Joint Frequency Offset and Channel Estimation Methods for Two-Way ...
Joint Frequency Offset and Channel Estimation Methods for Two-Way Relay Networks Gongpu Wang† , Feifei Gao∗ and Chintha Tellambura† †
Department of Electrical and Computer Engineering, University of Alberta, Edmonton, Canada,
∗
School of Engineering and Science, Jacobs University, Bremen, Germany Email: {gongpu}@ece.ualberta.ca GLOBECOM’09
Outline ■
Introduction
■
Previous Results
■
Problem Formulation
■
Estimation Methods
■
Simulation Results
■
Conclusion
Outline Introduction Previous Results Problem Formulation Estimation Methods Simulation Results Conclusions
2
Outline ■
Introduction
■
Previous Results
■
Problem Formulation
■
Estimation Methods
■
Simulation Results
■
Conclusion
Outline Introduction Previous Results Problem Formulation Estimation Methods Simulation Results Conclusions
2
Outline ■
Introduction
■
Previous Results
■
Problem Formulation
■
Estimation Methods
■
Simulation Results
■
Conclusion
Outline Introduction Previous Results Problem Formulation Estimation Methods Simulation Results Conclusions
2
Outline ■
Introduction
■
Previous Results
■
Problem Formulation
■
Estimation Methods
■
Simulation Results
■
Conclusion
Outline Introduction Previous Results Problem Formulation Estimation Methods Simulation Results Conclusions
2
Outline ■
Introduction
■
Previous Results
■
Problem Formulation
■
Estimation Methods
■
Simulation Results
■
Conclusion
Outline Introduction Previous Results Problem Formulation Estimation Methods Simulation Results Conclusions
2
Outline ■
Introduction
■
Previous Results
■
Problem Formulation
■
Estimation Methods
■
Simulation Results
■
Conclusion
Outline Introduction Previous Results Problem Formulation Estimation Methods Simulation Results Conclusions
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 Estimation Methods Simulation Results Conclusions
Sm 1 delay:
˜ 1 (t) h τ1
-
Rm
˜ 2 (t) h -
Sm 2
τ2
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).
■
Channel estimation problems in amplify-and-forward (AF) TWRN are different from those in traditional communication systems.
■
Flat-fading channel estimation and training design for AF TWRN has been done in [Feifei Gao, 2009].
■
Our paper will focus on joint frequency offset (CFO) and channel estimation for AF TWRN.
Outline Introduction Previous Results Problem Formulation Estimation Methods Simulation Results Conclusions
4
Problem Formulation ¿1 Outline Introduction
ej2¼f1t+
1
R
ej2¼f0t+
0
S2
ej2¼f2t+ 2
S1
Previous Results
¿p
Problem Formulation Estimation Methods Simulation Results Conclusions
¿2
Figure 2: Illustration of channel delay and signal processing delay in twoway relay transmission.
5
Problem Formulation ■
The passband signal sent by Si is
Outline Introduction
s˜i (t) =
Previous Results Problem Formulation
+∞ X
si [m]p(t − mTs )ej2πfi t .
(1)
m=−∞
Estimation Methods Simulation Results Conclusions
■
The received signal in R is r˜r (t) =
2 X
˜ i (t) ∗ s˜i (t) + n h ˜ r (t)
(2)
i=1
■
The passband signal sent out by relay is then r˜s (t − τp ) = α
2 X
˜ i (t) ∗ s˜i (t − τp ) + α˜ h nr (t − τp )
(3)
i=1
6
Problem Formulation ■
The data received at S1 is ˜ 1 (t) ∗ r˜s (t − τp ) + n y˜(t) = h ˜ 1 (t), 2 X −j2πf t i ˜ 1 (t) ∗ h ˜ i (t))e =α ((h ) ∗ si (t − τp )
Outline Introduction Previous Results Problem Formulation Estimation Methods Simulation Results
i=1
Conclusions
˜ 1 (t) ∗ n × ej2πfi (t−τp ) +αh ˜ r (t − τp )+ n ˜ 1 (t). ■
(4)
Then S1 down-convert y˜(t) to baseband by e−j2πf1 t y(t) = y˜(t)e−j2πf1 t −j2πf t 1 ˜ 1 (t) ∗ h ˜ 1 (t))e = α (h ∗ s1 (t − τp ) ejφ1 −j2πf t 2 ˜ 1 (t) ∗ h ˜ 2 (t))e + α (h ∗ s2 (t − τp ) ej2πvt+jφ2 ˜ 1 (t) ∗ n +α h ˜ r (t − τp ) e−j2πf1 t + n ˜ 1 (t)e−j2πf1 t . (5)
7
Problem Formulation ■ Outline
The baseband signal is rewritten as y(t) =a(t) ∗ s1 (t − τp ) + b(t) ∗ s2 (t − τp )ej2πvt
Introduction Previous Results
+ h1 (t) ∗ (˜ nr (t − τp )e−j2πf1 t ) + n ˜ 1 (t)e−j2πf1 t .
Problem Formulation
(6)
Estimation Methods Simulation Results Conclusions
■
Following the traditional approach [M.Morelli 2000], we can obtain y = S1 a + ΓS2 b + H1 nr + n
(7)
where Γ = diag 1, ej2πvTs , ..., ej2πv(N −1)Ts , h1 [L] . . . h1 [0] . . . 0 .. . .. .. .. H1 = α ... . . . . 0
. . . h1 [L] . . . h1 [0]
8
Estimation Methods
Outline Introduction
■
Approximated ML Estimation
■
Nulling Based LS Method
Previous Results Problem Formulation Estimation Methods Simulation Results Conclusions
9
Approximated ML Estimation ■
Maximizing the probability density function (pdf) of y: 1 × p(y|Θ) = N π det(H1 HH + I) 1 H H −1 exp (y − S1 a − ΓS2 b) (H1 H1 + I) (y − S1 a − ΓS2 b)
Outline Introduction Previous Results Problem Formulation Estimation Methods Simulation Results Conclusions
■
When the number of the channel taps is large, H1 HH 1 can be approximated by L X 2 H1 HH σh,l I (8) 1 ≈ l=0
■
Then the ML estimation is b vb} = arg min ky − S1 a − ΓS2 bk2 . {b a, b, a,b,v
(9)
10
Approximated ML Estimation ■ Outline
Denote C = [S1 , ΓS2 ] and d = [aT , bT ]T . As long as N > 4L + 2, d can be estimated as
Introduction
b = (CH C)−1 CH y. d
Previous Results Problem Formulation Estimation Methods Simulation Results Conclusions
■
(10)
CFO estimation is
=
b 2 arg min ||y − Cd||
=
arg max g(v).
vb =
v
arg max yH C(CH C)−1 CH y v v
(11)
11
´ Cramer-Rao Bound of CFO Estimation of AML
Outline
F
Introduction
=
Previous Results Problem Formulation Estimation Methods Simulation Results
T
r K V
F11 r P 2 l σh,l + 1 s 2
T
s VT , N
(12)
where
Conclusions
2 F11 = bH SH D = 2πTs diag{0, 1, . . . , (N − 1)}, 2 D S2 b, H −ℑ(SH DΓS b) −ℑ(S DS b) 2 2 1 2 r= , s= , H H ℜ(S1 DΓS2 b) ℜ(S2 DS2 b) H H H ℜ(SH S ) −ℑ(S S ) ℜ(S S ) −ℑ(S S ) 1 1 2 2 1 1 2 2 K= , N= , H H H H ℑ(S1 S1 ) ℜ(S1 S1 ) ℑ(S2 S2 ) ℜ(S2 S2 ) H H H H ℜ(S2 Γ S1 ) −ℑ(S2 Γ S1 ) V= . H H H ℑ(SH Γ S ) ℜ(S Γ S ) 1 1 2 2
CRB1 (v) =
P
l
2 σh,l +1 −1 [F11 − tT1 Q−1 , 1 t1 ] 2
(13)
where r , t1 = s
Q1 =
T
K, V V, N
.
12
Nulling Based Method ■ Outline
JH y = 0 + JH ΓS2 b + JH (Hnr + n), | {z }
Introduction Previous Results
(14)
G
Problem Formulation Estimation Methods Simulation Results Conclusions
Left-multiply both sides of (7) with JH , we obtain:
■
The estimation of b can be immediately found as ˆ = (GH G)−1 GH y. b
■
CFO is estimated as vˆ = arg max yH JG(GH G)−1 GH JH y v
■
(15)
(16)
The least square (LS) estimation of channel a is obtained from −1 H ˆ ˆ 2 b), ˆ = (SH a S ) S1 (y − ΓS 1 1
(17)
ˆ = diag{1, ej2πˆvTs , ..., ej2πˆv(N −1)Ts }. where Γ 13
CRB of CFO Estimation of the nulling based method The FIM is calculated as: Outline
F =
Introduction Previous Results Problem Formulation Estimation Methods Simulation Results
where
′ 2 F11 P 2 t2 l σh,l + 1
tT2 Q2
,
(18)
Conclusions
′ F11
t2 Q2
H H = bH SH 2 DΓ JJ DΓS2 b, H H H −ℑ(S2 Γ JJ DΓS2 b) = , H H ℜ(SH Γ JJ DΓS b) 2 2 H H ℜ(G G) −ℑ(G G) = . ℑ(GH G) ℜ(GH G)
The CRB of frequency offset estimation is: P 2 l σh,l + 1 ′ −1 CRB2 (v) = [F11 − tT2 Q−1 . 2 t2 ] 2
(19)
14
Simulation Results
0
10 Outline
−1
Introduction
10
Previous Results Problem Formulation −2
Estimation Methods
10
Channel a−AML Channel a−nulling CFO−AML CFO−nulling CRB−AML CRB−nulling
Simulation Results Conclusions
−3
MSE
10
−4
10
−5
10
−6
10
−7
10
0
5
10
15 SNR (dB)
20
25
30
Figure 3: MSEs of CFO and channel estimation versus SNR for AML and nulling based method with orthonormal J
15
Simulation Results
−2
10
FO−AML FO−nulling J1
Outline Introduction Previous Results Problem Formulation
FO−nulling J
−3
2
10
Estimation Methods
FO−nulling J
3
Simulation Results
CFO MSE
Conclusions −4
10
−5
10
−6
10
0
5
10
15 SNR (dB)
20
25
30
Figure 4: MSEs of CFO estimation versus SNR for AML and nulling based method with random J
16
Simulation Results
1
10
Channel a−AML Channel a−nulling J
Outline
1
Introduction Previous Results
Channel a−nulling J
Problem Formulation
Channel a−nulling J
3 2
Estimation Methods 0
Simulation Results
CE MSE (a b)
Conclusions
Channel b−AML Channel b−nulling J
10
1
Channel b−nulling J
3
Channel b−nulling J2 −1
10
−2
10
0
5
10
15 SNR (dB)
20
25
30
Figure 5: MSEs of channel estimation versus SNR for AML and nulling based method with random J
17
Conclusions ■
Formulate the signal model for two-way relay networks with frequency synchronization errors in a frequency selective environment.
■
Develop two joint CFO and channel estimations methods, i.e., AML and nulling-based methods.
■
Find CRB of CFO for each method and compare performance of the two methods.
■
Here, relay only acts as a repeater. If relay down converts the received signal, the problem will become more complex and interesting.
■
Our future work – WCNC and ICC 2010.
Outline Introduction Previous Results Problem Formulation Estimation Methods Simulation Results Conclusions
18
Outline Introduction Previous Results Problem Formulation Estimation Methods Simulation Results Conclusions
Questions and discussion
19
Department of Electrical and Computer Engineering, University of Alberta, Edmonton, Canada,
∗
School of Engineering and Science, Jacobs University, Bremen, Germany Email: {gongpu}@ece.ualberta.ca GLOBECOM’09
Outline ■
Introduction
■
Previous Results
■
Problem Formulation
■
Estimation Methods
■
Simulation Results
■
Conclusion
Outline Introduction Previous Results Problem Formulation Estimation Methods Simulation Results Conclusions
2
Outline ■
Introduction
■
Previous Results
■
Problem Formulation
■
Estimation Methods
■
Simulation Results
■
Conclusion
Outline Introduction Previous Results Problem Formulation Estimation Methods Simulation Results Conclusions
2
Outline ■
Introduction
■
Previous Results
■
Problem Formulation
■
Estimation Methods
■
Simulation Results
■
Conclusion
Outline Introduction Previous Results Problem Formulation Estimation Methods Simulation Results Conclusions
2
Outline ■
Introduction
■
Previous Results
■
Problem Formulation
■
Estimation Methods
■
Simulation Results
■
Conclusion
Outline Introduction Previous Results Problem Formulation Estimation Methods Simulation Results Conclusions
2
Outline ■
Introduction
■
Previous Results
■
Problem Formulation
■
Estimation Methods
■
Simulation Results
■
Conclusion
Outline Introduction Previous Results Problem Formulation Estimation Methods Simulation Results Conclusions
2
Outline ■
Introduction
■
Previous Results
■
Problem Formulation
■
Estimation Methods
■
Simulation Results
■
Conclusion
Outline Introduction Previous Results Problem Formulation Estimation Methods Simulation Results Conclusions
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 Estimation Methods Simulation Results Conclusions
Sm 1 delay:
˜ 1 (t) h τ1
-
Rm
˜ 2 (t) h -
Sm 2
τ2
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).
■
Channel estimation problems in amplify-and-forward (AF) TWRN are different from those in traditional communication systems.
■
Flat-fading channel estimation and training design for AF TWRN has been done in [Feifei Gao, 2009].
■
Our paper will focus on joint frequency offset (CFO) and channel estimation for AF TWRN.
Outline Introduction Previous Results Problem Formulation Estimation Methods Simulation Results Conclusions
4
Problem Formulation ¿1 Outline Introduction
ej2¼f1t+
1
R
ej2¼f0t+
0
S2
ej2¼f2t+ 2
S1
Previous Results
¿p
Problem Formulation Estimation Methods Simulation Results Conclusions
¿2
Figure 2: Illustration of channel delay and signal processing delay in twoway relay transmission.
5
Problem Formulation ■
The passband signal sent by Si is
Outline Introduction
s˜i (t) =
Previous Results Problem Formulation
+∞ X
si [m]p(t − mTs )ej2πfi t .
(1)
m=−∞
Estimation Methods Simulation Results Conclusions
■
The received signal in R is r˜r (t) =
2 X
˜ i (t) ∗ s˜i (t) + n h ˜ r (t)
(2)
i=1
■
The passband signal sent out by relay is then r˜s (t − τp ) = α
2 X
˜ i (t) ∗ s˜i (t − τp ) + α˜ h nr (t − τp )
(3)
i=1
6
Problem Formulation ■
The data received at S1 is ˜ 1 (t) ∗ r˜s (t − τp ) + n y˜(t) = h ˜ 1 (t), 2 X −j2πf t i ˜ 1 (t) ∗ h ˜ i (t))e =α ((h ) ∗ si (t − τp )
Outline Introduction Previous Results Problem Formulation Estimation Methods Simulation Results
i=1
Conclusions
˜ 1 (t) ∗ n × ej2πfi (t−τp ) +αh ˜ r (t − τp )+ n ˜ 1 (t). ■
(4)
Then S1 down-convert y˜(t) to baseband by e−j2πf1 t y(t) = y˜(t)e−j2πf1 t −j2πf t 1 ˜ 1 (t) ∗ h ˜ 1 (t))e = α (h ∗ s1 (t − τp ) ejφ1 −j2πf t 2 ˜ 1 (t) ∗ h ˜ 2 (t))e + α (h ∗ s2 (t − τp ) ej2πvt+jφ2 ˜ 1 (t) ∗ n +α h ˜ r (t − τp ) e−j2πf1 t + n ˜ 1 (t)e−j2πf1 t . (5)
7
Problem Formulation ■ Outline
The baseband signal is rewritten as y(t) =a(t) ∗ s1 (t − τp ) + b(t) ∗ s2 (t − τp )ej2πvt
Introduction Previous Results
+ h1 (t) ∗ (˜ nr (t − τp )e−j2πf1 t ) + n ˜ 1 (t)e−j2πf1 t .
Problem Formulation
(6)
Estimation Methods Simulation Results Conclusions
■
Following the traditional approach [M.Morelli 2000], we can obtain y = S1 a + ΓS2 b + H1 nr + n
(7)
where Γ = diag 1, ej2πvTs , ..., ej2πv(N −1)Ts , h1 [L] . . . h1 [0] . . . 0 .. . .. .. .. H1 = α ... . . . . 0
. . . h1 [L] . . . h1 [0]
8
Estimation Methods
Outline Introduction
■
Approximated ML Estimation
■
Nulling Based LS Method
Previous Results Problem Formulation Estimation Methods Simulation Results Conclusions
9
Approximated ML Estimation ■
Maximizing the probability density function (pdf) of y: 1 × p(y|Θ) = N π det(H1 HH + I) 1 H H −1 exp (y − S1 a − ΓS2 b) (H1 H1 + I) (y − S1 a − ΓS2 b)
Outline Introduction Previous Results Problem Formulation Estimation Methods Simulation Results Conclusions
■
When the number of the channel taps is large, H1 HH 1 can be approximated by L X 2 H1 HH σh,l I (8) 1 ≈ l=0
■
Then the ML estimation is b vb} = arg min ky − S1 a − ΓS2 bk2 . {b a, b, a,b,v
(9)
10
Approximated ML Estimation ■ Outline
Denote C = [S1 , ΓS2 ] and d = [aT , bT ]T . As long as N > 4L + 2, d can be estimated as
Introduction
b = (CH C)−1 CH y. d
Previous Results Problem Formulation Estimation Methods Simulation Results Conclusions
■
(10)
CFO estimation is
=
b 2 arg min ||y − Cd||
=
arg max g(v).
vb =
v
arg max yH C(CH C)−1 CH y v v
(11)
11
´ Cramer-Rao Bound of CFO Estimation of AML
Outline
F
Introduction
=
Previous Results Problem Formulation Estimation Methods Simulation Results
T
r K V
F11 r P 2 l σh,l + 1 s 2
T
s VT , N
(12)
where
Conclusions
2 F11 = bH SH D = 2πTs diag{0, 1, . . . , (N − 1)}, 2 D S2 b, H −ℑ(SH DΓS b) −ℑ(S DS b) 2 2 1 2 r= , s= , H H ℜ(S1 DΓS2 b) ℜ(S2 DS2 b) H H H ℜ(SH S ) −ℑ(S S ) ℜ(S S ) −ℑ(S S ) 1 1 2 2 1 1 2 2 K= , N= , H H H H ℑ(S1 S1 ) ℜ(S1 S1 ) ℑ(S2 S2 ) ℜ(S2 S2 ) H H H H ℜ(S2 Γ S1 ) −ℑ(S2 Γ S1 ) V= . H H H ℑ(SH Γ S ) ℜ(S Γ S ) 1 1 2 2
CRB1 (v) =
P
l
2 σh,l +1 −1 [F11 − tT1 Q−1 , 1 t1 ] 2
(13)
where r , t1 = s
Q1 =
T
K, V V, N
.
12
Nulling Based Method ■ Outline
JH y = 0 + JH ΓS2 b + JH (Hnr + n), | {z }
Introduction Previous Results
(14)
G
Problem Formulation Estimation Methods Simulation Results Conclusions
Left-multiply both sides of (7) with JH , we obtain:
■
The estimation of b can be immediately found as ˆ = (GH G)−1 GH y. b
■
CFO is estimated as vˆ = arg max yH JG(GH G)−1 GH JH y v
■
(15)
(16)
The least square (LS) estimation of channel a is obtained from −1 H ˆ ˆ 2 b), ˆ = (SH a S ) S1 (y − ΓS 1 1
(17)
ˆ = diag{1, ej2πˆvTs , ..., ej2πˆv(N −1)Ts }. where Γ 13
CRB of CFO Estimation of the nulling based method The FIM is calculated as: Outline
F =
Introduction Previous Results Problem Formulation Estimation Methods Simulation Results
where
′ 2 F11 P 2 t2 l σh,l + 1
tT2 Q2
,
(18)
Conclusions
′ F11
t2 Q2
H H = bH SH 2 DΓ JJ DΓS2 b, H H H −ℑ(S2 Γ JJ DΓS2 b) = , H H ℜ(SH Γ JJ DΓS b) 2 2 H H ℜ(G G) −ℑ(G G) = . ℑ(GH G) ℜ(GH G)
The CRB of frequency offset estimation is: P 2 l σh,l + 1 ′ −1 CRB2 (v) = [F11 − tT2 Q−1 . 2 t2 ] 2
(19)
14
Simulation Results
0
10 Outline
−1
Introduction
10
Previous Results Problem Formulation −2
Estimation Methods
10
Channel a−AML Channel a−nulling CFO−AML CFO−nulling CRB−AML CRB−nulling
Simulation Results Conclusions
−3
MSE
10
−4
10
−5
10
−6
10
−7
10
0
5
10
15 SNR (dB)
20
25
30
Figure 3: MSEs of CFO and channel estimation versus SNR for AML and nulling based method with orthonormal J
15
Simulation Results
−2
10
FO−AML FO−nulling J1
Outline Introduction Previous Results Problem Formulation
FO−nulling J
−3
2
10
Estimation Methods
FO−nulling J
3
Simulation Results
CFO MSE
Conclusions −4
10
−5
10
−6
10
0
5
10
15 SNR (dB)
20
25
30
Figure 4: MSEs of CFO estimation versus SNR for AML and nulling based method with random J
16
Simulation Results
1
10
Channel a−AML Channel a−nulling J
Outline
1
Introduction Previous Results
Channel a−nulling J
Problem Formulation
Channel a−nulling J
3 2
Estimation Methods 0
Simulation Results
CE MSE (a b)
Conclusions
Channel b−AML Channel b−nulling J
10
1
Channel b−nulling J
3
Channel b−nulling J2 −1
10
−2
10
0
5
10
15 SNR (dB)
20
25
30
Figure 5: MSEs of channel estimation versus SNR for AML and nulling based method with random J
17
Conclusions ■
Formulate the signal model for two-way relay networks with frequency synchronization errors in a frequency selective environment.
■
Develop two joint CFO and channel estimations methods, i.e., AML and nulling-based methods.
■
Find CRB of CFO for each method and compare performance of the two methods.
■
Here, relay only acts as a repeater. If relay down converts the received signal, the problem will become more complex and interesting.
■
Our future work – WCNC and ICC 2010.
Outline Introduction Previous Results Problem Formulation Estimation Methods Simulation Results Conclusions
18
Outline Introduction Previous Results Problem Formulation Estimation Methods Simulation Results Conclusions
Questions and discussion
19
