Dynamic Resource Allocation for Multi Source-Destination Relay Networks

Transcription

1 Dynamic Resource Allocation for Multi Source-Destination Relay Networks Onur Sahin, Elza Erkip Electrical and Computer Engineering, Polytechnic University, Brooklyn, New York, USA Abstract We consider a wireless network consisting of multiple sources communicating with their corresponding destinations utilizing a single half-duplex relay. The goal is to minimize the outage probability of the total rate in the network by allocating transmission powers and durations as well as rates to the nodes based on instantaneous channel gains, while satisfying a total average network power. The sources are allowed to use the relay opportunistically, that is only when relaying reduces overall power as opposed to direct transmission. We investigate the effect of interference when all sources transmit simultaneously, and the impact of time division among the sources.we show that dynamic allocation and opportunistic transmission based on instantaneous channel states provide signicant reduction in outage compared to any constant resource allocation scheme. I. INTRODUCTION Wireless ad-hoc network design is challenging because of the arbitrary and rapidly changing topology and interference created by simultaneous communications. Moreover, it is shown in [] that the interference severely impacts the throughput of a wireless ad-hoc network as the number of nodes increase. This necessitates more robust and scalable network model such as a hierarchical architecture, in which some of the nodes in the network are more capable than the others in terms of signal reception and transmission. These nodes act as relays in the network, receiving information from the source nodes and forwarding to another relay via multi-hop transmission or to the desired destinations. On the other hand, power consumption, as well as throughput, is a key design parameter for wireless ad-hoc networks, since each node may have a limited battery life. Hence optimal power allocation for a given network model with respect to different performance metrics such as minimum outage, or maximum throughput provides signicant power savings [2], [3] leading to an increased system performance. In this work we consider a network in which a dedicated relay helps a source pair that wishes to communicate with a destination pair. Relay operates in half-duplex decode-and-forward fashion such that the reception and transmission modes take place in orthogonal time slots and the received signals are decoded and forwarded. Each source wishes to communicate to a distinct destination. Hence the channel model becomes the interference relay channel [4]. Since the capacity of the interference channel is not known in general, our goal in this paper is to describe some simple communication and resource allocation strategies Supported by NSF under Grant No.: for the fading scenario. We assume centralized control utilizing partial channel state information (CSI of the whole network to dynamically allocate powers, rates and transmission durations of all nodes. The partial CSI of the network consists of the channel gains, as phase information is harder to obtain and maintain. Even though estimating and distributing this partial network CSI will create overhead, the gains we observe suggest the importance of CSI and motivate further analysis with limited CSI such as quantized channel states [5]. As the performance metric, we consider outage probability for the total rate from the sources to the destinations. The resource allocation aims at minimizing the total outage probability while the total average source and relay transmission powers is kept constant. Even though minimizing total source and relay powers may not guarantee optimal individual power allocation for each node, it has signicant implications systemwise. As an example, this will minimize the total interference caused to other nodes in the network. In [3], it is assumed that the relay helps to sources in an orthogonal fashion, such that each different Source-Relay- Destination transmission takes place in non-overlapping channels in time (TDMA or frequency (FDMA. Our model is more general, the sources are allowed to transmit simultaneously and the relay also broadcasts the aggregate information to the destinations. Even though each destination will observe interference from the other source in this case, this scheme will allow the system to operate at the optimal points of the multiple access channel connecting the sources to the relay. Moreover, since TDMA is not able to achieve the broadcast capacity region for any power and time allocation, simultaneous transmission from the relay to the destinations will increase the system performance. In order to understand the effect of interference from the sources to the destinations, we also consider TDMA only for source transmissions along with dynamic power and time allocations which removes the effect of the interference at the destinations. To further minimize the outage probability of the system, in addition to dynamically allocated resources based on instantaneous CSI, we also allow sources to transmit opportunistically [6]. In [6], it was shown that in a single Source- Relay-Destination system, allowing the source to transmit either directly without using the relay during the whole time slot or to transmit via relay with optimal time and power allocation provides outage performance very close to the cut-set bound.

2 This motives us to use the relay opportunistically for the two sources and their corresponding destinations as well. In Section II, the system model is given. In Section III, we assume the sources transmit simultaneously and the relay transmits to the destinations during the next slot using superposition. Section IV analyzes the case in which sources transmit in orthogonal fashion (TDMA, however the relay still transmits simultaneously to both destinations. For comparison, Section IV also considers TDMA for relay-to-destination links as well with xed transmission times for all nodes in the network. In Section V, we give numerical results, provide comparisons and conclude the paper in Section VI. II. SYSTEM MODEL We consider a relay network with a single relay and two Source-Destination pairs (S i, D i i =, 2 as shown in Figure. This channel is also referred to as interference relay channel [4]. Each source only wishes to communicate with its dedicated destination opportunistically [6], such that the relay helps the source or the source transmits directly without help of the relay depending on whichever is more power efcient. We consider a half-duplex relay which decodes the signal received from S, S 2 or both and forwards the decoded signals to their corresponding destinations. The AWGN noises at the relay and destinations are independent with variances. Channels have independent Rayleigh fading, that is the channel power gains α i, γ i, β i and ξ i, i =, 2, are iid exponential with means (m αi, m γi, m βi, m ξi, respectively, where the means include the distance and shadowing effect. Here α i represents the channel from S i to R, γ i from R to D i and β i from S i to D i for i =, 2. Also ξ (ξ 2 represents the interference link from S (S 2 to D 2 (D. We will denote the network state, which includes the instantaneous power gain of each channel, (α, α 2, γ, γ 2, β, β 2, ξ, ξ 2 by θ. We assume a slow fading (quasi-static environment in which the channels stay constant for the duration of a channel frame. Centralized resource allocation is assumed using the network CSI θ. The resource allocation consists of power, time and rate adaptation. The transmission powers of S, S 2, R and total network power are denoted by P S, P S2, P R and P tot respectively. There is a long term total network power constraint such that P tot P total. In this work, we consider the total achievable end-to-end rate R T = R + R 2, where R i denotes the transmission rate from S i to D i for i =, 2. Our goal is to minimize the outage on the total rate, that is the probability that a total rate R T U cannot be supported subject to long term total network power constraint, P total. Following the analysis in [6] for each communication strategy, we rst consider the minimum total power, P tot,min (θ, needed to guarantee total rate R T U for each channel state θ. We then apply a threshold, P th, to P tot,min (θ such that the sources transmit only if P tot,min (θ is less than this threshold power level P th. For the channel states where P tot,min (θ exceeds the threshold, the sources are not allowed to transmit and the system will be in outage. Using [6], we argue that the proper choice of P th minimizes the outage probability for the chosen transmission scheme while satisfying the total power constraint. Fig.. System Model. Each source S i wishes to communicate with its destination D i. (α i, β i, γ i, ξ i denote instantaneous power gains, i =, 2. Fig. 2. Channel Allocation for simultaneous transmission at the sources: (a corresponds to Section III Case, (b Section III Case 2, (c Section III Case 3. III. SIMULTANEOUS TRANSMISSION AT THE SOURCES In this section, we consider simultaneous transmission at the sources and superposition of decoded information at the relay as shown in Figure 2. Using the opportunistic transmission scheme [6] and considering the half-duplex limitation of the relay, S and S 2 can exploit the relay appropriately depending on the network channel conditions as argued in the following three cases. Note that as in [6], since the phase information of the fading coefcients are not available, beamforming is not possible and it is suboptimal for the sources to continue transmission along with the relay under total power constraint. In order to simplify the notation, without loss of generality, it is assumed that γ > γ 2 throughout the analysis. Case : This case corresponds to both sources utilizing the relay (Fig. 2-(a. During the rst time slot (0 t t m, S and S 2 transmit simultaneously to R, which constitutes a multiple access channel (MAC from the perspective of the relay. Each destination also receives the signal coming from its corresponding source as well as interference from the other source during this period. When the transmission of the sources is completed, in the second time slot (t m < t, the relay transmits the decoded signals to the destinations using superimposed independent Gaussian codebooks. Our power allocation strategy will ensure that decoding is always possible. The destinations then combine the signals received during the rst and the second time slot. Note that the overall received signal at the destinations

3 follows the interference channel model. However, rather that trying to cancel the interference, we will treat it as part of the noise. Even though this may cause degradation in achievable rates, it simplies the analysis and provides a lower bound on the performance. Furthermore when the interference is low, this strategy still gives relatively good performance [8]. We will denote the relay to destination transmissions combined with source signals in the rst slot as modied broadcast channel (MBC where we see increase in the achievable rates due to source transmissions in the rst time slot. Since the relay decodes-and-forwards the received signals, in order to achieve individual rates R and R 2 leading to total rate R T U = R +R 2, (R, R 2 should be supported both at the MAC and at the MBC. The conditions on the supported transmission rates will be; R (S R t log( + α P s R (S 2 R 2 t log( + α 2P s2 R (S R + R (S 2 R 2 t log( + α P s + α 2 P s2 R (S,R D t log( + β P s ξ 2 P s2 + +( t log( + γ P R R (S2,R D2 2 t log( + β 2P s2 ξ P s + +( t log( + γ 2 P R2 γ 2 P R + ( where t is the time allocated for simultaneous transmission of (S, S 2 R and P R and P R2 are the relay powers dedicated relaying the signals of S and S 2, respectively. The superscripts of the rates denote corresponding transmissions. At a given channel state θ, we will denote the minimum total network power to achieve total rate R T U for this case as P (C tot. Since the transmission rates satisfying total rate R T U depends on t, minimum total network power for this case is; P (C t,p S,P S2,P R,P R2 (t P s + t P s2 + ( t P R + ( t P R2 such that R + R 2 R T U (2 where R = min (R (S R, R (S,R D and R 2 = min (R (S 2 R 2, R (S 2,R D 2 2 satisfy eqn. (. Case 2: Similar to single Source-Relay-Destination scheme given in [6], only one of the sources, say S, may decide to use the relay and the other source (S 2 in this case directly transmits to its destination. This would happen, for example if β < α, β 2 > α 2 and ξ =ξ 2 =0. In this case, the source which doesn't use the relay, S 2, transmits during the whole time slot directly to its destination (Fig. 2-(b, while S and R utilize time division as in Case. Since the signal coming from S 2 is not decoded by R, it will cause interference at R during the rst time slot as well as at the destinations. Also the signal transmitted by R will interfere with S 2 during the second time slot. Using these observations, and again treating interference as noise, the end-to-end transmission rates from (S D and (S 2 D 2, assuming R helps S and S 2 transmits directly to D 2 can be written as follows; R (S2 D2 2 t log( + β 2P s2 ξ P s + +( t log( + β 2P s2 γ 2 P R + R (S R t log ( + α P s α 2 P s2 + R (S,R D t log ( + β P s ξ 2 P s2 + +( t log ( + γ P R ξ 2 P s2 + (3 where t is the channel time allocated for the transmission of S to R. Again denoting the minimum total network power that achieves total rate R T U for a given channel state θ as P (C2 tot, we have; P (C2 t,p s,p s2,p R (t P s + P s2 + ( t P R such that R + R 2 R T U (4 where R = min (R (S R, R (S,R D, R 2 = R (S2 D2 2 satisfying eqn (3. Note that Case 2 would also include the scenario in which S transmits directly while S 2 uses the relay. Case 3: When both of the sources choose to directly transmit to their destinations without utilizing the relay, (Fig. 2-(c, we have the classical interference channel, whose capacity region is not known for all channel gains [9]. Again for simplicity not allowing the destinations to successively cancel the signals coming directly from the interfering source (which obviously will be suboptimal, achievable transmission rates for this case can be written as; R log ( + β P s, R 2 log ( + β 2P s2 (5 ξ 2 P s2 + ξ P s + Minimum total network power for this case is; P (C3 (P s + P s2 such that R + R 2 R T U (6 The central controller will choose Case, 2, or 3 depending on which one results in smaller total power, that is Ptot sim = min{p (C tot, P (C 2 tot, P (C 3 tot } In Section V, we obtain the minimum total network power Ptot sim satisfying total rate R T U for each channel state θ numerically and as discussed in Section II, apply a properly chosen threshold, P th to Ptot sim, at each channel state θ to obtain minimum outage probability vs average total network power for the simultaneous transmission scheme. Discussion: Since the optimizations in (2, (4, (6 are nonconvex, it is hard to obtain an analytical solution that minimizes total power. However to gain insights, we discuss some special cases for xed network state θ. First, consider the situation where the sources and the destinations are far away from each other. Then

4 the effect of the interference and direct links will be small due to path loss and can be neglected, and ξ i = β i = 0, i =, 2. Eq. ( suggests that the transmission rates have to operate at the intersection of the MAC region, (S, S 2 R, and BC region, R (D, D 2. Also, since the sources do not have the opportunity to transmit directly without using the relay, only Case can occur. For the MAC from sources to the relay, in order to maximize rate sum, total source power P s +P s2 is minimized by only letting the user with the best α i, i =, 2 transmit [0]. Moreover, for the BC from the relay to the destinations, the optimal power allocation at the relay, PR, maximizing rate sum suggests transmitting to the destination with highest γ i, user in this case [0]. Hence, if α > α 2 and γ > γ 2 then the total network power P tot (θ is minimized by only letting S transmit. This can be interpreted as a type of multiuser diversity for this multihop channel, which requires a user to have the best hops. However, when α < α 2 and γ > γ 2, then optimal allocation gives power and time to both of the users depending their channel qualities. The analytical solution can be obtained by parameterizing the total power in terms of optimization variables, (P s, t. A more general case occurs when the interference links are weak, i.e. ξ i = 0, i =, 2, but the direct links to desired destinations are present. These conditions may be valid when the destinations are close to their sources but further away from the interfering transmitters. In this case, the intersection of the MAC and MBC regions constitute the capacity region []. For the situation where none of the sources use the relay as in Case 3, then the system parameters can be allocated optimally using water-lling among the parallel links (S D and (S 2 D 2 [7]. However, when only one source (say S exploits the relay, while the other one (S 2 transmits directly as analyzed in Case 2, due to interference free nature of the system, we obrain two parallel channels (S, R, D and (S 2, D 2. Based on the results of [2] obtained for the orthogonal relay channel, we expect a water-lling type power allocation among these two parallel links for optimal resource allocation. IV. ORTHOGONAL TRANSMISSION AT THE SOURCES We have seen in Section III that solving the most general resource allocation problem is quite complex. In this section, we consider an alternative to simultaneous transmission, which orthogonal transmission from sources to the relay using timedivision. When S and S 2 transmit to the R in orthogonal time slots, (Fig. 3, instead of simultaneous fashion as in Section III, the destinations will not receive any direct interference from the sources. We rst consider R transmitting to the destinations simultaneously, since time-division for BC channel may cause an achievable region much smaller than simultaneous transmission especially when the asymmetry among the transmitted nodes increases. As suggested in Section III, opportunistic transmission can also be used in the TDMA mode. However, since S and S 2 are not allowed to transmit during the whole time slot due to interference constraints, the advantage of opportunistic communication cannot be fully exploited. In the case when both 0 S S2 2 S(S2 S t (a 0 t (c t2 S2(S 2(D 0 t t2 (b S2 2 R,D2 R (D2 Fig. 3. Channel Allocation for orthogonal transmission at the sources: (a corresponds to Section IV Case, (b Section IV Case 2, (c Section IV Case 3. of sources transmit directly to their destinations, the system simplies to two source-destination TDMA system. Case : In this case, both sources utilize the relay for transmission to their corresponding destinations (Fig. 3-(a. S transmits to R for 0 t t, S 2 transmits to R for t < t t 2 and R transmits the aggregate information for t 2 < t to the destinations using independent Gaussian codebooks with respect to the sources. Since the sources transmit in orthogonal time slots, their corresponding destinations receive non-interfered signals, and BC region of the R (D, D 2 is shifted accordingly. Using this observation and considering the decode-and-forward strategy of the relay, the transmission rates can be written as, R (S R t log ( + α P s R (S 2 R 2 (t 2 t log ( + α 2P s2 R (S,R D t log ( + β P s +( t 2 log ( + γ P R R (S2,R D2 2 (t 2 t log ( + β 2P s2 γ 2 P R2 +( t 2 log ( + (7 γ 2 P R + where t is the time allocated for the transmission of S R, (t 2 t is the time allocated for the transmission of S 2 R, and P R and P R2 are the relay powers dedicated for the transmission of S, S 2, respectively. Minimum total power satisfying total end-to-end rate R T U requires optimizing t, t 2, P s, P s2, P R and P R2 and can be written as; P (C t,t 2,P s,p s2,p R,P R2 (t P s + (t 2 t P s2 + ( t 2 P R + ( t 2 P R2 such that R + R 2 R T U (8 (R, R 2 satisfy eqn (7 with R =min (R (S R, R (S,R D and R 2 =min (R (S 2 R 2, R (S 2,R D 2 2. Case 2: In this case, only one source, say S uses R and the other source, S 2, directly transmits its signal to D 2. The time allocation of the scheme can be seen at Fig. 3-(b. Due to

5 Fig. 4. Channel Allocation for constant time transmission This corresponds to Case in eq ( and (7. Similarly, we can formulate case 2 and 3 where only one source or both transmit directly for constant time allocation strategy and derive corresponding outage probabilities. Minimum total power in this case will be, P (C2 t,t 2,P s,p s2,p R (t P s + (t 2 t P s2 + ( t 2 P R such that R + R 2 R T U (0 where R = min (R (S R, R (S,R D, R 2 = R (S2 D2 2 satisfy eqn (9. Case 3: In this case neither S nor S 2 utilize R and transmit in optimized orthogonal time slots (Fig. 3-(c. Then, R t log ( + β P s R 2 ( t log ( + β 2P s2 ( Minimum total power will satisfy the following such that; P (C3 t,p s,p s2 (t P s + ( t P s2 such that R + R 2 R T U (2 where (R, R 2 satises eqn (. Similar to Section III, for the orthogonal transmission from (S, S 2 R, the minimum total power satisfying end-to-end total rate R T U will be the minimum of the total powers given in Case-Case3 such that, Ptot orth = min{p (C tot, P (C 2 tot, P (C 3 tot }. Then P th and the corresponding outage probability can be computed using Ptot orth and the total power constraint. In order to understand the performance gain obtained by opportunistic transmission, we next consider a transmitting strategy similar to the one given in [3], in which the sources and relay transmit in non-overlapping and equal length time-slots (Fig. 4. Each source (S i is allowed to transmit directly to its destination (D i if the channel gain in between, β i, is better than the channel gain to the relay, α i. We also allow the nodes to dynamically allocate their transmission powers depending on the channel conditions for a given total network power constraint. When both S and S 2 use R, the end-to-end transmission rates for i =, 2 will be; R (S i R i R (S i,r D i i 4 log ( + α ip si 4 log ( + γ ip Ri + 4 log ( + β ip si V. SIMULATION RESULTS orthogonal transmission, S 2 does not cause interference at R, and In this section, the outage probability performance of the D 2 does not observe interference from R during (t 2 < t. resource allocation schemes are compared via simulations for The end-to-end transmission rates can be written as follows; the desired total rate R T U = bits/channel use. R (S2 D2 2 (t 2 t log ( + β 2P s2 In Figure 5, we consider a symmetric network model with power gain means (m αi, m γi, m βi, m ξi = R (S R t log ( + α P s (, 0.2, , In this case, simultaneous transmission of the sources as in Section III has better outage performance R (S,R D t log ( + β P s + ( t 2 log ( + γ P R than orthogonal transmission and constant time allocation N schemes of Section IV. The performance difference between o (9 simultaneous and orthogonal transmission at the sources occur due to the fact that in simultaneous transmission, the directly transmitting source S 2 (S has full access to the channel during the whole time slot (0 t (Section III, Case2, whereas in orthogonal transmission, each source is still constrained to transmit in non-overlapping time slots (Section IV, Case2 restricting the advantage of opportunistic transmission. The gains are present despite interference caused by simultaneous transmission. On the other hand, the same gure shows that simultaneous transmission scheme outperforms constant time allocation around 3 db at P out = 0 3 and orthogonal transmission at the sources provides 2 db power gain compared to constant time allocation scheme. The gure also shows the outage performance of direct transmission without a relay. For direct transmission the source-destination power gain means are equal to the symmetric network, i.e. m βi =0.0625, and the channel model is idealized such that the interference links are removed, that is ξ = ξ 2 = 0. However, as can be seen from Figure 5, even this idealistic direct transmission performance is 4 5 db worse than opportunistic schemes using relay. Figure 6 shows the outage performance of an asymmetric network, where S has better average channel gains compared to S 2. The power gain means of S are (m α, m γ, m β, m ξ =(4, 2, 0.08, and S 2 are (m α2, m γ2, m β2, m ξ2 =(0.3, 0.2, 0.06, In this case, it can be seen that simultaneous and orthogonal transmissions have closer outage performances, compared to the symmetric network model, however both of the schemes still outperform constant time transmission signicantly. Idealized direct transmission without relay performs even worse than symmetric network. Figure 7 shows the power allocation of simultaneous transmission at the sources at each node for the asymmetric network considered in Figure 6, corresponding to (0 3.5dB range of total network power. It can be seen that in order to obtain total transmission rate R T U =, most of the network power is allocated to source, as it has better average channel gain. Since S 2 is muted or transmits with very low power compared to S, simultaneous and orthogonal transmission schemes, given in Section III and Section IV, become equivalent and have similar outage performances. However, when total network power is increased, more power is allocated to S 2 as

6 0 0 0 Opportunistic simultaneous transmission Opportunistic orthogonal transmission Constant time transmission Direct transmission w/o relay (No intf Opportunistic simultaneous transmission Opportunistic orthogonal transmission Constant time transmission Direct transmission w/o relay (No intf. Outage Probability 0 2 Outage Probability Average Power (db Average Power (db Fig. 5. Symmetric network: (m αi, m γi, m βi, m ξi = (, 0.2, , i =, 2. R= bit/channel use. Fig. 6. Asymmetric network: (m α, m γ, m β, m ξ = (4, 2, 0.08, 0.004, (m α2, m γ2, m β2, m ξ2 = (0.3, 0.2, 0.06, R= bit/channel use. well as R begins allocating power to help S 2 and simultaneous transmission outperforms orthogonal transmission as shown in Figures 6 and P s P s2 P R P R2 VI. CONCLUSION In this paper, the outage performance of a network consisting multiple sources, single relay and multiple destinations is analyzed considering a total average network power constraint. We consider simultaneous transmission of the sources as well as TDMA, followed by a broadcast relay signal. Both sources and relay allocate transmission powers, rate and channel access times dynamically. Moreover each source is allowed to transmit to its corresponding destination either directly or using the relay. While TDMA based scheme removes the interference among the sources, it limits the gains due to opportunistic transmission, since sources do not have channel access all the time. Simulations show that, for symmetric networks, i.e. when both sources have comparable overall average channel gains, simultaneous transmission outperforms TDMA whereas both schemes signicantly perform better than equal time allocation. However, when the network favors a specic source, i.e, when one source has a signicantly better overall channel state with respect to the other, the performance difference between simultaneous and orthogonal transmission vanishes for lower total network powers; however both opportunistic schemes still perform signicantly better than constant time allocation scheme. For a possible extension, performance of the considered schemes with nite rate CSI, as well as distributed resource allocation can be investigated. REFERENCES [] P. Gupta and P. R. Kumar, The capacity of wireless networks IEEE Tran. Info. Theory, Vol. 46, pp , March [2] Z. Yang, A. Host-Madsen, Minimum Outage Probability Routing and Power Allocation in Wireless Ad-hoc Networks, IWCMC [3] S. Serbetli, A. Yener, Optimum Power Allocation for Relay Assisted F/TDMA Ad Hoc Networks, WirelessCom [4] H. Bolcskei and R. U. Nabar, Realizing MIMO gains without user cooperation in large single-antenna wireless networks, ISIT Node Powers Total Power Fig. 7. Node Power Allocation for the asymmetric network of Fig. 6, simultaneous transmission at the sources. (P R, P R2 are the relay power used for S and S 2, respectively. [5] K. K. Mukkavilli, Sabharwal, E. Erkip and B. Aazhang, On beamforming with nite rate feedback in multiple antenna systems, IEEE Tran. on Info. Theory, vol. 49, no.0, pp , October [6] D. Gunduz, E. Erkip, Opportunistic cooperation by dynamic resource allocation, To appear, journal available at elza/publications/opp05.pdf [7] T. Cover, J. Thomas, Elements of Information Theory, Wiley & Sons, New York, 99. [8] A. Carleial, Interference Channels, IEEE Tran. on Info. Theory Volume 24, Issue, pp , Jan 978 [9] G. Kramer, Review of Rate Regions for Interference Channels, IZS 2006 [0] D. Tse, P. Viswanath, Fundamentals od Wireless Communications, Cambridge University Press, [] A. Tajer, A. Nosratinia, A broadcasting relay for orthogonal multiuser channels, IEEE Globecom, Nov 2006 [2] Yingbin Liang, V.V. Veeravalli, Gaussian Orthogonal Relay Channels: Optimal Resource Allocation and Capacity IEEE Tran. on Info. Theory, Volume 5, Issue 9, pp.: Sept. 2005