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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.