Outage Behavior of Cooperative Diversity with ... - Semantic Scholar

Outage Behavior of Cooperative Diversity with Relay Selection Kampol Woradit, Student Member, IEEE, Tony Q.S. Quek, Member, IEEE, Watcharapan Suwansantisuk, Student Member, IEEE, Henk Wymeersch, Member, IEEE, Lunchakorn Wuttisittikulkij, Member, IEEE, and Moe Z. Win, Fellow, IEEE

Abstract—Cooperative diversity is a useful technique to increase reliability and throughput of wireless networks. To analyze the performance gain from cooperative diversity, outage capacity is an important figure of merit that captures the inherent diversity-multiplexing trade-off in cooperative diversity schemes. In this paper we derive the outage capacity for several cooperative diversity schemes in decode-and-forward relay networks with a finite number of relay nodes. The obtained expressions are simple and applicable to arbitrary network topologies and signal-tonoise ratios in Rayleigh fading channels. The analysis shows that there exists a signal-to-noise ratio threshold, below which some cooperative diversity schemes are better than direct communication. We propose a new diversity scheme, which, compared to the conventional counterpart, offers improved performance and requires protocol overhead.

I. I NTRODUCTION Propagation effects, such as path loss and fading, limit the transmission rates and the communication ranges in wireless networks. These undesired effects can be mitigated by diversity techniques such as antenna diversity [1] and cooperative diversity [1]–[3]. Cooperative diversity has received considerably attention in recent years and various schemes have been explored in literature [2]–[9]. Here, we focus on decode-andforward (DF) relay networks. Initial work on DF considered single-relay cooperative diversity schemes [2], [3], where a system is composed of one source node, one destination node, and one relay node. More recent contributions addressed multiple relay nodes, either using all relay nodes [4]–[6] or with relay selection [7]–[9]. In the case of slow fading, where the communication time is short compared to the coherence time of the channel, outage capacity is an important figure of merit and is defined as the largest rate of transmission such that the outage probability (i.e., the probability that the mutual information between This research was supported, in part, by the National Science Foundation under Grants ANI-0335256 and ECS-0636519, DoCoMo USA Labs, the Belgian American Educational Foundations, and the Royal Golden Jubilee Ph.D. program (grant No.PHD/0010/2546). W. Suwansantisuk, H. Wymeersch, and M. Z. Win are with the Laboratory for Information and Decision Systems, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139. Email: [email protected], [email protected], [email protected] T. Q. S. Quek is with the Institute for Infocomm Research, 1 Fusionopolis Way, #21-01 Connexis, Singapore 138632. Email: [email protected] K. Woradit was with the Laboratory for Information and Decision Systems, Massachusetts Institute of Technology. K. Woradit and L. Wuttisittikulkij are with the Department of Electrical Engineering, Chulalongkorn University, 254 Phyathai Road, Bangkok, Thailand 10330. Email: [email protected], [email protected]

source and destination falls below a certain threshold) is less than a given tolerable value [10]. However, both outage capacity and outage probability are often difficult to derive in closed-form. As a result, many approximations have been made in the literature based on high signal-to-noise ratio (SNR) or low SNR assumptions [2]–[6]. The approximations may be invalid for practical relay networks operating a an arbitrary SNRs. Multiple-relay cooperative diversity schemes with relay selection were recently investigated [7]–[9]. Opportunistic cooperative diversity without maximum ratio combining (MRC) at the destination was analyzed in terms of outage probability [7]. In the following, we refer to the proactive DF opportunistic relaying in [7] with direct link as fixed selective decode-andforward (FSDF). In [9], a scheme was proposed which uses relay nodes only when it is beneficial. This scheme is referred to as smart selective decode-and-forward (SSDF). In this paper, we derive the exact outage capacity of several proactive relaying schemes. We first derive the exact outage probability and then perform a numerical inversion to determine the outage capacity. The obtained results are simple and applicable to arbitrary network topology, and SNR values. This paper is organized as follows. In Section II, we describe our system model. We consider different relaying schemes and derive the corresponding mutual information in Section III. In Section IV, we analyze the considered cooperative schemes in terms of outage probability and outage capacity. In Section V we present and discuss numerical results, and draw our conclusions in Section VI. II. S YSTEM M ODEL We consider a wireless relay network consisting of K + 2 single-antenna nodes: a source node, a destination node, and K relay nodes. The topology of the relay network is arbitrary but deterministic and static. Without loss of generality, the distance between the source and the destination nodes is normalized to one unit. The source and the destination nodes are denoted by S and D, respectively. The source node can send information directly to the destination node or via a relay. The relays operate in DF mode, and only a single relay can be selected to forward the signal. All nodes communicate in a common frequency band and use time-division-multiplexing (TDM) to avoid interference. The model for the received signal and the channel for a link between any pair of nodes “I” and “J” is given by yJ = hIJ xI + nJ ,

(1)

978-1-4244-2324-8/08/$25.00 © 2008 IEEE. This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE "GLOBECOM" 2008 proceedings.

where xI is the symbol transmitted over the channel by node I, hIJ ∼ CN (0, ΩIJ ) is the complex channel gain1 between the link I → J, where CN (μ, σ 2 ) denotes a complex circularly symmetric Gaussian distribution with mean μ and variance σ 2 , and nJ ∼ CN (0, N0 ) is the additive white Gaussian noise (AWGN) at the node J. The channel gain hIJ captures the effects of fading and path loss, and |hIJ |2 has an exponential distribution, with rate parameter λIJ = dα IJ , in which dIJ denotes the distance between node I and node J, and α is the path-loss exponent.2 We denote the channel gain from the source node to the k th relay by hB,k , in which subscript “B” stands for “backward hop.” Likewise, the channel gain from the k th relay node to the destination node is denoted by hF,k , in which subscript “F” stands for “forward hop.” Every node “I” transmits with the same average transmit signal power P E |xI |2 . Finally, we define SNR as the average signalto-noise ratio from the source node to the destination node, given by P , (2) SNR N0 W where W is the transmission bandwidth. III. C OOPERATIVE D IVERSITY S CHEMES In this section, we derive the mutual information3 for several cooperative diversity schemes: (1) direct communication, (2) FSDF, and (3) SSDF.

where K = {1, 2, . . . , K}, and Rflow (k) denotes the mutual information between the source and destination nodes that uses the k th relay node, and it is given by Rflow (k) = min{RB,k , RMRC (k)},

(5)

where RB,k is the maximum rate supported by the channel between the source and the k th relay node, and RMRC is the maximum rate achieved by combining the signal from the k th relay node and the signal from the direct source node: 1 log2 1 + |hB,k |2 SNR 2 1 RMRC (k) = log2 1 + |hSD |2 + |hF,k |2 SNR . 2 RB,k =

(6)

Note that a factor of 1/2 accounts for the fact that the transmission occurs over two time slots. C. Smart Selective Decode-and-Forward (SSDF) Dividing the transmission into two time slots results in a rate loss (i.e., the factor 1/2 in (6)), which may outweigh the benefit from performing cooperation. Note that this rate loss is particularly detrimental in the high SNR regime, where the degrees of freedom are important. Thus, we consider the SSDF scheme, which uses either FSDF or direct communication, based on the instantaneous channel conditions. The mutual information for this case is given by

A. Direct Communication Without any cooperative diversity, the source node transmits the signals directly to the destination node. In this case, the mutual information conditioned on the channel is given by (3) RDC = log2 1 + |hSD |2 SNR . B. Fixed Selective Decode-and-Forward (FSDF) This scheme is an enhanced version of proactive opportunistic DF relaying in [7] with the addition of direct link combining. Relay selection is based on maximizing the transmission rate from the source to the destination. The transmission for this scheme is divided into two time slots. In the first time slot, the source node (without exploiting any channel state information) transmits symbols, and the selected relay and the destination node listen. The selected relay decodes the received signal and, in the second time slot, transmits the signal to the destination node while the source node remains silent. The destination node finally combines the signals received from the source and relay nodes over the two time slots using MRC, which requires channel state information (CSI) of both the forward link and the direct link. The mutual information of FSDF is then given by RFSDF = max Rflow (k), k∈K

1 We

(4)

consider Rayleigh frequency-flat fading in our channel model. fading considered here is slow, so that the coherence-time of the channel is long enough for the system to establish cooperation and to complete the transmission. 3 Mutual information is expressed in b/s/Hz. 2 The

RSSDF = max{RFSDF , RDC },

(7)

where RDC is given in (3). D. Implementation Aspects None of the above three schemes requires CSI at the transmitter side (i.e., the source or any relay). Similar to [7], proactive relay selection can be performed during the network training phase. The destination selects and notifies the relays, based on the criteria (4) or (7). Upon notification, the selected relay (if any) takes part in the protocol, while the remaining K − 1 relays go to sleep mode, thus reducing the network energy consumption. In contrast to FSDF, the source needs to be notified whether or not a relay will be used in SSDF. This adds additional protocol overhead in SSDF as compared to FSDF. As we will see later, the outage capacity of SSDF converges to that of FSDF in the low SNR regime, and converges to that of direct communication in the high SNR regime. This observation motivates us to propose a hybrid scheme, which achieves performance comparable to SSDF, while exhibiting reduced protocol overhead. This hybrid scheme selects between FSDF and direct communication based on an optimal SNR threshold. Unlike SSDF, the selection criterion is not based on the instantaneous CSI, but only on the average SNR and the network topology. Hence, the selection can be made prior to communication and does not change as long as the average SNR and network topology are unchanged.


IV. O UTAGE C APACITY OF C OOPERATIVE C OMMUNICATION S CHEMES In this section, we derive the outage capacity for the communication schemes described in Section III. We only state the main results here and provide the derivations in the appendix. A. Direct Communication The outage probability at a rate R and a signal-to-noise ratio SNR is denoted by PDC out (R, SNR) P{RDC < R}. Given a target outage probability of 0 < ε < 1, the outage capacity is the maximum rate R such that PDC out (R, SNR) = ε. From [10, pp. 220, eq. (5.57)]), we know that 1 DC SNR . (8) Cε = log2 1 + ln 1−ε B. Fixed Selective Decode-and-Forward Proposition 1: At rate R and signal-to-noise ratio SNR, the outage probability of FSDF is given by PFSDF out (R, SNR) = 1

1+c1 (S) − wc1 (S)+c2 (S) l c2 (S)w , (−1) + −(1 − c2 (S))

w c1 (S)

2R 2 −1 , exp − SNR dα Sk ,

(10) (11)

k∈S

c2 (S)

dα kD .

SNRth =

2 ln(1 − ε) − ln(wε ) . (ln(1 − ε))2

(16)

Remark 2: We can improve the overall performance by adaptively switching between FSDF and direct communication depending on the SNR. When SNR is below SNRth , we use FSDF, which provides higher outage capacity. When SNR is above SNRth , we switch to use the direct communication. We will call this adaptive scheme the hybrid scheme. C. Smart Selective Decode-and-Forward

(9)

l∈K S⊆K |S|=l

where

Proof: By applying l’Hôpital’s rule, we obtain the desired result. Remark 1: In the high SNR regime, (14) indicates that FSDF is always two times worse than direct communication, irrespective of wε . This means that the loss in degrees of freedom due to signal repetition is detrimental compared to the gain in diversity using cooperation. In the low SNR regime, the topology of the relay network, which determines wε , can make the ratio in (15) to be greater or less than 1. From Theorem 1, we expect that it exists an SNR threshold at which both FSDF and direct communication provide equal outage capacity. Indeed, equating (8) and (13) yields

(12)

k∈S

Proposition 3: The outage probability of SSDF at a given rate R and SNR is given by PSSDF out (R, SNR) = 1 − v (−1)l wc1 (S)+c2 (S) v 1−c2 (S) − 1 , + −(1 − c2 (S))

(17)

l∈K S⊆K |S|=l

where w, c1 and c2 are defined in Proposition 1 and R 2 −1 . v exp − SNR

(18)

Proof: See Appendix A. From (9), it can be seen that R and SNR appear only through w. Since it can be shown that PFSDF out (w) is continuous and strictly decreasing between 0 and 1 as a function of w, we can determine wε uniquely for a fixed outage probability ε, by inverting PFSDF out (w) = ε numerically. Proposition 2: The outage capacity for FSDF is given by 1 1 FSDF SNR . (13) Cε = log2 1 + ln 2 wε

Proof: See Appendix B. Remark 3: From (10) and (18), we find that w = exp 2 ln v − (ln v)2 SNR . Since PSSDF out (v, SNR) is continuous and strictly decreasing between 1 to 0 as a function of v,we can determine vε,SNR uniquely for a fixed outage probability ε, by inverting PSSDF out (v, SNR) = ε numerically. Proposition 4: The outage capacity for SSDF is given by 1 SSDF Cε SNR . (19) = log2 1 + ln vε,SNR

Proof: Upon determining wε , we can use (10) to find R, which is the desired outage capacity. Theorem 1: The ratio between the outage capacities of FSDF and direct communication at a given outage probability ε in high SNR and low SNR regimes are given by

Proof: Upon determining vε,SNR for a given SNR, we can use (18) to find R. Theorem 2: The ratio between the outage capacities of the SSDF scheme and the direct communication scheme at a given outage probability ε in the high SNR and low SNR regimes are given by CεSSDF = 1, (20) lim SNR→∞ CεDC

1 CεFSDF = , SNR→∞ CεDC 2 lim

and

(14)

1 ln(wε ) CεFSDF , = SNR→0 CεDC 2 ln(1 − ε) lim

and (15)

respectively, where wε is the unique value w such that PFSDF out (w) = ε.

ln(vε,SNR ) CεSSDF ≥ 1, = SNR→0 CεDC ln(1 − ε) lim

(21)

respectively, where vε,SNR is the unique value v such that PSSDF out (v; SNR) = ε.


16 14

performance to SSDF in both low and high SNR regimes. Moreover, the performance difference between the hybrid and SSDF schemes is small for all SNRs. In contrast to SSDF, the hybrid scheme does not need knowledge of instantaneous CSI, implying that the hybrid scheme can be implemented in a distributed fashion, similar to FSDF. Therefore, the hybrid scheme enjoys the same benefits of both FSDF and SSDF.

SNRth

Cε=0.01 [b/s/Hz]

12 10 8

VI. C ONCLUSION 6 4

Direct Communication FSDF SSDF Hyrbrid Scheme

2 0 20

25

30

35

40 45 SNR [dB]

50

55

60

65

Fig. 1. Outage capacity of all schemes at high SNR regime as a function of the average SNR at an outage probability of 0.01 for 9-relay network with 3 × 3 grid topology. The threshold show the SNRs appearing the crossing curves.

Proof: From (4), we have RSSDF ≥ RDC . Thus, it follows DC that PSSDF out (R, SNR) ≤ Pout (R, SNR), and we obtain the SSDF ≥ CεDC . By applying l’Hôpital’s inequality such that Cε rule, we obtain the remaining results. Remark 4: In the high SNR regime, we see that the outage capacity of SSDF converges to that of direct communication, regardless of the relay network topology. This means that the loss in bandwidth from dividing the channel into two time slots for relaying dominates the improved received power from MRC. In the low SNR regime, the outage capacity of SSDF is greater than that of direct communication. V. N UMERICAL R ESULTS We consider K = 9 relay nodes in an evenly spaced 3-by3 grid, between the source and destination nodes, such that the horizontal or vertical distance between adjacent nodes is 1/4 of the distance between source and destination nodes. We consider high and low SNR regimes for a fixed outage probability ε = 10−2 . In the high SNR regime, Fig. 1 shows that the outage capacities of SSDF and direct communication are equal, which is consistent with Theorem 2. FSDF, on the other hand, is outperformed by both SSDF and direct communication. We also observe that there exists and SNR threshold SNRth at which the capacities of FSDF and direct communication coincide. This threshold is marked by a vertical line. Comparing to direct communication, all cooperative diversity schemes provide higher outage capacities when SNR is below SNRth . This indicates that the relay network topology in this example makes the ratios in (15) and (21) be greater than 1. In the low SNR regime (results not shown), the outage capacity of FSDF is the same as that of SSDF. This is expected since SSDF tends to use FSDF rather than direct communication in the low SNR regime. Fig. 1 also confirms that the hybrid scheme proposed in Sections III-D and described in Section IV-B has a similar

We have derived outage probabilities and outage capacities for several cooperative diversity schemes with relay selection. The derived formulas are simple, applicable to arbitrary network topologies and SNR values. The results indicate that FSDF improves the outage capacity compared to direct communication only when the SNR is below a certain threshold value, which is derived in this paper. We have also proposed a hybrid scheme, which has a performance comparable to SSDF, but does not require knowledge of instantaneous CSI for relay selection. More research is required to understand the different trade-offs in terms of power consumption, protocol overhead, and comparison with reactive relaying. A PPENDIX A O UTAGE P ROBABILITY OF FSDF We express the outage probability as

= P max min {R , R (k)} < R . PFSDF B,k MRC out k∈K

(22)

2

Conditioned on |hSD | , the r.v.’s min {RB,k , RMRC (k)} are independent, implying that

∞ K FSDF p|hSD |2 (x) αk (x) dx, Pout = 0

where

k=1

αk (x) P min {RB,k , RMRC (k)} < R

2 |hSD | = x ,

which equals 1 − (1 − P {RB,k < R}) 2 1 − P RMRC (k) < R |hSD | = x for x < (22R − 1)/SNR and equals P {RB,k < R} for other values of x. Substituting

22R − 1 2 P {RB,k < R} = P |hB,k | < SNR 2R 2 −1 , = 1 − exp −dα Sk SNR and

2 P RMRC (k) < R |hSD | = x 2R 2 −1 − x , = 1 − exp −dα kD SNR

into αk (x) gives (9).


A PPENDIX B O UTAGE P ROBABILITY OF SSDF

R EFERENCES

We write PSSDF out

= P max RSD , max min {RB,k , RMRC (k)} < R

= 0

=

0

k∈K

∞

K 2 p|hSD |2 (x)P RSD < R |hSD | = x αk (x)dx k=1

(2R −1)/SNR

p|hSD |2 (x)

K

αk (x)dx,

k=1

where αk (x) is defined in Appendix A and where the last equality follows from the fact that R −1 1 ; x < 2SNR 2 P RSD < R |hSD | = x = R −1 0 ; x ≥ 2SNR . Substituting the expression of αk (x) into the integrand and expanding the product into summations give (17).

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