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1 This document is donloaded from DR-NTU, Nanyang Technological University Library, Singapore. Title Amplify-and-forard based to-ay relay ARQ system ith relay combination Author(s) Luo, Sheng; Teh, Kah Chan Citation Luo, S., & Teh, K. C. (2015). Amplify-and-forard based to-ay relay ARQ system ith relay combination. IEEE communications letters, 19(2), Date 2014 URL Rights 2014 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating ne collective orks, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this ork in other orks. The published version is available at: [

2 1 Amplify-and-Forard Based To-Way Relay ARQ System ith Relay Combination Sheng Luo and Kah Chan Teh, Senior Member, IEEE Abstract In this letter, e propose an automatic repeat request (ARQ) protocol for the amplify-and-forard (AF) based to-ay relay system. In the proposed ARQ scheme, the ARQ process is carried out by the relay node (RN). We assume that the RN has one buffer and it can store the signals received. With the help of the buffer, the RN can combine the signals received from different slots and choose to transmit or receive adaptively. To combining methods, namely the imal sum signal-to-noise ratio combining (MSC) scheme and the imal minimum signal-tonoise ratio combining (MMC) scheme, are proposed, hich are shon to improve the reliability of the relaying systems. Index Terms Cooperative relaying, to-ay relay ARQ, bufferaided relaying. I. INTRODUCTION Recently, a lot of interest has been dran by the to-ay relaying scheme as it can effectively improve the efficiency of a relaying system by exploiting physical layer netork coding [1]. To enhance its reliability, automatic repeat request (ARQ) protocol as adapted in [2] [4]. In [2], the throughput of a toay relay ARQ system ith and ithout a direct link beteen the to users as compared. In [3] and [4], the throughput of different ARQ protocols for a to-ay relay system as investigated. In all these to-ay relay ARQ schemes, the decode-and-forard (DF) approach as used at the relay node (RN). In [5], the amplify-and-forard (AF) based to-ay relay ARQ protocol as investigated. The diversity multiplexing tradeoff of the AF-based to-ay relay ARQ scheme as obtained. In this letter, e propose an ARQ protocol for the AF-based to-ay relay system. Our scheme is different from the scheme of [5] from the folloing aspects: 1) The ARQ process of [5] is carried out by the users by feeding back positive or negative acknoledgement (ACK). In the proposed scheme, the ARQ process is carried out by the RN. 2) In our ork, e assume that the RN has one buffer and it can combine the signal received in several slots, thus our scheme can benefit from some poer gains. To the best of our knoledge, the combining strategies of the RN in a to-ay relay ARQ system have not been investigated before. The buffer-aided to-ay relaying as also investigated in [6] [8]. In these orks, the DF method as adopted and the RN sitches beteen different orking modes adaptively to imize the system throughput. Inspired by these orks, the RN in our proposed scheme chooses to receive or to transmit adaptively based on the channel quality at each S. Luo and K. C. Teh are ith School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore. SLU- O002@e.ntu.edu.sg; EKCTeh@ntu.edu.sg. Fig. 1. h1k h2k S R S2 1 A to-ay relay channel. time slot. Numerical results sho that our proposed scheme can achieve higher diversity order than the scheme of [5]. In addition, to combining methods, the imal sum signal-tonoise ratio (SNR) combining (MSC) scheme and the imal minimum SNR combining (MMC) scheme are proposed, hich are shon to improve the reliability of the system. II. SYSTEM MODEL AND ASSUMPTIONS A three-node system in hich to single-antenna users S 1, S 2 exchange information ith the help of a RN is considered. As there is no direct link beteen the users, S 1 and S 2 communicate only through R. We also assume that all nodes are half-duplex. Each time frame is divided into slots of equal length and block flat fading channel in hich the channel coefficients are constant during one time slot and change independently from one slot to another is assumed. We denote h 1k and h 2k as the channel gains beteen S 1, S 2 and R in the kth time slot, respectively. The channels are assumed to be reciprocal as time-division-duplex method is used. Moreover, similar to [6] [8], e assume that the RN has the knoledge of the channel coefficients. The received signals are corrupted by additive hite Gaussian noise (AWGN) ith a poer spectral density (PSD) of N 0. The transmitted poer of the users and the RN is P. The target rates of S 1 and S 2 are denoted as R 1 and R 2, respectively. III. ARQ STRATEGY AND SIGNAL MODEL The proposed ARQ strategy can be described as follos: Slot 1: S 1 and S 2 transmit signal to R simultaneously, the RN quantizes the received signals and stores them in the buffer. Note that in the AF-based half-duplex relaying netork, the RN receives and transmits in different time slots. Thus the received signals are actually alays quantized and stored in a buffer. The received signal can be expressed as y = P h 11 x 1 + P h 21 x 2 + n r, (1) here x 1 and x 2 are the transmitted symbols by S 1 and S 2, respectively, and n r is the noise received at the RN. It is assumed that the RN has an infinite buffer size [8]. Slot 2: As the RN knos the channel coefficients, it can calculate the end-to-end SNRs γ 1 and γ 2 of the to users. For

3 2 the AF-based to-ay relaying system, the end-to-end SNRs can be expressed as [5] γ i = ρ h i1 2 h i2 2 /( h i1 2 + h i2 2 + h i /ρ), (2) here ρ = P/N 0. i, i {1, 2} are the user indices hich satisfy that if i = 1(2), then i = 2(1). 1) If it satisfies γ 1 < λ 1, γ 2 < λ 2, here λ 1 = 2 R 2 1 and λ 2 = 2 R 1 1, then the RN sends a negative ACK to both users. The to users retransmit their packets to R, and the RN receives and stores it in the buffer. The system orks in a to-ay mode. 2) If one of the users can successfully receive the packet, for instance γ 1 > λ 1, γ 2 < λ 2, the RN amplifies the received signal and retransmits it to both users. The received signal of user i can be expressed as r i = β P h i1 h i2 x i + β P h i1 h i2 x i + βh i2 n r + n i, (3) here β = P/(P h P h N 0 ) is the amplification factor. As S 1 (S 2 ) knos its transmitted signal x 1 (x 2 ), it can subtract it from r 1 (r 2 ), thus the self-interference is cancelled. As only S 1 can successfully detect the message, a negative ACK and a positive ACK are sent to S 1 and S 2 by the RN, respectively. In the folloing time slots, only S 1 retransmits its packet to R and the system transforms into a one-ay mode. 3) If γ 1 > λ 1, γ 2 > λ 2, the RN amplifies the received signal and retransmits it to both users. As both users can successfully decode the messages, a positive ACK is transmitted to both users by the RN and the buffer is released to receive ne packets. Slot k (k > 2): As multiple copies of the signals are stored in the buffer, the RN firstly combines them. For the to-ay mode, the signal received by the RN during the previous k 1 slots can be expressed as y = P h 1k 1 x 1 + P h 2k 1 x 2 + n r, (4) here h 1k 1 = [h 11, h 1k 1 ] T, h 2k 1 = [h 21, h 2k 1 ] T are the channel vectors. The combined signal can be ritten as r = H y = P h eq 1k 1 x 1 + P h eq 2k 1 x 2 + ñ r, (5) here h eq ik 1 = H h ik 1, i = 1, 2 are the combined equivalent channel gains, stands for the combining vector, hich has a Frobenius norm = 1. ñ r is the equivalent noise after combining. The RN calculates γ 1, γ 2 using h eq ik 1 and h ik, i = 1, 2 to decide hether to receive or to transmit in the kth slot. Then similar operations as those in slot 2 are performed. If the system orks in the one-ay mode, for instance, e assume that S 2 has already successfully decoded x 1 in the nth (n k 1) time slot, the RN only needs to forard x 2 to S 1. Similar to the to-ay mode, the RN combines the received signals and calculates the end-to-end SNR of S 2 to decide hether to transmit or to receive. Hoever, in this case, h 1k 1 in (4) should be reritten as h 1k 1 = [h 11, h 12 h 1n, 0, 0]. In this paper, e set a imum delay of N time slots for the proposed ARQ scheme. If any one of the users fails to decode the message ithin N time slots, the system is in outage. Remark: In the proposed to-ay relay ARQ scheme, the users can retransmit one packet for at most N 1 times. Thus by increasing N, the reliability of the system can be improved, hich on the other hand decreases the overall throughput of the system as more time slots are used to deliver a packet. In addition, a signal packet is dropped if it is not successfully delivered ithin N time slots, thus the end-to-end delay of the proposed ARQ scheme is controlled ithin N time slots. Furthermore, as the ACK is sent by the RN, the users can receive it directly, hich is preferable for the to-ay relay system ithout a direct link beteen the users. IV. COMBINING APPROACH OF THE RELAY NODE In this section, the MSC scheme and the MMC scheme are investigated for the to-ay mode. The optimal combining method for the one-ay mode is also obtained. A. MSC Scheme In the MSC scheme, the relay node attempts to imize the sum SNR of the received signals and noise. As = 1, the PS- D of the combined noise is N 0. The sum poer of the signals received in k time slots is P sum = P H ( h 1k h H 1k + h 2kh H 2k), thus the folloing optimization problem can be formed: H H sum s.t. = 1, (6) here H sum = h 1k h H 1k + h 2kh H 2k is a positive definite Hermitian matrix. It has been shon in [9] that the optimal combining vector is the eigenvector corresponding to the imum eigenvalue of H sum. Thus the optimal can be obtained as = u, here u is the eigenvector associated to the imum eigenvalue of H sum. B. MMC Scheme In the MMC scheme, the RN attempts to imize the minimum SNR of the to received signals. The poer of the to combined signals can be ritten as P i = P H H i, i = 1, 2, here H i = h ik h H ik. Thus the SNRs of the to signals at the relay node are SNR i = ρ H H i. The MMC scheme solves the folloing problem min { H H 1, H H 2 } s.t. = 1. (7) As the objective function is a monotonously increasing function of, problem (7) is equivalent to the folloing problem min { H H 1, H H 2 } s.t. 1. (8) According to the results of [10], problem (8) is a non-convex problem hich can be solved by adapting a semidefinite relaxation (SDR) approach. Hoever, this approach has a high computational complexity. To reduce the complexity, e propose a suboptimal approach hich has much loer complexity. Lemma 1. The optimal lies in the set ψ = {ĥ1k, ĥ2k}, if it satisfies h H 1k h 2k > min{ h 1k 2, h 2k 2 }, here ĥ1k = h 1k / h 1k, ĥ2k = h 2k / h 2k.

4 3 Proof: As H h 1k 2 achieves the imum value hen = h 1k / h 1k, if it satisfies h H 1k h 2k > h 1k 2, problem (8) achieves the imum value hen = h 1k h 1k. If hh 1k h 2k > h 2k 2, the optimal combining vector is = h 2k / h 2k. Lemma 2. There alays exists = αĥ1k + βĥ2k, α, β C to make H H 1 = H H 2, if it satisfies h H 1k h 2k < min{ h 1k 2, h 2k 2 }. Proof: Set = αĥ1k + (1 α) ĥ2k. As h H 1k h 2k < min{ h 1k 2, h 2k 2 }, e have H h 1k 2 > H h 2k 2, hen α = 1. When α = 0, e have H h 1k 2 < H h 2k 2. As H h 1k, H h 2k are continuous functions of α, there must be a α (0, 1) hich makes H h 1k 2 = H h 2k 2. In the scenario h H 1k h 2k < min{ h 1k 2, h 2k 2 }, e use = a 1 ĥ 1k +a 2 ĥ 2k, here a 1 and a 2 are complex scalars to approximate the optimal combining vector. Denote a = [a 1, a 2 ] T, Ĥ = [ĥ1k, ĥ2k], and let R 1 = ĤH H 1 Ĥ, R 2 = ĤH H 2 Ĥ, then problem (8) can be transformed as min { a H R 1 a, a H R 2 a } a s.t. a = 1. (9) Based on lemma 2, it can be concluded that the optimal a alays satisfies a H R 1 a = a H R 2 a, thus e firstly find the a that satisfies a H Ra = 0, here R = R 1 R 2 is a 2 by 2 Hermitian matrix. The eigenvalue decomposition of R can be expressed as R = V H ΣV, here V is a 2 by 2 unitary matrix that consists of the eigenvectors of R, Σ is a diagonal matrix ith diagonal elements λ 1 and λ 2, hich are the eigenvalues of R. Denote b H = a H V H, then e let b 1 2 λ 1 + b 2 2 λ 2 = 0. (10) It can be observed that any vector b = [b 1, b 2 ] T that satisfies (10) ill result in a satisfying a H Ra = 0. Without loss of generality, let b = e [ jφ λ 2, λ 1 e jθ], φ, θ [0, 2π]. It can be observed that the value of φ has no influence on the value of a H R 1 a, so e only need to find the optimal θ that imizes a H R 1 a. By using a one-dimensional search, the optimal θ can be found. After finding the optimal a, e can obtain and normalize its Frobenius norm to 1. C. Combining in the one-ay mode If one of the transmitted signals is successfully detected, then the system orks in the one-ay mode. We assume that S 1 has correctly received x 2, then the RN only aims to effectively deliver x 1 in this scenario. The optimal combining vector of the RN should imize the end-to-end SNR of S 2, i.e., SNR 2 s.t. = 1, (11) in hich SNR 2 = ρ H h 1k 2 g 2k 2 /( H h 1k 2 + H h 2k 2 + g 2k ρ ). Denote D = H 1 +H 2 +[ g 2k 2 +ρ]i, here I is the TABLE I COMPLEXITY COMPARISONS FOR DIFFERENT SCHEMES. Scheme Complexity MSC 72k 3 + 7k 2 + 7k MMC-proposed 24k π/ε MMC-SDR O ( 6k 4.5 log 10 (1/e 1 ) ) log 2 (t/e 2 ) identity matrix, then e get the folloing optimization problem H H 1 H D s.t. = 1. (12) As D is a positive definite matrix, it has been proved that the optimal value of the objective function of (12) is the imum eigenvalue of the matrix D 1 2 H 1 D 1 2 [10]. Denote the eigenvector of D 1 2 H 1 D 1 2 associated ith the imum eigenvalue to be v, then let ŵ = D 1 2 v. By Normalizing ith ŵ, e get the optimal solution of (12) to be = ŵ/ ŵ. D. Complexity Comparison The complexity of obtaining the combining vector is evaluated using the number of flops. A flop is defined to be a real floating point operation. A complex addition and multiplication have to and six flops, respectively. After k (k 2) retransmissions, as H sum is a Hermitian matrix, the MSC scheme needs 7k 2 +7k flops to get H sum and 72k 3 flops to perform eigenvalue decomposition of H sum [11]. Thus, the total complexity of the MSC scheme can be expressed as 72k 3 + 7k 2 + 7k. For the proposed suboptimal MMC scheme, it takes 12k flops to get Ĥ H h 1k, thus the number of flops needed to get R is 24k As R is a 2 by 2 matrix, its eigenvalue decomposition needs flops. For the one-dimensional searching ith step size ϵ, 2π/ε iterations are needed, and each iteration needs 36 flops to get the value of a H R 1 a. As a result, the total complexity of the proposed suboptimal MMC scheme can be ritten as 24k π/ε. For the SDR-based MMC scheme, based on the result of [12], the complexity is provided in Table I here e 1 and e 2 denote the solution accuracies of the SDR process and bisection searching process, respectively. It can be observed that the proposed suboptimal MMC scheme has the loest complexity hen k is small. Both the proposed MSC scheme and the MMC scheme have loer complexity as compared to the SDR-based MMC scheme. V. NUMERICAL RESULTS AND DISCUSSION In Fig. 2, the outage probabilities of different schemes are provided. We define that the system is in outage if any one of the users fails to deliver the packet to the destination ithin a imum number of N time slots. The target information rate of the to users is set to be R = 1 bit/s/hz and all channel gains are independently Rayleigh distributed random variables ith unit variance. The performance of the to-ay relay ARQ scheme of [5] is also presented as a comparison. In [5], each ARQ round takes to time slots. In the first slot, both users

5 4 Outage probability d=2 d=3 d=5 ARQ of [5], N=4 slots ARQ of [5], N=6 slots proposed ARQ, N=4 slots, no combining proposed ARQ, N=6 slots, no combining proposed ARQ, N=6 slots, MSC proposed ARQ, N=6 slots, suboptimal MMC proposed ARQ, N=6 slots, SDR MMC SNR(dB) Fig. 2. Outage probabilities of the proposed ARQ scheme and the existing ARQ scheme in [5] ith N = 4 and 6. transmit signal to the RN and the RN broadcasts the received signal in the second slot. We present the achieved diversity order of different schemes by comparing the slopes of the outage curves ith the reference outage curves plotted as P ref out = A SNR d, here A is a constant. From Fig. 2, it can be observed that the proposed ARQ scheme achieves a diversity order of N 1 hile the diversity order of the scheme in [5] is N/2 hen the imum delay for both systems is set as N. This improvement results from the fact that in our scheme, the RN chooses to receive or transmit adaptively in each slot. The performance gains of the proposed combining schemes are illustrated by comparing ith the scheme ithout combining at the RN. In the proposed ARQ scheme ithout relay combining, the RN releases the packets received in the previous slots and only stores the packet received in the current slot. As there is only one packet in the buffer, no combining is performed. It can be observed that the MSC scheme achieves more than 1dB SNR gain compared ith the system ithout combination at the RN. The MMC scheme outperforms the MSC scheme by about 1dB if the SDR approach is used. Negligible performance loss is observed for the proposed suboptimal algorithm as compared ith the SDRbased approach. Outage probability proposed ARQ, suboptimal MMC, infinite buffer proposed ARQ, SDR MMC, infinite buffer proposed ARQ, no combining proposed ARQ, suboptimal MMC, to packets buffer proposed ARQ, suboptimal MMC, one packet buffer SNR(dB) In Fig. 3, the effect of the buffer size on system performance is investigated. The outage probabilities of the MMC scheme ith different RN buffer sizes are presented. For the one packet buffer case, the RN can only store one packet of information, thus the combining method at the RN becomes the selective combination. Specifically, the RN decides hether to keep the existing packet or to receive the ne packet based on the channel response of different ARQ rounds. For the MMC scheme, the nth packet is stored if it satisfies n = arg {min [ h 1i, h 2i ]}. For the to packets buffer case, i the RN can store to packets. From the third ARQ round, half of the buffer is used to store the combined packet of the previous ARQ rounds, the other half is used to receive the ne packet; then these to packets are combined and forarded to the destination. It can be observed from Fig. 3 that as the buffer size increases, the system can achieve loer outage probability. VI. CONCLUSION In this letter, e have proposed an ARQ scheme for the toay relaying system. The combining technique as also applied at the RN as it has a buffer. To combining schemes, i.e., the MSC scheme and the MMC scheme, have been proposed and the optimal combining vectors have been obtained. Numerical results have shon that the proposed ARQ method and the combining schemes can improve the system reliability. REFERENCES [1] Y. Zhang, Z. Zhang, and R. Yin, On the capacity bounds of non-restricted Gaussian amplify-and-forard to-ay relay channel, IEEE Commun. Lett., vol. 16, pp , Oct [2] W. Guan and K. J. R. 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