Physical Layer Network Coding for Uni-Cast Applications Hans-Martin Zimmermann
Ying-Chang Liang
Institute of Communication Networks Munich University of Technology Munich, Germany email:
[email protected]
Institute of Infocomm Research A*STAR Singapore email:
[email protected]
Abstract— Network coding has created tremendous research interest over the last few years. Latest research works even propose a transition from the digital message encoding mechanism on the network layer to a corresponding analog encoding approach on the physical layer. This technique is also known as physical layer network coding and allows the simultaneous reception of several signals at a common receiver. Coding gains by network coding are generally only achieved for multicast problem formulations, as network coding is not directly applicable to uni-cast scenarios. However, research has shown that some uni-cast problems can be re-formulated as multi-cast problems to take advantage of the network coding paradigm. This paper presents an analytical cross-layer optimization approach for network coding problems and analyzes the network coding solution of exemplarily chosen uni-cast problems.
I. INTRODUCTION Network coding has created strong research interest within the last years. Typically, point-to-multi-point connections are considered for network coding, as the application of network coding results in significant performance gains compared to conventional routing in these scenarios. However, the problem is that many scenarios in a network’s daily operation consist of uni-cast flows. The multi-cast or broadcast case is thereby only one of many different scenarios and applications. The broadcast case might be even of minor interest compared to the uni-cast case. Nevertheless, it can be shown that there are some uni-cast scenarios (e.g. [1]), where network coding can indeed be applied. A relaying scenario with bidirectional uplink/ downlink connection from an access point over a relay station to a mobile terminal is one very prominent example for such a case. The interesting question is now, whether this problem statement can be put into a more general framework, i.e. whether it is possible to derive a model or solution approach, how to transform uni-cast problems to multi-cast subproblems in order to successfully apply network coding as solution strategy. Linear network coding has been introduced some years ago in the pioneering work of Ahlswede [2]. This technique includes logical flow operations on the network layer in order to approach and also to achieve the capacity boundary. It also offers new insights toward a better understanding of the operation and management of networks [3]. Later research proposes the introduction of physical layer network coding
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as e.g. described in [4]. In the simplest case, physical layer network coding can be seen as an extension of network coding. Physical layer network coding performs network coding operations in combining physical transmission signals at the receiver air interface. Concerning this transmission scheme, capacity issues are still under investigation, especially for complex channel models. However, assuming all network operations at the transmitter as well as the receiver to be subjected to the same cyclic and therefore limited symbol alphabet, any physical layer network coding operation can be interpreted as a simple network coding operation, which is optimized in e.g. the time domain due to the parallel signal reception. Mathematically, the simultaneous reception process can be described by additions within a Galois field. The interested reader may refer to literature as [4] and [5] for a deeper insight into the physical layer issues of this technique. The principle of physical layer network coding is also investigated in combination with simple relaying scenarios. A performance analysis for these cases can be found in [6] and [7]. The contribution of this work is characterized by the following aspects: By analyzing various uni-cast examples, the importance of a generalized theory for this research field shall be emphasized. In doing so, exemplary solutions are provided for small, but typical relaying scenarios. These examples impressively illustrate the potential of the described uni-cast to multi-cast transformations for the general network coding case as well as for the physical layer network coding case. The remainder of this work is as follows: Section II gives an introductive example to explain the principles and differences between network coding, physical layer network coding and conventional routing. After that, an analytical approach to network coding is shortly wrapped up in Section III, before presenting various uni-cast examples with their corresponding network coding solution in Section IV. The paper closes with a final summary of the previous sections. II. MOTIVATION Network coding can achieve performance gains in multicast scenarios. Each multi-cast connection is thereby defined by a single source node and a set of destination nodes. In the simplest case, however, there is only one destination node. Clearly, the case is referred as uni-cast problem and is a special
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case of the multi-cast problem. Given a set of different unicast connections, the standard capacity-achieving approaches would model this as different sessions. A network coding gain can not be achieved in this case, because the encoding step of network coding can be interpreted as a resource sharing mechanism, which is only applicable within the same session. Nevertheless, it can be shown that in some cases, network coding can even be applied successfully to uni-cast scenarios. Let us explain this issue with an introductory example, where a set of three nodes {V1 , V2 , V3 } is given as depicted in Fig. 1. These nodes are deployed in a straight line such that each node has a physical connection to its direct neighbor only. Imagine the case, where V1 and V3 want to exchange two messages x1 and x2 . Accordingly, V1 and V3 will have to rely on the intermediate node V2 as relaying node. It turns out that collision-free routing (see Fig. 1(a)) will require four time slots to perform the four necessary independent signal transmissions of messages x1 and x2 . The relay node marks the bottleneck for the transmissions in this case. Routing implies that the intermediate relay node will only forward the messages, but not change its contents. The number of transmissions can be reduced by one, in combining both messages to a joint message x1 ⊕ x2 as shown in Fig. 1(b). This encoding step is generally referred as linear network coding. In doing so, network coding requires three time slots. Going one step further on to physical layer network coding, both incoming messages can be received simultaneously at the relay node as shown in Fig. 1(c). Provided an adequate modulation scheme, the encoding step will be done at the relay node’s air interface by the additive signal reception. This also means that the relay node is no more able to decode the messages x1 and x2 separately. However, V1 and V3 will be able to decode the joint message after its reception. This is due to the fact V1 and V3 have a priori knowledge, as V1 and V3 can use x1 and x2 respectively for the decoding of the joint message. In this case, the number of required transmissions equals three and the number of time slots is reduced to two slots. This section has outlined the basic ideas of network coding and physical layer network coding. It may already give you a feeling for the enormous potential of those techniques. You may also confirm these examples by reading further literature, e.g. [1] or [5]. The example represents a very simple case, where it is indeed possible to perform network coding in a uni-cast scenario. III. N ETWORK C ODING A PPROACH Assume the wireless transmission processes to be modeled by a directed graph G = {V, E, c} with nodes Vi ∈ V, edges ei ∈ E and finite transmission capacities c on each of the edges. Thereby, the edges ei represent noise-free channels, which are subjected to a finite symbol alphabet. The graphtheoretical model of the wireless transmission device is equivalent to a previous work in [8] and extended by virtual receiver nodes to make it suitable for physical layer network coding. In doing so, each wireless transmission device is modeled by three graph nodes, namely a core node Vic connected to a
V1
V2
x1 x2
V3
x1 x2
(a) relaying by routing.
V1
V2
x1
x2
V3
x1 + x2 (b) relaying by network coding.
V1
V2
x1
x2
V3
x1 + x2 (c) relaying by physical layer network coding. Fig. 1.
simple relaying example with 3 nodes.
Vr
Vc
Fig. 2.
Vs
graph-theoretical node model.
virtual receiver node Vir and a virtual transmitter node Vis as illustrated in Fig. 2. Publication [8] introduces the virtual transmitter node to achieve an adequate graph-theoretical description of a wireless network from an information-theoretic perspective for the broadcast case. Similarly, we add a virtual receiver node in order to deal with multiple, additive and simultaneous signal receptions at a single receiver. Based upon this graph model, let us assume a multi-cast session m ∈ {1, . . . , M } ⊂ N+ to be defined by a set of source-destination flows fjm ∈ F m , whereby j ∈ N+ is an index for the number of individual source-destination pairs from source node Vs to a destination node Vdj within each session. According to the approach in [9], the solution to the optimization problem is defined by a subgraph of the original graph, S = {V, E, g} ⊂ G. Following this notation, the subgraph is constrained by ∀e ∈ E : M g m (e) ≤ c(e, t), (1) T
m=1
where - in contrast to [9] - binary capacity elements c(e, t) ∈ {0, 1} in a slotted system with T = {1, . . . , T } time slots are assumed. The required resources gm (e) for each multi-cast flow represent the envelope of all source-destination flows fjm ∀e ∈ E, ∀j, ∀m : 0 ≤ fjm (e) ≤ g m (e).
(2)
A flow of rate r is established from the source for each source-destination node pair (Vs , Vdj ) by ∀Vs ∈ V, ∀j, ∀m :
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m
E (in) (Vs )
fjm (e
(in)
)−
E (out) (Vs )
fjm (e
(out)
) = rm
(3)
and preserved at each intermediate node by ∀V ∈ V\{Vs , Vdj }, ∀j, ∀m :
fjm (e
(in)
)−
E (in) (V )
fjm (e
(out)
) = 0.
(4)
E (out) (V )
∀E (in) (V r )
In consequence, these equations guarantee that a flow of rate rm arrives at each destination node Vdj . While the equations above define a mathematical model to calculate a network coding solution on top of a given topology, we still have to establish a set of equations to consider the physical layer characteristics. The physical resources for a transmission process are provided by so-called elementary capacity graphs as presented in [8]. Any elementary capacity graph represents a feasible capacity allocation for a single transmission process, preventing events like interference or full-duplex operation. Instead of a heuristic determination, we want to rely on a description of elementary capacity graphs by a set of linear equations. Please recall that each transmission process between a core node V1c and a node V2c requires the allocation of at least 3 −−−→ −−−→ simultaneous capacity elements: c(V1c V1s , t), c(V1s V2r , t) and −−r−→c c(V2 V2 , t). In mathematical terms, this mutual dependency is expressed by ∀V s ∈ V, ∀t ∈ T : c(e
(out)
(V s ), t) ≤ c(e
(in)
(V s ), t)
(5)
at the transmitter node V s and ∀V r ∈ V, ∀t ∈ T : c(e
(in)
(out)
(V r , t) ≤ c(e
(V c ), t) + c(e
(out)
(V r ), t)
(6)
(in)
(V c ), t) ≤ 1.
(7)
Furthermore, mutual interference of simultaneous transmissions shall be avoided. As commonly known, interference is caused by simultaneous transmissions of multiple signals to a common receiver node, whereby the reception of at least one transmission signal is not intended, but harms the correct reception of the other transmissions. Concerning the graphtheoretical model, interference is considered as an infeasible allocation of capacity resources. To deal with interference, define a transmitter to be active at time t, if and only if ∀V s ∈ V, ∀t ∈ T : ∀E (out) (V s )
c(e
(out)
In order to prevent interference, it is postulated that ∀t ∈ T , ∀V s ∈ V, ∀V r ∈ V:
−−−→ ∃e = V1s V2r ∧ V1s ⇔ active ∧ V2r ⇔ active −−−−→ ⇐⇒ c(V1s , V2r , t) = 1. (10) In words, a capacity element has to be allocated for any edge between an active transmitter and an active receiver node. If the transmitter is not active, no interference will be done to the receiver. If the receiver is not active, potential interference does not matter. Thus, this scheme will successfully prevent interference. In [10], it is proved that the maximum achievable sum rate for a multiple access finite field adder channel with two transmitter nodes equals the capacity ”1”, provided an optimum code rate and a normalized maximum transmission rate ”1” on each of the two channels. Following this proof, the time-dependent reception capacity of the receiver node has to be restricted by ∀t ∈ T , ∀V r ∈ V : c e (in) (V r ), t ≤ µ. (11) ∀E
at the receiver node V r . The device-related property of half-duplex operation means that any device can not transmit and receive simultaneously. In other words, resources c(e, t) can not be aggregated on the core node’s incoming and outgoing edge at the same time. As operating with binary capacities, this is equal to ∀V c ∈ V, ∀t ∈ T : c(e
and accordingly a receiver to be active at time t, if and only if ∀V r ∈ V, ∀t ∈ T : c(e (in) (V r ), t) ≥ 1. (9)
(V s ), t) ≥ 1
(8)
(in) (V r )
The reception coefficient µ ∈ N+ constrains the number of simultaneously allocated capacity elements on incoming edges at the receiver. µ = 1 means that exactly one transmitter allocates transmission capacities exclusively. Given time as transmission resource, this corresponds to a typical TDMA transmission scheme, which we will assume for the general network coding and routing scheme. In contrast, µ = 2 allows the simultaneous transmission of several transmitters to a common receiver. According to the proof in [10], it is possible to achieve a sum rate ”1” for a simultaneous transmission of two transmitters to a common receiver. As - to the authors’ best knowledge - there is currently no proof for the model being correct for µ > 2, µ = 2 shall be assumed to describe the combination of two incoming signals at the receiver for physical layer network coding. An objective function Ω is defined as numerical optimization goal and given by the linear combination of the number of transmissions and the maximum frame duration
Ω := α ·
T
:=n(t)
c(e
(out)
, t) +β · T.
(12)
V c ∈V E (out) (V c )
A transmission is performed, whenever a core node allocates capacity resources on its outgoing edge such that it is sufficient
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MT1
to add up all outgoing capacity elements to determine the number of transmissions. The weighting coefficients α = 104 and β = 1 are used to define the required capacity elements as primary objective, while the maximum number of required transmission slots T is just a desirable optimization goal of minor importance. The number of transmissions n(t) within a slot t thus adds up to the overall number of transmissions within a time frame n := T n(t) and can be written as a vector n(T ) := (n(t1 ), n(t2 ), . . . , n(T )).
AP
RS
MT2
(a) physical connectivity in uplink/ downlink scenario.
MT1
IV. SIMULATION RESULTS The proposed solutions of the different scenarios are calculated by an implementation of the previously described equations, which in turn can be solved using numerical optimization software. For implementation purpose, the proprietary C++ library GRAPH [11] is used, which allows an easy modeling of graph-based problem statements. GRAPH is able to interact with the C++ interface of the commercial optimization software CPLEX v9.1 [12] such that an optimization model can be established using both libraries. As the overall problem statement is defined by a set of linear equations, CPLEX guarantees the optimality of the solution, provided the convergence of the optimization algorithm, of course. As the algorithm indeed converges for all scenarios, the optimality of the later presented simulation results is granted.
AP
RS
MT2
(b) routing in uplink/ downlink scenario.
MT1
AP
RS
MT2
(c) network coding in uplink/ downlink scenario.
VS
VMT1
A. Uplink / Downlink Relaying Example Let us consider a very simple scenario given by two mobile terminals VM T1 and VM T2 , which are connected over a relay node VRS to an access point VAP as shown in Fig. 3(a). It is assumed that the resources are shared by a separated uplink connection (VM T1 , VAP ) and a downlink connection (VAP , VM T2 ). Assuming simple routing, four transmissions are required to transmit two messages over the relay to the determined destination (see Fig. 3(b)). Given the network coding approach, each uni-cast connection would be interpreted as separated session such that network coding becomes inapplicable on the first view. In a similar way as done in [1], let us now assume that an artificially introduced virtual source node Vs is connected with VM T1 and VAP by two links e1 and e2 with an overall capacity c(e1 , e2 ) = 1 (shown in Fig. 3(d)). In doing so, the original problem can indeed be re-formulated as a multi-cast problem {Vs , (VAP , VM T2 )} and r = 2. The corresponding problem can be solved by three transmissions as illustrated in Fig. 3(c). It is clearly visible that the new problem is an extension of the original problem and thus represents a valid solution to the overall problem statement. This example shows that it may make sense to additionally constrain the unicast problem into a multi-cast problem to achieve performance gains.
VMT2
(d) graph-theoretical model of uplink/ downlink scenario. Fig. 3.
relaying example with uplink and downlink flow.
and destination nodes. A capacity element c = 1 is sufficient to transmit a single data unit between two graph nodes in each transmission frame. Three different flow schemes shall be compared in the following, namely a triple multi-cast flow scenario 3MC, a circular flow scenario CIRC and multi-cast downlink uni-cast uplink flow scenario ULDL. Starting with the triple multi-cast scenario, it is assumed that three multi-cast flows {V1 , (V5 , V7 )}, {V5 , (V1 , V7 )} and {V7 , (V1 , V5 )} have to be transmitted. Introducing again a virtual source node and three virtual edges of capacity c = 1, the problem can be considered as a multi-cast problem {VS , (V1 , V5 , V7 )} with r = 3. The scenario has been investigated for routing (RT), network coding (NET) and physical layer network coding (PHY). The results are as follows: scheme n T n(T )
B. Star-Shaped Relaying Example With 7 Nodes Let us now consider a star-shaped relaying problem with 7 nodes as illustrated in Figure 4. We assume that continuous flows are fragmented into r = 1 data unit per transmission frame T and shall be transmitted between different source
VRS
VAP
NET 11 6 (3,3,2,1,1,1)
PHY 11 4 (4,3,3,1)
RT 15 7 (3,3,2,2,2,2,1)
The results demonstrate impressively, how network coding
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can achieve performance gains, especially wherever obvious bottlenecks as exemplarily V3 are present. Not only the decreased amount of transmissions (-26.7%) can be considered to be advantageous, but especially the efficient resource allocation in terms of time slots (-42.9%) for physical layer network coding. For the CIRC scenario, we assume three uni-cast flows {V1 , V5 }, {V5 , V7 } and {V7 , V1 }. The ULDL scenario is given by a multi-cast flow {V5 , (V1 , V7 )} and two opposite uni-cast flows {V1 , V5 } and {V7 , V5 }. A closer look on the CIRC and ULDL scenario reveals that both scenarios are included in the 3MC scenario. The following table compares the amount of required transmissions in all three cases: scenario scheme n
3MC NET/ PHY 11
RT 15
CIRC NET/ PHY 8
RT 12
ULDL NET/ PHY 9
RT 13
It can be derived from these results that the network coding solution of 3MC still requires less transmissions than the routing solution of CIRC and ULDL. Nevertheless, it is not the optimum case. Considering ULDL, the network coding solution can be found by considering two separate UL/DL pairs (V7 , V1 ) and (V7 , V5 ). Both of these flow sets can be encoded and transmitted with 5 messages each according to the example with a relaying chain of 5 consecutive nodes in [1]. As the downlink message from V7 to V6 , which is part of the multi-cast flow, has to be transmitted only once on the first hop, the number of transmissions is reduced by 1 from 10 to 9. In case of CIRC, the central crossing node V3 can be included in all three uni-cast sets such is extended into that {V1 , V7 }, {V7 , V5 }, {V5 , V1 } The new {V1 , (V3 , V7 )}, {V7 , (V3 , V5 )}, {V5 , (V3 , V1 )}. problem formulation includes a uplink as well as a downlink flow from each of the outer nodes (V1 , V5 , V7 ) to the central node V3 . For each of those pairs, a multi-cast problem can be formulated similar to the relaying example presented in Section II. Using the solution of the multi-cast problems as part of the overall solution, only 8 transmissions per frame are required. This example teaches two lessons: Overhead can be indeed reduced dramatically by applying the network coding mechanism for uni-cast scenarios. It also shows that the additional definition of non-destination nodes as V3 may help to find a sub-problem, which can be solved by network coding. However, this helper nodes have to be determined very carefully, otherwise the optimum performance can not be achieved. V. C ONCLUSION The paper presents thoughts about the determination of an optimum transmission strategy for uni-cast routing problems. It is exemplary shown that in some small, but very relevant scenarios a reformulation of a uni-cast problem as multi-cast scenario may indeed help to save transmission resources. Although the multi-cast problem description is more restrictive,
V7
V6
V1 Fig. 4.
V2
V3
V4
V5
physical connectivity in star-shaped relaying problem with 7 nodes.
it will allow the application of network coding and therefore overcompensate the additional resource demand caused by the more restrictive problem formulation. Future work will include the establishment of a refined theory how to suitably select the potential set of multi-cast nodes in order to reformulate the uni-cast as efficient multi-cast problems. ACKNOWLEDGMENT This work has been done in a joint research project when the first author visited Infocomm Research Institute, Singapore, in 2007. Special thanks goes to the Lothar and Sigrid RohdeFoundation and Prof. J¨org Ebersp¨acher (TUM, Germany) for enabling this research stay. In addition to that, the authors would like to thank Lijuan Geng for the fruitful discussions. R EFERENCES [1] Yunnan Wu, Philip A. Chou, and Sun-Yuan Kung. Information exchange in wireless networks with network coding and physical-layer broadcast. In Proceedings of 39th Annual Conference On Information Sciences and Systems, Mar 2005. [2] Rudolf Ahlswede, Ning Cai, Shuo-Yen Robert Li, and Raymond W. Yeung. Network information flow. In IEEE Transactions on Information Theory, volume 46, pages 1204–1216, 2000. [3] Tracey Ho, Muriel Medard, and Ralf Koetter. An information-theoretic view of network management. In IEEE Transactions on Information Theory, volume 51, pages 1295–1311, Apr 2005. [4] Shengli Zhang, Soung C. Liew, and Patrick P. Lam. Physical-layer network coding. In Proceedings of MobiComm 2006, pages 358–365, 2006. [5] Sachin Katti, Shyamnath Gollakota, and Dina Katabi. Embracing wireless interference: Analog network coding. In Proceedings of SIGCOMM 2007, pages 257–269, 2007. [6] Peter Larsson, Niklas Johansson, and Kai-Erik Sunell. Coded bidirectional relaying. In Vehicular Technology Conference, pages 851–855, Melbourne, Australia, May 2006. [7] Petar Popovski and Yomo Hiroyuki. Bi-directional amplification of throughput in a wireless multi-hop network. In Vehicular Technology Conference, volume 2, pages 588–593, Melbourne, Australia, May 2006. [8] Yunnan Wu, Philip A. Chou, Qian Zhang, Kamal Jain, Wenwu Zhu, and Sun-Yuan Kung. Network planning in wireless ad hoc networks: a crosslayer approach. In IEEE Journal on Selected Areas in Communications, volume 23, pages 136–150, Jan 2005. [9] Yunnan Wu, Philip A. Chou, and Sun-Yuan Kung. Minimum-energy multicast in mobile ad hoc networks using network coding. In IEEE Transactions on Communications, volume 53, pages 1906–1918, Nov 2005. [10] Siddharth Ray, Muriel Medard, and Jinana Abounadi. Random coding in noise-free multiple access networks over finite fields. In Globcomm, volume 4, pages 1898–1902, Dec 2003. [11] Claus Gruber. C++ library graph. In http://www.layonics.com, 2007. [12] ILOG Inc. Ilog cplex. In http://www.ilog.com, 2007.
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