The Tenth International Symposium on Wireless Communication Systems 2013
Robust MIMO Relay Precoder Design for Multiple
Operator One-Way Relaying with Imperfect Channel
State Information
Jianhui Li and Martin Haardt
Communications Research Laboratory, Ilmenau University of Technology
P.O. Box 100565, D-98684 Ilmenau, Germany, http://tu-ilmenau.de/crl
Email: {jianhui.li, martin.haardt}@tu-ilmenau.de
Abstract—In this paper, we propose a robust relay precoder in
the multiple operator relay sharing system, where multiple base
stations transmit to their target user terminals simultaneously
with the assistance of an amplify-and-forward relay. The base
stations and the user terminals are equipped with single antennas
while the relay employs multiple antennas. The channel state
information at the relay is assumed to be imperfect, where the
additive channel state information error is modeled as Gaussian
distributed. Based on this model, we propose a robust relay
precoder design that minimizes the total average relay transmit
power under a signal-to-interference-plus-noise ratio constraint
at each user terminal. Simulation results show that the proposed
robust method outperforms the non-robust solution significantly
in terms of the outage probability of the signal to interference
plus noise ratio. Furthermore, we demonstrate the sharing gain
achieved by this robust method via spectrum and relay sharing
between multiple operators compared to the case where the
spectrum and the relay are accessed exclusively by each operator.
I.
I NTRODUCTION
The use of relays has drawn enormous attention due to its
promising capability in achieving reliable communications and
coverage extension in wireless networks [1]. Recently, it has
attracted more and more interest of the research community
to employ a relay in multi-point to multi-point transmission.
In this work, an amplify-and-forward (AF) relay is employed
to assist the transmission between multiple pairs of base
stations (BSs) and user terminals (UTs) belonging to different
operators, namely a multiple operator AF relay sharing system
as depicted in Fig. 1. The direct link between the source and
the destination is neglected due to the assumption of a large
path loss.
In this paper, we study the relay precoder to achieve
power efficient transmission, where imperfect channel state
information (CSI) is assumed at the relay. In the state-of-the-art
work [2], the authors design a relay precoder to minimize the
relay transmit power in the worst case subject to the worst case
signal-to-interference-plus-noise ratio (SINR). The worst case
of the relay transmit power and SINR refer to the maximum
relay power and the minimum SINR for the largest possible
channel errors, respectively. In contrast to that reference, our
work is focused on a novel robust relay precoder design in
order to minimize the average relay transmit power under the
SINR constraints for each operator. Monte-Carlo simulation
results show that the proposed robust method outperforms
.
Fig. 1.
.
System Model for relay sharing between multiple operators
the non-robust solution significantly in terms of the outage
probability of the SINR. Furthermore, we demonstrate the
sharing gain achieved by this robust method via spectrum and
relay sharing between multiple operators compared to the case
where the spectrum and the relay are accessed exclusively by
each operator.
II.
S YSTEM M ODEL
A multi-user amplify-and-forward (AF) relay sharing system is considered as shown in Fig. 1, where K base stations
(BSs) transmit data to their respective target user terminals
(UTs) with the assistance of a shared AF relay which operates
in half-duplex mode. Each BS and UT is equipped with a
single antenna and the relay has MR antennas. The direct links
between BSs and UTs are not used since it is assumed that
they are weak due to a large path loss.
The transmission process consists of two phases. In the
multiple access (MAC) phase, both BSs transmit to the relay.
In the broadcasting (BC) phase, the relay amplifies the received
signal from the MAC phase and forwards it to the UTs. The
received signals received at UTs are expressed as
yk =
gkT FR hk sk
+
K
gkT FR hj sj
j=1,j=k
+ gkT FR nR + nk ,
where hk ∈ CMR denotes the channel between each BS and
the relay gk ∈ CMR denotes the channel between the relay and
each UT. The relay amplification matrix is FR ∈ CMR ×MR .
The transmit signal at each BS is sk and the transmit power
ISBN 978-3-8007-3529-7
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522
The Tenth International Symposium on Wireless Communication Systems 2013
at each BS is constrained by PT , i.e., E{|sk |}2 ≤ PT . The
first term denotes the desired signal while the second term
stands for the interference that needs to be mitigated. All the
left terms are the effective noise. The noise at the relay nR
and that at UTs nk for k = 1, . . . , K contain independent,
identically distributed complex additive white Gaussian noise
samples.
III.
PR
As stated in [3], the channel uncertainties are modeled as
= ĥk + ek ,
= ĝkT + fkT ,
k = 1, 2, . . . , K.
(1)
H =
G =
Ĥ + E,
Ĝ + F ,
H
E{EE } =
E{F F H } =
2
KσE
IMR ,
MR σF2 IK .
=
PT tr(FR Ĥ Ĥ H FRH ) + PT E tr(FR EE H FRH )
=
+σn2 tr(FR FRH )
2
+ σn2 )tr(FR FRH )
PT tr(FR Ĥ Ĥ H FRH ) + (PT KσE
=
PT tr (Ĥ T ⊗ IMR ) vec(FR ) vec(FR )H (Ĥ T ⊗ IMR )H
P
+
f
R
σn2 )tr
vec(FR )vec(FR )H
2
PT tr(fRH P H P fR ) + (PT KσE
+ σn2 )tr(fRH fR )
=
2
2
fRH PT P H P + (PT KσE
+ σn2 )IM
fR .
R
(5)
gkT FR hj sj + gkT FR nR + nk ,
(6)
j=k
where the first term is the desired signal and the second term
represents the inter-operator interference caused to user k. The
effective noise is given by the remaining terms on the right
hand side. In the following, we express the SINR of UTk as
a function of fR .
The power of the desired signal in (6) is calculated as
E
T
2 gk FR hk sk =
H ∗
PT E tr(gkT FR hk hH
k FR gk )
=
H
H
PT E tr (ĝkT + fkT )FR (ĥk + ek )(ĥH
k + ek )FR
(ĝk∗ + fk∗ )
.
(7)
Due to the assumption that ĝkT and fˆk are uncorrelated with
fˆkT and êk , we could further write (7) as follows
H
H
PT E tr (ĝkT + fkT )FR (ĥk + ek )(ĥH
k + ek )FR
(ĝk∗ + fk∗ )
H ∗
2 T
H ∗
= PT tr(ĝkT FR ĥk ĥH
k FR ĝk + σE ĝk FR FR ĝk
H
2 2
H
+σF2 FR ĥk ĥH
k FR + σE σF FR FR )
(3)
T
T
T H
= PT tr (ĥT
k ⊗ ĝk ) vec(FR ) vec(FR )(ĥk ⊗ ĝk )
aT
2
+σE
(IMR
⊗
fR
T
ĝk )vec(FR )vec(FR )H (IMR
⊗ ĝkT )H
B
The relay transmit power is expressed as
H
T
H
+σF2 (ĥT
k ⊗ IMR )vec(FR )vec(FR ) (ĥk ⊗ IMR )
C
2 2
+σE
σF vec(FR )vec(FR )H
= PT E tr(FR HH H FRH ) + σn2 tr(FR FRH ). (4)
2
2 2
= PT fRH (a∗ aT+σE
B H B+σF2 C H C+σE
σF IMR2 )fR . (8)
ISBN 978-3-8007-3529-7
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523
=
yk = gkT FR hk sk +
Furthermore, we assume that E and F are uncorrelated with
Ĥ and Ĝ, respectively.
= E tr(xR xH
R)
+ σn2 tr(FR FRH )
With respect to the SINR constraint at the UTs, we first derive
the received signal at UTk as
(2)
where the matrices H = [ h1 , · · · , hK ] and G =
T
[ g1 , · · · , gK ] represent the true CSI between the BSs
and the relay and that between the relay and the UTs. The
channel estimation error matrices E and F are assumed Gaussian distributed with zero mean and E{vec(E)vec(E H )} =
2
σE
IKMR , E{vec(F )vec(F H )} = σF2 IKMR . Hence, it is
obtained that
PT E tr FR (Ĥ + E)(Ĥ + E)H FRH
2
+(PT KσE
The vectors hk and gkT represent the true CSI between the
BSk and the relay and that between the relay and the UTk .
The imperfect CSI available at the relay are denoted by ĥk and
ĝkT . The corresponding CSI error are ek and fkT , respectively.
Equivalently, we write the equation (1) in a compact form,
PR
=
ROBUST R ELAY P RECODER D ESIGN
As stated in [3], the CSI errors can be modeled in two ways.
One is named stochastic error (SE) model, where the probability distribution of the CSI error is Gaussian. This model
is applicable when the channel estimation error dominates
compared to the quantization errors. The other is the normbounded error (NBE) model, where the CSI error is specified
by an uncertainty set. This model is used when the CSI error
is mainly due to the quantization errors. We apply the SE error
model in this work. The channel uncertainties is modeled as
follows
hk
gkT
By inserting (2) and (3) into (4), PR is further written as
The Tenth International Symposium on Wireless Communication Systems 2013
The problem is to minimize the average relay transmit
power PR = PT tr(FR HH H FRH ) + σn2 tr(FR FRH ) under the
SINR constraint at each UT, which is formulated as
Based on (6), the power of the inter-operator interference is
obtained as
E
2
T
F
h
s
g
= PT E tr(gkT FR H̃k H̃kH FRH gk∗ ) ,
k R j j
j=k
(9)
where H̃k = [ h1 , · · · , hk−1 , hk+1 , · · · , hK ] and
ˆ + Ẽ . The matrix Ẽ is the channel estimation
H̃k = H̃
k
k
k
error matrix excluding user k. Similarly as in (7) and (8), (9)
is further calculated as
FR
E
Rˆ k k R E F R R ˆ
T
T
T
T H
= PT tr (H̃
k ⊗ ĝk) vec(F
R) vec(FR )(H̃k ⊗ ĝk )
fR
W
s.t.
tr
2
+σE
(IMR ⊗ ĝkT )vec(FR )vec(FR )H (IMR ⊗ ĝkT )H
= tr(Rk W ) ≥ γk σn2 ,
k = 1, 2, . . . , K
rank(W ) = 1.
2
2 2
= PT fRH (AH A+σE
B H B+σF2 D H D+σE
σF IMR2 )fR(.10)
T
2 gk FR nR + nk = σn2 E tr(gkT FR FRH gk∗ ) + σn2
= σn2 E tr (ĝkT + fkT )FR FRH (ĝk∗ + fk∗ )
=
=
P
aT
B
C
+ σn2
σn2 tr(ĝkT FR FRH ĝk∗ + σF2 FR FRH ) + σn2
σn2 tr (IMR ⊗ ĝkT ) vec(FR ) vec(FR )(IMR
B
f
(11)
Combing (8), (10) and (11), the SINR constraint for the k-th
user is expressed as
+ σn2
≥ γk ,
(12)
R1
R2
2
2 2
= a∗ aT + σE
B H B + σF2 C H C + σE
σF IMR2 ,
2
2 2
= AH A + σE
B H B + σF2 D H D + σE
σF IMR2 ,
= B H B + σF2 IMR2 .
The equation (12) can be written as
2
2
fRH PT a∗ aT + (PT σE
− γk PT σE
− γk σn2 )B H B
2 2
2 2
σF − γk PT σE
σF − γk σn2 σF2 )IMR2
+PT σF2 C H C + (PT σE
−γk PT AH A − γk PT σF2 D H D fR ≥ γk σn2 .
k
Consider the randomization method, the singular value
decomposition (SVD) of W is first computed W = U ·Σ·V H .
2
1
To initialize, w(i) is set to w(i) = U Σ 2 x, where x ∈ CMR is
a randomly generated zero-mean circular symmetric complex
Gaussian (ZMCSCG) vector and i is the iteration index.
The corresponding W (i) at the i-th iteration is obtained as
W (i) = w(i) w(i)H . Note that we need to find a scaling factor
(i)
αk for each user to fulfill its SINR requirement exactly,
where k, i represents the index of user and the iteration. The
(i)
coefficient αk is calculated as
where
R0
k
The original problem in (14) is a non-convex quadratically
constrained quadratic program (QCQP). By relaxing the nonconvex constraint rank(W ) = 1 in (14), the original problem
turns out to be convex in W and can be solved effectively
by semi-definite relaxation (SDR) [4], [5] using the convex
optimization toolbox cvx [6], [7]. To retrieve w from W , the
rank-1 approximation is performed using the randomization
method, which is introduced briefly in the following.
+ σn2
PT fRH R0 fR
H
PT fR R1 fR + σn2 fRH R2 fR
Ĥ T ⊗ IMR ,
T
ĥT
k ⊗ ĝk ,
IMR ⊗ ĝkT ,
ĥT
k ⊗ IMR ,
ˆ
H̃ T ⊗ ĝ T
ˆ = [ h , ··· , h
with H̃
1
k−1 , hk+1 , · · · , hK ] dek
noting the estimated channel matrix excluding user k, D =
ˆ T⊗I .
H̃
MR
k
⊗ ĝkT )H
= σn2 fRH (B H B + σF2 IMR2 )fR + σn2 .
=
=
=
=
A =
R
+σF2 vec(FR )vec(FR )H
(14)
Here
Concerning the power of the effective noise,
E
2
2
PT a∗ aT + (PT σE
− γk PT σE
− γk σn2 )B H B
−γk PT AH A − γk PT σF2 D H D W
ˆ T ⊗ I )vec(F )vec(F )H (H̃
ˆ T ⊗ I )H
+σF2 (H̃
MR
R
R
MR
k
k
2 2
+σE
σF vec(FR )vec(FR )H
B
2 2
2 2
σF − γk PT σE
σF − γk σn2 σF2 )IMR2
+PT σF2 C H C + (PT σE
D
2
tr (PT P H P + (PT KσE
+ σn2 )IMR2 )W
min
F
2 ≥ γk .
T F h s +E g T F n + n 2
g
j=k k R j j k
k R R
By defining fR = vec(FR ) and W = fR fRH , the problem can
finally be formulated as By defining W = fR fRH , the problem
of minimizing the relay transmit power under SINR constraint
is formulated as
ˆ H̃
ˆ H F H ĝ ∗ + σ 2 ĝ T F F H ĝ ∗
= PT tr(ĝkT FR H̃
k
k R k
E k R R k
ˆ
ˆ
2
H H
2 2
+σ F H̃ H̃ F + σ σ F F H )
2
E gkT FR hk sk s.t.
PT E tr(gkT FR H̃k H̃kH FRH gk∗ )
A
min PR
(13)
ISBN 978-3-8007-3529-7
(i)
αk =
γk σn2
.
tr(Rk W (i) )
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The Tenth International Symposium on Wireless Communication Systems 2013
TABLE I.
R ANK -1 RELAXATION OF THE ROBUST RELAY PRECODER
DESIGN FOR SINGLE STREAM TRANSMISSION
5
Robust iCSI
non−Robust iCSI
1
UΣ 2
(i)
w (i) =
W (i) = w
x, x ∈
w (i)H
M2
C R
Relay transmit power [dBW]
Compute SVD W = U · Σ · V H
(0)
Set PR = δ > 0
for i = 1 : L
is a randomly generated ZMCSCG vector
2
γk σ n
=
tr(Rk W (i) )
2
2
2
2
Rk = PT a∗ aT+ (PT σE
− γk PT σE
− γk σn
)B H B + PT σF
CHC
2 2
2 2
2 2
+(PT σE
σF − γk PT σE
σF − γk σn
σF )IM 2 − γk PT AH A
R
2
−γk PT σF
DH D
(i)
→ α(i) = max(αk )
(i)
(i)
2
2
PR = α tr
PT P H P + (PT KσE
+ σn
)IM 2 W (i)
R
(i)
(i−1)
if PR < P
√R
(i) (i)
(i)
αk
else
end
end
w=
α
(i)
PR
(i−1)
PR
,
=
1
0
(i)
PR
= α(i) tr
2
3
4
5
2
PT P H P + (PT KσE
+
(i−1)
(i)
S IMULATION R ESULTS
A two operator system with a shared AF relay is considered. Each element of all channel matrices is a zero mean
circularly symmetric complex Gaussian random variable with
unit variance CN (0, 1). The simulation results are obtained
over 1000 channel realizations.
Fig. 2 gives the consumed relay transmit power versus
the SINR threshold γ for the robust and non-robust methods.
The relay employs MR = 4 antennas. The simulation runs
for SNR = 20 dB and unit noise variance is assumed. The
CSI error variance is set to σE = σF = 0.1. It is observed
that the relay transmit power for both cases increases with
an increasing SINR threshold. This is because more power
has to be payed in order to meet higher QoS requirement.
Both robust and non-robust methods consume almost the same
relay power. However, the robust solution outperforms the nonrobust method significantly in terms of the outage probability
of SINR, as can be seen from Fig. 3. By using the robust
method, the SINR requirements are much more often satisfied
compared to the non-robust method.
In order to evaluate the benefits of advanced signal processing algorithms brought by sharing the spectrum and relays
PR versus SINR threshold γ at SNR = 20 dB
Robust iCSI
non−Robust iCSI
0.9
. If PR is smaller than PR
, w = w(i) and PR is
PR
used as a new threshold for the next iteration for comparison.
(i−1)
is set as the
Otherwise, w does not change and PR
benchmark for the next iteration. At initialization, a predefined
(0)
value PR = δ > 0 is used to start the iterative process.
The iteration continues until all the number of iterations L
is complete. In the simulation, we set L = 50. Table I
gives a summary on the randomization method for the rank-1
approximation.
IV.
1
1
and compared to that from the last iteration
(i)
0
SINR threshold [dB]
Outage probablity of SINR
tion is obtained as
(i−1)
2
−1
w = w (i−1)
Then the coefficient α(i) at the i-th iteration, is selected
(i)
from αk such that the minimum SINR
for all the users is
√
satisfied. After rescaling w(i) = α(i) w(i) and W (i) =
α(i) w(i) w(i)H , the relay transmit power at the i-th iteraσn2 )IMR2 W (i)
3
w
Fig. 2.
4
0.8
0.7
0.6
0.5
0.4
0
1
2
3
4
5
SINR threshold [dB]
Fig. 3.
dB
Outage probability of SINR versus SINR threshold γ at SNR = 20
in multi-operator environments, we define the sharing gain
as the SAPHYRE gain, which is obtained for the sharing
scenario compared to the exclusive use of the spectrum and the
infrastructure (i.e., the relay) by a single operator (time division
case, in this case, the users are multiplexed via TDMA). We
can interpret the SAPHYRE gain in terms of the relay transmit
power, i.e., the required relay transmit power consumed in
the sharing scenario is compared to that consumed by the
exclusive use of the spectrum and infrastructure for a single
operator (TDMA access). The SAPHYRE gain in terms of
relay transmit power is defined as
K
PkSU
K
ΞF,power =
k=1
K
,
(15)
Pk
k=1
where the relay transmit power of the k-th user in the sharing
scenario and the time division case are denoted by Pk and
PkSU [8].
ISBN 978-3-8007-3529-7
© VDE VERLAG GMBH · Berlin · Offenbach, Germany
525
The Tenth International Symposium on Wireless Communication Systems 2013
[4] Y. Huang and D. P. Palomar, “Rank-Constrained Separable Semidefinite
Programming with Applications to Optimal Beamforming,” IEEE Trans.
Signal Processing, vol. 58, no. 2, pp. 664 – 678, Feb. 2010.
[5] Z. Luo, W. Ma, A. M. So, Y. Ye, and S. Zhang, “Semidefinite Relaxation
of Quadratic Optimization Problems,” IEEE Signal Processing Magazine,
vol. 20, May 2010.
[6] S. Boyd and L. Vandenberghe, Convex Optimization.
Cambridge
University Press, 2004.
[7] M. Grant, S. Boyd, and Y. Ye, “Matlab software for disciplined convex
programming,” Jan. 2009.
[8] J. Li, F. Roemer, and M. Haardt, “Efficient Relay Sharing (EReSh)
between Multiple Operators in Amplify-and-Forward Relaying Systems,”
in Proc. IEEE 4th Int. Workshop on Computational Advances in MultiSensor Adaptive Processing (CAMSAP 2011), San Juan, Puerto Rico,
Dec. 2011, pp. 249 – 252.
8
Relay transmit power [dBW]
7
6
5
4
3
Robust iCSI M = 8
R
2
Robust iCSI TDMA M = 8
R
1
Robust iCSI TDMA MR = 4
0
−1
−2
5
10
15
20
Power of each transmitter [dBW]
Fig. 4.
SAPHYRE gain in terms of power for the robust design
The required relay transmit power using the robust method
for the sharing case and that for the TDMA access in the nonsharing case is plotted in Fig. 4. The gap between the blue
curve obtained by robust method and the red one for TDMA
scenario with MR = 8 denote the spectrum sharing gain. There
is a gain of around 5 dB by the shared use of the spectrum
between the two operators at high SNRs. An additional 2 dB
gain is obtained by an additional sharing of the relay compared
to the exclusive access of the relay with half of the number of
relay antennas MR = 4 for each operator.
V.
C ONCLUSIONS
In this paper, we propose a robust relay precoder in the
multiple operator AF relay sharing system to achieve a power
efficient transmission for the multiple operator AF relay sharing system. We study the special case that the BSs and the UTs
are equipped with single antennas, where only single stream
transmission is possible. The channel state information at the
relay is assumed to be imperfect, where the additive channel
state information error is modeled as Gaussian distributed.
Based on this model, we propose a robust relay precoder
design that minimizes the total average relay transmit power
under the SINR constraint at each user terminal. Simulation
results show that the proposed robust method outperforms
the non-robust solution significantly in terms of the outage
probability of SINR. Following that, the SAPHYRE sharing
gain is investigated in terms of the required relay transmit
power. There is 5 dB gain by sharing the spectrum and a 7
dB gain is observed by both the spectrum and relay sharing
compared to the exclusive use of these physical resources.
R EFERENCES
[1]
[2]
[3]
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1927–1938, Nov. 2003.
B. K. Chalise and L. Vandendorpe, “MIMO Relay Design for Multipointto-Multipoint Communications With Imperfect Channel State Information,” IEEE Trans. Signal Processing, vol. 57, no. 7, pp. 2785–2796, Jul.
2009.
P. Ubaidulla and A. Chockalingam, “Relay Precoder Optimization in
MIMO-Relay Networks With Imperfect CSI,” IEEE Trans. on Signal
Processing, vol. 59, no. 11, pp. 5473–5484, Nov. 2011.
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