Joint Power and Rate Adaptation aided Network-Coded PSK for Two-way Relaying over Fading Channels

Transcription

1 IEEE ICC Wireless Communications Symposium Joint Power and Rate Adaptation aided Network-Coded PSK for Two-way Relaying over Fading Channels Yanping Yang, Wei Chen, Ou Li, Ke Ke and Lajos Hanzo National Digital Switching System Engineering and Technological R&D Center, Zhengzhou, China, y.p.yang1986@gmail.com, ouli219@gmail.com, keke219@gmail.com Department of Electronic Engineering, Tsinghua University, Beijing, China, wchen@tsinghua.edu.cn School of Electronics and Computer Science, University of Southampton, UK, lh@ecs.soton.ac.uk Abstract Advanced downlink (DL decode-and-forward twoway relaying (DF-TWR is developed for the sake of maximizing the attainable spectral efficiency. Network-coded phase-shift keying (NC-PSK holds the potential of significantly increasing the data rate of the network, while adaptive modulation is a powerful technique of improving both the energy and spectral efficiency. Hence power-controlled adaptive NC-PSK has the potential of achieving performance enhancements over fading channels. Given this framework, based on the bit-error-ratio (BER bounds developed, we derive the closed-form spectral efficiency for a continuous-rate, continuous-power adaptive NC-PSK scheme, where both the transmit power and the transmit rate are optimized subject to specific power- and BER- constraints. We then conceive and investigate a discrete-rate scheme relying on a pair of solutions proposed for maximizing the throughput of the network. Our simulation results reveal that the proposed schemes are capable of achieving a higher spectral efficiency than their fixed-power counterparts. I. INTRODUCTION Wireless communication networks rely on energy- and bandwidth- efficient communication techniques for communicating over fading channels. Two-way relaying (TWR, also known as bi-directional relaying, constitutes an appealing technique of improving the throughput of the existing communication systems. Li, Yeung and Cai [1] proposed linear Network Coding (NC and showed that the multicast rate of a network can be increased, if efficient channel- and/or NC coding is invoked by the network nodes. Inspired by this work, a large variety of NC-aided techniques have been invoked at the physical layer conceived for the TWR scenario [2 6]. Wu et al. [2] exploited the combination of NC and the broadcast nature of the wireless medium for improving the bandwidth-efficiency, with the aid of superposition assisted win-win cooperation of the source and relay. Xie [3] extended the NC philosophy to multiuser coding problems, where the achievable rate regions of both a two-way relay channel and of a three-way broadcast This paper is partially supported by the National Basic Research Program of China (973 Program No. 213CB3366, NSFC Excellent Young Investigator Award No , NSFC project No , the National High Technology Research and Development Program of China (863 Program No. 212AA12166, 973 Program No. 212CB316, and Beijing nova program No.Z channel were determined. Specifically, both Wu et al. [2] and Xie [3] investigated the DL capacity of asymmetric DF-TWR. Most recently, inter-relay interference cancellation and subcarrier matching techniques have been exploited to further improve the spectral efficiency of cooperative relaying, respectively in [] and [5]. These contributions laid a foundation to the research of asymmetric NC-aided transmissions. As a further advance, both Network-Coded Modulation (NCM and a NC oriented maximum ratio combining (MRC receiver technique were conceived for DF-TWR for improving both the throughput as well as the achievable spatial diversity gain at a low complexity [6], which circumvented the asymmetric transmission problems of DF-TWR. More specifically, the authors of [6] proposed constant-power, variable-rate adaptive NCM for the sake of maximising the attainable data rate of DF-TWR subject to certain BER constraints. The above-mentioned attractive techniques motivate us to study variable-power, variable-rate NCM conceived for DF- TWR over fading channels. The performance of wireless links is severely degraded by the deleterious effects of channel fading, which considerably limits the overall system performance. An alternative technique of mitigating the effects of interference and fading is to rely on adaptive modulation [7 11]. Hanzo et al. designed near-instantaneously adaptive modulation techniques in [7 9]. Based on Shannon s capacity and on BER bounds, Goldsmith et al. [1, 11] investigated adaptive modulation schemes for single-user flat-fading channels, where both the data rate and the transmit power were varied so as to maximize the achievable spectral efficiency, whilst maintaining a constant BER. These adaptive modulation contributions studied single-user transmissions, which prompt us to develop adaptive modulation for DF-TWR. Since adaptive modulation is quite effective in terms of improving the achievable system throughput of conventional NCM [6], we intrinsically and inherently amalgamate these techniques. The authors of [12] investigated the achievable channel capacity relying on careful power allocation in the DL of time-varying TWR, where both the source-relay (SR and relay-destination (RD links are referred to as DL. Therefore we intend to investigate the resource allocation of DF-TWR, relying on the /15/$ IEEE 1976

2 IEEE ICC Wireless Communications Symposium Uplink: Multiple Access (MA Stage Downlink: Broadcast (BC Stage The 1st time-slot The 2nd time-slot The 3rd time-slot The 3rd time-slot DN1 Relay DN2 w 1 w 2 TRANSMITTER Encoder (Network Coded Modulation Power Control S(r 1,r 2 [t] x [t] x [t] FEEDBACK CHANNEL 1 CHANNEL 1 y 1 [t] RECEIVER1 Channel Estimator 1 Decoder 1 2 Decoder 2 1 DN1 Destination Node 1 DN2 Destination Node 2 Fig. 1. System Model of Bi-Directional Relaying x [t] CHANNEL 2 y 2 [t] FEEDBACK CHANNEL 2 Channel Estimator 2 RECEIVER2 BER bounds developed in [13] and the results of [11]. Inspired by the idea of intrinsically amalgamating NCM [6] and adaptive modulation [11], we propose nearinstantaneously adaptive NC-PSK for the DL of DF-TWR, which can be regarded as the joint optimization of the transmit power and rate, subject to specific average power and BER constraints. The key challenge of this design lies in the fact that the relay node (RN broadcasts its signals to two receiver nodes simultaneously, therefore the same transmit power has to adapt to a pair of DL channel conditions at the same time, instead of a single one. Thus the primary design goal of the proposed scheme is to optimize both the transmit power and transmission rates for the sake of maximizing the attainable bandwidth efficiency, while satisfying both the average power and BER constraints. Based on the power allocation policy of [12] and on the BER bounds [13], we derive the closedform solution of bandwidth efficiency for our continuousrate, continuous-power NC-PSK scheme. We then conceive a practical discrete-rate, continuous-power NC-PSK scheme, relying on the set-partitioning philosophy of [11]. Compared to NC-PSK applied to fixed-power schemes, a higher bandwidth efficiency is achieved, because our adaptive design exploits the receiver s channel estimates at the transmitter. Therefore, it may be concluded that adaptive NC-PSK modulation holds the potential of significantly improving the bandwidth efficiency of the system for transmission over fading channels. II. SYSTEM MODEL Consider a typical DF-TWR system designed for bidirectional relaying as seen in Fig. 1, where a common RN broadcasts its signals to a pair of destination nodes (DN1 and DN2 during the DL process. Each DN has a prior knowledge of its own message intended for the other. Based on the abovementioned TWR system model, we conceive the adaptive NC- PSK system model of Fig. 2, where the transmitter dynamically adjusts both its power and constellation sizes to cope with the pair of DL channel variations. The receiver design for the NC-PSK has already been well studied in [6]. Before introducing our system s structure, we first present the simplifying assumptions adopted in this paper: A1 We consider slowly-varying non-dispersive fading channels. If the channel is changing faster than it can be estimated and fed back to the transmitter, adaptive techniques Fig. 2. System Model of Adaptive NC-QAM/PSK for DF-TWR will perform poorly. A main emphasis in this paperis on Rayleigh fading channels. A2 Perfect channel state information is available both at the RN and DNs. The feedback channel is assumed to be error free and has no latency, which could be at least approximately satisfied by using a low-delay feedback link relying on powerful error control. Thus the feedback path delays are set to τ i =, i = 1, 2, as in [11]. A3 The system employs linear modulation, where the adaptation takes place at integer multiples of the symbol rate of R s = 1/T s, with T s denoting the symbol duration. It is also assumed that the system uses ideal Nyquist data pulses, where the bandwidth is B = 1/T s. Furthermore, we assume having a discrete-time DL channel model associated with stationary and ergodic time-varying gains of g i [t], i = 1, 2 and additive white Gaussian noise (AWGN n i [t]. We now describe the fundamental framework of adaptive NC-PSK scheme and characterize its operation for transmission over time-varying fading channels. In the static asymmetric DF-TWR DL [6], the equivalent baseband signals received at the coherent receiver of the DNs are represented by Y i = h i X + Z i, i = 1, 2, (1 where the channel gains are denoted by h i 2 = g i, with g i representing the power gains. The transmit symbol at the RN is denoted by X, while Z i denotes the AWGN at each DN. We consider a discrete-time DL channel, where t denotes discrete time instants. Then the previous symbol X will be represented as x [t], while Y i will be represented by y i [t], i = 1, 2. The messages x [t] received at the RN will be processed and then broadcast to both DNs using NC-PSK. Let us now describe our adaptive transmission scheme seen in Fig. 2. Again, we consider discrete-time flat fading channels obeying the assumptions A1-A3, where the transmitter dynamically adjust both its transmit power and transmit rates according to the power gains g 1 [t] and g 2 [t] signalled to it from the pair of DL receivers (DN1 and DN2. Let us denote the average transmit power by S, the noise density of n i [t] by N /2, the channel gain by g i [t] and the average 1977

3 IEEE ICC Wireless Communications Symposium channel gain by g. For a constant transmit power S, the pair of instantaneous received signal-to-noise ratios (SNR is γ i [t] = Sg i [t]/(n i B. Upon normalization by S, we can assume that g = 1. Then the average received SNRs are γ i = S/(N i B. We denote the probability distribution of the received SNR by p(γ i = p(γ i [t] = γ i. In this paper, the fading distributions p(γ i are assumed to be exponential (Rayleigh fading. When the context is unambiguous, we will omit the time reference t related to n i, g i, γ i and γ i. III. ADAPTIVE NC-PSK MODULATION FOR DF-TWR In this section we will further present the specific form of adaptive NC-PSK designed for the DF-TWR DL. A. The Generation of NC-PSK Symbols at the Transmitter Without loss of generality, we assume that the transmit constellation sizes are denoted by M 1, M 2, let M 2 M 1, M 2 /M 1 N. The messages w 1, w 2 to be transmitted from the pair of source nodes will be merged into a single signal denoted by X using the modulo two operation at the RN [6]. For NC-PSK, w 1, w 2 will be mapped to the symbols χ 1, χ 2 chosen from a normalised M-ary PSK (MPSK constellation as in χ i = cos θ i + j sin θ i : θ i Θ i }, where we have Θ i =, 2π M i,..., 2 (M i 1 π M i }, i = 1, 2. (2 Given the phases θ 1 and θ 2, the transmitter generates an NC- PSK symbol formulated as X = E s (cos θ+j sin θ, (3 where E s denotes the symbol energy, while the symbol s phase θ is given by θ =θ 1 +θ 2 mod 2π. ( For time-varying fading channels, the modulated signals will be represented by the signal sequence. B. Continuous-Rate, Continuous-Power Adaptive NC-PSK 1 Description of Power and BER Constraints: The capacity of fading channels encountered in DF-TWR is limited by the available transmit power and bandwidth. Let S(γ 1, γ 2 denote the transmit power, which is a function of the instantaneous SNR γ 1 and γ 2, subject to the average power constraint of S(γ 1, γ 2 p(γ 1 p(γ 2 dγ 1 dγ 2 =, (5 where p(γ 1 is independent of p(γ 2. When considering a Rayleigh fading channel, for example, we have p(γ i = 1 γ i e γi/γi, i = 1, 2, (6 where γ i = ST s / N = E s / N denotes the average SNR per symbol, T s = 1/B. The theoretical BER expressions of [12] contain the Gaussian Q-function [13], which is hard to invert. Therefore in contrast to the information theoretic discussions of [12], our proposed adaptive NC-PSK schemes rely on BER bounds [13, Eq. 9.9]. The BER performance of NC-PSK matches well with that of the theoretical BER expressions provided in [6]. Thus the BER bounds have the form of P b1 (γ 1 c 1 exp P b2 (γ 2 c 1 exp [ ] S(γ c 2γ 1,γ k(γ 1 c [ ] S(γ c 2 γ 1,γ 2 2, 2 k(γ 2 c where c 1, c 2 and are positive constants, while c is a real constant. The SNRs at the DN s receivers now become γ i S (γ 1, γ 2 /, i = 1, 2. The transmit rates k(γ1, k(γ 2 hence obey: k(γ i = log 2M i (γ i, i = 1, 2, (8 where M i (γ i represents the constellation sizes. For MPSK, we have c 1 =.5, c 2 = 6, = 1.9, c = 1 [13, Eq. 9.9]. In order to facilitate our forthcoming discussions and calculations, Eq. (7 may be reformulated as S(γ 1, γ 2 M i (γ i c + K i γ i, i = 1, 2, (9 where we have K i = c 2 /ln (P bi /c 1. We can readily obtain M i (γ i or k (γ i as a function of P bi and S(γ 1, γ 2. For our continuous-rate NC-PSK scheme, the pair of inequalities in Eq. (9 are capable of simultaneously satisfying the equality conditions. We then have M 1 (γ 1 = 1 + K 1 γ 1 S(γ 1,γ 2 M 2 (γ 2 = 1 + K 2 γ 2 S(γ 1,γ 2. (7 (1 2 Formulation of the Continuous-Rate Optimization Problem: In the downlink of DF-TWR, where the RN broadcasts its signals to both DN1 and DN2, the receiver-side SNR γ 1, γ 2 fluctuates as a function of time. We adjust the transmit power S(γ 1, γ 2 according to γ 1, γ 2, under the average power constraint of. Thus, our optimization problem can be formulated as that of maximizing the bandwidth efficiency of adaptive NCM. Let R [γ 1, γ 2, S (γ 1, γ 2 ] denote the available rate as a function of both γ 1, γ 2 and of S(γ 1, γ 2, which yields: R [γ 1, γ 2, S (γ 1, γ 2 ] = 2 i=1 ω i log 2 M i (γ i, i = 1, 2, (11 where M i (γ i is given by Eq. (1 and ω 1 denotes the significance of the DN1 channel, whilst that of the DN2 channel is given by ω 2. Naturally, we have ω 1 +ω 2 = 1 and ω i 1, i = 1, 2. The achievable bandwidth efficiency is obtained by integrating the rate function over the fading region D. We then formulate the optimization problem as follows: maximize R B = R [γ 1, γ 2, S (γ 1, γ 2 ] p (γ 1 p (γ 2 dγ 1 dγ 2 D subject to S(γ 1, γ 2 p(γ 1 p(γ 2 dγ 1 dγ 2 = D S(γ 1, γ 2, (12 where D = R 2,, S(γ1, γ 2, γ i, B, p(γ i were defined in Section II as part of our system model. 1978

4 IEEE ICC Wireless Communications Symposium 3 Solution to the Continuous-Rate Optimization Problem: The logarithmic functions in Eq. (11 are concave and so is their sum, therefore Eq. (12 constitutes a convex optimization problem. We form the Lagrangian by exploiting the Karush- Kuhn-Tucker (KKT condition yielding: J[S(γ 1,γ 2 ]= 2 ( S(γ 1,γ 2 ω i log 2 1+K i γ i p(γ 1 p(γ 2 dγ 1 dγ 2 i=1 ( + υ S(γ 1,γ 2 p(γ 1 p(γ 2 dγ 1 dγ 2 + µ S(γ 1,γ 2 p(γ 1 p(γ 2 dγ 1 dγ 2, (13 where υ and µ are Lagrange multipliers. Upon differentiating the Lagrangian and setting the resultant derivative to zero, we arrive at: [ 2 ( 1/ ln 2 K ω iγ i i S(γ i=1 1+K iγ 1,γ 2 i υ +µ ]p(γ 1 p(γ 2 = µ S(γ 1, γ 2 = S(γ 1, γ 2 µ. (1 Solving Eq. (1 for S(γ 1, γ 2 under the relevant power constraint yields the complementary slack condition υ 1 and the power adaptation policy that maximizes Eq. (11 as seen in Eq. (15. Upon substituting the channel estimates and the power adaptation policy of Eq. (15 back into Eq. (12, we arrive at the jointly-optimized cutoff fade depth υ, below which the transmissions are disabled. As a result the maximum bandwidth efficiency can be achieved for the parameters γ 1, γ 2, p(γ 1, p(γ 2, ω 1, ω 2 and P bi. C. Discrete-Rate, Continuous-Power Adaptive NC-PSK 1 Power Control Policy of the Discrete-Rate Scheme: Based on the optimization problem formulated in previous subsection, we continue our study by conceiving a discreterate, continuous-power NC-PSK scheme. Explicitly, in the traditional single-user continuous-rate adaptation scheme we have to find the optimal cutoff fade depth parameter υ [11], whilst in the proposed discrete-rate scheme, our goal is that of finding the joint optimal power and rate for the pair of independent fading distributions of the RN-DN1 and RN- DN2 links. In this joint-optimization scheme destined for the receivers DN1 and DN2, the transmit rates are denoted by k 1,η and k 2,δ, which directly depend on the constellation sizes M 1,η and M 2,δ as follows: k1,η = log2m1,η k 2,δ = log 2 M (16 2,δ. 1 Firstly, let µ >, S (γ 1, γ 2 =, we have ω 1Kγ 1 + ω 2Kγ 2 υ + µ = υ > ω 1Kγ 1 +ω 2 Kγ 2. Secondly, let µ =, S (γ 1, γ 2 >, we have υ ω = ( 1 Kγ 1 + ( ω 2 Kγ 2 < ω 1 Kγ 1 +ω 2 Kγ 2 1+Kγ 1 S(γ 1,γ 2 S 1+Kγ 2 S(γ 1,γ 2 S. Finally, the critical value is classified into the first case. For each receiver side DN1 and DN2, we adopt the single-user partitioning method of [11]. Therefore we obtain the discrete sets of MPSK constellations as M 1 = M 1,,..., M 1,N1 1}, M 2 = M 2,,..., M 2,N2 1}, with M 1, = and M 2, =. The receiver-side SNR distributions are then divided into N 1 and N 2 fading magnitude regions denoted by R 1,n1 = [γ 1,n1 1, γ 1,n1, n 1 =,..., N 1 1, R 2,n2 = [γ 2,n2 1, γ 2,n2, n 2 =,..., N 2 1. We accordingly activate the fixed constellation sizes of M 1,n1, M 2,n2 when the receiver side SNRs obey γ 1 R 1,n1, γ 2 R 2,n2. Based on the classic fading-magnitude partitioning method [11] and on the basic optimization problem of Eq. (12, we then formulated our basic discrete-rate scheme for DF-TWR DL as follows. We first discuss the associated power control policy conceived for this joint-optimization scheme. Let S ηδ (γ 1, γ 2, η, 1,, n 1 }, δ, 1,, n 2 } denote the RN s transmit power for γ 1 R 1,n1, γ 2 R 2,n2. From Eq. (9 we arrive at: Sηδ (γ 1,γ 2 M 1,η c K 1 γ 1 S ηδ (γ 1,γ 2 M 2,δ c K 2 γ 2, (17 where γ 1, γ 2, c, K 1 and K 2 were derived in the previous section. In contrast to our continuous-rate scheme of Section III B, the inequalities in Eq. (17 cannot assume equality at the same time. Since the rates only have discrete values, a fixed S ηδ (γ 1, γ 2 fails to satisfy both equations simultaneously, except when we have γ 1 =γ 2, which is practically impossible in time-varying fading channels. This indicates that there is either an inevitable power-loss or a rate-loss. Let us now continue by making some reasonable adjustments to our power control policy. Let S ηδ (γ 1, γ 2 = max M1,η c, M } 2,δ c,. (18 K 1 γ 1 K 2 γ 2 2 Formulation of the Discrete-Rate Optimization Problem: According to Eq. (12, the optimization problem can be distilled down to maximizing N1 1 R B = N 2 1 (ω 1 k 1,η +ω 2 k 2,δ p (γ 1 dγ 1 p (γ 2 dγ 2 γ 1,η 1 γ 2,δ 1 η= δ= subject to N 1 1 η= N 2 1 δ= D η,δ S ηδ (γ 1,γ 2 S D η,δ Dη,δ = ϕ D η,δ = R 2, η δ γ1,η γ2,δ p (γ 1 p (γ 2 dγ 1 dγ 2 = 1 (19 (2 where D η,δ and D η,δ denote the different regions corresponding to the different transmit rates of k 1,η, k 2,δ and k 1,η, k 2,δ. To find the optimal fading-magnitude switching thresholds for each DN, we may also formulate the Lagrangian with the aid of the KKT conditions. However, the shape of D η,δ obeys arbitrary quadrilaterals, therefore the discrete-rate optimization problem becomes excessively complex to be solved with the aid of general optimization methods. Inspired by the basic 1979

5 IEEE ICC Wireless Communications Symposium S(γ 1, γ 2 ( c 2 ( K 1γ 1 + c K 2γ 2 1 υ ln 2 υ K 1K 2γ 1γ 2 υ c 2 ω1ck1γ1+ω2ck2γ2 1 = 2, υ ω 1K 1 γ 1 +ω 2 K 2 γ 2 c ln 2. 1 ln 2 + 2υ ln 2 c 2K 1γ 1 c 2K 2γ 2, υ < ω1k1γ1+ω2k2γ2 c ln 2 (15 set-partition algorithm of [11], without changing the nature of the problem, we make some adjustments for our problem by restricting the fading-magnitude regions corresponding to different MPSK mode into rectangular areas, as shown in Fig. 3. In each rectangular area, we use the constellation sizes M 1,η, η N for DN1 and M 1,δ, δ N for DN2, which determine the attainable transmission rates. Therefore the constraint conditions of Eq. (2 can be rewritten as: N 1 1N 2 1 γ1,η η= δ= γ 1,η 1 γ2,δ γ 2,δ 1 S ηδ (γ 1,γ 2 p(γ 1 p(γ 2 dγ 1 dγ 2 =1 < γ 1, < < γ 1,η 1 < γ 1,η < < γ 1,N1 1 < γ 2, < < γ 2,δ 1 < γ 2,δ < < γ 2,N2 1, (21 where γ 1,η and γ 1,j denote the rectangular fading region boundaries, where again k 1,η, k 2,δ, p (γ 1, p (γ 2, ω 1 and ω 2 were derived in the first part of Section III C. 3 Solution to the Discrete-Rate Optimization Problem: The corresponding power adaptation policy is the same as that of Eq. (18. Upon substituting Eq. (18 into Eq. (21, we arrive at the power constraint for our discrete-rate scheme. An intuitive interpretation of this constraint is as follows. Throughout the entire fading-magnitude region, given γ 1, γ 2 and S ηδ (γ 1, γ 2, the discrete transmit rates destined for DN1 and DN2 cannot reach their maximum value at the same time. However, in the context of joint optimization, Eq. (18 facilitates that at least one equality is satisfied in Eq. (17, which implies that if one of the user s rate and power achieves the optimal match, the other user will have a rate determined by the maximum constellation size it can reach. The key point of our discrete-rate scheme is to obtain the optimal MPSK partitions R 1,η = [γ 1,η 1, γ 1,η, η =,..., N 1 1 and R 2,δ = [γ 2,δ 1, γ 2,δ, δ =,..., N 2 1, which are jointly determined by the average power constraint and the fading distributions. There is no closed-form solution to this kind of problem [11]. However, similarly to the approach of [11], we conceive a numerical search algorithm for finding the optimal MPSK boundaries R 1,η and R 2,δ. This may require a large amount of calculations. However, once the optimal Fig. 3. The different-throughput MPSK regions boundaries have been found, they can be used without realtime calculations. IV. PERFORMANCE RESULTS In this section, a range of representative numerical results are presented for validating our theoretical analysis. Our emphasis is on the spectral efficiency of variable-rate, variable-power NC-PSK. Therefore both the continuous-rate and discrete-rate adaptive NC-PSK schemes are compared to their respective benchmark schemes for demonstrating its potential. Specifically, we invoke the adaptive single-user MPSK scheme of [13] and the Shannon capacity based jointoptimization schemes [1, 12] as our benchmarks, which are described as Scenario 1-7 in Table I. We unify the simulation parameters for both the continuousrate and the discrete-rate schemes as follows: B1 Let us focus on Rayleigh fading channels, whose distributions p (γ i are given by Eq. (6. The near-instantaneous SNR fluctuations within a dynamic range, which are set to be 1 times the average SNR, that is γ i [, 1 γ i ]. B2 Let the unified parameters be γ i = [1, 2, 3,, 5, 1, 15, 3, 5, 1, 2, 316], i = 1, 2, P b1 = P b2 = 1 3, B = 1, S = 1 and ω 1 = ω 2 =.5. B3 For our discrete-rate scheme, we divide the dynamic range of the fading into four regions associated with TABLE I SCENARIOS AND UNIFIED PARAMETERS FOR PSK Scenarios Rate and Power Strategies System Model Bound BER Unified Parameters Scenario 1 AWGN Channel Capacity Scenario 2 Optimal Rate and Power Adaptation [12] Single-User p(γ i = 1 γ e γ i/γ i, i = 1, 2. i Scenario 3 Optimal Rate and Power Adaptation DF-TWR Shannon Bound S = 1, B = 1 Scenario Optimal Rate and Constant Power [12] Single-User ω 1 = ω 2 =.5 Scenario 5 Optimal Rate and Power Adaptive MPSK [12] Single-User P b =1 3 γ i [, 1 γ i ], i = 1, 2 Scenario 6 Optimal Rate and Power Adaptive M-ary NC-PSK DF-TWR BER Bound P b1 =P b2 =1 3 γ i = 1,2,3,,5,1,15,3,1,2,316} Scenario 7 Optimal Rate and Constant Power Single-User P b =

6 IEEE ICC Wireless Communications Symposium Throughput (bps/hz Throughput(bps/Hz Scenario 1 Scenario 2 Scenario 3 Scenario Scenario 5 Scenario 6 Scenario Average SNR(dB Fig Comparison of Scenarios 1-7 in terms of their throughput (PSK Scenario 5 Scenario 6 Discrete-Rate Adaptive MPSK(Single-User Discrete-Rate Adaptive NC-PSK(DF-TWR Average SNR(dB Fig. 5. Throughput of our continuous-rate adaptive NC-PSK, discrete-rate adaptive NC-PSK (Scenario 5, 6 of Fig., M =, 2,, 8} M i =, 2,, 8}, i = 1, 2. Fig. includes the benchmarks of [1], [12], [13], as well as Eq. (15 derived for our NC-PSK scheme as a function of the average received SNR for transmission over Rayleigh fading channels. Several observations are worth discussing. Firstly, our adaptive NC-PSK is capable of approaching both the capacities of our proposed continuous-rate adaptive scheme, as well as of the schemes proposed in [12] and those of the single-user adaptation proposed in [13]. This is quite valuable, because instead of a single-user link we are supporting a bidirectional network-coded scenario. Secondly, both our schemes and the scheme proposed in [12] perform better than MPSK operating without power adaptation observe Scenario 7 in Fig.. Finally, upon increasing the average SNR, the discrepancy between our proposed schemes and the single-user adaptive modulation schemes of [13] tends to narrow. Fig. 5 characterizes the performance of both our discreterate NC-PSK scheme and of the single-user adaptive scheme of [13]. From Fig. 5 we confirm that the performance of our discrete-rate scheme approaches that of the single-user discrete-rate MPSK schemes proposed in [13], whilst supporting bi-directional NC. Our simulation results also indicate that the gaps between our proposed scheme and the single-user adaptation methods of [13] tend to decrease upon increasing of the average SNR. Moreover, by increasing the number of discrete signal constellations N i, a better match with the continuous-rate adaptation scheme will be obtained, hence resulting in a higher bandwidth efficiency. Let us now conclude by considering both the powerallocation and rate-adaptation policy for a specific scenario. Using the parameters of γ i = 1, i = 1, 2, γ i [, 1], w i =.5, P bi = 1 3, S = 1, B = 1, M i =, 2,, 8}, i = 1, 2 for MPSK, we may characterize the discrete-rate adaption policies as functions of γ 1 and γ 2 for four fading regions corresponding to four adaptive MPSK strategies. We then obtained R 1 = [,.5, 1.1, 2.5, 1], R 2 = [,.5, 1.1, 2.5, 1] through solving Eqs. (18, (19 and (21. The corresponding maximum rate is.5375 bps/hz. V. CONCLUSIONS In this paper, a near-instantaneously adaptive NC-PSK design was conceived for the DL of DF-TWR systems, relying on perfect CSI. Both continuous-rate and discrete-rate adaptation schemes were proposed for the sake of maximising the bandwidth efficiency of the system. It is shown that the performance of our designs is close to those of single-user transmission, despite the fact that our proposed schemes are capable of supporting more complicated TWR scenarios. Our simulation results demonstrated that the proposed variablerate, variable-power NC-PSK DF-TWR schemes are capable of obtaining a higher spectral efficiency compared to the benchmark scheme operating without power adaptation. REFERENCES [1] S.-Y. Li, R. W. Yeung, and N. Cai, Linear network coding, IEEE Trans. Inf. Theory., vol. 9, no. 2, pp , Feb. 23. [2] Y. Wu, P. A. Chou, S.-Y. Kung et al., Information exchange in wireless networks with network coding and physical-layer broadcast, Microsoft Research, Redmond, WA, USA, Tech. Rep. MSR-TR-2, 2. [3] L.-L. Xie, Network coding and random binning for multi-user channels, in Proc. 1th Can. Workshop on Inf. 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