Decentralized Two-Hop Opportunistic Relaying With Limited Channel State Information

Decentralized Two-Hop Opportunistic Relaying With Limited Channel State Information

\authorblockNShengshan Cui and Alexander M. Haimovich \authorblockADepartment of Electrical and Computer Engineering
New Jersey Institute of Technology, Newark, NJ 07102, USA
Email: {shengshan.cui, alexander.m.haimovich}@njit.edu
\authorblockNOren Somekh and H. Vincent Poor \authorblockADepartment of Electrical Engineering
Princeton University, Princeton, NJ 08544, USA
Email: {orens, poor}@princeton.edu
Abstract

A network consisting of source-destination pairs and relays with no direct link between source and destination nodes, is considered. Focusing on the large system limit (large ), the throughput scaling laws of two-hop relaying protocols are studied for Rayleigh fading channels. It is shown that, under the practical constraints of single-user encoding-decoding scheme, and partial channel state information (CSI) at the transmitters (via integer-value feedback from the receivers), the maximal throughput scales as even if joint scheduling among relays is allowed. Furthermore, a novel opportunistic relaying scheme with receiver CSI, partial transmitter CSI, and decentralized relay scheduling, is shown to achieve the optimal throughput scaling law of .

I Introduction

The ever growing demand for ubiquitous access to high data rate services necessitates new network architectures, such as ad hoc and relay networks. Over the last decade, a large body of work analyzing the fundamental system throughput limits of such networks has been reported. In particular, numerous communication schemes approaching these limits under various settings have been proposed, e.g. [1, 2, 3, 4, 5].

Notably, Gowaikar et al. [2] proposed a new wireless ad hoc network model, whereby the strengths of the connections between nodes are drawn independently from a common distribution, and analyzed the maximum system throughput under different fading distributions. Such a model is appropriate for environments with rich scattering but small physical size, so that the connections are governed by random fading instead of deterministic path loss attenuations (i.e., dense network). When the random channel strengths follow a Rayleigh fading model, the system throughput scales as .111Throughout the paper indicates the natural logarithm. For two functions and , means that , and the notation means that . We write to denote and . This result is achievable through a multihop scheme that requires central coordination of the routing between nodes.

In this work, we focus on dense networks and two-hop relaying schemes, in which source nodes communicate with destination nodes via relay nodes (no direct connection is allowed between sources and destinations). Dana and Hassibi have proposed an amplify-and-forward protocol in [4] and shown that a throughput of bits/s/Hz is achievable with relay nodes. It is assumed that each relay node has full local channel state information (CSI) (backward channels from all source nodes, and forward channels to all destination nodes), so that the relays can perform distributed beamforming. In [5], Morgenshtern and Bölcskei showed a similar distributed beamforming scheme which demonstrates tradeoffs between the level of available CSI and the system throughput. In particular, using a scheme with relays partitioned into groups, where relays assigned in the same group require knowledge of backward and forward channels of only one source-destination (S–D) pair, the number of relays required to support a throughput is . Hence, restricting the CSI in such a way increases the number of required relays from to to support throughput of .

While the two-hop schemes reported in [4] and [5] do not require central coordination among relays (central coordination is required for the multihop schemes of [1, 2, 3]), some level of transmitter CSI (channel amplitude and/or phase) is still required. In a large system, obtaining this level of CSI, especially at the transmit side, may not be feasible. This consideration leads to the following questions: How does the throughput scaling change under a practical, partial CSI assumption? Can the throughput scaling bounds be approached with any specific schemes?

In the sequel, we give partial answers to the questions above by restricting ourselves to decode-and-forward protocols. In Section II, an upper bound on the throughput is calculated in the large system regime. It is shown that with only partial CSI at the transmitters, the throughput scaling of any two-hop scheme is upper-bounded by . In Section III, an opportunistic relaying scheme that can achieve the optimal scaling is proposed. This scheme operates in a completely decentralized fashion and requires only receiver CSI knowledge and a low-rate feedback to the transmitters. Finally, Section IV concludes the paper.

Ii Throughput Scaling Upper Bound for Two-Hop Protocols

In this section, we establish an upper bound on the throughput scaling of two-hop protocols. We adopt the random connection model of [2] and specifically assume a Rayleigh fading model, i.e., the connections between any source-to-relay (S–R) pair and between any relay-to-destination (R–D) pair follow independent and identically distributed (i.i.d.) flat Rayleigh fading. We assume that in each hop the receivers have perfect CSI knowledge of the channel realizations, but the transmitters do not have full CSI knowledge. We assume a single-user encoding-decoding scheme, i.e, mutual interfering signals are treated as additive noise. Furthermore, we assume the transmission rate is fixed, i.e., the transmission rate of each scheduled link is not adaptive to instantaneous signal-to-interference-plus-noise ratio (SINR). Accordingly, a transmission is deemed successful only if the SINR is not below a prescribed threshold.

We have the following throughput upper bound.

Theorem 1

Under the aforementioned assumptions, the throughput of each hop scales at most as .

{proof}

(Outline) We begin with the first hop. Since the transmission rate of each link is a fixed number, finding the throughput upper bound is equivalent to finding the maximum number of concurrent successful transmissions. To this end, we consider following genie scheme. For any channel realization of the network, the genie scheme is assumed to have the full CSI of the network, and thus is able to schedule in every time-slot the largest set of concurrent successful S–R pairs. Specifically, in testing whether concurrent successful transmissions are supported or not, the genie scheme will deploy relays and test whether there exists an -element subset of source nodes whose transmissions to relays are all successful. In doing so, the genie scheme will test all ways of choosing sources for transmission. Moreover, for each combination of sources, the genie scheme tests possible ways of associating S–R pairs. If the genie scheme can find a combination, among all possible combinations, such that all transmissions are successful, we claim that simultaneous transmissions are achievable. By a probabilistic argument, it is shown in [6, Th. 2] that with probability approaching , one cannot find a set of nodes whose simultaneous transmissions to the relays are all successful. Conversely, with probability approaching , and for any , there exists a set of nodes whose simultaneous transmissions to relays are all successful. Since the genie scheme executes an exhaustive search for maximum number of concurrent successful transmissions, it sets the upper bound for any decentralized scheme.

Upper bound for the second hop can be derived similarly to the first hop. There, we seek to find the existence of an -element destination set such that all concurrent R–D transmissions are successful.

The reader is referred to [6, Th. 2] for the complete proof.

Theorem 1 implies that the throughput scaling of the two-hop scheme is upper-bounded by the order of . Thus, we are able to answer the first question raised at the outset of the paper: the lack of full transmitter CSI reduces the throughput scaling from a power law ( [4] and [5]) to a logarithmic law if is interpreted as the total number of nodes in the network. When implementation is concerned, however, we note that the genie scheme in the proof requires joint scheduling among relays and thus is not readily implementable in practice. We are left with the question as to whether the same throughput scaling is achievable with practical constraints such as decentralized relay operation and low rate feedback.

Iii Opportunistic Relaying Scheme

Assuming decentralized relay operation, the relays cannot cancel mutual interference and have to contend with single-user encoding-decoding in the two hops. Nevertheless, multiuser diversity gain, an innate feature of fading channels, is still available and lends itself to distributed operation. It is shown in the sequel that, somewhat surprisingly, by exploiting the multiuser diversity, the throughput scaling of can be achieved with decentralized relay operations. To enable the scheduling, the scheme requires an index-valued (integer) CSI from the receivers via low-rate feedback.

Iii-a Scheduling

As illustrated in Fig. 1, the proposed opportunistic relaying scheme is a two-hop, decode-and-forward-based communication protocol. In the first hop, a subset of sources is scheduled for transmission to the relays. Then, the relays decode and buffer the packets received in the first hop. During the second hop, the relays forward packets to a subset of destinations (not necessarily the same set of destinations associated with the sources set in the first hop). The two phases (hops) are time-interleaved: Phase and Phase take place in even and odd-indexed time-slots, respectively.

We assume that the channel gains are dominated by the effects of small-scale fading. In particular, it is assumed that the wireless network consists of i.i.d. flat Rayleigh channels. Accordingly, the channel gain between the th source node () and the th relay node (), and the channel gain between the th relay () and the th destination node (), are exponentially distributed random variables, i.e., . Quasi-static fading is assumed, in which channels are fixed during the transmission of each hop, and take on independent values at different time-slots. We also assume that, in both hops, the receivers are aware of their backward channel information, and allow for an integer-value CSI feedback from receivers to transmitters (relays to sources in Phase , and destinations to relays in Phase ).

Iii-A1 First Hop Scheduling

The first hop scheduling can be thought of as a natural generalization of the classic multiuser-diversity-scheme with single receiver antenna [7] to multiple, decentralized antennas. Specifically, all relays operate independently, and each relay schedules its best source by feeding back the index of the source. For example, relay compares the channels , , and schedules the transmission of the strongest source node, say , by feeding back the index . The overhead of this phase of the protocol is a single integer per relay node. Suppose the scheduled nodes constitute a set , then since there are relays, up to source nodes can be scheduled, i.e., (a source can be scheduled by multiple relays). The scheduled source nodes transmit simultaneously at the same rate of bit/s/Hz. The communication from source to relay is successful if the corresponding , i.e,

 SINRP1i,r=γi,r1/ρ+∑t∈Kt≠iγt,r≥1, (1)

where is the average signal-to-noise ratio (SNR) of the S–R link.

Iii-A2 Second Hop Scheduling

In the second hop, the transmitters are the relay nodes, and the multiuser diversity is achieved by scheduling the destination nodes via a criterion. In particular, each destination node , , with the assumption of knowing the forward channel strengths, , , computes SINRs by assuming that relay is the desired sender and the other relays are interference:

 SINRP2k,j=ξk,j1/ρR+∑1≤ℓ≤mℓ≠kξℓ,j, (2)

where denotes the average SNR of the R–D link. If the destination node captures one good SINR, say, for some , it instructs relay to send data by feeding back the relay index . Otherwise, the node does not provide feedback. It follows that the overhead of the second hop is also at most an index value per destination node. When scheduled by a feedback message, relay transmits the data to the destination node at rate bit/s/Hz. In case a relay receives multiple feedback messages, it randomly chooses one destination for transmission.

It is noted that in the steady state operation of the system, the relays are assumed to buffer the data received from all source nodes, such that it is available when the opportunity arises to transmit it to the intended destination nodes over the second hop of the protocol. This ensures that relays always have packets destined to the nodes that are scheduled. It should also be noted that, due to the opportunistic nature of scheduling, the received packets at the destinations are possibly out of order and therefore each destination is assumed to have capability of buffering data.

Iii-B Throughput Analysis

In this subsection, we first derive analytical expressions of the throughput for each hop assuming the system has a finite number of nodes. Then, we extract the scaling laws when the system size increases, i.e., , and compare those to the upper bounds established in the previous section. For the sake of brevity, we provide here only an outline of the derivation, and the reader is referred to [6] for more details.

Iii-B1 Finite n and m

In the first hop, relays independently schedule sources. The number of scheduled sources could be any integer between and . Accounting only for the case in which exactly sources are scheduled, the average throughput of the first hop can be lower-bounded as follows,

 R1≥m⋅Pr[Nm]⋅Pr[Sm], (3)

where is the probability of having exactly sources scheduled, implying a total transmission rate of bits/s/Hz. is the probability for a successful S–R transmission.

By symmetry, each source node has a probability of to be the best node with respect to a relay. Thus, . For finite values of and , exact characterization of is mathematically involved. This is because the numerator (the maximum of i.i.d. random variables) and the denominator (summation of some non-maximum random variables) are not independent. Fortunately, it is possible to further lower-bound as follows,

 Pr[Sm] =Pr[SINRP1≥1]=Pr[X1/ρ+Y≥1] =Pr[X≥s]⋅Pr[X1/ρ+Y≥1∣∣X≥s] ≥Pr[X≥s]⋅Pr[X1/ρ+Y≥1∣∣X≥s] ≥Pr[X≥s]⋅Pr[s1/ρ+Y≥1] =(1−FX(s))FY(s−1/ρ), (4)

where represents the maximum of i.i.d. exponential random variables, whose cumulative distribution function (CDF) can be written explicitly as . The term denotes the CDF of the aggregate interference, which is shown in [6] to be well approximated to a chi-square random variable with degrees-of-freedom with CDF , when is sufficiently large, e.g., . Note that the lower bound (4) suggests a suboptimal scheduling scheme according to which, each relay schedules the transmission of the “strongest” source only if the source’s power gain exceeds a prescribed threshold . The probability of such event is given by , and is a lower bound on the probability of a successful communication with the relay at a rate of bit/s/Hz.

Substituting the lower bound of into (3), we get a lower bound on , as expressed in the following lemma.

Lemma 1

For any , and , the achievable throughput of the opportunistic relay scheme of the first hop is lower-bounded by

 R1≥mn(n−1)⋯(n−m+1)nm(1−(1−e−s)n)FY(s−1ρ). (5)

Turning to the second hop and recalling that its scheduling is based on SINR instead of SNR, all transmissions are successful by definition. Thus, the throughput of the second hop depends on how many relays receive feedback and therefore transmit data packets to the destinations. Furthermore, a relay is scheduled when at least one destination measures its channel with SINR greater than or equal to one. Therefore, the average throughput can be characterized in a closed form expression, as formulated in the lemma.

Lemma 2

For any , and , the achievable throughput of the opportunistic relay scheme in the second hop is given by

 R2=m(1−(1−e−1/ρR2m−1)n). (6)

Iii-B2 Large n and Finite m

With the closed-form expressions of (5) and (6) at hand, we proceed to the regime of large , but fixed . The discussion of this regime is of practical importance in that as communication devices become pervasive, the number of infrastructure nodes (here the relays) is not likely to keep pace.

As mentioned above, the parameter in (5) can be interpreted as a scheduling threshold. Note that in a system with sources and Rayleigh fading channels, the maximum channel gain seen by each relay is of the order of [8], we empirically set in (5). Then, it is easy to show that with . Similarly, letting in (6), results in . Now, since the average throughput of the two-hop scheme is , we conclude that for . The results for large and finite are summarized in the following theorem.

Theorem 2

For fixed , the two-hop opportunistic relaying scheme achieves a system throughput of bits/s/Hz as .

In the opportunistic scheme, we make practical assumptions of decentralized relays and partial CSI. Thus, it is instructive to compare the throughput of the opportunistic scheme with that of an unconstrained scheme. In fact, it is straightforward to show that, the information-theoretical sum-rate for any two-hop scheme is upper-bounded by [6, Lemma 3], even if relay cooperation and full CSI at the relays are assumed. This upper bound (with cooperation and full CSI) can be interpreted as a multiple antenna system, which is well-known to be able to support parallel channels. Moreover, each of the parallel channels enjoys multiuser diversity gain of that translates into a throughput of . In contrast, the opportunistic scheme, with simplified network operation (decentralized operation and partial CSI assumption), has no such freedom to support parallel channels with rate . However, it succeeds in preserving the pre-log factor of the upper bound. Intuitively, the inherent multiuser diversity gain, which is of the order of , is applied to compensate for the mutual interference stemming from concurrent transmissions and to make the scheduled links reliable.

Iii-B3 Large n and m

Theorem 2 shows that when the number of S-D pairs is large and the number of relay nodes is fixed, the average system throughput scales linearly with . This implies that one can increase the number of relays to increase system throughput. However, both (5) and (6) present a tradeoff of throughput in : by making large, one increases the number of transmissions, but as a consequence the reliability of each link degrades. Therefore, there exists an optimal value of such that the throughput scaling is maximized. Finding the optimal order of is equivalent to finding the throughput scaling of the proposed opportunistic relaying scheme. Specifically, we are interested in finding whether the proposed scheme can achieve the throughput scaling upper bound of established in Section II.

To prove that the average throughput of the first hop indeed scales as , it is sufficient to show that the lower bound (5) achieves scaling of order . To this end, consider the case of and . With , it follows that and . Furthermore, for , the interference term can be approximated by a Gaussian random variable with mean and variance both equal to . Due to the symmetry of the Gaussian distribution, we have . This result implies that if we deploy relays, with high probability, sources will be scheduled for transmission, and half of them will be, on average, successful. This yields an average throughput of for the first hop.

Examining the asymptotic behavior of (6) with respect to and , it is straightforward to show that the maximum throughput scaling of the second hop also scales as [6, Th. 3]:

Theorem 3

For the second hop of the two-hop opportunistic relaying scheme, if the number of relays , then . Conversely, if , then .

By considering two hops as a whole, we get the following:

Theorem 4

Under the setup of Section III, the proposed two-hop opportunistic relaying scheme yields a maximum achievable throughput of .

Interestingly, we see that the proposed opportunistic relaying scheme, which assumes decentralized relay operations and practical CSI assumption, incurs no loss in achieving the optimal throughput scaling upper bound. This gives an affirmative answer to the second question posed at the outset of the paper.

The achievability of is also substantiated by Monte Carlo simulations. In the simulations, the average SNR of each hop is assumed to be dB and the simulation curve was obtained by averaging throughput over channel realizations. In Fig. 2, the average system throughput of the two-hop opportunistic relaying scheme is shown as a function of the number of S–D pairs . (Note that the throughput depends on both and . For each value of , optimal throughput (by maximizing over ) is plotted.) We observe that the throughput exhibits the trend, as predicted by Theorem 4. It is also found in simulation that the system throughput is always limited by Phase , i.e., . Thus, we also plot of the upper bound and lower bound of for reference. Recall that the average throughput of is upper-bounded by the genie bound (cf. Theorem 1) and lower-bounded by .

According to the opportunistic relaying scheme, a feedback mechanism is needed to schedule the good nodes enjoying multiuser diversity. By direct computation, it can be shown that, in the limiting operation regime of , the feedback overhead per fading block is for the first hop and for the second hop. The overhead of feedback is negligible when the block length is large.

Iv Conclusion

In this paper, we have considered a network having S–D pairs and relay nodes, operating in the presence of Rayleigh fading. The emphasis is on characterizing the throughput scaling under the assumption of practical CSI requirement. It has been shown that the lack of full CSI at the relays reduces the throughput scaling drastically from a power law (e.g., [4]) to a logarithmic law in the total number of nodes in the network.

Furthermore, an opportunistic relaying scheme that operates in a completely decentralized fashion and assumes only CSI at receivers and partial CSI at the transmitters, has been proposed and shown to achieve a throughput scaling of . Thus, the lack of joint scheduling among relays causes no loss of optimality as far as throughput scaling is concerned.

An interesting subject for further research is the performance analysis of opportunistic relaying schemes employed in more general system models. In particular, models that include both small-scale fading and geographical attenuation (e.g. the model presented in [9]) are of interest.

Acknowledgment

This research was supported in part by the U.S. National Science Foundation under Grants CNS-06-26611, CNS-06-25637 and ANI-03-38807, and by a Marie Curie Outgoing International Fellowship within the 6th European Community Framework Programme.

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