOFDM with CFO
0 2 4 6 8 10 12 14 16 18 2010
−4
10−3
10−2
10−1
100
EbN0
BE
R
Performance Comparison of OFDM and SC−FDMA with CFO = 0.1
OFDM Conventional MMSEOFDM OverSampling MMSESC−FDMA Conventional MMSESC−FDMA Oversampling MMSE
YunXiang LIU, Leo ([email protected]) May 14, 2013 23 / 34
.. BER Performance of SC-FDMA under different CFO
0 2 4 6 8 10 12 14 16 18 2010
−7
10−6
10−5
10−4
10−3
10−2
10−1
100
EbN0
BE
R
BER Performance of SC−FDMA
SC−FDMA Conventional MMSE with CFO = 0SC−FDMA Oversampling MMSE with CFO = 0SC−FDMA Conventional MMSE with CFO = 0.05SC−FDMA Oversampling MMSE with CFO = 0.05SC−FDMA Conventional MMSE with CFO = 0SC−FDMA Oversampling MMSE with CFO = 0.1
YunXiang LIU, Leo ([email protected]) May 14, 2013 24 / 34
.. Effect of Oversampling Factor
1 1.5 2 2.5 3 3.5 4 4.5 510
−7
10−6
10−5
10−4
10−3
10−2
10−1
100
EbN0
BE
R
Effect of oversampling factor
S = 77/64S = 78/64S = 79/64
YunXiang LIU, Leo ([email protected]) May 14, 2013 25 / 34
.. Oversampling Gain
0 2 4 6 8 10 12 14 16 18 2010
−6
10−5
10−4
10−3
10−2
10−1
100
EbN0
BE
R
BER Performance of Different Paths
Conv. OFDM − 1 PathConv. SC−FDMA − 1 PathOverSampling SC−FDMA − 1 PathConv. SC−FDMA − 4 PathOverSampling SC−FDMA − 4 PathConv. SC−FDMA − 6 PathOverSampling SC−FDMA − 6 PathConv. SC−FDMA − 8 PathOverSampling SC−FDMA − 8 Path
YunXiang LIU, Leo ([email protected]) May 14, 2013 26 / 34
.. One Path Channel with Different CFO
0 2 4 6 8 10 12 14 16 18 2010
−3
10−2
10−1
100
EbN0
BE
R
BER Performance of One Paths with different CFO
Conv. SC−FDMA − 1 PathConv SC−FDMA − 1 Path with CFO = 0.05Oversampling SC−FDMA 1 path with CFO = 0.05Conv. SC−FDMA − 1 Path with CFO = 0.1OverSampling SC−FDMA − 1 Path with CFO = 0.1Conv. SC−FDMA − 1 Path with CFO = 0,2OverSampling SC−FDMA − 1 Path with CFO = 0.2
YunXiang LIU, Leo ([email protected]) May 14, 2013 27 / 34
.. Outline
• Background
• Introduction to Oversampling MMSE Equalizer SC-FDMA System
• Simulation Results
• Conclusions
• Future Work
YunXiang LIU, Leo ([email protected]) May 14, 2013 28 / 34
.. Conclusion
• Uncoded SC-FDMA is always better than uncoded OFDM
• SC-FDMA has a remarkable improvement with Oversampling MMSEEqualizer, which extract more frequency diversity than conventionalSC-FDMA
• Oversampling MMSE Equalizer improves the performance with CFO
• Oversampling factor S = 1+(Q-1)/N could get the optimal performance
• Oversampling Gain gets from frequency diversity, which could compensatedeep fading.
YunXiang LIU, Leo ([email protected]) May 14, 2013 29 / 34
.. Conclusion
• Uncoded SC-FDMA is always better than uncoded OFDM
• SC-FDMA has a remarkable improvement with Oversampling MMSEEqualizer, which extract more frequency diversity than conventionalSC-FDMA
• Oversampling MMSE Equalizer improves the performance with CFO
• Oversampling factor S = 1+(Q-1)/N could get the optimal performance
• Oversampling Gain gets from frequency diversity, which could compensatedeep fading.
YunXiang LIU, Leo ([email protected]) May 14, 2013 29 / 34
.. Conclusion
• Uncoded SC-FDMA is always better than uncoded OFDM
• SC-FDMA has a remarkable improvement with Oversampling MMSEEqualizer, which extract more frequency diversity than conventionalSC-FDMA
• Oversampling MMSE Equalizer improves the performance with CFO
• Oversampling factor S = 1+(Q-1)/N could get the optimal performance
• Oversampling Gain gets from frequency diversity, which could compensatedeep fading.
YunXiang LIU, Leo ([email protected]) May 14, 2013 29 / 34
.. Conclusion
• Uncoded SC-FDMA is always better than uncoded OFDM
• SC-FDMA has a remarkable improvement with Oversampling MMSEEqualizer, which extract more frequency diversity than conventionalSC-FDMA
• Oversampling MMSE Equalizer improves the performance with CFO
• Oversampling factor S = 1+(Q-1)/N could get the optimal performance
• Oversampling Gain gets from frequency diversity, which could compensatedeep fading.
YunXiang LIU, Leo ([email protected]) May 14, 2013 29 / 34
.. Conclusion
• Uncoded SC-FDMA is always better than uncoded OFDM
• SC-FDMA has a remarkable improvement with Oversampling MMSEEqualizer, which extract more frequency diversity than conventionalSC-FDMA
• Oversampling MMSE Equalizer improves the performance with CFO
• Oversampling factor S = 1+(Q-1)/N could get the optimal performance
• Oversampling Gain gets from frequency diversity, which could compensatedeep fading.
YunXiang LIU, Leo ([email protected]) May 14, 2013 29 / 34
.. Outline
• Background
• Introduction to Oversampling MMSE Equalizer SC-FDMA System
• Simulation Results
• Conclusions
• Future Work
YunXiang LIU, Leo ([email protected]) May 14, 2013 30 / 34
.. Future Work
• USRP Implementation
−10 −5 0 5 10
−10
−8
−6
−4
−2
0
2
4
6
8
10
Qua
drat
ure
In−Phase
Scatter plot
−8 −6 −4 −2 0 2 4 6 8
−8
−6
−4
−2
0
2
4
6
8
Qua
drat
ure
In−Phase
Scatter plot
YunXiang LIU, Leo ([email protected]) May 14, 2013 31 / 34
.. Future Work
• USRP Implementation
−10 −5 0 5 10
−10
−8
−6
−4
−2
0
2
4
6
8
10
Qua
drat
ure
In−Phase
Scatter plot
−8 −6 −4 −2 0 2 4 6 8
−8
−6
−4
−2
0
2
4
6
8
Qua
drat
ure
In−Phase
Scatter plot
YunXiang LIU, Leo ([email protected]) May 14, 2013 31 / 34
.. Future Work
• USRP Implementation
−10 −5 0 5 10
−10
−8
−6
−4
−2
0
2
4
6
8
10
Qua
drat
ure
In−Phase
Scatter plot
−8 −6 −4 −2 0 2 4 6 8
−8
−6
−4
−2
0
2
4
6
8
Qua
drat
ure
In−Phase
Scatter plot
YunXiang LIU, Leo ([email protected]) May 14, 2013 31 / 34
.. Future Work
• Effect of Oversampling Factor
• Theorem Analysis of Oversampling MMSE Equalizer• Python Programming for Numerical Analysis
YunXiang LIU, Leo ([email protected]) May 14, 2013 32 / 34
.. Future Work
• Effect of Oversampling Factor• Theorem Analysis of Oversampling MMSE Equalizer
• Python Programming for Numerical Analysis
YunXiang LIU, Leo ([email protected]) May 14, 2013 32 / 34
.. Future Work
• Effect of Oversampling Factor• Theorem Analysis of Oversampling MMSE Equalizer• Python Programming for Numerical Analysis
YunXiang LIU, Leo ([email protected]) May 14, 2013 32 / 34
.. Future Work
• Effect of Oversampling Factor• Theorem Analysis of Oversampling MMSE Equalizer• Python Programming for Numerical Analysis
YunXiang LIU, Leo ([email protected]) May 14, 2013 32 / 34
.. Reference
• 1. Shi, Q., Liu, L., Member, S., and Guan, Y. L. (2010). Fractionally SpacedFrequency-Domain MMSE Receiver for OFDM Systems, 59(9), 4400-4407.
YunXiang LIU, Leo ([email protected]) May 14, 2013 33 / 34