Sum-rate maximization of two-way amplify-and-forward relay networks with imperfect channel state information

Abstract

Considering a two-way amplify-and-forward (AF) relay network and aiming to simultaneously maximize the two users' mutual information lower bounds in the presence of channel estimation errors, we study the Pareto-front of users' mutual information lower bounds. Based on the Pareto-front we investigate the optimal power allocation among the two users and the relay, as well as the optimal power allotment between training and data symbols that maximize the average sum-rate lower bound, and explore the variations of these optimal power allocation as the relay position changes. We also show that the mean square error (MSE) of channel estimation is minimized when training vectors transmitted by the two users are orthogonal.

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SUM-RATE MAXIMIZATION OF TWO-WAY AMPLIFY-AND-FORWARD RELAY
NETWORKS WITH IMPERFECT CHANNEL STATE INFORMATION
Yupeng Jia and Azadeh Vosoughi
Department of Electrical and Computer Engineering, University of Rochester, NY14627, USA
ABSTRACT
Consideringatwo-wayamplify-and-forward (AF)relaynetworkand
aiming to simultaneously maximize the two users’ mutual informa-
tion lower bounds in the presence of channel estimation errors, we
study the Pareto-front of users’ mutual information lower bounds.
Based on the Pareto-front we investigate the optimal power alloca-
tion among the two users and the relay, as well as the optimal power
allotment between training and data symbols that maximize the av-
erage sum-rate lower bound, and explore the variations of these op-
timal power allocation as the relay position changes. We also show
thatthemean square error (MSE)of channel estimationisminimized
when training vectors transmitted by the two users are orthogonal.
Index Terms— Two-way relay channel, channel estimation,
bi-objective optimization, Pareto-front, power allocation, sum-rate
lower bound maximization, training design
1. INTRODUCTION
Two-way relay networks have gained a great attention due to its
ability to reduce the spectral loss caused by half-duplex relay-
ing [1]. Current literature have studied several problems concern-
ing with two-way amplify-and-froward (AF) [1] and decode-and-
forward (DF) relaying, including sum-rate maximization [1] [2] [3],
power consumption minimization and network lifetime maximiza-
tion [4], and beamforming design for multi-antenna two-way relay-
ing [5] [6] [7]. These works have mainly assumed perfect channel
state information (CSI) at relay and users’ terminals.
The literature on two-way relaying system with imperfect CSI
is sparse. We note that the channel estimation problem and the neg-
ative impact of channel estimation errors on the information rate in
two-way relaying systems are different from those of point-to-point
and one-way relaying communications, since in the former systems
channel estimation errors lead to self-interference and thus further
reduce information rate. For DF two-way relaying systems [8] pro-
poses an iterative directed self-interference aided channel estimation
method, that does not require training symbols transmission. The
advantage of this method, however, is offset by the increasing com-
plexity required for decoding and iterative self-interference suppres-
sion. In [9], the authors propose optimal training sequence design
for AF two-way relaying via minimizing the channel mean square
error (MSE). [10] considers the optimal training sequence design for
AF two-way relaying and analyzes the received signal-to-noise ratio
(SNR) in the presence of imperfect CSI. In our previous work [11],
we considered the effect of channel estimation errors on the sum-
rate in an AF two-relaying network, where the relay appends an ex-
tra training symbol to the transmit packet, to facilitate the channel
estimation at the terminals.
Considering a different and more bandwidth efficient training
transmission strategy from [11], we study the simultaneous max-
Fig. 1. System model of two-way AF relaying system
imization of the two users’ mutual information lower bounds in
the presence of channel estimation errors and find the Pareto-front.
Based on the Pareto-front we investigate the optimal power alloca-
tion among the three nodes, as well as the optimal power allotment
between training and data symbols that maximize the average sum-
rate lower bound, and explore the variations of these optimal power
allocation as the relay position changes.
2. SYSTEM MODEL
We consider a two-way relay network, consisting of terminals TA
and TBand a half duplex relay node R, where each node isequipped
with a single antenna (Fig.1). The complex baseband channel be-
tween TA and R is denoted as ha and the one between TB and R
is denoted as hb. To capture the effect of pathloss and Rayleigh flat
fading we assume ha ∼ CN(0,σ2
variances are σ2
notes the distance between Tiand R, ε is the pathloss exponent, and
G is a constant that depends on antenna gains and the wavelength.
Although ε and G may vary for each channel link, throughout this
paper it is assumed that ε and G are identical for all links. We as-
sume block fading model, in which ha, hb are constant within one
transmission block, after which they change to different independent
values that hold for another block.
Let si = [si1 si2...siN] be the block transmitted by Ti for i ∈
{A,B}. Also, let Ntand Nddenote the number of training and data
symbols, respectively, in a transmission block of N symbols. We
assume that each transmission block is augmented as si = [st
in which st
the 1×Ntand 1×Ndrow vectors including training and data sym-
bols, respectively, and sd
Suppose sd
Bare independent. We note that Gaussian input
has been adopted before [12] [13] to study the effect of channel es-
timation on capacity, although it may not be the optimal distribution
in terms of maximizing mutual information given imperfect CSI.
We consider a transmission protocol consisting of two phases
(Fig. 1). During phase I, TAand TB transmit, respectively, vectors
sA and sB to R and R receive yR = hasA+ hbsB + nR. During
phase II, R amplifies yR and forwards x = fyR to TA and TB,
where f is the amplifying factor. Assuming channel reciprocity for
haand hbterminals TAand TB receive
ha) and hb ∼ CN(0,σ2
A,Rand σ2
hb). The
ha= G/dε
hb= G/dε
B,R, where di,R de-
i,sd
i]
i= [si1 si2 ... siNt] and sd
i= [si1 si2 ... siNd] denote
i∼ CN(0,ρd
iINd) with ρd
idefined in (1).
Aand sd
yA
=
=
hax + nA = fgsB+ fhAsA+ fhanR+ nA
hbx + nB = fgsA+ fhBsB+ fhbnR+ nB
yB
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in which g = hahb is the overall complex relay channel between
TA and TB, hA = h2
nj ∼ CN(0,σ2
ing channels are mutually independent. To keep the complexity
of training assisted AF two-way relay system low, we assume that
only the terminals are equipped with channel estimator. Terminal
TA (TB) first uses received training symbols to estimate the un-
known channels hA,g (hB,g). It uses these channel estimates to
suppress self-interference, caused by back-propagation, and then re-
covers the data symbols sd
A). Due to channel estimation errors
self-interference at the terminals cannot be completely canceled and
the residual self-interference reduces the rate, compared to the case
when self-interference is perfectly canceled.
Let PA,PB,PRbe the average power constraints per transmis-
sion block at TA,TB, and TR. Suppose the total transmit power of
the system P = PA + PB + PR is fixed. Let ρt
{A,B} denote the average transmit power per training symbol, per
data symbol, and per symbol in a transmission block, and Pt
be the training power and the average data power per transmission
block. Conservation of time and energy yields N = Nt+ Ndand
Pt
Pt
ii
α, 0 < α < 1 be the fraction of the total transmission power that Ti
assigns to its Nddata symbols. We have
aand hB = h2
njIN) for j ∈ {R,A,B}. The noises and fad-
b. The noise vectors are
B(sd
i, ρd
i, ρi for i ∈
i, Pd
i
i+ Pd
i = ρt
i = Pi for i ∈ {A,B} where Pi = ρiN = E{sisH
iNt = st
= ρd
i},
ist
H, and Pd
iNd = E{sd
isd
i
H}. Let
Pd
ρi =Pi
i = αPi
Pt
i= (1 − α)Pi
αPi
N − Nt
(1)
N
ρd
i=ρt
i=(1 − α)Pi
Nt
Considering the average power constraint PR = E{xxH} =
f2(PAσ2
ha+ PBσ2
hb+ σ2
nRN) at R we find
f =
?
PR
PAσ2
ha+ PBσ2
hb+ σ2
nRN
We assume that R does not estimate the channels, hence it can-
not utilize ha,hbin choosing f. Given the knowledge of the chan-
nel variances, however, R can incorporate σ2
f. The received signal vectors at TA and TB can be partitioned as
yA = [yt
are the received training and the received data row vectors
haand σ2
hbin designing
A, yd
A] and yB = [yt
B, yd
B], where yt
i, yd
i, i ∈ {A,B}
yt
yd
yt
yd
A
=
=
=
=
fqASt+ fhant
fqASd+ fhand
fqBSt+ fhbnt
fqBSd+ fhbnd
R+ nt
R+ nd
R+ nt
R+ nd
A
AA
BB
BB
where we define the 1 × 2 effective channel vectors qi = [hi g],
the 2 × Nt training matrix St= [st
matrix Sd= [sd
terms during training and data transmission, and are mutually inde-
pendent. TerminalTiusesyt
itoestimatetheeffectivechannel vector
qi and suppress self-interference caused by the term fhisd
the estimate ˆ qi and the self-interference suppressed received vector
zd
A; st
A,nt
B] and the 2 × Nd data
Band nd
A; sd
B]. The vectors nt
A,nd
Bare noise
i. Using
i= yd
i−fˆhisd
i, terminal Tirecovers sd
jfor i,j ∈ {A,B},i = j.
3. CHANNEL ESTIMATION
We model the channel vectors at terminals Tias [14]
qi = ˆ qi+ ˜ qi, ˆ qi = [ˆhi ˆ gi], ˜ qi = [˜hi ˜ gi], i ∈ {A,B}
(2)
where ˆ qiistheestimation vector and ˜ qiisthezero mean error vector
with covariance matrix R˜ qi= E{˜ qH
be the channel covariance matrix, where RqAis a diagonal ma-
trix with diagonal elements of σ2
LMMSE estimate ˆ qAis [14]
i˜ qi}. Let RqA= E{qH
AqA}
hAand σ2
g. Given yA
t at TA, the
ˆ qA
=
yt
AE{yt
?
A
Hyt
A}−1E{yt
A
HqA}
(3)
=
yt
A
StHRqASt+ (f2σ2
haσ2
nR+ σ2
nA)INt
?−1StHRqA
and the estimation error covariance matrix is [14]
R˜ qA=
?
R−1
qA+
f2StStH
haσ2
(f2σ2
nR+ σ2
nA)Nt
?−1
(4)
We can find ˆ qB and R˜ qBby substituting yt
and (4) with yt
the training matrix Stwe wish to design Stsuch that tr(R˜ qA) is
minimized. Schwartz’s inequality states that for a positive definite
matrix A we have [A−1]ii ≥
Aiiin which the equality holds when
A is diagonal [13]. Noting that R−1
tr(RqA) is minimized if and only if StStHis diagonal. However
A,RqA,σ2
ha,σ2
nAin (3)
B,RqB,σ2
hb,σ2
nB. Under the power constraint on
1
qAis diagonal, we realize that
StStH=
?
st
Ast
ρ∗
A
H
ρ
st
Bst
B
H
?
=
?
Pt
ρ∗
A
ρ
Pt
B
?
implying that orthogonal training vectors for which ρ = 0 minimize
tr(R˜ qA). When ρ = 0, R˜ qAis diagonal and tr(R˜ qA) attains its
minimum value of
tr(R˜ qA)=σ2
˜hA+ σ2
˜ gA= σ2
hA
?
1 +
f2ρt
Aσ2
nR+ σ2
hA
f2σ2
haσ2
?−1
nA
?−1
+σ2
g
?
1 +
f2ρt
Bσ2
nR+ σ2
g
f2σ2
haσ2
nA
Therefore, we conclude that the design that training vectors sent
from TAand TBare orthogonal is the optimal, in the sense of mini-
mizing MSE.
4. MUTUAL INFORMATION LOWER BOUND
In this section, we obtain the mutual information lower bound at
each terminal given imperfect CSI. Substituting (2) into yd
and subtracting self-interference at each terminal, we obtain
Aand yd
B
zd
A= yd
A− fˆhAsd
A= fˆ gAsd
B+ f˜ gAsd
?
?
B+ f˜hAsd
A+ fhand
??
??
R+ nd
A
?
B
wd
A
zd
B= yd
B− fˆhBsd
B= fˆ gBsd
A+ f˜ gBsd
A+ f˜hBsd
B+ fhbnd
R+ nd
?
wd
B
,
where wd
trices Rwd
σ2
wd
iare zero mean noise vectors with diagonal covariance ma-
i= E{wd
i
wd
Bare given below
iwd
H} = σ2
iINdand the variances σ2
wd
Aand
σ2
wd
A
=f2σ2
˜ gAρd
B+ f2σ2
˜hAρd
A+ f2σ2
haσ2
nR+ σ2
nA
σ2
wd
B
=f2σ2
˜ gBρd
A+ f2σ2
˜hBρd
B+ f2σ2
hbσ2
nR+ σ2
nB
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ρA
eff=f2ρd
Aσ2
σ2
wd
ˆ gA
A
=
PRPd
Bσ2
ˆ gA
PRPd
Bσ2
˜ gA+ PRPd
Aσ2
˜hA+ PRσ2
haσ2
nRNd+ PAσ2
haσ2
nANd+ PBσ2
hbσ2
nANd+ σ2
nAσ2
nRNd
(7)
ρB
eff=f2ρd
Bσ2
σ2
wd
ˆ gB
B
=
PRPd
Aσ2
ˆ gB
PRPd
Aσ2
˜ gB+ PRPd
Bσ2
˜hB+ PRσ2
hbσ2
nRNd+ PAσ2
haσ2
nBNd+ PBσ2
hbσ2
nBNd+ σ2
nBσ2
nRNd
(8)
Note that due to channel estimation error perfect self-interference
cancelation is not feasible. One can show that for LMMSE chan-
nel estimates where E{˜ qA|ˆ qA} = 0 [14], the additive noises wd
and wd
Bare conditionally uncorrelated.
E{sd
BA
A
Furthermore, we have
Hwd
A|ˆ qA} = 0 and E{sd
Hwd
B|ˆ qB} = 0, since
E{sd
B
Hwd
A|ˆ qA} = fE{sd
B
Hsd
B|ˆ qA}E{˜ gA|ˆ qA}
+fE{sd
B
Hsd
A|ˆ qA}E{˜hA|ˆ qA} + E{sd
B
H(fhand
R+ nd
A)|ˆ qA} = 0
Given ˆ qj at terminal Tj, we denote the conditional mutual informa-
tionbetween data vector sd
itransmitted by terminal Tiand the corre-
sponding received data vector zd
jat terminal Tj as I(sd
i;zd
j|ˆ qj) for
i,j ∈ {A,B} and j = i. The quantity I(sd
i,zd
j|ˆ qj) is the mutual
information of a known channel system ˆ qj, subject to the additive
noise vector wd
tual information lower bounds, given the LMMSE channel estimates
at the terminals.
Lemma 1: The conditional mutual information per transmis-
sion block at terminals TA and TB can be lower bounded as
I(sd
jthat is uncorrelated with sd
i. Lemma 1 provides mu-
B;zd
A|ˆ qA)≥IA
lowerand I(sd
A;zd
B|ˆ qB)≥IB
lowerwith
IA
lower=Nd
2NE¯ gA{log(1 + ρA
lower=Nd
2NE¯ gB{log(1 + ρB
eff|¯ gA|2)}
(5)
IB
eff|¯ gB|2)}
(6)
where ¯ gA =
ˆ gA
σˆ gA, ¯ gB =
effare shown in (7) (8) and the variances of the LMMSE
channel estimates can be found using orthogonality principle [14]
σ2
ˆ gB
σˆ gBare the normalized channel estimates,
ρA
effand ρB
ˆhA=σ2
Proof of Lemma 1 is omitted due to space limitation, we refer
interested readers to [11] [12] [15].
hA− σ2
˜hAand σ2
ˆhB=σ2
hB− σ2
˜hB.
5. PARETO-FRONT AND SUM-RATE MAXIMIZATION
We aim to maximize the mutual information lower bounds simul-
taneously under total transmit power constraint. This task can be
accomplished via scalarizing the bi-objective optimization problem
[16] and finding the Pareto-front. Using the Pareto-front we can
solve the average sum-rate lower bound maximization problem and
find the effect of channel estimation errors upon the solution. The
optimization problem is
max
(α,PA,PB,PR)
{IA
lower,IB
lower}
(9)
subject to
PA+ PB+ PR ≤ P, α ∈ (0,1)
We aim to find the optimal trade-off surface (Pareto-front). To
simplifytheoptimization, werelaxtheconstraints andlet αtobeany
given constant between (0,1). We can rewrite (9) as the following
constrained weighted-sum maximization problem [16]
max
(PA,PB,PR)
subject to
λ1IA
lower+ λ2IB
lower
(10)
PA+ PB+ PR ≤ P
λ1+ λ2 = 1, α = constant ∈ (0,1)
Fig. 2. Geometric view of Pareto-front and average sum-rate lower bound
maximization
00.511.5
0
0.5
1
1.5
Mutual information lower bound at TA (bps/Hz)
Fig. 3. Pareto-front with various η and fixed 1 − α = 0.2
Mutual information lower bound at TB (bps/Hz)


η=0.3, perfect CSI
η=0.5, perfect CSI
η=0.7, perfect CSI
η=0.3, imperfect CSI
η=0.5, imperfect CSI
η=0.7, imperfect CSI
PA
*=0.25P, PB
*=0.25P, PR
*=0.5P
PA
*=0.14P, PB
*=0.31P, PR
*=0.55P
PA
*=0.31P, PB
*=0.14P, PR
*=0.55P
Given α and P the Pareto-front can be obtained by taking the union
corresponding to different pairs of (λ1,λ2). Due to mathematical
intractability, we use numerical methods to obtain the Pareto-front.
To obtain the maximum average sum-rate lower bound at each
relay location and given α, we can examine the Pareto-front curve
from a geometric point of view. The point on this curve that maxi-
mizes the average sum-rate lower bound is the point that the straight
line of IA
lower= constant just touches the curve, as it is
shown in Fig.2. The maximum average sum-rate lower bound is the
summation of the coordinates of this point.
lower+ IB
6. NUMERICAL RESULTS
In this section, we present numerical results to evaluate the effects
of channel estimation errors upon Pareto-front and average sum-rate
lower bound maximization. We consider a linear network model and
define dA,R/dA,B = η and dR,B/dA,B = 1−η, where dA,Bis the
distance between TA and TB. The noise variances σ2
σ2
The block length N = 20 and the training length Nt = 2 [15].
The pathloss exponent ε = 3 and G = 1. We run 100 channel
realizations to measure the average performance.
Fig. 3 shows the Pareto-front with three different relay locations
η = 0.3, 0.5, 0.7 and fixed 1 − α = 0.2. The Pareto-front for two-
way relaying system with perfect CSI is plotted to serve as a bench-
nA= σ2
nB=
nR= 1 and the system SNR is defined as SNR = P = 15dB.
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00.20.40.60.81 1.21.41.6
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Mutual Information Lower Bound at TA (bps/Hz)
Fig. 4. Pareto-front with fixed η = 0.5 and various 1 − α
Mutual Information Lower Bound at TB (bps/Hz)


1−α=0.1
1−α=0.2
1−α=0.3
1−α=0.4
Perfect CSI
0.6
0.5
0.4
03
0.2
0.1
0
0.2
0.4
0.6
0.8
1
0
0.5
1
1.5
2

η
1−α

Maximum Sum−Rate (bps/Hz)
Fig. 5. Maximum average sum-rate lower bound versus 1−α and η
mark. The optimal triplet (P∗
each relay location. The symmetric characteristics of two-way re-
lay networks are captured by the relationship between η = 0.3 and
η = 0.7. In Fig. 4, we fix η = 0.5 and vary 1 − α and note that the
optimal 1 − α∗is between 0.2 and 0.3. Fig. 5 demonstrates the 3-D
display of maximum average sum-rate lower bound versus 1−α and
η. From this plot, we conclude that the optimal relay position is at
the midpoint between TA and TB and the optimal training fraction
(1 − α)∗≈ 0.2.
A,P∗
B,P∗
R) is marked on the curves for
7. CONCLUSIONS
Considering a two-way AF relay network and aiming to simulta-
neously maximize the two users’ mutual information lower bounds
in the presence of channel estimation errors, we studied the Pareto-
front of users’ mutual information lower bounds.
Pareto-front we investigated the optimal power allocation among
the two users and the relay, as well as the optimal power allotment
between training and data symbols that maximize the average sum-
rate lower bound. Our simulations show that the optimal training
power fraction is about 0.2 and this value stays constant as the relay
position changes, while the optimal relay location is at the midpoint
between TAand TB.
Based on the
8. REFERENCES
[1] B. Rankov and A. Wittneben, “Spectral efficient protocols for
half-duplex fading relay channels,” IEEE J. Sel. Areas Com-
mun., vol. 25, no. 2, pp. 379–389, 2007.
[2] S. Shahbazpanahi and M. Dong,
and sum-rate maximization for network beamforming for bi-
directional relay networks,” in Proc. IEEE Int. Conf. Acoust.
Speech and Signal Process., 2010, pp. 2510–2513.
“Achievable rate region
[3] H.Q. Ngo, T.Q.S. Quek, and H. Shin, “Amplify-and-Forward
Two-Way Relay Networks: Error Exponents and Resource Al-
location,” IEEE Trans. Commun., vol. 58, no. 9, pp. 2653–
2666, 2010.
[4] Y. Jia and A. Vosoughi, “Two-way relaying for energy con-
strained systems: Joint transmit power optimization,” in Proc.
IEEE Int. Conf. Acoust. Speech and Signal Process., 2010, pp.
3026–3029.
[5] V. Havary-Nassab, S. Shahbazpanahi, and A. Grami, “Optimal
distributed beamforming for two-way relay networks,” IEEE
Trans. Signal Process., vol. 58, no. 3, pp. 1238–1250, 2010.
[6] M. Zeng, R. Zhang, and S. Cui, “On Design of Distributed
Beamforming for Two-Way Relay Networks,” Arxiv preprint
arXiv:1005.1711, 2010.
[7] R. Zhang, Y.C. Liang, C.C. Chai, and S. Cui, “Optimal beam-
forming for two-way multi-antenna relay channel with ana-
logue network coding,” IEEE J. Sel. Areas Commun., vol. 27,
no. 5, pp. 699–712, 2009.
[8] TH Pham, YC Liang, A. Nallanathan, and GH Krishna, “On
the design of optimal training sequence for bi-directional relay
networks,” IEEE Signal Process. Lett., vol. 16, pp. 200–203,
2009.
[9] J. Zhao, M. Kuhn, A. Wittneben, and G. Bauch,
interference aided channel estimation in two-way relaying sys-
tems,” in IEEE Global Tele. Conf., 2008, pp. 1–6.
“Self-
[10] F.Gao, R.Zhang, andY.C.Liang, “Optimalchannel estimation
and training design for two-way relay networks,” IEEE Trans.
Commun., vol. 57, no. 10, pp. 3024–3033, 2009.
[11] Y. Jia and A. Vosoughi, “Impact of channel estimation error
upon sum-rate in amplify-and-forward two-way relaying sys-
tems,” in Proc. IEEE Int. Workshop Signal Process. Advances
in Wireless Commun., 2010, pp. 1–5.
[12] B. Hassibi and B.M. Hochwald, “How much training is needed
in multiple-antenna wireless links?,” IEEE Trans. Inf. Theory,
vol. 49, no. 4, pp. 951–963, 2003.
[13] A. Vosoughi and A. Scaglione, “On the effect of receiver esti-
mation error upon channel mutual information,” IEEE Trans.
Signal Process., vol. 54, no. 2, pp. 459–472, 2006.
[14] T. Kailath, A.H. Sayed, and B. Hassibi, Linear estimation,
Prentice Hall Upper Saddle River, NJ, 2000.
[15] Y. Jia and A. Vosoughi, “How does channel estimation affect
average sum-rate in two-way amply-and-forward relay net-
works?,” IEEE Trans. Wireless Commun., submitted for publi-
cation, Dec.2010.
[16] S.P. Boyd and L. Vandenberghe, Convex optimization, Cam-
bridge Univ Pr, 2004.
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