Joint CFO and Channel Estimation for ZP-OFDM Modulated Two-Way Relay Networks
Gongpu Wang† , Feifei Gao‡ , Yik-Chung Wu∗ , and Chintha Tellambura† †
University of Alberta, Edmonton, Canada, ‡ Jacobs University, Bremen, Germany ∗ The University of Hong Kong, Hong Kong Email:
[email protected] WCNC’10
Outline ■
Introduction
■
Previous Results
■
Problem Formulation
■
Proposed Solution
■
Performance Analysis
■
Simulation Results
■
Conclusion
Outline Introduction Previous Results Problem Formulation Proposed Solution Performance Analysis Simulation Results Conclusion
2
Introduction ■
Outline Introduction
Two-way relay networks (TWRN) can enhance the overall communication rate [Boris Rankov, 2006], [J.Ponniah, 2008].
Previous Results Problem Formulation Proposed Solution Performance Analysis Simulation Results Conclusion
T1 f1
h1
- R fr
h2
T2 - f2
Figure 1: System configuration for two-way relay network.
3
Previous Results ■
Most existing works in TWRN assumed perfect synchronization and channel state information (CSI).
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Channel estimation problems in amplify-and-forward (AF) TWRN are different from those in traditional communication systems.
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Flat-fading and frequency-selective channel estimation and training design for AF TWRN has been done in [Feifei Gao, 2009].
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Our paper will focus on joint frequency offset (CFO) and channel estimation for AF-based OFDM-Modulated TWRN.
Outline Introduction Previous Results Problem Formulation Proposed Solution Performance Analysis Simulation Results Conclusion
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Joint CFO and Channel Estimation Problems in TWRN ■
With CFOs, the orthogonality between subcarriers will be destroyed in TWRN.
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Even with completed estimation, data detection is not simple as circular convolution no longer exists.
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How to estimate the mixed CFOs and channels and how to faciliate data detection?
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We introduce some redundancy and modify the OFDM TWRN system to facilitate both the joint estimation and detection.
Outline Introduction Previous Results Problem Formulation Proposed Solution Performance Analysis Simulation Results Conclusion
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Signals at Relay ■ Outline
The relay R will down-convert the passband signal by e−2πfr t and obtain
Introduction Previous Results
rzp =
Problem Formulation Proposed Solution
2 X
(N ) Γ(N +L) [fi − fr ]Hzp [hi ]si + nr ,
(1)
i=1
Performance Analysis Simulation Results Conclusion
where Γ(K) [f ] = diag{1, ej2πf Ts , . . . , ej2πf (K−1)Ts } and x0 . . . 0 .. .. .. si,0 . . . si,1 .. [x] , si = FH ˜si = . H(K) . x0 zp xP .. . .. .. . . si,N −1 . . 0 . . . xP {z } | K columns
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Next, R adds L zeros to the end of r and scales it by the factor of αzp to keep the average power constraint. 6
Signals at Terminal T1 ■
T1 will down-convert the passband signal by e−2πf1 t and get
Outline
(N +L) yzp =αzp Γ(N +2L) [fr − f1 ]Hzp [h1 ]rzp + n1
Introduction Previous Results
(N +L) =αzp Γ(N +2L) [fr − f1 ]Hzp [h1 ] 2 X (N ) × Γ(N +L) [fi − fr ]Hzp [hi ]si
Problem Formulation Proposed Solution Performance Analysis Simulation Results Conclusion
j=1
(N +L) + αzp Γ(N +2L) [fr − f1 ]Hzp [h1 ]nr + n1 {z } |
(2)
ne
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Next, using the following equalities h
i
(K) H(K) [f ] = Γ(K+P ) [f ]H(K) Γ(K) [−f ]x , zp [x] Γ zp
(3)
h i (P +1) Γ [f ]x Γ(K) [f ].
(4)
and Γ
(K+P )
[f ]H(K) zp [x]
=
H(K) zp
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Signals at Terminal T1 ■ Outline
yzp can be rewritten as (N +L) (L+1) (N ) yzp =αzp Hzp [Γ [fr − f1 ]h1 ]Hzp [h1 ] s1 + ne
Introduction Previous Results
(N +L) (L+1) (N ) + αzp Γ(N +2L) [f2 − f1 ]Hzp [Γ [fr − f2 ]h1 ] × Hzp [h2 ] (5)
Problem Formulation Proposed Solution Performance Analysis Simulation Results Conclusion
(N +L)
(N )
(N )
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We further note that Hzp [x1 ]Hzp [x2 ] = Hzp [x1 ⊗ x2 ] where ⊗ denotes the linear convolution.
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Hence yzp is finally written as h i (N ) (L+1) yzp =αzp Hzp (Γ [fr − f1 ]h1 ) ⊗ h1 s1 + ne | {z } azp
+ αzp Γ
(N +2L)
(N ) f1 ]Hzp
[f2 − | {z } v
h
(Γ |
(L+1)
i
[fr − f2 ]h1 ) ⊗ h2 s2 , (6) {z } bzp
where azp , bzp are the (2L + 1) × 1 equivalent channel vectors and v is the equivalent CFO.
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Joint CFO and Channel Estimation ■
We then obtain
Outline
y = S1 a + ΓS2 b + ne .
Introduction
(7)
Previous Results Problem Formulation Proposed Solution Performance Analysis
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Since S1 is a tall matrix, it is possible to find a matrix J such that JH S1 = 0.
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Left-multiplying y by JH gives
Simulation Results Conclusion
JH y = 0 + JH ΓS2 b + JH ne . | {z } | {z } G
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(8)
n
Joint CFO estimation and channel estimation vˆ = arg max yH JG(GH G)−1 GH JH y, v
ˆ =(GH G)−1 GH JH y, b ˆ ˆ 2 b). ˆ =(SH S1 )−1 SH (y − ΓS a 1
1
(9) (10) (11) 9
Performance Analysis ■ Outline
At high SNR, the perturbation of the estimated CFO can be approximated by
Introduction Previous Results
∆v , vˆ0 − v0 ≈ −
Problem Formulation Proposed Solution
g(v ˙ 0) , E{¨ g (v0 )}
(12)
Performance Analysis Simulation Results
where g(v) = yH JG(GH G)−1 GH JH y.
Conclusion
■
The NLS estimation of CFO is unbiased and its MSE is 2 σne . E{∆v } = H H −1 H H ˙ ˙ 2b G [I − G(G G) G ]Gb 2
■
(13)
ˆ is unbiased and its MSE is The channel estimation b H ˙ H ˙ MSE{b} = (GH G)−1 GH Gbb G G(GH G)−1 E{∆v 2 } 2 + σne (GH G)−1 .
(14)
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Simulation Results
−3
10
Outline
v numerical MSE N=16 v theoretical MSE N=16 v numerical MSE N=32 v theoretical MSE N=32
Introduction Previous Results Problem Formulation
−4
10
Proposed Solution Performance Analysis Simulation Results
−5
Conclusion
CFO MSE
10
−6
10
−7
10
−8
10
−9
10
0
5
10
15 SNR (dB)
20
25
30
Figure 2: Numerical and Theoretical MSEs of CFO versus SNR 11
Simulation Results
0
10
Outline
a numerical MSE N=16 a numerical MSE N=32 b numerical MSE N=16 b theoretical MSE N=16 b numerical MSE N=32 b theoretical MSE N=32
Introduction Previous Results Problem Formulation Proposed Solution Performance Analysis
−1
10
Conclusion
Channel Estimation MSE
Simulation Results
−2
10
−3
10
−4
10
0
5
10
15 SNR (dB)
20
25
30
Figure 3: Numerical and Theoretical MSEs of Channel Estimation versus SNR 12
Conclusion
Outline Introduction Previous Results Problem Formulation
1. Adapt ZP-based OFDM transmission scheme.
Proposed Solution Performance Analysis Simulation Results Conclusion
2. Suggest joint estimation method of CFO and channels. 3. Performance analysis: prove unbiasedness and give closed-form MSE expression.
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Conclusion
Outline Introduction Previous Results Problem Formulation
1. Adapt ZP-based OFDM transmission scheme.
Proposed Solution Performance Analysis Simulation Results Conclusion
2. Suggest joint estimation method of CFO and channels. 3. Performance analysis: prove unbiasedness and give closed-form MSE expression. Problem: How to obtain individual frequency and channel parameters? (Globecom 2010)
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Outline Introduction Previous Results Problem Formulation Proposed Solution Performance Analysis Simulation Results Conclusion
Questions and discussion? Email:
[email protected] 14