Diversity and degrees of freedom of cooperative wireless networks

Sreeram, K.; Birenjith, S.; Kumar, P. Vijay

doi:10.1109/isit.2008.4595188

Cited by 6 publications

(5 citation statements)

References 19 publications

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“…Furthermore, as P s increases, the SN R D in (18) approaches the SNR in the cut-set bound (17). The comparison of the analog network coding rate with the MAC cut-set bound is shown in Fig.…”

Section: Examples Example 1: Diamond Networkmentioning

confidence: 96%

“…Channel gains are fixed as given by the network topology. From (17) and (18), the difference between the two bounds for any value of gains, and for large powers evaluates to a constant…”

Section: Examples Example 1: Diamond Networkmentioning

confidence: 99%

“…The diversity-multiplexing tradeoff (DMT) of multihop amplifyand-forward when relays have multiple antennas was characterized in [16]. By considering a deterministic wireless network, the diversity and degrees-of-freedom were analyzed in [17]. For a special type of networks, DMT of this scheme was also considered in [18], [19].…”

Section: Introductionmentioning

confidence: 99%

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Analog Network Coding in the High-SNR Regime

Marić

Goldsmith

Médard³

2010

2010 Third IEEE International Workshop on Wireless Network Coding

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Abstract-A node performing analog network coding simply forwards a signal it receives over a wireless channel. This allows for a (noisy) linear combination of signals simultaneously sent from multiple sources to be forwarded in the network. As such, analog network coding extends the idea of network coding to wireless networks. However, the analog network coding performance is limited by propagated noise, and we expect this strategy to perform well only in high SNR. In this paper, we formalize this intuition and determine high-SNR conditions under which analog network coding approaches capacity in a layered relay network. By relating the received SNR at the nodes with the propagated noise, we determine the rate achievable with analog network coding. In particular, when all the received powers are lower bounded by 1/δ, the propagated noise power in a network with L layers is of the order Lδ. The result demonstrates that the analog network coding approaches the cut-set bound as the received powers at relays increase. As all powers in the network increase, the analog network coding rate is within a constant gap from the upper bound. The gap depends on number of nodes. We further demonstrate by an example that analog network coding can perform close to sum-capacity also in the multicast case.

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Section: Examples Example 1: Diamond Networkmentioning

confidence: 96%

“…Channel gains are fixed as given by the network topology. From (17) and (18), the difference between the two bounds for any value of gains, and for large powers evaluates to a constant…”

Section: Examples Example 1: Diamond Networkmentioning

confidence: 99%

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Analog Network Coding in the High-SNR Regime

Marić

Goldsmith

Médard³

2010

2010 Third IEEE International Workshop on Wireless Network Coding

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“…In this paper, we characterize maximum diversity, maximum multiplexing gain and achievable DMT for arbitrary cooperative networks. Some of these results were presented in conference versions of this paper [21]- [24] (see also [25], [26]). Special classes of networks are considered in the second part of this two-part paper, [1].…”

Section: Resultsmentioning

confidence: 99%

DMT of multi-hop cooperative networks

Sreeram

Birenjith²,

Kumar

2010

IEEE Information Theory Workshop 2010 (ITW 2010)

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Some basic results that help in determining the Diversity-Multiplexing Tradeoff (DMT) of cooperative multihop networks are first identified. As examples, the maximum achiev able diversity gain is shown to equal the min-cut between source and sink, whereas the maximal multiplexing gain is shown to equal the minimum rank of the matrix characterizing the MIMO channel appearing across a cut in the network.Two multi-hop generalizations of the two-hop network are then considered, namely layered networks as well as a class of networks introduced here and termed as K -parallel-path (KPP) networks. The DMT of KPP networks is characterized for K > 3.It is shown that a linear DMT between the maximum diversity dmax and the maximum multiplexing gain of 1 is achievable for fully-connected, layered networks. Explicit coding schemes achieving the DMT that make use of cyclic-division-algebra-based distributed space-time codes underlie the above results.Two key implications of the results in the paper are that the half-duplex constraint does not entail any rate loss for a large class of cooperative networks and that simple, amplify and-forward protocols are often sufficient to attain the optimal DMT. I. IN TRODUCTIONWe present an overview of the results contained in [1], [2]. The setting for these results is a single-source, single-sink (ss ss) wireless network where each link experiences Rayleigh fading and where the additive noise on each link is CN(O, 1).The focus is on performance as measured by the diversity multiplexing tradeoff (DMT).The results are of two kinds: basic results that apply to such networks in general as well as results that provide information about the DMT of specific classes of networks. A. Prior Work While two-hop cooperative relay networks have been stud ied in considerable detail in the literature, (see for example, [4]-[8], [17]-[19], [27]-[31]), in terms of the DMT, few results are available for more general networks. Yang and Belfiore in [33]consider Amplify-and-Forward (AF) protocols for a family of MIMO multi-hop networks (which are termed as multi-antenna layered networks in the current paper). They derive the optimal DMT for the Rayleigh-product channel which they prove is equal to the DMT of the AF protocol applied to this channel. They also propose AF protocols to achieve the optimal diversity of these multi-antenna layered networks.Oggier and Hassibi [23] have proposed distributed space time codes for multi-antenna layered networks that achieve diversity gain equal to the minimum number of relay nodes among the hops. Vaze and Heath [24] construct distributed space time codes based on orthogonal designs that achieve the optimal diversity of the multi-antenna layered network. In [25], the same authors study the question of the circumstances under which coding is required to achieve full diversity in a layered network.Borade, Zheng and Gallager in [26] consider AF schemes on a class of multi-hop layered networks where each layer has the same number of relays (termed as regular networks in the curr...

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“…However, it must be noted that they consider only full-duplex networks. The degrees of freedom of arbitrary fullduplex ss-ss and multicast wireless networks is established in [21] using a connection with deterministic wireless networks.…”

Section: Introduction a Prior Workmentioning

confidence: 99%

Multi-hop Cooperative Wireless Networks: Diversity Multiplexing Tradeoff and Optimal Code Design

Sreeram,

Birenjith,

Kumar

2008

Preprint

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We consider single-source single-sink (ss-ss) multi-hop relay networks, with slow-fading links and single-antenna half-duplex relay nodes. While two-hop cooperative relay networks have been studied in great detail in terms of the diversity-multiplexing tradeoff (DMT), few results are available for more general networks. In this paper, we identify two families of networks that are multi-hop generalizations of the two-hop network: K-Parallel-Path (KPP) networks and layered networks.KPP networks, can be viewed as the union of K node-disjoint parallel relaying paths, each of length greater than one. KPP networks are then generalized to KPP(I) networks, which permit interference between paths and to KPP(D) networks, which possess a direct link from source to sink. We characterize the DMT of these families of networks completely for K > 3. Layered networks are networks comprising of layers of relays with edges existing only between adjacent layers, with more than one relay in each layer. We prove that a linear DMT between the maximum diversity d max and the maximum multiplexing gain of 1 is achievable for single-antenna fully-connected layered networks. This is shown to be equal to the optimal DMT if the number of relaying layers is less than 4. For multiple-antenna KPP and layered networks, we provide an achievable DMT, which is significantly better than known lower bounds for half duplex networks.For arbitrary multi-terminal wireless networks with multiple source-sink pairs, the maximum achievable diversity is shown to be equal to the min-cut between the corresponding source and the sink, irrespective of whether the network has half-duplex or full-duplex relays. For arbitrary ss-ss single-antenna directed acyclic networks with full-duplex relays, we prove that a linear tradeoff between maximum diversity and maximum multiplexing gain is achievable.Along the way, we derive the optimal DMT of a generalized parallel channel and derive lower bounds for the DMT of triangular channel matrices, which are useful in DMT computation of various protocols. We also give alternative and often simpler proofs of several existing results and show that codes achieving full diversity on a MIMO Rayleigh fading channel achieve full diversity on arbitrary fading channels. All protocols in this paper are explicit and use only amplify-and-forward (AF) relaying. We also construct codes with short block-lengths based on cyclic division algebras that achieve the optimal DMT for all the proposed schemes.Two key implications of the results in the paper are that the half-duplex constraint does not entail any rate loss for a large class of cooperative networks and that simple AF protocols are often sufficient to attain the optimal DMT.

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Diversity and degrees of freedom of cooperative wireless networks

Cited by 6 publications

References 19 publications

Analog Network Coding in the High-SNR Regime

Analog Network Coding in the High-SNR Regime

DMT of multi-hop cooperative networks

Multi-hop Cooperative Wireless Networks: Diversity Multiplexing Tradeoff and Optimal Code Design

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