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Opportunistic Cooperation with ReceiverBased
Ratio Combining Strategy
Qian Wang
, Kui Ren
, Yanchao Zhang
, and Bo Zhu
Illinois Institute of Technology, Chicago, IL , USA
kren,qwangece.iit.edu
New Jersey Institute of Technology, Newark, NJ , USA
yczhangnjit.edu
Concordia University, MontreaHG M, Canadal, QC
zhubociise.concordia.ca
Abstract. In cooperative wireless communication systems, many com
bining techniques could be employed at the receiver, such as maximal
ratio combining MRC, equal gain combining EGC, etc. To address
the eect of receiver diversity combining on optimum energy allocation,
we analyze the problem of minimizing average total transmit energy un
der a SNR constraint when dierent ratio combining methods are utilized
at destination. For maximal ratio combining MRC, based on the ex
plicit analytical solution an asymptotic solution for normalized optimum
total energy in terms of and was derived in the highSNR scenario.
For xed ratio combining FRC, we nd that there does not exist an
explicit analytical solution to the optimum energy allocation problem.
However, the convexity proof for the energy function provides a way of
using numerical convex optimization methods to nd the unique solu
tion. Our results also show that, while direct transmission E
r
is
optimum for certain channel states when the destination uses MRC, the
relay should always transmit, i.e. E
r
gt for all channel states, when the
combining ratio is a xed number.
Introduction
In cooperative wireless communication, each user is assumed to transmit data
as well as act as a cooperative agent for another user. The transmitters or re
ceivers can collectively act as an antenna array and create a virtual or distributed
multipleinput multipleoutput MIMO system. The basic ideas behind coop
erative communication can be traced back to the work of Cover and El Gamal
on the information theoretic properties of the relay channel . However, the
earliest work specically on user cooperation is due to Sedonaris et al. in
for cellular networks and Laneman et al. in for ad hoc networks.
It has been shown that the cooperative transmission strategy provides power
ful benets of multiantenna systems without the need for physical arrays, e.g. an
increased capacity, a robustness to fading and reduced outage probability. Re
cent results in implementation of dierent cooperative signaling methods such
Y. Li et al. Eds. WASA , LNCS , pp. , .
c SpringerVerlag Berlin Heidelberg
Q. Wang et al.
as amplifyandforward and decodeandforward , indicate that coop
erative communication has a promising future. These results also demonstrated
that while knowledge of channel state information at the transmitters CSIT
is benecial, it is not necessary to achieve signicant gains in energy eciency
with respect to direct noncooperative transmission.
While recent work in this area has focused on the goal of minimizing BER,
minimizing total power to a rate constraint, minimizing total power subject to
xed SNR and outage probability constraints, the problem of how the diversity
combining methods aect the optimum energy allocation has not been fully
investigated.
In this paper, we consider the problem of optimum energy allocation and
weighted total energy minimization under SNR constraint in two scenarios i
mrc
, i.e. maximal ratio combining MRC is used at the receiver and ii
is a xed number, i.e. xed ratio combining FRC is used at the receiver.
In both cases, we derive the optimum opportunistic energy allocation strategies
and explicitly describe the set of channel conditions under which the objective
can be realized. Our analysis shows that, when MRC is utilized at destination,
cooperative transmission is more energy ecient than direct transmission except
when the relaydestination channel is not advantaged. The asymptotic solution
we derived for the highSNR scenario can best illustrate this. We also show that,
while direct transmission E
r
is optimum for certain channel states when
the destination uses MRC, the relay should always transmit when xed ratio
combining FRC is utilized at destination, i.e. E
r
gt for all channel states.
The impact of channel state information on AF cooperative transmission using
MRC and EGC has been studied in and , respectively, the intuitive results
in this paper could be regarded as an extension of prior works.
System Model
To facilitate analysis, we consider the same system model as in Figure .
In the network, each source is both a user and a relay, one source communi
cates directly to a destination and another source acts as a relay under certain
channel conditions. The channels in the system are all assumed to be frequency
nonselective and the channel magnitudes g
s
,g
r
, and h are assumed to be in
dependent Rayleigh distributed random variables. We also assume the channels
stay roughly constant for several timeslots, i.e., in the process of cooperation.
In this paper, we use Amplifyandforward as our signaling method in co
operative communication sytem. Amplifyandforward is a simple method that
lends itself to analysis, and thus has been very useful in furthering our under
standing of cooperative communication systems. This method was proposed and
analyzed by Laneman et al. . It has been shown that for the twosource case,
this method achieves diversity order of two, which is the best possible outcome
at high SNR. In AF, each source receives a noisy version of the signal transmit
ted by its partner relay. The relay then amplies and retransmits this noisy
Opportunistic Cooperation with ReceiverBased Ratio Combining Strategy
S
R
D
gs
Gs
gr
Gr
h
H
Fig. . System model
version. The destination combines the information sent by the source and relay,
and makes a nal decision on the transmitted bit.
In the rst timeslot, the source transmits the symbol x to the destination.
The signals received by the destination and relay in this timeslot are as
follows
y
sd
g
s
a
s
xw
sd
y
sr
ha
s
xw
sr
,
where a
s
is the amplitude of the sources transmission and w
sd
and w
sr
are
additive white Gaussian noise at the receivers of the destination and relay ,
respectively.
In the second timeslot, the relay retransmits the signal that it observed in
the rst timeslot to the destination. The signal received by the destination
in this slot is
y
rd
g
r
a
r
y
sr
w
rd
g
r
a
r
ha
s
xg
r
a
r
w
sr
w
rd
where a
r
is the amplitude of the relays transmission and w
rd
denotes the
receiver noise at the destination in the second timeslot.
The destination makes a nal decision on x based on the observations in the
two timeslots
y
d
y
sd
y
rd
where
and
are nonnegative combining ratios.
SNR Analysis
We model AWGN as independent normal random variables with zero mean and
unit variance. The instantaneous SNR of the nal decision can be written as
SNR
,
g
s
g
r
a
r
h
a
s
Ex
H
x
g
r
a
r
Q. Wang et al.
By setting
/
and plugging G
s
g
s
,G
r
g
r
,Hh
, can be
rewritten as
SNR
G
s
E
s
HE
s
G
r
E
r
HE
s
E
s
G
r
E
r
G
s
HHE
s
G
r
E
r
HE
s
where E
s
a
s
Ex
H
x and E
r
a
r
HE
s
.
Note that the relay transmission energy is conditioned on HE
s
and includes
both a signal component and a noise component. The noise component is a
consequence of the fact that the relay transmission is simply an amplied copy
of the noisy signal received in the rst timeslot.
When the destination has full access to the channel state information CSI
and transmit energies, maximal ratio combining MRC can be used to maxi
mize the SNR of the decision statistic. The resulting instantaneous SNR at the
destination, after MRC, can be expressed as
SNR
mrc
G
s
E
s
G
r
E
r
HE
s
G
r
E
r
HE
s
.
where
mrc
G
s
HE
s
G
r
E
r
G
r
HE
r
HE
s
.
Note that the rst part of is the SNR of direct transmission.
When the destination does not have access to the channel state, equal gain
combining EGC can be used i.e.
egc
. The resulting instantaneous SNR
at the destination, after EGC, can be expressed as
SNR
egc
G
s
E
s
G
r
E
r
E
s
HG
s
/E
s
G
r
E
r
G
s
HHE
s
G
r
E
r
HE
s
.
In this paper, to establish a framework for optimum energy allocation, we
dene
E
tot
E
s
E
r
as the weighted total transmission energy used in the cooperative transmission
interval. The parameter allows for a weighting of the cost of the relays
energy relative to the cost of the sources energy. The following sections derive
the optimum energy allocation strategies for an AF cooperative transmission
under MRC and FRC using the weighted total transmission energy metric .
Optimum Energy Allocation
In this section, for a given channel state s G
s
,G
r
, H, we consider the prob
lem nding the optimum energy allocation E
s
,E
r
that minimizes the weighted
total energy under a minimum SNR constraint , i.e.,
E
tot
min
Es,ErB
E
tot
Opportunistic Cooperation with ReceiverBased Ratio Combining Strategy
where B is the set of energy allocations satisfying E
s
,E
r
, and the SNR
constraint SNRs, , E
s
,E
r
.
The SNR of the sources information at the destination is determined not only
by the channel states and the transmission energies but also by how the destina
tion forms its decision statistic from the received source and relay transmissions.
In the following sections, we rst analyze the energy minimization problem when
MRC technique is utilized at the destination, where the destination has full ac
cess to the channel states and transmit energies of both sources in both timeslots.
In this case, is a function of channel states and transmit power. In the second
part of this section, we will discuss another situation, when xed ratio combining
FRC is utilized at the destination, i.e., is a xed number.
. Maximal Ratio Combining
To facilitate analysis of in the case of a destination using MRC, we dene
two nonnegative quantities
G
s
G
r
G
s
H
and
H
HG
s
.
The explicit solution to the total energy minimization problem for a destina
tion using MRC is given in the following proposition.
Proposition . When
mrc
, the normalized minimum weighted total energy
E
tot
G
s
can be expressed as
GsGrGsHGsH
GsHGsHH
GsGsH
GrGsH
lt ,
.
The proof of Proposition is provided in Appendix A.
We note that when the SNR constraint , G
s
/G
r
can be con
sidered an indicator of source or relay channel advantage, i.e. gt indicates
that the source has an advantaged channel to the destination and lt indi
cates that the relay is advantaged. Similarly, can be considered an indicator
of sourcerelay or source advantage. When H is large with respect to G
s
, the
quantity , which means the source and relay are much closer in proximity
than the source and destination.
Without loss of generality, we consider the problem in highSNR scenario.
When , the normalized minimum weighted total energy can be expressed
Q. Wang et al.
in terms of and as
E
tot
/G
s
when lt ,
when .
We can also dene the total energy gain of optimum cooperative transmission
as the ratio of the E
tot
achieved with direct transmission, i.e.
Gs
to the E
tot
achieved with optimum AF cooperative transmission.
Similarly, we can show that the asymptotic solution for normalized optimum
source energy in the highSNR scenario can be expressed as
E
s
/E
tot
when lt ,
when .
. Fixed Ratio Combining
This section analyzes the scenario when FRC is used at the destination is a
xed number, i.e. it is not dependent on the channel states and transmit power.
Note that equal gain combining EGC can be considered as a special case of
FRC where
egc
.
The relay node energy E
r
can be written as a function of and E
s
by solving
for E
r
when SNR . The solution yields two roots for E
r
. When E
r
,
by solving the equation SNR we have E
s
Gs
. The correct root
should satisfy this condition and can be written as
E
r
HE
s
G
s
HE
s
HE
s
G
s
E
s
G
r
HE
s
E
s
G
s
H
HE
s
G
s
E
s
G
r
HE
s
The admissible range of instantaneous energy allocations that satisfy SNR
can be described as the region in R
where E
r
and
H
Gs
E
s
Gs
. The case E
r
establishes the upper limit on the interval of admissible
solutions for E
s
. The lower limit on the interval is established by the requirement
for total energy to be a realvalued quantity. The square root in the numerator of
reveals that E
r
R only if E
s
H
Gs
.
Denote the admissible range
H
Gs
,
Gs
of E
s
as A. Given and
the squared channel amplitudes G
s
,G
r
, and H, implies that E
r
is dependent
on E
s
. It can be shown that it is hard to nd an explicit analytical solution to
. Numerical solutions to , however, are aided by the following result.
Opportunistic Cooperation with ReceiverBased Ratio Combining Strategy
Proposition . When FRC is used at the destination, the total energy E
tot
is
still a convex function of E
s
on A.
The proof of Proposition is provided in Appendix B.
Proposition implies that standard numerical convex optimization methods
can be used to nd the unique solution to .
Denote E
s
as the value of E
s
that attains the minimum in and note that E
r
is implied by . Given the convexity of E
tot
on A, we can determine whether
the unique minimum of E
tot
on A occurs at the point E
s
Gs
by evaluating
Etot
Es
at this point. If
Etot
Es
gt at E
s
Gs
, then the minimum of E
tot
on A
must occur at E
s
lt
Gs
corresponding to E
r
gt , otherwise the minimum
occurs at E
s
Gs
corresponding to E
r
. It can be shown that
E
tot
E
s
Es
Gs
gt ,
hence the unique minimum of E
tot
on A must occur at E
s
lt
Gs
. This implies
that E
r
gt for all G
s
,G
r
, H, . Thus in the case of FRC, the relay should always
transmit, i.e. E
r
gt for all channel states. This is in contrast to the result in
Section . showing that direct transmission E
r
is optimum for certain
channel states when the destination uses MRC.
Simulation Results
In this section, we present the performance of the optimum energy allocation
scheme and show how the optimum total energy gain are aected by this scheme.
Proposition implies when the relay does not have an advantaged channel
to the destination, the total energy is minimized when all of the transmission
energy is allocated to the source and the relay does not transmit. Figure shows
the total energy gain of optimum AF cooperative transmission when .
Similarly, the largest gains occur when , which corresponds to the case
where the relay has a advantaged channel to the destination and , which
corresponds to the case where the source and relay are much closer in proximity
than the source and destination.
In gure , it can be shown that
E
s
E
tot
when , i.e. the relay does not
have an advantaged channel to the destination, all of the transmission energy
is allocated to the source. Only when lt , i.e. the relay has an advan
taged channel that the total energy could be minimized through cooperation
transmission. As expected, for a xed ,
E
s
E
tot
decreases when increases, which
corresponds to the case when the sourcerelay channel are more favorable, more
transmission energy is allocated to the relay. Note that for a xed , when ,
i.e. the relay has a much advantaged channel to the destination, the relay only
needs a small amount of transmission energy to satisfy the SNR requirement,
thus
E
s
E
tot
becomes larger.
Q. Wang et al.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.........
.
.
.
.
.
.
.
.
.
Fig. . Etot gain, in dB with respect to direct transmission, of AF cooperative trans
mission with optimum energy allocation E
s
,E
r
as a function of the parameters
and
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
..
.
.
.........
.
.
.
.
.
.
.
.
.
Fig. . Normalized optimum source energy allocation E
s
/E
tot
as a function of the pa
rameters and
Conclusion
This paper examines optimum energy allocation for amplifyandforward coop
eration with the goal of minimizing average total transmit energy under a SNR
constraint in two scenarios i maximal ratio combining MRC and ii xed
ratio combining FRC. For MRC, based on the explicit analytical solution an
asymptotic solution for normalized optimum total energy in terms of and
Opportunistic Cooperation with ReceiverBased Ratio Combining Strategy
was derived in the highSNR scenario. For FRC, we nd that though it is hard
to nd an explicit analytical solution, standard numerical convex optimization
methods can be used to nd the unique solution to the problem. Based on these
analysis, we explicitly describe the set of channel conditions under which the
optimum energy allocation strategy can be realized.
Acknowledgement
This research is supported in part by ERIF, IIT, National Science Foundation
Grant CNS, and National Sciences and Engineering Research Council
of Canada NSERC.
References
. Cover, T.M., Gamal, A.A.E. Capacity Theorems for the Relay Channel. IEEE
Trans. Info. Theory ,
. Sendonaris, A., Erkip, E., Aazhang, B. User cooperation diversity part i System
description. IEEE Trans. Commun. ,
. Sendonaris, A., Erkip, E., Aazhang, B. User cooperation diversity part ii Imple
mentation aspects and performance analysis. IEEE Trans. Commun. ,
. Laneman, J.N., Wornell, G.W., Tse, D.N.C. An Ecient Protocol for Realizing
Cooperative Diversity in Wireless Networks. In Proc. IEEE ISIT, Washington, DC,
June , p.
. Scaglione, A., Goeckel, D.L., Laneman, J.N. Cooperative communications in mobile
ad hoc networks. IEEE Signal Processing Magazine ,
. Hunter, T.E., Nosratinia, A. Cooperative Diversity through Coding. In Proc. IEEE
ISIT, Laussane, Switzerland, July , p.
. Hunter, T.E., Nosratinia, A. Diversity through Coded Cooperation. IEEE Trans
actions on Wireless Communications ,
. Yang, J., Brown III, D.R. The eect of channel state information on optimum
energy allocation and energy eciency of cooperative wireless transmission systems.
In Conference on Info. Sciences and Systems CISS, pp.
. Yang, J., Brown III, D.R. The Eect of Receiver Diversity Combining on Optimum
Energy Allocation and Energy Eciency of Cooperative Wireless Transmission Sys
tems. In Conference on Acoustics, Speech and Signal Processing ICASSP, vol. ,
pp. IIIIII
A Proof of Proposition
Proof. Before deriving the minimum weighted total energy under a minimum
SNR constraint, we rst determine the conditions for direct transmission and
cooperative transmission. From , we note that the space of admissible energy
allocations satisfying SNR
mrc
can be described as the region in R
where
E
r
and
HGs
lt E
s
Gs
, where the upper limit to E
s
corresponds to the
case when E
r
and the lower limit corresponds to the case when E
r
.
Q. Wang et al.
Using , the total energy required to satisfy the constraint SNR
mrc
can
be written as
E
tot
E
s
E
r
E
s
HE
s
G
s
G
s
HE
s
G
r
HG
s
E
s
.
Dene the interval A
HGs
,
Gs
. If
E
tot
arg min
EsA
E
tot
G
s
then E
r
and E
tot
is minimized with direct transmission. Otherwise, E
r
gt
and cooperative transmission minimizes E
tot
.
In order to determine if the minimum of on Aoccurs at the point E
s
Gs
,
we rst establish that can have only one minimum on A by proving that
is a strictly convex function of E
s
on A. The second derivative of with
respect to E
s
can be written as
E
tot
E
s
E
tot
HHG
s
G
r
HG
s
E
s
Note that the numerator of is a negative quantity not dependent on E
s
.
Since E
s
HG
s
gt and G
r
gt , the denominator of is also negative on
the interval E
s
A, hence E
tot
is always positive on A. This implies that E
tot
is
a strictly convex function of E
s
on A.
Given the convexity of E
tot
on A, we can determine whether the unique mini
mum of on A occurs at the point E
s
Gs
by evaluating the rst derivative
of at this point. If the rst derivative is positive, then the minimum of
on A must occur at E
s
lt
Gs
corresponding to cooperative transmission, oth
erwise the minimum occurs at E
s
Gs
corresponding to direct transmission.
The rst derivative of evaluated at E
s
Gs
can be written as
E
tot
G
s
E
s
E
tot
G
s
G
s
HG
s
G
r
H
This quantity is positive if and only if the condition of
Gs
Gr
Gs
H
lt are
satised, i.e. lt , hence the unique minimum of on A must occur at
E
s
lt
Gs
when lt . Otherwise, when the minimum of on A
must occur at E
s
Gs
and direct transmission is optimum.
We now derive the explicit solution to the total energy minimization prob
lem. The optimal source energy allocation can be found by solving
Es
E
tot
.
Computation of the partial derivative and algebraic simplication yields
HG
s
E
s
G
s
G
r
HG
s
E
s
G
s
E
s
H HE
s
G
r
HG
s
E
s
By solving this equation for E
s
, the correct root which satises
E
s
E
tot
E
s
is
E
s
HG
s
HG
s
H
HG
s
HG
r
G
s
G
s
G
r
Opportunistic Cooperation with ReceiverBased Ratio Combining Strategy
In the case lt , the total energy can be minimized through coopera
tive transmission. By plugging E
s
into , the minimized total energy can be
expressed as
E
tot
G
r
G
s
HG
s
H
HG
s
HH
G
s
H
G
r
G
s
H
B Proof of Proposition
Proof. To prove E
tot
is convex, and hence has a unique minimum on A
H
Gs
,
Gs
, we will show that
Etot
E
s
gt a.s. Here,
Etot
E
s
is a function
of E
s
. Substitute E
s
with y, we have
E
tot
E
s
FyGy.
The function
Fy y
G
s
y
G
s
H
G
s
G
s
Hy
H
G
s
HG
s
H
G
s
y
G
s
H
G
s
H
H
G
s
.
and
Gy
G
s
H
G
s
HH
G
r
yG
s
y
where y
G
s
H
HE
s
G
s
E
s
.
Note that the squared channel amplitudes G
s
,G
r
and H are exponentially
distributed, thus lim
P X , where X denotes the squared channel
amplitudes. Thus Gy gt a.s. on A Note that y . Hence, the condition
Etot
E
s
gt a.s. on A Fy gt a.s. on C, where C
,
H
. Observe that
only the y
term has negative coecient. The function Fy can be written as
Fy Ry Sy Ty,
where
Ry
G
s
GsHy
HG
s
H
Gsy
Sy
Gsy
GsHy
H
Gsy
Ty y
G
s
H
G
s
H
H
G
s
.
Note that Ty for
,
H
. We will consider the behavior of Ry
and Sy in following claims.
Q. Wang et al.
Claim Ry gt a.s. on C.
proof Observe that Ry is a quadratic equation of one variable. It can be written
as
Ry yry,
where
ry
G
s
G
s
Hy
HG
s
H
G
s
.
First, we consider the case when gt . Observe that y gt and r
H
HG
s
H
G
s
gt a.s. Thus, to prove Ry gt a.s. on C,
it is only necessary to prove that ry is decreasing on C. It can be shown that
ry
y
G
s
G
s
H lt a.s.
Thus, ry gt a.s. on C, this result implies Ry gt a.s. on C. When ,
Ry y
H
G
s
gt a.s. on C.
Claim Sy gt a.s. on C.
proof Observe that Sy is a cubic equation of one variable. It can be written as
Sy ysy,
where
sy
G
s
y
G
s
Hy
H
G
s
.
First, we consider the case when gt . Observe that y gt and sy is a
quadratic equation. To prove that Sy gt a.s. on C, it is only necessary to
prove that sy gt a.s. on C. It can be shown that
sy
y
G
s
gt a.s.,
hence sy is convex on C. Thus, to prove sy gt a.s. on C, it is only necessary
to prove that sy has no real root a.s. It can be shown
G
s
H
G
s
H
G
s
G
s
H
lt a.s.
This implies that sy gt a.s. on C, which implies Sy gt a.s. on C. When
, Sy y
H
G
s
gt a.s. on C.
Amplifyandforward is a simple method that lends itself to analysis. minimizing total power
subject to xed SNR and outage probability constraints. minimizing total power to a rate
constraint. we consider the same system model as in Figure . the intuitive results in this
paper could be regarded as an extension of prior works. maximal ratio combining MRC is
used at the receiver and ii is a xed number.e. The asymptotic solution we derived for the
highSNR scenario can best illustrate this. In this paper.e. While recent work in this area has
focused on the goal of minimizing BER. Our analysis shows that. we consider the problem of
optimum energy allocation and weighted total energy minimization under SNR constraint in
two scenarios i mrc . cooperative transmission is more energy ecient than direct transmission
except when the relaydestination channel is not advantaged. Wang et al. which is the best
possible outcome at high SNR. We also show that. the problem of how the diversity
combining methods aect the optimum energy allocation has not been fully investigated. i. Q.
We also assume the channels stay roughly constant for several timeslots. while direct
transmission Er is optimum for certain channel states when the destination uses MRC. in the
process of cooperation. each source receives a noisy version of the signal transmitted by its
partner relay. In this paper. i. xed ratio combining FRC is used at the receiver. each source is
both a user and a relay. The channels in the system are all assumed to be frequency
nonselective and the channel magnitudes gs . as amplifyandforward and decodeandforward .
and h are assumed to be independent Rayleigh distributed random variables. gr .
respectively. i.e. . the relay should always transmit when xed ratio combining FRC is utilized
at destination. In AF. In the network. indicate that cooperative communication has a
promising future. In both cases.e. one source communicates directly to a destination and
another source acts as a relay under certain channel conditions. we use Amplifyandforward
as our signaling method in cooperative communication sytem. i. System Model To facilitate
analysis. These results also demonstrated that while knowledge of channel state information
at the transmitters CSIT is benecial. this method achieves diversity order of two. and thus
has been very useful in furthering our understanding of cooperative communication systems.
it is not necessary to achieve signicant gains in energy eciency with respect to direct
noncooperative transmission. The relay then amplies and retransmits this noisy . when MRC
is utilized at destination. Er gt for all channel states. we derive the optimum opportunistic
energy allocation strategies and explicitly describe the set of channel conditions under which
the objective can be realized. The impact of channel state information on AF cooperative
transmission using MRC and EGC has been studied in and .. This method was proposed and
analyzed by Laneman et al. It has been shown that for the twosource case.
. where as is the amplitude of the sources transmission and wsd and wsr are additive white
Gaussian noise at the receivers of the destination and relay .Opportunistic Cooperation with
ReceiverBased Ratio Combining Strategy S h H gs Gs D R gr Gr Fig. gr a r . The signals
received by the destination and relay in this timeslot are as follows ysd gs as x wsd ysr has x
wsr . and makes a nal decision on the transmitted bit. The destination combines the
information sent by the source and relay. The instantaneous SNR of the nal decision can be
written as gs gr ar h a ExH x s SNR . the source transmits the symbol x to the destination. In
the second timeslot. the relay retransmits the signal that it observed in the rst timeslot to the
destination. The destination makes a nal decision on x based on the observations in the two
timeslots yd ysd yrd where and are nonnegative combining ratios. SNR Analysis We model
AWGN as independent normal random variables with zero mean and unit variance. The
signal received by the destination in this slot is yrd gr ar ysr wrd gr ar has x gr ar wsr wrd
where ar is the amplitude of the relays transmission and wrd denotes the receiver noise at
the destination in the second timeslot. In the rst timeslot. respectively. System model version.
for a given channel state s Gs . Er that minimizes the weighted total energy under a
minimum SNR constraint . egc . Optimum Energy Allocation In this section. When the
destination does not have access to the channel state. can be rewritten as SNR Gs Es HEs
Gr Er HEs Es Gr Er Gs HHEs Gr Er HEs where Es a ExH x and Er a HEs . we consider the
prob lem nding the optimum energy allocation Es . Gr Er HEs In this paper. The resulting
instantaneous SNR at the destination. H h .. The parameter allows for a weighting of the cost
of the relays energy relative to the cost of the sources energy. The resulting instantaneous
SNR at the destination. Wang et al. Gr Er HEs Gs HEs Gr Er . s r Note that the relay
transmission energy is conditioned on HEs and includes both a signal component and a
noise component. The following sections derive the optimum energy allocation strategies for
an AF cooperative transmission under MRC and FRC using the weighted total transmission
energy metric .Er B min Etot . maximal ratio combining MRC can be used to maximize the
SNR of the decision statistic.e. When the destination has full access to the channel state
information CSI and transmit energies. we dene Etot Es Er as the weighted total
transmission energy used in the cooperative transmission interval. By setting / and plugging
Gs gs . The noise component is a consequence of the fact that the relay transmission is
simply an amplied copy of the noisy signal received in the rst timeslot.e. Gr HEr HEs Note
that the rst part of is the SNR of direct transmission. to establish a framework for optimum
energy allocation. H. Gr gr . i. Q. Etot Es . can be expressed as SNRegc Gr Er Es H Gs / Es
Gr Er Gs HHEs Gs Es . after MRC. Gr . can be expressed as SNRmrc Gs Es where mrc Gr
Er HEs . equal gain combining EGC can be used i. after EGC.
The SNR of the sources information at the destination is determined not only by the channel
states and the transmission energies but also by how the destination forms its decision
statistic from the received source and relay transmissions. We note that when the SNR
constraint . we rst analyze the energy minimization problem when MRC technique is utilized
at the destination.Opportunistic Cooperation with ReceiverBased Ratio Combining Strategy
where B is the set of energy allocations satisfying Es .e. When . . When H is large with
respect to Gs . In this case. Proposition . Er . Similarly. H Gs The explicit solution to the total
energy minimization problem for a destination using MRC is given in the following
proposition.. . The proof of Proposition is provided in Appendix A. is a xed number. When
mrc . Gs /Gr can be considered an indicator of source or relay channel advantage. i. can be
considered an indicator of sourcerelay or source advantage. is a function of channel states
and transmit power. Es . i. we dene two nonnegative quantities Gs Gr Gs H and H . and the
SNR constraint SNRs. Maximal Ratio Combining To facilitate analysis of in the case of a
destination using MRC.e. Without loss of generality. when xed ratio combining FRC is
utilized at the destination. where the destination has full access to the channel states and
transmit energies of both sources in both timeslots. which means the source and relay are
much closer in proximity than the source and destination. In the second part of this section.
we consider the problem in highSNR scenario. . the quantity . Er . gt indicates that the source
has an advantaged channel to the destination and lt indicates that the relay is advantaged.
the normalized minimum weighted total energy Etot can be expressed as G s Gs Gr Gs HGs
H Gs HGs HH Gr Gs H Gs Gs H lt . we will discuss another situation. In the following
sections. the normalized minimum weighted total energy can be expressed .
The solution yields two roots for Er . of Es as A. when .e.e. It can be shown that it is hard to
nd an explicit analytical solution to . The correct root Gs should satisfy this condition and can
be written as HEs Gs HEs HEs Gs Es Gr HEs Er Es Gs H HEs Gs Es Gr HEs The
admissible range of instantaneous energy allocations that satisfy SNR can be described as
the region in R where Er and H Gs Es The case Er establishes the upper limit on the interval
of admissible solutions for Es . Numerical solutions to . When Er . however. it is not
dependent on the channel states and transmit power. Given and Gs the squared channel
amplitudes Gs . Wang et al. . we can show that the asymptotic solution for normalized
optimum source energy in the highSNR scenario can be expressed as Es /Etot when lt . i.
are aided by the following result. Denote the admissible range H Gs . The lower limit on the
interval is established by the requirement for total energy to be a realvalued quantity. Fixed
Ratio Combining This section analyzes the scenario when FRC is used at the destination is a
xed number. The square root in the numerator of reveals that Er R only if Es H Gs . . Note
that equal gain combining EGC can be considered as a special case of FRC where egc . Gs
. We can also dene the total energy gain of optimum cooperative transmission as the ratio of
the Etot achieved with direct transmission. by solving the equation SNR we have Es . i. Gs to
the Etot achieved with optimum AF cooperative transmission. The relay node energy Er can
be written as a function of and Es by solving for Er when SNR . in terms of and as Etot /G s
when lt . Similarly. Gr . implies that Er is dependent on Es . and H. Q. when .
which corresponds to the case where the relay has a advantaged channel to the destination
and . As expected. for a xed . i. the relay should always transmit. It can be shown that Gs
Etot Es at this point. showing that direct transmission Er is optimum for certain channel
states when the destination uses MRC. Es In gure .e. which tot corresponds to the case
when the sourcerelay channel are more favorable.e. Similarly. the relay does not tot have an
advantaged channel to the destination. Er gt for all channel states. the total energy is
minimized when all of the transmission energy is allocated to the source and the relay does
not transmit.e. hence the unique minimum of Etot on A must occur at Es lt . the total energy
Etot is still a convex function of Es on A. the relay has an advantaged channel that the total
energy could be minimized through cooperation Es transmission. If Etot Es Es G s gt . we
can determine whether the unique minimum of Etot on A occurs at the point Es by evaluating
Gs Etot Es gt at Es Gs . Only when lt . E decreases when increases. Proposition implies that
standard numerical convex optimization methods can be used to nd the unique solution to .e.
i.Opportunistic Cooperation with ReceiverBased Ratio Combining Strategy Proposition . the
relay has a much advantaged channel to the destination. i. then the minimum of Etot on A
must occur at Es lt corresponding to Er gt . all of the transmission energy is allocated to the
source. H. tot . . we present the performance of the optimum energy allocation scheme and
show how the optimum total energy gain are aected by this scheme. This implies Gs that Er
gt for all Gs . i. the relay only needs a small amount of transmission energy to satisfy the
SNR requirement. when . Given the convexity of Etot on A. Simulation Results In this
section. more transmission energy is allocated to the relay. This is in contrast to the result in
Section . it can be shown that E when . Figure shows the total energy gain of optimum AF
cooperative transmission when . Denote Es as the value of Es that attains the minimum in
and note that Er is implied by . Gr . otherwise the minimum Gs occurs at Es corresponding to
Er . which corresponds to the case where the source and relay are much closer in proximity
than the source and destination. Es thus E becomes larger. Proposition implies when the
relay does not have an advantaged channel to the destination. When FRC is used at the
destination. Thus in the case of FRC. the largest gains occur when . The proof of Proposition
is provided in Appendix B. Note that for a xed .
. . . Fig. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Er as a function of the parameters
and . . . . . . . . . . . . . Normalized optimum source energy allocation Es /Etot as a function of
the parameters and Conclusion This paper examines optimum energy allocation for
amplifyandforward cooperation with the goal of minimizing average total transmit energy
under a SNR constraint in two scenarios i maximal ratio combining MRC and ii xed ratio
combining FRC. . Wang et al. . . . . . . . . . . . Fig. of AF cooperative trans mission with
optimum energy allocation Es . . . . . . . . . . . . Etot gain. . based on the explicit analytical
solution an asymptotic solution for normalized optimum total energy in terms of and . . . . . .
For MRC. . . . . in dB with respect to direct transmission. Q. . . . .
D. . Erkip.E.. An Ecient Protocol for Realizing Cooperative Diversity in Wireless Networks.
Laussane. Yang. . pp.. DC.. . Brown III.R. Yang. IEEE ISIT.Opportunistic Cooperation with
ReceiverBased Ratio Combining Strategy was derived in the highSNR scenario. T. J.
Goeckel. User cooperation diversity part ii Implementation aspects and performance
analysis. .. . In Conference on Info. Laneman.E. . Info.N. Theory . J. Wornell. D. . The Eect of
Receiver Diversity Combining on Optimum Energy Allocation and Energy Eciency of
Cooperative Wireless Transmission Systems. . From . In Conference on Acoustics. IIT.A..
IEEE Trans. IEEE Trans. standard numerical convex optimization methods can be used to
nd the unique solution to the problem. Erkip. . T. . Commun. B. Gamal. B. Sendonaris. J. A.
Sciences and Systems CISS. . Capacity Theorems for the Relay Channel. vol. pp..
Cooperative Diversity through Coding. D. The eect of channel state information on optimum
energy allocation and energy eciency of cooperative wireless transmission systems. Before
deriving the minimum weighted total energy under a minimum SNR constraint.N. Based on
these analysis. IEEE Trans.. T. Cooperative communications in mobile ad hoc networks.
Switzerland. Nosratinia... IEEE ISIT.N.W. June . Sendonaris. E. we nd that though it is hard
to nd an explicit analytical solution. we explicitly describe the set of channel conditions under
which the optimum energy allocation strategy can be realized. E.C. July .. Laneman. IIIIII A
Proof of Proposition Proof. A. A. we note that the space of admissible energy allocations
satisfying SNRmrc can be described as the region in R where Er and HGs lt Es Gs . Hunter.
and National Sciences and Engineering Research Council of Canada NSERC. IEEE Signal
Processing Magazine . A. A.. IEEE Transactions on Wireless Communications .
Acknowledgement This research is supported in part by ERIF.. Washington.R. D. .E. we rst
determine the conditions for direct transmission and cooperative transmission. National
Science Foundation Grant CNS. p. A. Aazhang. Brown III. Cover..M. In Proc. p. Scaglione.
Tse. User cooperation diversity part i System description. Aazhang. where the upper limit to
Es corresponds to the case when Er and the lower limit corresponds to the case when Er . G.
In Proc. Diversity through Coded Cooperation. Hunter. For FRC. Nosratinia. Speech and
Signal Processing ICASSP. J.L. References . Commun.
This implies that Etot is a strictly convex function of Es on A. the correct root which satises is
Es Es HGs H H Gs H Gs HGr Gs Gs Gr . Given the convexity of Etot on A. If Es A Gs then
Er and Etot is minimized with direct transmission. then the minimum of on A must occur at Es
lt Gs corresponding to cooperative transmission. Q. the denominator of is also negative on
the interval Es A. In order to determine if the minimum of on A occurs at the point Es Gs . i.
we rst establish that can have only one minimum on A by proving that is a strictly convex
function of Es on A. oth erwise the minimum occurs at Es Gs corresponding to direct
transmission. The second derivative of with respect to Es can be written as Etot arg min Etot
Etot H H Gs E tot Es Gr H Gs Es Note that the numerator of is a negative quantity not
dependent on Es . The optimal source energy allocation can be found by solving Es Etot .
the total energy required to satisfy the constraint SNRmrc can be written as HEs Gs Gs HEs
. hence Etot is always positive on A. Since Es H Gs gt and Gr gt . Gs . Using . The rst
derivative of evaluated at Es Gs can be written as Etot Gs Etot Es Gs Gs H Gs Gr H Gs Gr
Gs H This quantity is positive if and only if the condition of lt are satised. lt . we can
determine whether the unique mini mum of on A occurs at the point Es Gs by evaluating the
rst derivative of at this point. Wang et al.e. Otherwise. when the minimum of on A must occur
at Es Gs and direct transmission is optimum. Etot Es Er Es Gr H Gs Es Dene the interval A
HGs . Er gt and cooperative transmission minimizes Etot . hence the unique minimum of on
A must occur at Es lt Gs when lt . Computation of the partial derivative and algebraic
simplication yields HGs Es Gs Gs Es H HEs Gr H Gs Es Gr H Gs Es Es Etot By solving this
equation for Es . If the rst derivative is positive. We now derive the explicit solution to the
total energy minimization prob lem. Otherwise.
. Thus Gy gt a. the total energy can be minimized through coopera tive transmission. s s s
and Gy G H Gs H H s Gr y Gs y where y Gs H HEs Gs Es .s. where Ry G Gs Hy HG H Gs y
s s Sy Gs y Gs Hy H Gs y Ty y G H G H H G .Opportunistic Cooperation with ReceiverBased
Ratio Combining Strategy In the case lt . we have Etot FyGy. thus lim P X . Here. the
minimized total energy can be expressed as Gr Gs H Gs H HGs H H Etot Gs H Gr Gs H B
Proof of Proposition Proof. Substitute Es with y.s. Es The function Fy y Gs y Gs H G Gs Hy s
H Gs HG H Gs y s G H G H H G . Gr and H are exponentially distributed. the condition only
the y term has negative coecient. and hence has a unique minimum on A Etot E H Gs .
Hence.s.s. on A Fy gt a. The function Fy can be written as Etot Es gt a. To prove Etot is
convex. E tot is a function Gs s s of Es . where X denotes the squared channel amplitudes. s
s s Note that Ty for . H . By plugging Es into . on A Note that y . We will consider the
behavior of Ry . we will show that E gt a. H . Observe that Fy Ry Sy Ty. Note that the
squared channel amplitudes Gs . and Sy in following claims. where C . on C.
Thus. Sy y H Gs gt a.s.s. on C.s. it is only necessary to sy prove that sy gt a. we consider
the case when gt . When .s.s.s. Thus. Observe that y gt and sy is a quadratic equation. To
prove that Sy gt a.s. Wang et al. proof Observe that Sy is a cubic equation of one variable.
on C. It can be shown Gs H Gs H Gs G H lt a. s it is only necessary to prove that ry is
decreasing on C.s. on C. on C. When . First.. s y Thus. which implies Sy gt a. Observe that y
gt and r H HG H Gs gt a. Ry y H Gs gt a. on C.s. to prove Ry gt a. we consider the case
when gt . it is only necessary to prove that sy has no real root a. on C. ry gt a. on C.s. on C.
.s. It can be written as Sy ysy. hence sy is convex on C. on C. on C. It can be shown that ry
G Gs H lt a. where sy Gs y Gs Hy H Gs .s. where ry G Gs Hy HG H Gs . Claim Sy gt a.s. s s
First. It can be shown that y Gs gt a.s.s. Q. to prove sy gt a.s. Claim Ry gt a. It can be written
as Ry yry. s This implies that sy gt a. proof Observe that Ry is a quadratic equation of one
variable. this result implies Ry gt a.s. on C. on C.