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Nov 9, 2015 - exploiting the trust degrees between nodes on the beamforming design. Index Terms—Trust degree, beamforming, MISO cooperative.
IEEE COMMUNICATIONS LETTERS, VOL. 19, NO. 11, NOVEMBER 2015

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Trust Degree Based Beamforming for MISO Cooperative Communication System Jong Yeol Ryu, Member, IEEE, Jemin Lee, Member, IEEE, and Tony Q. S. Quek, Senior Member, IEEE Abstract—For a multiple-input-single-output (MISO) cooperative communication system, we consider a beamforming design problem by taking into account trust degrees between nodes. For given trust degrees, we design the optimal transmit beamformer in terms of an expected achievable rate as a linear combination of channel vectors weighted by trust degrees. In the numerical results, it is shown that the achievable rate can be increased by exploiting the trust degrees between nodes on the beamforming design. Index Terms—Trust degree, beamforming, MISO cooperative communication.

I. I NTRODUCTION

C

OOPERATIVE communications have been regarded as a promising technology to meet the rapidly increasing mobile traffic demands [1]. In the cooperative communications, the cooperative nodes, i.e., relay node (RNs), are often selected based on the quality of physical channels, e.g., signalto-interference-plus-noise ratio (SINR). However, we cannot be sure whether the selected RNs will cooperate with transmitter (Tx) although they have good qualities of channels. The RN may not forward the data to receiver (Rx) due to either the selfish behavior to save its own resource or the malicious purpose to disconnect the communication between Tx and Rx. As a result, cooperation between nodes should also take into account the relationship of nodes, i.e., trust, as well as physical channel conditions [2]. The relationship between nodes has been exploited to develop communication strategies [3]–[6]. By taking into account both social relationship and physical channels of nodes, social group utility maximization (SGUM) game frameworks were investigated to maximize the weighted sum of individual utilities [3]. For device-to-device (D2D) communication networks, the relationship of nodes has also been investigated [4]–[6]. Specifically, by exploiting the characteristic of relationship, social-aware D2D communication were provided in [4]. Based on social reciprocity, which is achieved by exchanging the altruistic actions among nodes, the D2D relay selection strategy was proposed in [5]. In [6], online and offline social relationshipbased traffic offloading algorithms were proposed for D2D communications. However, most of these works developed

Manuscript received May 7, 2015; revised August 5, 2015; accepted August 18, 2015. Date of publication August 26, 2015; date of current version November 9, 2015. This research was supported in part by the A∗ STAR SERC under Grant 1224104048 and the Temasek Research Fellowship. The associate editor coordinating the review of this paper and approving it for publication was A. Ikhlef. J. Y. Ryu and T. Q. S. Quek are with Information Systems Technology and Design Pillar, Singapore University of Technology and Design, Singapore 487372 (e-mail: [email protected]; [email protected]). J. Lee is with iTrust, Centre for Research in Cyber Security, Singapore University of Technology and Design, Singapore 487372 (e-mail: jemin_lee@ sutd.edu.sg). Digital Object Identifier 10.1109/LCOMM.2015.2473160

Fig. 1. MISO cooperative communication network with relaying users: (a) physical wireless links and (b) trust degree links between users.

communication strategies based on a simple system, consisted of nodes with a single antenna. In this letter, for the MISO cooperative communication system, we provide a novel framework for trust degree based beamforming design. Specifically, as a new design parameter in beamforming design, we consider a trust degree, which quantifies the trustworthiness between nodes on relaying action, as well as wireless channels. To maximize an expected achievable rate, we first derive an optimal structure of the beamformer, which is shown as a linear combination of weighted channel vectors. For an approximated expected achievable rate, we obtain the transmit beamformer in closed form as a function of trust degrees, which can explicitly show an effect of trust degrees on the beamforming design. Notation: ΠX  X(XH X)−1 XH represents the orthogonal projection onto the column space of X and Π⊥ X  I − ΠX denotes the orthogonal projection onto the orthogonal complement of the column space of X. II. S YSTEM M ODEL We consider a cooperative communication system consisting of a Tx-Rx pair and a single RN as shown in Fig. 1. We assume that Tx is equipped with M (≥ 2)-transmit antennas but Rx and RN have a single antenna, respectively. Hence, MISO channels are built from Tx to Rx and RN. Here, h0 and h1 denote the channel vectors from Tx to Rx and Tx to RN, respectively, and are modeled by M × 1 circularly symmetric complex Gaussian vectors whose entries are independent and identical distributed (i.i.d.) Gaussian random variables with zero means and variances σ02 and σ12 , respectively. The single-input-singleoutput (SISO) channel from RN to Rx is denoted by h10 , which is assumed to follow complex Gaussian distribution with zero 2 . In our system, we consider virtual mean and variance σ10 trust degree links between nodes ((b) in Fig. 1) as well as physical wireless channels ((a) in Fig. 1). The tie in the virtual trust degree link, trust degree, is defined as a belief level that one node can put on another node for a specific action such as relaying [2].1 Hence, in our cooperative communication system, RN would be willing to participate in the cooperation 1 Such trust degree can be evaluated and quantified based on the information of the previous interactions between nodes [2].

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IEEE COMMUNICATIONS LETTERS, VOL. 19, NO. 11, NOVEMBER 2015

to help Tx if trust degree between Tx and RN is high, and otherwise, RN may not cooperate with Tx. In addition, RN would be willing to contribute more power to assist the data receiving at Rx if there is a high trust degree between RN and Rx. Therefore, the trust degrees of Tx-Rx, Tx-RN, and RNRx are defined as the pairs (α0 , β0 ), (α1 , β1 ), and (α2 , β2 ). Here, αi denotes a probability that one node of link i helps the transmission of the other node of the link i as a relay node. Once a relay node decides to forward it’s received data to a receiver, which is connected with the relay node by link j, the relay node determines the portion of transmission power βj to use for relaying transmission based on the trust degree between the relay and receiver of link j, similar to [7]. Hence, αi and βj are in the ranges of 0 ≤ αi ≤ 1, ∀i, and 0 ≤ βj ≤ 1, ∀j, which can be determined based on accumulated information of previous interactions of nodes. Consequently, in our system, RN participates in the cooperation with probability α1 , and in the cooperation, RN relays data from Tx to Rx by using transmission power β2 P1 , where P1 is the maximum power budget of RN. As a cooperative transmission strategy, we consider fullduplex decode-and-forward (DF) relaying [1]. In this letter, all trust degrees are assumed to be bidirectional between nodes [2].2 We consider the beamformer design at Tx based on α1 and β2 as well as wireless channels to maximize an expected rate achieved by cooperative transmission from RN. Since the trust degree is estimated from the historical information, the estimation error of trust degree can exist due to insufficient amount of information or inaccurate information. However, to give insight on beamforming based on trust degrees, we assume that Tx knows perfect information of both the trust degrees as well as instantaneous channels.3 For given β2 and channels, the expected achievable rate with respect to α1 is defined by Rc (wc )  E [Rc (wc )], where Rc is an achievable α1

rate of cooperative transmission via RN and wc ∈ CM×1 is a transmit beamformer at Tx. Then, the beamforming problem to maximize the expected achievable rate is formulated by Rc (wc ). P : max 2 wc  ≤1

(1)

III. T RUST D EGREE BASED B EAMFORMING In this section, based on the problem P in (1), we design the optimal transmit beamformer. By considering both events that RN helps and does not help transmission of Tx, the expected achievable rate is derived by       † 2 Rc (wc ) = α1 min log 1 + ρ0 h1 wc  ,    2   log 1 + ρ0 h†0 wc  + β2 ρ1 |h10 |2   2    + (1 − α1 ) log 1 + ρ0 h†0 wc  , (2) where ρ0 = σP2t and ρ1 = Pσ21 . Here, Pt and σn2 are transmisn n sion power of Tx and variance of background noise, respec2 When the trust relationship is built based on the direct interactions between two nodes, the trust of nodes can be symmetric and bidirectional. 3 Tx can obtain the information of channels by feedback from RN and Rx or direct estimation using channel reciprocity.

tively. First term of (2) denotes the expected achievable rate when RN helps transmission of Tx with probability α1 , and it is bounded by the minimum of achievable rates at RN and Rx due to the constraint of DF relaying [1]. The second term of (2) denotes the expected rate achieved by direct transmission from Tx to Rx when RN does not help transmission of Tx with probability 1 − α1 . We define the constant values for given channels, ν1 , ν2 , ν3 , and ν4 as follows: 2   , ν4  ρ1 /ρ0 |h10 |2 . ν1  h1 2 , ν2  Πh1 h0 2 , ν3  Π⊥ h 1 h0 Then, an optimal structure of the beamformer is obtained by the following lemma. Lemma 1 (Necessary Condition): The optimal transmit beamformer that maximizes the expected achievable rate can be represented by  √ wcopt = γ1 w1 + (1 − γ1 )w1⊥ , (3) where w1 =

Πh 1 h 0 ⊥ Πh1 h0  , w1 ν2 ν2 +ν3 ≤ γ1 ≤

=

Π⊥ h0 h 1

Π⊥ h0  h

and γ1 is a constant in

1

the range of 1. Proof: Similar to [8], we can prove the necessary condition of the optimal beamformer by contradiction. Let us define ˜ ∈ CM×1 = wcopt be an optimal beamformer the beamformer w ˜ can be that maximizes the expected achievable rate. Then, w represented by linear combination of M -orthonormal bases as √ √ √ √ ˜ = η1 w1 + η2 w1⊥ + η3 f 3 + · · · + ηM f M , (4) w where w1† · f i = 0 and (w1⊥ )† · f i = 0 for i = 3, . . . , M and ηi is a real value with ηi = 0 for some i. Then, expected achievable rate can be represented by ˜ = α1 min log (1 + ρ0 η1 ν1 ) , Rc (w) √ 

√ log 1 + ρ0 ( η1 ν2 + η2 ν3 )2 + ν4 β2

√ √ (5) + (1 − α1 ) log 1 + ρ0 ( η1 ν2 + η2 ν3 )2 . ˆ that satisfies w ˆ 2 = w ˜ 2 as We can define w  √ ˆ = η1 w1 + ηˆ2 w1⊥ , w

(6)

where ηˆ2 = η2 + · · · + ηM . Since (5) is an increasing function with η2 and ηˆ2 > η2 , we have ˜ < Rc (w), ˆ Rc (w)

(7)

and it is contradiction. Therefore, for optimal beamformer, ηi , ∀i = 3, . . . , M , should be 0 and hence, the optimal beamformer is represented by (3). Now, we can represent wc -related terms in (2) using wcopt in (3) as follows: 1) |h†1 wc |2 becomes γ1 ν1 , which increases √ with γ1 for 0 ≤ γ1 ≤ 1; and 2) |h†0 wc |2 becomes ( γ1 ν2 +  2 (1−γ1 )ν3 )2 , which increases up to γ1 = ν2ν+ν and after that, 3 decreases with γ1 . Hence, as γ1 increases, both |h†1 wc |2 and |h†0 wc |2 (consequently Rc (wcopt ) as well) increase up to γ1 = opt ν2 ν2 ν2 +ν3 and thus, for 0 ≤ γ1 ≤ ν2 +ν3 , Rc (wc ) is maximized 2 2 . For ν2ν+ν ≤ γ1 ≤ 1, Rc (wcopt ) may increase when γ1 = ν2ν+ν 3 3

with γ1 since |h†1 wc |2 increases. Therefore, γ1opt in wcopt , which 2 maximizes Rc (wc ), exists in the range of ν2ν+ν ≤ γ1opt ≤ 1.  3

RYU et al.: TRUST DEGREE BASED BEAMFORMING FOR MISO COOPERATIVE COMMUNICATION SYSTEM

Remark 1: Since h1 = Πh1 h0 for some scalar and h0 = Πh1 h0 + Π⊥ h1 h0 , the structure of the optimal beamformer in (3) can be represented by a linear combination of h1 and h0 opt such as wc = ξ1 h1 + ξ0 h0 , where ξ1 and ξ0 are determined to satisfy wcopt 2 = 1. Therefore, the beamformer design can be interpreted as a steering direction of the beamformer between directions of h1 and h0 based on trust degrees and channel gains. In Lemma 1, it is difficult to present γ1 that maximizes Rc (wc ) in closed form, but we can find it by solving the problem numerically or by exhaustive search. However, from the numerically obtained beamformer, we cannot clearly see the effect of trust degrees on Rc (wc ). Thus, in order to obtain the transmit beamformer in closed form, we use the approximated expected achievable rate, obtained by high signal-to-noise ratio (SNR) approximation, as c (wc ) ≈ Rc (wc ) R  

        † 2  † 2 2 = α1 min log ρ0 h1 wc  , log ρ0 h0 wc  +β2 ρ1 |h10 |   2    + (1 − α1 ) log ρ0 h†0 wc  . (8)

In the following theorem, we obtain the transmit beamformer that maximizes the approximated expected achievable rate. Theorem 1: For given trust degrees, α1 and β2 , the transmit c (wc ) is obtained by beamformer that maximizes R   (9) wc∗ = γ1∗ w1 + (1 − γ1∗ )w1⊥ , where

⎧ 2 ν2 2 +ν3 ) ⎪ if β2 < β = ν1 νν24−(ν ⎨ ν2 +ν3 (ν2 +ν3 ) 2 γ1∗ = γα , if β2 > β = ν1ν−ν 4 ⎪ ⎩ min(γα ,γβ ) otherwise

and γα is given by γα =

ν2 + 2ν3 α1 +

 ν22 + 4ν2 ν3 α1 (1 − α1 ) , 2(ν2 + ν3 )

(10)

(11)

and γβ is given by (12), shown at the bottom of the page. c (wcopt ) in (8) can be reProof: By using Lemma 1, R written as c (γ1 ) = α1 log (ρ0 {min (f (γ1 ), g(γ1 ) + ν4 β2 )}) R + (1 − α1 ) log (ρ0 g(γ1 )) , (13) where f (γ1 ) and g(γ1 ) are given by  2   f (γ1 )  γ1 ν1 = h†1 wc  , 2 2  √    γ1 ν2 + (1 − γ1 )ν3 = h†0 wc  . g(γ1 ) 

(14) (15)

First, for β2 < β, we have f (γ1 ) > g(γ1 ) + ν4 β2 for any γ1 2 c (γ1 ) in (13) can be in ν2ν+ν ≤ γ1 ≤ 1. Thus, in this case, R 3

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represented by c (γ1) = α1 log(ρ0 {g(γ1)+ν4 β2})+(1−α1) log(ρ0 g(γ1)). R (16) 2 Since (16) is a decreasing function with γ1 in ν2ν+ν ≤ γ ≤ 1, 1 3 2 we can obtain γ1∗ that maximizes (16) as γ1∗ = ν2ν+ν . 3 For β2 > β, we have f (γ1 ) < g(γ1 ) + ν4 β2 for any γ1 in ν2 ν2 +ν3 ≤ γ1 ≤ 1. In this case, (13) can be represented by c (γ1 ) = α1 log (ρ0 f (γ1 )) + (1 − α1 ) log (ρ0 g(γ1 )) . (17) R Since (17) is concave with respect to γ1 , we obtain γ1∗ to c (γ1 ) = 0 as γ1∗ = γα , where γα maximize (17) by solving ∂ R∂γ 1 is given in (11). c (γ1 ) in (13) can be repFor β ≤ β2 ≤ β, according to γ1 , R resented by either (16) or (17). First, let us derive γβ , which is the value of γ1 satisfying f (γβ ) = g(γβ ) + ν4 β2 , given in (12). c (γ1 ) is represented by (16), which is a For γβ ≤ γ1 ≤ 1, R decreasing function of γ1 , thus, in this case, we can obtain γ ∗ as 2 c (γ1 ) is represented by (17), γ ∗ = γβ . For ν2ν+ν ≤ γ1 ≤ γβ , R 3 which is a concave function of γ1 achieving maximum value at 2 ≤ γα ≤ γβ , we γ1 = γα . Hence, if γα is in the range of ν2ν+ν 3 ∗ 4 can obtain γ = γα . Otherwise, if γα > γβ , we obtain γ ∗ = γβ 2 since (17) is an increasing function of γ1 for ν2ν+ν ≤ γ1 ≤ γβ . 3 Consequently, for β ≤ β2 ≤ β, if γα ≤ γβ , we obtain γ ∗ = γα , and otherwise, we have γ ∗ = γβ . Therefore, γ1∗ that maxc (γ1 ) is presented by γ ∗ = min(γα , γβ ).  imizes R 1 Remark 2: From Theorem 1, we note that the direction of the transmit beamformer is affected by both α1 and β2 . We first observe that for given channels, if β2 is very low, i.e., β2 < β, the beamformer is designed independently with α1 h0 2 as wc∗ = w0,mrt = h (γ ∗ = ν2ν+ν ). In this case, due to low 0 3 trust degree between RN and Rx, the expected rate of RNRx link is small and hence, the bottleneck of achievable rate of DF relaying is always caused at relaying channel between RN and Rx, h10 , rather than channel between Tx and RN, h1 . Therefore, in this case, the direction of the beamformer does not have to be steered toward h1 even if α1 is high. On the other hand, for high β2 , i.e., β2 > β, the direction of the transmit beamformer mainly depends on trust degree between Tx and RN, α1 . When α1 is high, since Tx expects a cooperation of RN with high probability, to increase the expected rate from RN, the direction of the beamformer is steered toward h1 based on γα , which is an increasing function with α1 . For β ≤ β2 ≤ β, the direction of the transmit beamformer is steered to be balanced properly based on α1 and β2 . When α1 is relatively high such as γα > γβ , the beamformer is designed based on β2 and vice versa.

(17) is an increasing function with γ1 in 0 ≤ γ1 < 2 cannot be in 0 ≤ γα < ν ν+ν . 4 Since

2

ν2 , ν2 +ν3

γα

3

 ν3 (ν1 + ν2 + ν3 ) + ν4 (ν1 − ν2 + ν3 )β2 + 2 ν2 ν3 {ν1 ν3 + ν4 (ν1 − ν2 − ν3 − ν4 β2 )β2 } γβ = (ν1 − ν2 )2 + ν3 (2ν1 + 2ν2 + ν3 )

(12)

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IEEE COMMUNICATIONS LETTERS, VOL. 19, NO. 11, NOVEMBER 2015

Fig. 2. Average expected achievable rate versus transmit SNR (β2 = 0.6).

IV. N UMERICAL R ESULTS We evaluate the performance of the proposed trust degree based beamforming, which maximizes the high SNR approximated expected achievable rate with respect to α1 . We compare the performance of the proposed beamforming with those of the optimal beamforming, which is obtained based on Lemma 1 and corresponding coefficient γ1 is obtained by an exhaustive search, and the conventional beamforming for DF based relaying, which is obtained by regarding that RN always relays data from Tx to Rx with P1 [8]. The number of antennas at Tx is assumed as M = 4 and the average channel gains are 2 {σ02 , σ12 , σ10 } = {−5, 0, 10} dB. In Fig. 2, for β2 = 0.6, we plot the average expected achievable rates, Eh [Rc (wc )], for α1 = 0.3 and α1 = 0.7 according to the transmit SNR, ρ0 (= ρ1 ). For this, the expected achievable rate is averaged over 5 × 104 channel realizations. In Fig. 2, we can observe that the performance of the proposed beamforming, which is designed to maximize high SNR approximated achievable rate, approaches to that of the optimal beamforming for whole SNR regime including low SNR regime. By comparing with direct transmission case (α1 = 0), the expected achievable rates are increased by exploiting trust degrees of RN. We can also see that the proposed beamforming achieves higher expected rate than the conventional beamforming, especially when trust degree is low, e.g., α1 = 0.3. This is because in the design of the conventional beamforming, it does not consider the case that RN can do not cooperative with Tx. In Fig. 3, for α1 = 0.7, the expected achievable rates are plotted according to the transmit power portion of Rx, β2 , for given channels. The transmit SNRs of Tx and RN are given by ρ0 = ρ1 = 10 dB. As shown in Theorem 1, when β2 is very low such as β2 < β, the proposed beamformer is designed as wc∗ = w0,mrt because the expected rate achieved by relaying is low due to the low transmit power of RN, β2 P1 . Otherwise, the proposed beamformer is designed to properly adjust the direction of the beamformer based on both α1 and β2 . For instance, when 0.1 ≤ β2 < 0.6, the proposed beamformer is designed based on β2 , equivalently the coefficient is determined by γ1∗ = γβ . When β2 is high such as β2 ≥ 0.6, the coefficient of proposed beamformer is determined by γ1∗ = γα , which is a function of α1 only. When β2 is high, the rate achieved by relaying of RN is sufficiently high and thus, the achievable rate of DF relaying is

Fig. 3. Expected achievable rate versus β2 (α1 = 0.7).

bounded by the rate achieved at RN. Therefore, the beamformer should be designed to maximize the trust degree α1 weighted sum of approximated rates at RN and Rx such as (17). V. C ONCLUSION In this letter, we develop a framework for trust degree based beamforming in the MISO cooperative communication. We first show that the optimal beamformer that maximizes the expected achievable rate is represented by the linear combination of weighted channel vectors. Furthermore, the closed form beamformer that maximizes the approximated expected achievable rate is provided, which properly adjusts the directions of direct and relaying channels based on trust degrees of Tx-RN and RN-Rx as well as channel qualities. Comparing to the conventional schemes, the performance enhancement of the proposed beamforming is verified by numerical results. This letter opens several issues for future research on the trust degree based beamforming including the analysis of the effect of inaccurate trust degree and robust communication protocol design using imperfect trust degree. R EFERENCES [1] J. N. Laneman, D. N. C. Tse, and G. W. Wornell, “Cooperative diversity in wireless networks: Efficient protocols and outage behavior,” IEEE Trans. Inf. Theory, vol. 50, no. 12, pp. 3062–3080, Dec. 2004. [2] J. P. Koon, “Modelling trust in random wireless networks,” in Proc. IEEE Int. Symp. Wireless Commun. Syst., Barcelona, Spain, Aug. 2014, pp. 976–981. [3] X. Gong, X. Chen, and J. Zhang, “Social group utility maximization game with applications in mobile social networks,” in Proc. Asilomar Conf. Signals, Syst., Comput., Monticello, IL, USA, Oct. 2013, pp. 1496–1500. [4] Y. Li, T. Wu, P. Hui, D. Jin, and S. Chen, “Social-aware D2D communications: Qualitative insights and quantitative analysis,” IEEE Commun. Mag., vol. 52, no. 6, pp. 150–158, Jun. 2014. [5] X. Chen, B. Proulx, X. Gong, and J. Zhang, “Exploiting social ties for cooperative D2D communications: A mobile social networking case,” IEEE/ACM Trans. Netw., vol. 23, no. 5, pp. 1471–1484, Oct. 2015. [6] Y. Zhang, L. Song, W. Saad, Z. Dawy, and Z. Han, “Exploring social ties for enhanced device-to-device communications in wireless networks,” in Proc. IEEE Global Telecommun. Conf., Atlanta, GA, USA, Dec. 2013, pp. 4597–4602. [7] M. Zhang, X. Chen, and J. Zhang, “Social-aware relay selection for cooperative networking: An optimal stopping approach,” in Proc. IEEE Int. Conf. Commun., Sydney, Australia, Jun. 2014, pp. 2257–2262. [8] J. Y. Ryu and W. Choi, “Balance linear precoding in decode-and-forward besed MIMO relay communications,” IEEE Trans. Wireless Commun., vol. 10, no. 7, pp. 2390–2400, Jul. 2011.