888 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 62, NO. 3, MARCH 2014 Selective Relay-Activation for Conditional DF Relaying S.H. Song, Member, IEEE, Q.T. Zhang, Fellow, IEEE, and K. B. Letaief, Fellow, IEEE Abstract—This paper considers a conditional decode-and- forward (DF) based cooperative system where a source (S) with multiple (M) antennas transmits information to a single- antenna destination (D) with the help of multiple (L ≤ M - 1) single-antenna relays ({Ri }). The optimal transmit weighting vector at the source is not available in the literature due to the non-linear conditional DF operation, which renders the problem non-convex. To solve this problem, we first show that the optimal transmit vector for the single-relay system can be determined by comparing the S-D beamformer (maximum-ratio- transmit beamforming vector for the S-D link) with the one that utilizes “just sufficient” energy to activate the relay-link. However, it is difficult to directly apply the above idea to the multi-relay system, due to the fact that the S-R and S-D links are normally not orthogonal. To tackle this issue, we propose to utilize basis functions that are orthogonal to the S-D and S-R links, respectively, which enables the activating of one S-R link without considering the S-D link and the other S-R links. We then apply the new basis functions to the multi-relay system and propose a selective relay-activation algorithm, where the optimal solution is obtained by comparing the S-D beamformer with schemes that selectively activate different combinations of the relay-links. The selective relay-activation algorithm is different from the conventional water-filling in the sense that the energy is filled to discrete levels to activate the S-R links, a unique feature arising from the conditional DF operation. Index Terms—Cooperative system, decode-and-forward, selec- tive water-filling. I. I NTRODUCTION C OOPERATIVE systems provide a promising method to achieve the space diversity gain by utilizing the distributed antennas of multiple users [1]. There are two popular cooperative diversity techniques, namely, the amplify- and-forward (AF) [2], [3], [4], [5] and the decode-and-forward (DF) schemes [6], [7], [8]. The optimal DF rate for the single MIMO relay channel [9], [10] and the multiple-relay channel [11] have been investigated from the information theoretic point of view. The capacity analysis for a general multiple- input multiple-output (MIMO) relay channel in cooperative systems is available in the literature [12], [13], [14]. Results have also been obtained for the outage performance evaluation of DF systems [7], [8], [15]. One problem with DF systems Manuscript received January 28, 2013; revised July 11 and December 6, 2013. The editor coordinating the review of this paper and approving it for publication was G. Bauch. This work was partially supported by the Hong Kong Research Grant Council under Grant No. 610311. S. H. Song and K. B. Letaief are with the Electronic and Computer Engi- neering Department, the Hong Kong University of Science and Technology (e-mail: shsong@ieee.org, eekhaled@ece.ust.hk). Q. T. Zhang is with the Dept of Electrical Engineering, City University of Hong Kong (e-mail: qtzhang@ieee.org). Digital Object Identifier 10.1109/TCOMM.2014.020314.130082 is the possible propagation of decoding errors in their relays. As a remedy, the conditional DF relaying [16] scheme was proposed in which a relay forwards its received information only when a codeword is correctly decoded. In many practical systems, the base station is equipped with multiple antennas whereas their relay nodes usually employ a single antenna. In this paper, we shall investigate the system design for a general cooperative system with conditional DF scheme over vector channels. In particular, we consider the case where a source (S) with multiple (M ) antennas transmits information to a single-antenna destination (D) with the help of multiple (L ≤ M - 1) single-antenna relays ({R i }). Our objective is to optimize the transmit weighting vector at the source so that the received SNR at the destination is maximized. Note that such a conditional DF scheme, though not optimal, avoids error propagation and is much easier to implement. To the best knowledge of the authors, the optimal design for such a cooperative system is not available, no matter the AF or DF mode is adopted by the relays [17]. The difficulty arises from the nonlinear unit-step function involved in the DF operation, lending the conventional optimization techniques no longer useful. Specifically, the S-R i -D link can be either active or not, based on whether the information is correctly decoded at R i . Obviously, pouring more power to an already active S-R i -D link does not make any sense for a practical fixed rate transmission, except for pushing the relay further to a saturate region. To obtain a clear physical insight, we start from the system with only one relay [18]. It is shown that the optimal transmit vector is a linear combination of the MRT (maximum ratio transmit) beamformers for the S-R and S-D channels. A similar phenomenon is observed in the design of the transmit vectors in MISO (multiple-input-single-output) interference channels [19], [20]. In this paper, we will refer to the MRT beamforming vectors for the S-D and S-R links as the S-D beamformer and S-R beamformer, respectively. It is shown that, if the S-D beamformer is sufficient to activate the S-R link, the S-D beamformer is optimal. On the other hand, if the S-R beamformer fails to activate the S-R link, we also choose the S-D beamformer, where all energy is utilized to maximize the received SNR for the S-D link. However, there are situations in the middle, where the S-R beamformer is able to activate the relay link but the S-D beamformer cannot. Under such circumstances, we compare the performance of the S-D beamformer to its counterpart with minimum energy to activate the S-R link while leaving the remaining power for the S-D link. If the loss from the S-D link due to not utilizing the S-D beamformer is larger than the gain from the S-R-D link, the S-D beamformer is selected. Otherwise, we should 0090-6778/14$31.00 c  2014 IEEE