A Joint Network and Channel Coding Strategy for Wireless Decode ...

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 59, NO. 1, JANUARY 2011

181

A Joint Network and Channel Coding Strategy for Wireless Decode-and-Forward Relay Networks Qiang Li, Student Member, IEEE, See Ho Ting, Member, IEEE, and Chin Keong Ho, Member, IEEE Abstract—In this paper, we consider a wireless multicast network with multiple sources, relays, and destinations. We adopt a multi-hop decode-and-forward relay protocol such that two canonical subnetworks are relevant, namely broadcast channel with receiver side information (BC-RSI) and orthogonal multiple access channel with correlated sources and receiver side information (MAC-CS-RSI). A joint network and channel coding (JNCC) strategy is proposed by exploiting ARQ, RSI, and correlated sources. The proposed JNCC strategy does not require the knowledge of RSI nor any transmit channel state information, such that each transmitter simply performs retransmissions until the intended receivers have accumulated enough mutual information to successfully decode all desired messages. This successful decoding is then conveyed from the receivers to the respective transmitters via an acknowledgement message. To measure the performance of the proposed JNCC strategy with ARQ, we derive closed-form expressions of network throughput by applying the renewal-reward theorem. Analytical results show that the proposed JNCC strategy outperforms, in terms of network throughput, the conventional separate network and channel coding strategy with random linear network coding. Index Terms—Wireless network coding, joint network and channel coding, broadcast channel, multiple access channel.

A. Background

I. I NTRODUCTION

I

N the seminal work by Shannon [1], it was proven that information can be transmitted with an arbitrarily small probability of error if and only if the entropy rate of the information source is less than the capacity of the channel, and this can always be achieved asymptotically by separating source and channel coding. As an extension, a separation principle for network and channel coding [2] was proven in a single-source communication network with discrete memoryless channels (DMC). Originally proposed for wired networks, network coding [3]-[6] was also shown to be able to achieve the min-cut capacity [7] for a class of multicast problems with a separation of network and channel coding. Network coding is an attractive choice for wireless networks, because the broadcast nature of wireless transmissions allows for opportunistic listening [8]-[10] to exist between neighboring nodes, which gives rise to many network coding Paper approved by J. Kliewer, the Editor for Iterative Methods and CrossLayer Design of the IEEE Communications Society. Manuscript received September 7, 2009; revised April 27, 2010, August 14, 2010, and September 1, 2010. This work has been partly presented in conference proceedings [46] and [47]. This work is supported by the Singapore Ministry of Education Academic Research Fund Tier 2, MOE2009-T2-2-059. Q. Li and S. H. Ting are with the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639801 (e-mail: {liqi0008, shting}@ntu.edu.sg). C. K. Ho is with the Institute for Infocomm Research, A*STAR, Singapore 138632 (e-mail: [email protected]). Digital Object Identifier 10.1109/TCOMM.2011.110910.090545

opportunities [11]. However, most of the existing works directly extended the separation of network and channel coding from wired networks to the wireless realm [8]-[10], [12]-[18]. That is, as shown in Fig. 1(a), channel coding is used in the physical layer for each transmission to transform the noisy channels into erasure channels [19]. Then at the network layer, network coding is performed on top of these erasure channels to combine the error-free messages. A natural question would be whether such a separation is still optimal for wireless networks. In contrast to the optimality of separate network and channel coding in networks with point-to-point DMC [2], [20], this separation was shown to be suboptimal in wireless deterministic relay networks and discrete memoryless broadcast channels [22], [21]. An unified framework for source, channel, and network coding based on linear codes was proposed in [23], where the suboptimality of separate coding strategies across canonical wireless subnetworks, i.e. broadcast channel (BC) [24] and multiple access channel (MAC) [25], was proven. That is, for general wireless networks, although codes optimal for canonical subnetworks can be separately implemented, the resulting solution is in general not optimal for the network as a whole, which makes the case for the need to perform endto-end joint coding across the network. However, for a general multi-terminal wireless network (multiple sources, multiple destinations) [7], a comprehensive end-to-end coding strategy may be beyond current research achievements. Thus, most current works [26]-[34] mainly focus on the canonical wireless subnetworks, i.e. BC and MAC, instead. A network-channel coding scheme, based on turbo codes, was proposed in [26] and [27] for the two-way relay channel [12]-[15] and multiple-access relay channel respectively. The main idea is that the redundancy in the network code can be used to augment the channel code for better error protection at the destination. However, to recover the source messages, separate network and channel coding was performed at the intermediate relay. An alternative network-channel coding design was proposed for the binary symmetric channel and additive white Gaussian noise (AWGN) channel in [28]. By exploiting the linear property of both the channel and network codes, network coding is performed prior to channel decoding by directly combining the hard or soft decisions of each received packet. A nested coding scheme [29], [30], where multiple information words are first separately channel encoded and then XORed together (i.e. network encoding), was implemented in [31] as a way of achieving joint channel and network coding. A two-hop multi-source multi-destination network scenario was considered. To ensure reliable message delivery,

c 2011 IEEE 0090-6778/11$25.00 ⃝

182


Receiver Side Information . . .

Orthogonal Multiple Access Channel

. . .

. . .

Channel. Decoder

. . .

. .

Channel Decoder

Network Coding

. . . . . .

Channel Encoder

Broadcast Channel

(a) A separate network and channel coding (SNCC) strategy at an intermediate node. Receiver Side Information

Orthogonal Multiple Access Channel

. . .

. . .

. . .

Joint Network and Channel Decoder

. . .

Source Messages

. . .

Joint Network and Channel Encoder

. . . . . .

Broadcast Channel

(b) A joint network and channel coding (JNCC) strategy at an intermediate node. Fig. 1.

System models for SNCC and the proposed JNCC strategies. earlier transmissions current transmission

message A message B message C

Tx2 Tx2

earlier transmissions current transmission message A

Tx1

Rx1

RSI

RSI

message A

message A

Rx message A message B message C

Tx3

message A

Tx1

RSI message B

Rx2

message A message C

Tx2'

message B

Tx2 Rx3 RSI=Ø

(a) A broadcast channel with receiver side information (BC-RSI). Fig. 2.

(b) A multiple access channel with correlated sources and receiver side information (MAC-CS-RSI).

Examples for the two canonical subnetworks in wireless networks.

retransmissions and feedbacks were performed based on an erasure channel where only correctly received codewords are stored. A BC where each receiver may possess some receiver side information1 (RSI) was analyzed in [32]-[34] where the capacity regions were derived for a degraded broadcast channel with two receivers. To achieve the respective capacity regions, different coding strategies were proposed independently in [32]-[34], where explicit code designs consisting of superposition coding, nested coding, and linear network coding were performed, depending on the exact RSI available at each receiver. B. Our Contributions In order to implement the end-to-end joint coding across the network, there needs to exist a “genie” who has knowledge of the whole network topology as well as the global channel state information (CSI). These pieces of information may be very difficult to obtain in practice, especially for large and 1 Receiver side information refers to the a priori information available at a receiver about the messages being transmitted.

dynamic networks. Thus, instead of an ambitious end-to-end joint coding scheme, it will be pragmatic to focus on canonical subnetworks, i.e. BC and MAC, and see what are the gains that can be achieved if joint coding is applied, as compared to a simplistic separate network and channel coding (SNCC) strategy. As an example, in Fig. 2(a), due to opportunistic listening, Rx1 and Rx2 have already received message A and B respectively from Tx1 and Tx2. Suppose Tx3 now broadcasts messages {A, B, C} to Rx1 and Rx2, this will constitute a BC with receiver side information (BC-RSI) [29]-[35]. Similarly, as shown in Fig. 2(b), Rx has already received message A from Tx1. Suppose both Tx2 and Tx2’ now transmit to Rx with common messages {A, C}, this will constitute a MAC with correlated sources and receiver side information (MACCS-RSI) [37]-[39]. In this paper, we consider a wireless multicast network with discrete memoryless fading channels where a group of sources wish to transmit independent sets of messages to a group of destinations via intermediate nodes and each destination desires the messages from all sources. To exploit the existing information-theoretic results for BC-RSI and MAC-CS-RSI,

LI et al.: A JOINT NETWORK AND CHANNEL CODING STRATEGY FOR WIRELESS DECODE-AND-FORWARD RELAY NETWORKS

we adopt a decode-and-forward (DF) relay protocol [40] at each intermediate node. A joint network and channel coding (JNCC) strategy, as shown in Fig. 1(b), is proposed. Our contributions are as follows. ∙

∙

∙

∙

When a transmitter wishes to broadcast to a group of receivers, due to the independent wireless fading channels between the transmitter and multiple receivers, it is highly probable that some receivers are not able to decode all desired messages after a single transmission. To ensure reliable message delivery, we generalize the results in [46], [47] with an automatic repeat request (ARQ) scheme [41] where the transmitter simply performs retransmissions until an acknowledgement2 (ACK) is sent back from each of the intended receivers indicating successful decoding of all desired messages. In [32]-[35], either RSI or transmit CSI is assumed to be available at the transmitter such that explicit code designs3 and rate adaptation can be performed. In contrast, for the proposed JNCC strategy, without any knowledge of the available RSI at each receiver, the transmitter simply uses independent and identically generated codebooks for each retransmission and at the same time, optimal rate adaptation is achieved without the need for transmit CSI or any explicit rate adaptation algorithm4. In contrast to [32]-[34] where generalization to a broadcast channel with more than two receivers is highly nontrivial, our proposed JNCC strategy is applicable to a BC with an arbitrary number of receivers. As explained later, the encoder for the proposed JNCC strategy is independent of the number of receivers and the RSI they possess. The receiver that experiences a better channel and with more RSI available is able to decode the desired messages faster. Taking into account the ARQ protocol, we quantify the performance of the proposed JNCC strategy in terms of network throughput by applying the renewal-reward theorem [42], [43].

Analytical and simulation results show that the proposed JNCC strategy substantially outperforms the conventional SNCC strategy with random linear network coding (RLNC) [5], [6] in terms of network throughput, thus proving that joint coding is better than separate coding in wireless networks. The organization of this paper is as follows. In Section II, we present the network model for the considered wireless multicast network. In Section III, a JNCC strategy is proposed and the achievable rate region is presented. In Section IV, we derive the network throughput for both the proposed JNCC strategy and SNCC with RLNC. In Section V, closed-form expressions of network throughput and simulation results are presented for a toy example. Finally, Section VI summarizes the main results of this paper. 2 For simplicity, we assume that the ACK feedback channel is delay-free and error-free [43]. 3 Although RSI and CSI can be obtained via feedback, explicit code design is a NP-hard problem and is usually very complex [31]-[34] in general. 4 After the submission of this manuscript, a similar idea was presented in [36] where the same message is sent multiple times using independent codebooks and the decoder performs joint typicality decoding on the received signals.

183

II. N ETWORK M ODEL AND ARQ P ROTOCOL We consider a wireless multicast network with multiple sources where all nodes are assumed to operate in half-duplex mode. To avoid interference among the different incoming links to a common receiver, we assume an orthogonal MAC where FDMA is employed. ∙

∙

∙

∙

The considered wireless multicast network is represented by a finite directed graph 𝒢 = {𝒱, ℰ}, where 𝒱 is the set of vertices representing the communication nodes, and ℰ is the set of edges representing the wireless channels. A communication channel exists between node 𝑢 and node 𝑣 if 𝑒𝑢𝑣 ∈ ℰ, where 𝑒𝑢𝑣 is an edge from 𝑢 to 𝑣. To model a BC, we assume that all outgoing edges from a transmitter form a hyperarc carrying the same transmit information. To model an orthogonal MAC, we assume that the receiving node receives non-interfering signals from each of the incoming edges. We define the set of sources as 𝒱𝑠 ⊂ 𝒱, and the set of destinations as 𝒱𝑑 ⊂ 𝒱. We assume that each source has a set of independent messages to transmit and each destination desires the messages from all sources. All nodes in 𝒱 are grouped into disjoint level sets 𝐿1 , ⋅ ⋅ ⋅ , 𝐿𝐾+1 . Level sets 𝐿1 ≜ 𝒱𝑠 and 𝐿𝐾+1 ≜ 𝒱𝑑 . Nodes in 𝐿𝑘 receive messages only from nodes in levels 𝐿𝑘′ where 𝑘 ′ < 𝑘. Then nodes in 𝐿𝑘 will send messages only to nodes in levels 𝐿𝑘′′ where 𝑘 ′′ > 𝑘, after they have successfully decoded all desired messages from 𝐿𝑘′ . This model constitutes a feedforward flowgraph [44] that only has edges of the form {𝑒𝑢𝑣 ∣𝑢 ∈ 𝐿𝑘 , 𝑣 ∈ 𝐿𝑘′′ }, ∀ 𝑘 ∈ {1, ⋅ ⋅ ⋅ , 𝐾}, 𝑘 ′′ ∈ {2, ⋅ ⋅ ⋅ , 𝐾 + 1}, and 𝑘 ′′ > 𝑘. All nodes in 𝐿𝑘 , 𝑘 ∈ {1, ⋅ ⋅ ⋅ , 𝐾} are assumed to be time synchronized and they transmit simultaneously. Transmission phase 𝑘 is defined as the period when nodes in 𝐿𝑘 start transmitting until all intended receivers, denoted as 𝑉𝑘 where 𝑉𝑘 ⊆ {𝐿𝑘′′ ∣𝑘 ′′ > 𝑘}, have successfully decoded all desired messages from 𝐿𝑘 . A simple ARQ scheme is adopted within transmission phase 𝑘 where each transmitter performs retransmissions until all intended receivers have successfully decoded all desired messages and sent back an ACK. Immediately after, transmission phase 𝑘 + 1 proceeds where nodes in level 𝐿𝑘+1 start transmitting5 . Thus, information flow across the network can be visualized as a series of time-orthogonal multi-hop transmissions where each hop corresponds to a transmission phase. In transmission phase 𝑘, a transmitter 𝑢 ∈ 𝐿𝑘 transmits to a group of intended receivers denoted as ℛ𝑢 ⊆ 𝑉𝑘 , such that {𝑒𝑢𝑣 ∣𝑣 ∈ ℛ𝑢 } ⊆ ℰ. The intended receivers in ℛ𝑢 may have some RSI. Thus from the perspective of transmitter 𝑢, we have a BC-RSI. On the other hand, a receiver 𝑣 ∈ 𝑉𝑘 receives from a group of intended transmitters denoted as 𝒯𝑣 ⊆ 𝐿𝑘 , such that {𝑒𝑢𝑣 ∣𝑢 ∈ 𝒯𝑣 } ⊆ ℰ. Since all transmitters 𝑢 ∈ 𝐿𝑘 operate under FDMA, we have an orthogonal MAC-CS-RSI from the perspective of receiver 𝑣.

5 In a typical wireless fading channel, although some receivers will be able to finish decoding earlier, they will have to wait for all receivers in 𝑉𝑘 to finish decoding before transmission phase 𝑘 + 1 proceeds.

184


Transmission phase 1 Transmission phase 2 Transmission phase 3 D1 S1

Ws

1

S2

S1 R1 Ws

2

S2

S3

L1

D1 R3

R2 Transmission phase 1 Transmission phase 2 Transmission phase 3

S1 S2

D2 S3

R2

R2

D2

L2

Ws

D1 R3

R3

D2

V2={L3}

L3

V3={L4}

V1={L2 , L 4 }

R1 R2 L2

D1 R3 D2 L3

L4

L1

3

(b) The feedforward flowgraph of the toy example.

A toy example for the considered multicast network.

In Fig. 3, we show a toy example where all nodes in the network are grouped into four disjoint level sets and the endto-end information flow can thus be characterized by three separate transmission phases. III. JNCC S TRATEGY IN A T RANSMISSION P HASE Since end-to-end information flow across the network is divided into separate transmission phases, we can focus on each transmission phase separately. In this section, we present the mathematical formulations and propose a JNCC strategy for the network model described in Section II and then derive the achievable rate region for an arbitrary transmission phase 𝑘. We first define the global message set as 𝒲𝐺 = {𝒲𝑖 ∣𝑖 ∈ ℐ𝐺 }, which contains all source messages to be transmitted across the network 𝒢, and ℐ𝐺 is the corresponding set of message identifiers which is unique for each message. Each message 𝒲𝑖 is drawn from an index set {1, 2, ⋅ ⋅ ⋅ , 2𝑛𝑅𝑖 } where 𝑅𝑖 is the information rate6 of message 𝒲𝑖 . Without loss of generality, we assume that a transmitter 𝑢 ∈ 𝐿𝑘 has a set of independent messages 𝒲𝑢 = {𝒲𝑖 ∣𝑖 ∈ sum ℐ𝑢 } = {1, 2, ⋅ ⋅ ⋅ , 2𝑛𝑅𝑢 } to transmit, where ℐ𝑢 ⊆ ℐ𝐺 is the corresponding set of message identifiers at 𝑢 and ∑ 𝑅𝑢sum = 𝑅 . Thus for a receiver 𝑣 ∈ 𝑉𝑘 , a set of 𝑖 𝑖∈ℐ𝑢 messages 𝒲𝑣 = ∪𝑢∈𝒯𝑣 𝒲𝑢 from all intended transmitters in 𝒯𝑣 is desired. Receiver 𝑣 attempts to decode for 𝒲𝑣 with the assistance of available7 RSI 𝒮𝑣 ⊆ 𝒲𝑣 and possible correlation between the intended transmitters in 𝒯𝑣 . A. Proposed JNCC strategy ∙

R1

S3

(a) A wireless multicast network with multiple sources. Fig. 3.

R1

Encoding: For a transmitter 𝑢 ∈ 𝐿𝑘 , consider a 𝑛-length code of rate 𝑅𝑢sum with an encoder ( ) sum (1) 𝑋𝑢 : {1, 2, ⋅ ⋅ ⋅ , 2𝑛𝑅𝑢 } −→ 𝒳𝑢𝑛 ,

6 Here 𝑅 = 𝑘 /𝑛 is defined as the ratio of the number of information bits 𝑖 𝑖 𝑘𝑖 to the number of channel uses 𝑛 per ARQ transmission [41]. 7 Note that the proposed JNCC strategy includes the special case of 𝒮 = ∅ 𝑣 when no RSI is available at 𝑣.

∙

∙

where 𝒳𝑢 denotes the finite input alphabet set of transmitter 𝑢. Equation (1) indicates that the encoder of 𝑢 performs joint encoding8 of all messages to be transmitted. This is where the notion of network coding comes in as “mixing” of messages is performed at the intermediate nodes. It is obvious from (1) that the proposed JNCC strategy is an arbitrary mapping of messages to a joint network channel codeword. Thus in general, this joint network and channel code is nonlinear in nature. sum For codebook generation, 2𝑛𝑅𝑢 codewords are independently and identically generated according to some distribution 𝑝(𝑥𝑢 ) on 𝒳𝑢 and indexed as 𝒙𝑢 (𝑤𝑢 ), where sum 𝑤𝑢 ∈ {1, 2, ⋅ ⋅ ⋅ , 2𝑛𝑅𝑢 }. This joint network channel codeword is then broadcasted by transmitter 𝑢. It is clear that each index 𝑤𝑢 corresponds to a message index in 𝒲𝑢 . ARQ: As shown in Fig. 4, to ensure reliable message delivery, a transmitter 𝑢 performs retransmissions such that for each retransmission 𝑡 = 1, 2, ⋅ ⋅ ⋅ , 𝑇 , we have an encoder ( ) sum 𝑋𝑢,𝑡 : {1, 2, ⋅ ⋅ ⋅ , 2𝑛𝑅𝑢 } −→ 𝒳𝑢𝑛 , and codeword 𝒙𝑢,𝑡 (𝑤𝑢 ) is broadcasted from independently and identically generated codebooks9 until an ACK has been received from each intended receiver 𝑣 ∈ ℛ𝑢 indicating successful decoding of all desired messages. Decoding: For receiver 𝑣 ∈ 𝑉𝑘 , as shown in Fig. 4, joint typical-set decoding [7] is performed using all received codewords 𝒚 𝑢𝑣,𝑡 ∀ 𝑡 = 1, 2, ⋅ ⋅ ⋅ , 𝑇 such that with each retransmission a decreasing set of possible transmitted codeword candidates is produced. The correct transmitted codeword can then be recovered with high

8 A similar idea was proposed in [31] where the encoder jointly encodes all its messages and the overall code rate increases with the number of messages that are jointly encoded, unlike in RLNC [5], [6] where the overall code rate remains constant. 9 Without loss of generality, in Fig. 4, we assume that message index 𝑤 = 𝑢 1 is being transmitted, i.e. codeword 𝒙𝑢,𝑡 (1) is sent for 𝑡 = 1, 2, ⋅ ⋅ ⋅ , 𝑇 .


Transmitter u

. . .

xu, 1 (2

Receiver v

ARQ

Codebooks xu, 1(1) xu, 1(2) nRsum u

Possible set of codewords xu, 1(1) t=1

w^ u yuv,1 →

3

2

4

1

xu, 2 (1) t=2

w^ u yuv,2 →

2 6 1 5

)

Joint typical-set decoding

→

3 2

→

3 4

4 1

X u,1 X u,2

Wu 1, 2, 3, sum ..., 2nR u

xu, 2 (1) x u, 2 (2) . . .

u ) xu, 2 (2nRsum

X u,t

. . .

X u,T

xu, T (1) x u, T(2) . . .

u ) xu, T (2nRsum

. . .

. . .

xu, T (1) t=T

6 5

. . .

. . .

1 w^ u yuv,T → 4 5

2 1

3

→

7

4

2 1

6 5

7

Fig. 4. The proposed JNCC strategy where message index 𝑤𝑢 = 1 is transmitted, i.e. codeword 𝒙𝑢,𝑡 (1) is transmitted for 𝑡 = 1, 2, ⋅ ⋅ ⋅ , 𝑇 .

probability after the 𝑇 th transmission when only one unique codeword is found in the intersection set across all retransmissions. This decoding can be further assisted by exploiting available RSI 𝒮𝑣 and possible correlation between the intended transmitters 𝒯𝑣 . We will defer the details to the proof for Theorem 1. B. Achievable Rate Region Theorem 1: For a receiver 𝑣, suppose that each intended transmitter 𝑢 ∈ 𝒯𝑣 has a set of messages, 𝒲𝑢 = {𝒲𝑖 ∣𝑖 ∈ ℐ𝑢 }, to transmit to 𝑣 over 𝑇 retransmissions, with each retransmission consisting of 𝑛 → ∞ channel uses. Thus, in total, a set of messages 𝒲𝑣 = ∪𝑢∈𝒯𝑣 𝒲𝑢 from all intended transmitters in 𝒯𝑣 is desired at 𝑣. Assuming that receiver 𝑣 possesses some RSI 𝒮𝑣 ⊆ 𝒲𝑣 , the set of messages that is effectively transmitted to 𝑣 becomes 𝒲𝑣 ∖𝒮𝑣 . Then {𝑅𝑖 ∣𝒲𝑖 ∈ 𝒲𝑣 ∖𝒮𝑣 } satisfying ∑ 𝑖∈{𝑖∣𝒲𝑖 ∈𝒞}

𝑅𝑖
𝑅𝑢sum )}

log2 (1 + 𝜌𝛾𝑢𝑣,𝑡 ) > 3𝑅𝑢sum

. (25)

For the specific derivation and expression for Pr(𝒜𝑇2 ), please refer to Appendix B. We denote 𝒜𝑇3 as the event that both D1 and D2 finish decoding within 𝑇3 retransmissions in transmission phase 3. Due to the available RSI 𝒮D1 = {𝒲S1 } and 𝒮D2 = {𝒲S3 } obtained in transmission phase 1, we have from (13)

=

Pr

≈

𝑄

) log2 (1 + 𝜌𝛾𝑢𝑣,𝑡 ) > 2𝑅𝑢sum

𝑡=1

𝑢=R3 , 𝑣∈{D1 ,D2 }

(

(𝑇 3 ∑

2𝑀 𝑅0 − 𝑇3 𝜇 √ 𝑇3 𝜎 2

)2

.

(26)

Substituting (24), (25), and (26) into (9) and (11), we can thus obtain the network throughput for the proposed JNCC strategy as /{ ∞ ∑ 𝑇1 [Pr(𝒜𝑇1 ) − Pr(𝒜𝑇1 −1 )] 𝜂 = ℜ 3 + +

2

∞ ∑

𝑇2 [Pr(𝒜𝑇2 ) − Pr(𝒜𝑇2 −1 )]

𝑇2 =1 ∞ ∑

}

𝑇3 [Pr(𝒜𝑇3 ) − Pr(𝒜𝑇3 −1 )] ,

(27)

∑

𝑢∈{S1 ,S2 ,S3 }

𝑅𝑢sum = 3𝑀 𝑅0 .

Pr(𝒜rlnc D1 ,𝑇1 )

(20)

,

(21)

)2

(22) (23)

= Pr(𝒜rlnc ) ( ⌈D2 ,𝑇⌉1 ) 𝑀 − 𝑇1 (1 − 𝑝𝑢𝑣 ) 𝑏 ≈ 𝑄 √ , 𝑇1 𝑝𝑢𝑣 (1 − 𝑝𝑢𝑣 )

(28)

(29)

where 𝑝𝑢𝑣 = Pr (log2 (1 + 𝜌𝛾𝑢𝑣,𝑡 ) < 𝑅rlnc = 𝑏𝑅0 ). From (16), we have rlnc rlnc rlnc rlnc Pr(𝒜rlnc 𝑇1 ) = Pr(𝒜R1 ,𝑇1 ) Pr(𝒜R2 ,𝑇1 ) Pr(𝒜D1 ,𝑇1 ) Pr(𝒜D2 ,𝑇1 ). (30) In transmission phase 2, for R3 to successfully decode ⌈ ⌉ ¯ = 3𝑀 transmis{𝒲S1 , 𝒲S2 , 𝒲S3 }, it has to receive 𝑁 2𝑏 sions correctly (best-case scenario) from each of the two (16), incoming links R1 → R3 and R2 { → R3 . From ⌈ 3𝑀 ⌉} rlnc rlnc ∩ = 𝒜 = 𝑠 ≥ we have event 𝒜 R1 R3 ,𝑇2 𝑇2 R3 ,𝑇2 2𝑏 { ⌈ ⌉} 𝑠R2 R3 ,𝑇2 ≥ 3𝑀 , which occurs with probability 2𝑏

(⌈ Pr(𝒜rlnc 𝑇2 ) ≈ 𝑄

3𝑀 2𝑏

⌉

− 𝑇2 (1 − 𝑝𝑢𝑣 )

)2

√ 𝑇2 𝑝𝑢𝑣 (1 − 𝑝𝑢𝑣 )

.

(31)

In transmission phase 3, for D1 , D2 to successfully decode ¯ {𝒲 S1 , 𝒲⌉S2 , 𝒲S3 }, they both only have to receive 𝑁 = ⌈ 3𝑀−𝑀 transmissions correctly (best-case scenario) due to 𝑏 the RSI 𝒮D1 = {𝒲S1 } and 𝒮D2 = {𝒲S3 } obtained in = transmission have event 𝒜rlnc 𝑇3 ⌈phase { ⌉} 1.{ From (16), ⌈we ⌉} 2𝑀 2𝑀 𝑠R3 D1 ,𝑇3 ≥ 𝑏 ∩ 𝑠R3 D2 ,𝑇3 ≥ 𝑏 , which occurs with probability (⌈

C. SNCC Strategy with RLNC In transmission phase 1, for R1 , R2 , D1 , D2 to successfully decode {𝒲S1 , 𝒲S2 }, {𝒲S2 , 𝒲S3 }, 𝒲S1 , and 𝒲S3 respectively,⌈ as⌉ a best-case scenario, they each have to receive ¯ = 𝑀 transmissions correctly from every incoming link. 𝑁 𝑏

,

= Pr(𝒜rlnc R2 ,𝑇1 ) )2 (⌈ ⌉ 𝑀 − 𝑇1 (1 − 𝑝𝑢𝑣 ) 𝑏 , ≈ 𝑄 √ 𝑇1 𝑝𝑢𝑣 (1 − 𝑝𝑢𝑣 )

𝑇1 =1

𝑇3 =1

where ℜ =

(

)2

Again we assume that all receivers finish decoding within 𝑇1 retransmissions. From (15), for large 𝑇1 , the binomial distribution ℬ (𝑇1 , 1 − 𝑝𝑢𝑣 ) can be approximated [45] as a Gaussian distribution 𝒩 (𝑇1 (1 − 𝑝𝑢𝑣 ), 𝑇1 𝑝𝑢𝑣 (1 − 𝑝𝑢𝑣 )). Thus we have for each intended receiver Pr(𝒜rlnc R1 ,𝑇1 )

𝑢∈{R1 ,R2 },𝑣=R3 𝑡=1

Pr(𝒜𝑇3 ) ∏

≈𝑄

𝑀 𝑅0 − 𝑇1 𝜇 √ 𝑇1 𝜎 2

𝑀 𝑅0 − 𝑇1 𝜇 √ ≈𝑄 Pr log2 (1 + 𝜌𝛾𝑢𝑣,𝑡 ) > 𝑇1 𝜎 2 𝑡=1 𝑢∈{S2 ,S3 }, 𝑣=R2 ) (𝑇 ( ) 1 ∑ 𝑀 𝑅0 − 𝑇1 𝜇 sum √ ≈𝑄 Pr log2 (1 + 𝜌𝛾S1 D1 ,𝑡 ) > 𝑅S1 , 𝑇1 𝜎 2 𝑡=1 (𝑇 ) ) ( 1 ∑ 𝑀 𝑅0 − 𝑇1 𝜇 sum √ , Pr log2 (1 + 𝜌𝛾S3 D2 ,𝑡 ) > 𝑅S3 ≈𝑄 𝑇1 𝜎 2 𝑡=1

𝑡=1

𝑢∈{R1 ,R2 },𝑣=R3

(

)

(𝑇 1 ∑

Similarly, we denote 𝒜𝑇2 as the event that R3 finishes decoding within 𝑇2 retransmissions in transmission phase 2. From (13), we have Pr (𝒜𝑇2 ) {

𝑅𝑢sum

Pr(𝒜rlnc 𝑇3 )

≈𝑄

2𝑀 𝑏

⌉

− 𝑇3 (1 − 𝑝𝑢𝑣 )

√ 𝑇3 𝑝𝑢𝑣 (1 − 𝑝𝑢𝑣 )

)2 .

(32)

Substituting (30), (31), and (32) into (9) and (11), we can


[bits/time-frequency resource]

Network Throughput

2.5

JNCC (analytical, eq. (27)) JNCC (simulation) SNCC (analytical, eq. (33)) SNCC (simulation)

2

b=5

JNCC

1.5 1

SNCC b=2

0.5

b=1

0 0 Fig. 5.

5

10

15 ρ [dB]

20

25

30

Network throughput for the toy example with 𝑀 = 10, 𝑅0 = 1.

also obtain the network throughput for SNCC with RLNC as /{ ∞ ∑ [ ] rlnc rlnc 𝜂 = ℜ 3 𝑇1 Pr(𝒜rlnc 𝑇1 ) − Pr(𝒜𝑇1 −1 ) ⌈ ⌉ 𝑀 𝑏

𝑇1 =

+

∞ ∑

2

𝑇2 =

+

⌈

3𝑀 2𝑏

∞ ∑ 𝑇3 =

⌈

2𝑀 𝑏

⌉

] [ rlnc 𝑇2 Pr(𝒜rlnc 𝑇2 ) − Pr(𝒜𝑇2 −1 ) [

⌉

𝑇3 Pr(𝒜rlnc 𝑇3 )

−

]

Pr(𝒜rlnc 𝑇3 −1 )

} .

(33)

D. Simulation Results We assume that each source has a set of 𝑀 = 10 independent messages to transmit, 𝑅0 = 1 and that all the channels in the network experience i.i.d. Rayleigh fading such that 𝐸[𝛾𝑢𝑣,𝑡 ] = 1 ∀ 𝑒𝑢𝑣 ∈ ℰ. Thus for RLNC, the outage probability is given by 𝑝𝑢𝑣,𝑡

=

Pr (log2 (1 + 𝜌𝛾𝑢𝑣,𝑡 ) < 𝑅rlnc ) ( 𝑅rlnc ) −1 2 = 1 − exp − ∀ 𝑒𝑢𝑣 ∈ ℰ, ∀ 𝑡. (34) 𝜌 In Fig. 5, the network throughput 𝜂 is shown for both the proposed JNCC and SNCC with RLNC under different values of 𝜌. It can be observed that the simulation results denoted by markers agree well with the analytical results which are denoted by lines, and the proposed JNCC strategy always outperforms SNCC with RLNC. This result is mainly due to the fact that in the proposed JNCC, we can take advantage of the accumulation of mutual information and this means that no received codeword is wasted, regardless of whether it can be correctly decoded or not. In contrast, SNCC with RLNC is originally designed based on erasure channels and thus only codewords that are successfully decoded will contribute to the final decoding for source messages {𝒲S1 , 𝒲S2 , 𝒲S3 }. For SNCC with RLNC, different rates of 𝑅rlnc = 𝑏𝑅0 , 𝑏 = 1, 2, 5 are adopted at the transmitters15 . From Fig. 5, it can 15 We assume that all transmitters in the network (i.e. sources and the intermediate nodes) adopt the same rate of 𝑏𝑅0 .

189

be observed that the throughput performance of SNCC is limited by the choice of 𝑏. Although a higher throughput can be achieved in the high SNR region with a larger 𝑏, the corresponding throughput performance is severely reduced in the low SNR region due to the high outage probability. Conversely, for a smaller 𝑏, although a better performance is achieved in the low SNR region, the corresponding throughput in the high SNR region is limited. This is because without transmit CSI at the transmitters, SNCC with RLNC is not able to perform any meaningful rate adaptation to take advantage of the fluctuating channel conditions. However for the proposed JNCC strategy, as shown in Theorem 1 and Example 13, optimal rate adaptation is passively achieved without the requirement for transmit CSI and this translates into good throughput performance throughout the whole SNR range. E. Random Networks In addition to the toy example shown in Fig. 3, the proposed JNCC strategy is also applied to a random multihop multicast network shown in Fig. 6(a). In 𝐿1 , each of the three sources S1 , S2 , S3 (with the same transmit power 𝑃 ) has 𝑀 = 10 independent messages to transmit and each message is of the same rate 𝑅0 = 1. In 𝐿4 , three destinations D1 , D2 , D3 desire the messages from all sources. 10 relay nodes are then randomly distributed among two levels 𝐿2 and 𝐿3 . Communication links between neighbouring levels are randomly chosen under the constraint that there is at least a path from each source to each destination. Under the same channel conditions as the toy example above (i.e. i.i.d. Rayleigh fading channel and Gaussian noise 𝜎𝑛2 ), we obtain the simulation results for the corresponding network throughput averaged over 1000 network realizations in Fig. 6(b). To better illustrate the spread of the network throughput with respect to the average value, we also plot the standard deviation and the 10th percentile throughput. To reduce clutter in the figure, we have just shown one representative standard deviation bar for each graph. Again we can see that the proposed JNCC strategy substantially outperforms SNCC with RLNC (different rates of 𝑅rlnc = 𝑏𝑅0 , 𝑏 = 1, 2, 5 are adopted at the transmitters) in terms of network throughput. VI. C ONCLUSION In this paper, we considered a wireless multicast network with multiple sources, relays, and destinations. For tractable analysis, we considered a feedforward flowgraph where all nodes in the network are divided into disjoint level sets such that the end-to-end information flow can be characterized by separate transmission phases. A joint network and channel coding strategy was proposed by exploiting ARQ, receiver side information, and correlated sources. Without requiring receiver side information for explicit code design, nor transmit channel state information for rate adaptation, a transmitter simply performs retransmissions blindly until each of the intended receivers has successfully decoded all desired messages and sent back an ACK. To measure the performance of the proposed joint network and channel coding strategy, we quantified the network throughput over retransmissions by applying the renewal-reward theorem. Our results show that the proposed joint network and channel coding strategy

190


10 relay nodes

{

A PPENDIX A. Proof for Theorem 1

D1

S1 . . .

S2

. . .

D2 D3

S3 L1

L3

L2

L4

(a) The feedforward flowgraph of a random multi-hop multicast network. 0.9 JNCC (simulation) 10th percentile

Network Throughput

0.8

SNCC, b=5 (simulation) 10th percentile SNCC, b=2 (simulation) 10th percentile SNCC, b=1 (simulation) 10th percentile

[bits/time-frequency resource]

0.7 0.6

}

SNCC

0.5 0.4

b=5

JNCC

0.3 0.2

}

0.1

}b=1

b=2

0

0

5

10

15

ρ [dB]

20

25

30

(b) Network throughput. Fig. 6.

A random multi-hop multicast network.

W1 W2

Common messages

W5 W6

W3 W4

u1

u2 v

Sv={W , W , W } 2

4

6

Fig. 7. An orthogonal MAC-CS-RSI at receiver 𝑣 with two intended transmitters 𝒯𝑣 = {𝑢1 , 𝑢2 }.

substantially outperforms the conventional separate network and channel coding strategy with random linear network coding.

To prove Theorem 1, for ease of exposition, we consider an orthogonal MAC-CS-RSI with two intended transmitters. The results can then be easily generalized to an orthogonal MAC-CS-RSI for any receiver with an arbitrary set of intended transmitters. As shown in Fig. 7, we consider an orthogonal MAC-CSRSI with DMC for a receiver 𝑣 with two intended transmitters 𝒯𝑣 = {𝑢1 , 𝑢2 }: ⎛ ⎞ ∏ ⎝𝒳𝑢1 × 𝒳𝑢2 , 𝑝𝑌𝑢𝑣 ∣𝑋𝑢 (𝑦𝑢𝑣 ∣𝑥𝑢 ) , 𝒴𝑢1 𝑣 × 𝒴𝑢2 𝑣 ⎠ , 𝑢∈{𝑢1 ,𝑢2 }

(35) where 𝒳𝑢1 and 𝒳𝑢2 denote the input alphabet sets of transmitters 𝑢1 and 𝑢2 , 𝒴𝑢1 𝑣 and ∏ 𝒴𝑢2 𝑣 denote the corresponding output alphabet sets at 𝑣, and 𝑢∈{𝑢1 ,𝑢2 } 𝑝𝑌𝑢𝑣 ∣𝑋𝑢 (𝑦𝑢𝑣 ∣𝑥𝑢 ) is the channel transition probability. All alphabets are assumed to be finite. We seek to derive the achievable rate region and the corresponding coding strategy for a scenario where transmitters 𝑢1 , 𝑢2 each has a set of independent messages 𝒲𝑢1 = {𝒲1 , 𝒲2 , 𝒲3 , 𝒲4 }, and 𝒲𝑢2 = {𝒲3 , 𝒲4 , 𝒲5 , 𝒲6 } to transmit. The correlation between the two intended transmitters, via 𝒲3 and 𝒲4 , may come from an earlier transmission phase from other links. Thus, in total a set of independent messages 𝒲𝑣 = ∪𝑢∈{𝑢1 ,𝑢2 } 𝒲𝑢 = {𝒲1 , 𝒲2 , 𝒲3 , 𝒲4 , 𝒲5 , 𝒲6 } is desired at 𝑣. Without loss of generality, we assume that 𝒮𝑣 = {𝒲2 , 𝒲4 , 𝒲6 } is known a priori at 𝑣 as RSI. Thus the set of messages that is effectively transmitted to 𝑣 becomes 𝒲𝑣 ∖𝒮𝑣 = {𝒲1 , 𝒲3 , 𝒲5 }. Lemma 1: We consider the orthogonal MAC-CS-RSI in (35) where 𝒲𝑢1 = {𝒲1 , 𝒲2 , 𝒲3 , 𝒲4 }, 𝒲𝑢2 = {𝒲3 , 𝒲4 , 𝒲5 , 𝒲6 }, and 𝒮𝑣 = {𝒲2 , 𝒲4 , 𝒲6 }. Retransmissions are performed over 𝑇 time slots, with each time slot consisting of 𝑛 → ∞ channel uses. Then {𝑅1 , 𝑅3 , 𝑅5 } satisfying 𝑅5 < 𝐼2sum (𝑇 ), 𝑅1 < 𝐼1sum (𝑇 ), 𝑅1 + 𝑅3 + 𝑅5 < 𝐼1sum (𝑇 ) + 𝐼2sum (𝑇 ),

(36)

is achievable for some product distribution 𝑝(𝑥𝑢1 )𝑝(𝑥𝑢2 ) on 𝒳𝑢1 × 𝒳𝑢1 , where 𝐼1sum (𝑇 ) (𝐼2sum (𝑇 )) is the sum mutual information offered by the channel 𝑢1 → 𝑣 (𝑢2 → 𝑣) across 𝑇 retransmissions. Remark 2: If the two transmitters 𝑢1 , 𝑢2 know 𝒮𝑣 and simply transmit {𝒲1 , 𝒲3 } and {𝒲3 , 𝒲5 } respectively, then clearly (36) is the achievable rate region. However, we will show in the following proof that the transmitters do not need to explicitly know the RSI in order to achieve (36). Achievability Proof: Employing the proposed JNCC strategy in Section III-A, the codebooks for encoding by 𝑢1 and 𝑢2 are independently and identically generated for every transmission. We denote the transmitted and received codewords in the 𝑡th transmission as 𝒙𝑢,𝑡 and 𝒚 𝑢𝑣,𝑡 respectively, where 𝑢 ∈ {𝑢1 , 𝑢2 }, 𝑡 = 1, 2, ⋅ ⋅ ⋅ , 𝑇 . (𝑛) Let 𝐴𝜖 (𝑋𝑢,𝑡 , 𝑌𝑢𝑣,𝑡 ) be the set of joint typical (𝒙𝑢,𝑡 , 𝒚 𝑢𝑣,𝑡 ) sequences. The decoder of 𝑣 chooses the transmitted codeword ˆ3 ) and (𝑤 ˆ3 , 𝑤 ˆ5 ) satisfying the joint as the unique (𝑤 ˆ1 , 𝑤 typicality condition in (37).


𝑇 ∩ 𝑡=1

{

191

} } {( } {( ) ) (𝑛) (𝑛) ˆ1 , 𝑤2 , 𝑤 ˆ3 , 𝑤4 ) , 𝒚 𝑢1 𝑣,𝑡 ∈ 𝐴𝜖 (𝑋𝑢1 ,𝑡 , 𝑌𝑢1 𝑣,𝑡 ) ∩ 𝒙𝑢2 ,𝑡 (𝑤 ˆ3 , 𝑤4 , 𝑤 ˆ5 , 𝑤6 ) , 𝒚 𝑢2 𝑣,𝑡 ∈ 𝐴𝜖 (𝑋𝑢2 ,𝑡 , 𝑌𝑢2 𝑣,𝑡 ) . 𝒙𝑢1 ,𝑡 (𝑤 (37)

That is, for every transmission, joint typical-set decoding is employed to produce a set of possible codewords, then ˆ3 , 𝑤 ˆ5 } that are present in all we choose the unique {𝑤 ˆ1 , 𝑤 sets across 𝑇 retransmissions. If none or more than one {𝑤 ˆ1 , 𝑤 ˆ3 , 𝑤 ˆ5 } is found, an error is declared. Analysis of Probability of Error: By the symmetry of the code construction, the probability of error does not depend on the particular codeword that was sent. Thus, without loss of generality, we assume that codewords 𝒙𝑢1 ,𝑡 (1, 𝑤2 , 1, 𝑤4 ) and 𝒙𝑢2 ,𝑡 (1, 𝑤4 , 1, 𝑤6 ) are transmitted for 𝑡 = 1, 2, ⋅ ⋅ ⋅ , 𝑇 . We note that due to the RSI 𝒮𝑣 = {𝒲2 , 𝒲4 , 𝒲6 }, the decoder of 𝑣 is able to narrow down number of incor( 𝑛(𝑅 the ) 1 +𝑅2 +𝑅3 +𝑅4 ) from 2 rect codewords for 𝒙 𝑢 ,𝑡 1 ) ( 𝑛(𝑅 +𝑅 +𝑅 +𝑅−)1 to) ( 𝑛(𝑅 +𝑅 ) 3 4 5 6 −1 2 ( 1 3 − 1 , and ) for 𝒙𝑢2 ,𝑡 from 2 to 2𝑛(𝑅3 +𝑅5 ) − 1 [29]-[35]. For ease of exposition, we define 𝒲𝑒1 , 𝒲𝑒3 , and 𝒲𝑒5 as the sets of incorrect codewords for messages 𝒲1 , 𝒲3 , and 𝒲5 respectively. An error occurs when the incorrect codewords are jointly typical with the received sequences. ˆ3 , 𝑤4 ) = We define the events 𝐸𝑡1 (𝑤ˆ1 , 𝑤2 , 𝑤 (𝑛) {(𝒙𝑢1 ,𝑡 (𝑤 ˆ1 , 𝑤2 , 𝑤 ˆ3 , 𝑤4 ), 𝒚 𝑢1 𝑣,𝑡 ) ∈ 𝐴𝜖 (𝑋𝑢1 ,𝑡 , 𝑌𝑢1 𝑣,𝑡 )}, 𝐸𝑡2 (𝑤ˆ3 , 𝑤4 , 𝑤 ˆ5 , 𝑤6 ) = {(𝒙𝑢2 ,𝑡 (𝑤 ˆ3 , 𝑤4 , 𝑤 ˆ5 , 𝑤6 ), 𝒚 𝑢2 𝑣,𝑡 ) (𝑛) and 𝐸𝑡1,𝑐 (𝑤 ˆ1 , 𝑤2 , 𝑤 ˆ3 , 𝑤4 ), ∈ 𝐴𝜖 (𝑋𝑢2 ,𝑡 , 𝑌𝑢2 𝑣,𝑡 )}, 𝐸𝑡2,𝑐 (𝑤 ˆ3 , 𝑤4 , 𝑤 ˆ5 , 𝑤6 ) as the corresponding complementary events. In total, we have the following four cases for the error events. 1: Correct codewords are not jointly typical with received sequences. 𝜉1 =

𝑇 ∪

𝐸𝑡1,𝑐 (1, 𝑤2 , 1, 𝑤4 )

𝑇 ∪

∪

𝑡=1

𝐸𝑡2,𝑐 (1, 𝑤4 , 1, 𝑤6 ).

(38)

𝑡=1

2: For codewords that satisfy the typicality test, one of {𝑤 ˆ1 , 𝑤 ˆ3 , 𝑤 ˆ5 } is wrong. ( 𝑇 ∩ ∪ 𝐸𝑡1 (1, 𝑤2 , 1, 𝑤4 ) 𝜉2 = 𝑡=1

𝑤 ˆ 5 ∈𝒲𝑒5

∩ 𝜉3 =

∪

𝑡=1 𝑇 ∩

(

𝑡=1


∩ 𝜉4 =

∪

𝑇 ∩

𝑇 ∩ 𝑡=1 𝑇 ∩

(

𝑡=1


∩

𝑇 ∩ 𝑡=1

) 𝐸𝑡2 (1, 𝑤4 , 𝑤 ˆ5 , 𝑤6 )

,

(39)

𝑡=1

𝑤 ˆ3 ∈𝒲𝑒3 ,𝑤 ˆ 5 ∈𝒲𝑒5

∩

𝑇 ∩

) 𝐸𝑡2 (𝑤 ˆ3 , 𝑤4 , 𝑤 ˆ5 , 𝑤6 ) , (42)

𝑡=1

(

∪

𝜉6 =

𝑇 ∩ 𝑡=1

𝑤 ˆ 1 ∈𝒲𝑒1 ,𝑤 ˆ5 ∈𝒲𝑒5

∩ 𝜉7 =

𝑇 ∩ 𝑡=1 𝑇 ∩

(

∪

𝐸𝑡1 (𝑤 ˆ1 , 𝑤2 , 1, 𝑤4 ) ) 𝐸𝑡2 (1, 𝑤4 , 𝑤 ˆ5 , 𝑤6 ) 𝐸𝑡1 (𝑤 ˆ1 , 𝑤2 , 𝑤 ˆ3 , 𝑤4 )

𝑡=1

𝑤 ˆ 1 ∈𝒲𝑒1 ,𝑤 ˆ3 ∈𝒲𝑒3

∩

𝑇 ∩

, (43)

) 𝐸𝑡2 (𝑤 ˆ3 , 𝑤4 , 1, 𝑤6 )

. (44)

𝑡=1

4: For codewords that satisfy the typicality test, none of {𝑤 ˆ1 , 𝑤 ˆ3 , 𝑤 ˆ5 } is correct. ( 𝑇 ∩ ∪ 𝜉8 = 𝐸𝑡1 (𝑤ˆ1 , 𝑤2 , 𝑤 ˆ3 , 𝑤4 ) 𝑤 ˆ1 ∈𝒲𝑒1 ,𝑤 ˆ 3 ∈𝒲𝑒3 ,𝑤 ˆ5 ∈𝒲𝑒5

∩

𝑡=1 𝑇 ∩

)

𝐸𝑡2 (𝑤 ˆ3 , 𝑤4 , 𝑤 ˆ5 , 𝑤6 )

. (45)

𝑡=1

From [7], by the union bound and the use ∑8 of i.i.d. codebooks, the error probability is given as 𝑃𝑒 = 𝑖=1 Pr(𝜉𝑖 ), where Pr(𝜉1 ) Pr(𝜉2 ) Pr(𝜉3 ) Pr(𝜉4 ) Pr(𝜉5 ) Pr(𝜉6 ) Pr(𝜉7 ) Pr(𝜉8 )

< < < < < < <
𝑀 𝑅0

𝑡=1

𝑢=R1 ,𝑣=R3

∩

∑

𝑢=R2 ,𝑣=R3

𝑇2 ∑

𝑢∈{R1 ,R2 }, 𝑣=R3 𝑡=1

⎫) ⎬ log2 (1 + 𝜌𝛾𝑢𝑣,𝑡 ) > 3𝑀 𝑅0 . ⎭

where

∫ 𝑃𝐷𝐴𝐸 ˆ

D

= =

3MR0

2MR0

B

P

BAC

MR0

A

O

MR0

E

C

2MR0

3MR0

X

Probability region for event 𝒜𝑇2 in (48).

Hence, with the proposed JNCC strategy, we can choose 𝜖 and 𝑛 such that the rates in (46) can be achieved asymptotically. ■ The results of Lemma 1 can be readily generalized to any receiver with arbitrary RSI and an arbitrary set of intended transmitters and messages to obtain Theorem 1. Since the encoding at all transmitters 𝑢 ∈ 𝐿𝑘 and the decoding at all receivers 𝑣 ∈ 𝑉𝑘 are independent, it is clear that Remark 1 holds true. B. Derivation of Pr(𝒜𝑇2 ) From (25), since the message set 𝒲S1 is transmitted solely through channel R1 → R3 , 𝒲S3 is transmitted solely through channel R2 → R3 , and 𝒲S2 is transmitted through both channels, we can obtain Pr(𝒜𝑇2 ) in (47). ease of exposition, we denote 𝑥 = ∑For 𝑇2 log (1 + 𝜌𝛾 ) where 𝑢 = R and 𝑣 = R 𝑢𝑣,𝑡 1 3, 2 𝑡=1 ( ) ∑ 2 ′ where 𝑢 = R2 and log2 1 + 𝜌𝛾𝑢𝑣,𝑡 and denote16 𝑦 = 𝑇𝑡=1 𝑣 = R3 . Thus, ( Pr(𝒜𝑇2 ) = Pr {𝑥 > 𝑀 𝑅0 } ∩ {𝑦 > 𝑀 𝑅0 } ) (48) ∩ {𝑥 + 𝑦 > 3𝑀 𝑅0 } , ˆ shown in which is denoted by the shaded region 𝐷𝐵𝐶𝐸 Fig. 8. Then we have Pr(𝒜𝑇2 ) = 𝑃𝐷𝐵𝐶𝐸 ˆ = 𝑃𝐷𝐴𝐸 ˆ − 𝑃𝐵𝐴𝐶 ˆ, 16 Note

0 ∫𝑀𝑅 ∞

𝑀𝑅0

≈ 𝑄

DAE

′ that here 𝛾𝑢𝑣,𝑡 and 𝛾𝑢𝑣,𝑡 are i.i.d. random variables.

∫

∞

(

P

Fig. 8.

{𝑇 2 ∑

(47)

∞

𝑓𝑋,𝑌 (𝑥, 𝑦)𝑑𝑦𝑑𝑥 ∫ ∞ 𝑓𝑋 (𝑥)𝑑𝑥 𝑓𝑌 (𝑦)𝑑𝑦 𝑀𝑅0

𝑀 𝑅0 − 𝑇2 𝜇 √ 𝑇2 𝜎

𝑀𝑅0 )2

,

(49)

𝑓𝑋 (𝑥) and 𝑓𝑌 (𝑦) are the probability density functions for 𝑥 and 𝑦 respectively. The approximation in (49) comes from the Central Limit Theorem where 𝑥 ∼ 𝒩 (𝑇2 𝜇, 𝑇2 𝜎 2 ) and 𝑦 ∼ 𝒩 (𝑇2 𝜇, 𝑇2 𝜎 2 ) approximately[for large 𝑇2 , where 𝜇 = ] 𝐸 [log2 (1 + 𝜌𝛾𝑢𝑣,𝑡 )] and 𝜎 2 = 𝐸 ∣ log2 (1 + 𝜌𝛾𝑢𝑣,𝑡 ) − 𝜇∣2 . Since the integration for the triangular area ˆ 𝐵𝐴𝐶 is intractable, we can approximate 𝑃𝐵𝐴𝐶 by calculating the sum ˆ probability of a set of smaller non-overlapping square areas within ˆ 𝐵𝐴𝐶. From (49), we can see that for a square area defined by (𝑥, 𝑦) and (𝑥 + 𝑐, 𝑦 + 𝑐) where 𝑥, 𝑦, 𝑐 ≥ 0, the corresponding probability can be obtained by (50). Thus the probability of 𝑃𝐵𝐴𝐶 ˆ can be derived in terms of a summation of (50) for different 𝑥, 𝑦, and 𝑐. In our derivations, we use 7 smaller squares to approximate the triangular area ˆ as shown in Fig. 8, and simulation results show that 𝐵𝐴𝐶 it is a good approximation. We omit the details here due to space constraints. R EFERENCES [1] C. E. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J., vol. 27, pp. 379–423, 1948. [2] L. Song, R. W. Yeung, and N. Cai, “A separation theorem for singlesource network coding,” IEEE Trans. Inf. Theory, vol. 52, no. 5, pp. 1861–1871, May 2006. [3] R. Ahlswede, N. Cai, S.-Y. R. Li, and R. W. Yeung, “Network information flow,” IEEE Trans. Inf. Theory, vol. 46, no. 4, pp. 1204–1216, July 2000. [4] S.-Y. R. Li, R. W. Yeung, and N. Cai, “Linear network coding,” IEEE Trans. Inf. Theory, vol. 49, no. 2, pp. 371–381, Feb. 2003. [5] R. Koetter and M. Medard, “An algebraic approach to network coding,” IEEE/ACM Trans. Networking, vol. 11, no. 5, pp. 782–795, Oct. 2003. [6] T. Ho, M. Medard, J. Shi, M. Effros, and D. R. Karger, “On randomized network coding,” in Proc. 41st Annual Allerton Conf. Commun. Control Computing, Oct. 2003. [7] T. M. Cover and J. A. Thomas, Elements of Information Theory, 2nd edition. Wiley, 2006. [8] S. Katti, D. Katabi, W. Hu, H. Rahul, and M. Medard, “The importance of being opportunistic: practical network coding for wireless environments,” in Proc. Allerton Conf. Commun., Control, Computing, 2005. [9] S. Katti, H. Rahul, W. Hu, D. Katabi, M. Medard, and J. Crowcroft, “XORs in the air: practical wireless network coding,” IEEE/ACM Trans. Networking, vol. 16, no. 3, pp. 497–510, June 2008. [10] A. Argyriou, “Wireless network coding with improved opportunistic listening,” IEEE Trans. Wireless Commun., vol. 8, no. 4, pp. 2014–2023, Apr. 2009. [11] T. Ho and D. S. Lun, Network Coding: An Introduction. Cambridge University Press, 2008.


) ( )] [ ( ) ( )] [ ( 𝑥 + 𝑐 − 𝑇2 𝜇 𝑦 − 𝑇2 𝜇 𝑦 + 𝑐 − 𝑇2 𝜇 𝑥 − 𝑇2 𝜇 √ √ √ −𝑄 ⋅ 𝑄 √ −𝑄 . 𝑄 𝑇2 𝜎 𝑇2 𝜎 𝑇2 𝜎 𝑇2 𝜎

[12] Y. Wu, P. A. Chou, and S.-Y. Kung, “Information exchange in wireless networks with network coding and physical-layer broadcast,” in Proc. CISS, Mar. 2005. [13] P. Larsson, N. Johansson, and K.-E. Sunell, “Coded bi-directional relaying,” in Proc. ADHOC, May 2005. [14] Y. Han, S. H. Ting, C. K. Ho, and W. H. Chin, “Performance bounds for two-way amplify-and-forward relaying,” IEEE Trans. Wireless Commun., vol. 8, no. 1, pp. 432–439, Jan. 2009. [15] Q. Li, S. H. Ting, A. Pandharipande, and Y. Han, “Adaptive two-way relaying and outage analysis,” IEEE Trans. Wireless Commun., vol. 8, no. 6, pp. 3288–3299, June 2009. [16] X. Bao and J. Li, “Adaptive network coded cooperation (ANCC) for wireless relay networks: matching code-on-graph with network-ongraph,” IEEE Trans. Wireless Commun., vol. 7, no. 2, pp. 574–583, Feb. 2008. [17] M. Xiao and T. Aulin, “Optimal decoding and performance analysis of a noisy channel network with network coding,” IEEE Trans. Commun., vol. 57, no. 5, pp. 1402–1412, May 2009. [18] M. Xiao and M. Skoglund, “Design of network codes for multiple-user multiple-relay wireless networks,” in Proc. IEEE ISIT, June 2009, pp. 2562–2566. [19] A. F. Dana, R. Gowaikar, R. Palanki, B. Hassibi, and M. Effros, “Capacity of wireless erasure networks,” IEEE Trans. Inf. Theory, vol. 52, no. 3, pp. 789–804, Mar. 2006. [20] S. P. Borade, “Network information flow: limits and achievability,” in Proc. IEEE ISIT, June 2002, p. 139. [21] M. R. Aref, “Information flow in relay networks,” Ph.D. dissertation, Stanford Univ., Stanford, CA, 1980. [22] N. Ratnakar and G. Kramer, “The multicast capacity of deterministic relay networks with no interference,” IEEE Trans. Inf. Theory, vol. 52, no. 6, pp. 2425–2432, June 2006. [23] M. Effros, M. Medard, T. Ho, S. Ray, D. Karger, R. Koetter, and B. Hassibi, “Linear network codes: a unified framework for source, channel, and network coding,” in Proc. DIMACS Workshop Network Inf. Theory, invited paper, 2003. [24] T. M. Cover, “Broadcast channels,” IEEE Trans. Inf. Theory, vol. 18, no. 1, pp. 2–14, Jan. 1972. [25] H. Liao, “Multiple access channels,” Ph.D. dissertation, University of Hawaii, Honolulu, 1972. [26] C. Hausl and J. Hagenauer, “Iterative network and channel decoding for the two-way relay channel,” in Proc. IEEE ICC, June 2006, pp. 1568–1573. [27] C. Hausl, “Joint network-channel coding for the multiple-access relay channel based on turbo codes,” European Trans. Telecommun., vol. 20, no. 2, pp. 175–181, June 2009. [28] S. Zhang, Y. Zhu, S. Liew, and K. B. Letaief, “Joint design of network coding and channel decoding for wireless networks,” in Proc. IEEE WCNC, Mar. 2007, pp. 779–784. [29] L. Xiao, T. E. Fuja, J. Kliewer, and D. J. Costello, Jr., “Nested codes with multiple interpretations,” in Proc. CISS, Mar. 2006. [30] L. Xiao, T. E. Fuja, J. Kliewer, and D. J. Costello, Jr., “A network coding approach to cooperative diversity,” IEEE Trans. Inf. Theory, vol. 53, no. 10, pp. 3714–3722, Oct. 2007. [31] J. Kliewer, T. Dikaliotis, and T. Ho, “On the performance of joint and separate channel and network coding in wireless fading networks,” in Proc. IEEE Inf. Theory Workshop Inf. Theory Wireless Netw., July 2007. [32] Y. Wu, “Broadcasting when receivers know some messages a priori,” in Proc. IEEE ISIT, June 2007, pp. 1141–1145. [33] F. Xue and S. Sandhu, “PHY-layer network coding for broadcast channel with side information,” in Proc. IEEE ITW, Sep. 2007, pp. 108–113. [34] G. Kramer and S. Shamai, “Capacity for classes of broadcast channels with receiver side information,” in Proc. IEEE ITW, Sep. 2007, pp. 313– 318. [35] H. Yomo and P. Popovski, “Opportunistic scheduling for wireless network coding,” IEEE Trans. Wireless Commun., vol. 8, no. 6, pp. 2766–2770, June 2009. [36] S. H. Lim, Y. Kim, A. El Gamal, and S. Chung, “Noisy network coding,” in Proc. IEEE ITW, Jan. 2010. [37] D. Slepian and J. K. Wolf, “Noiseless coding of correlated information sources,” IEEE Trans. Inf. Theory, vol. 19, no. 4, pp. 471–480, July 1973.

193

(50)

[38] D. Slepian and J. K. Wolf, “A coding theorem for multiple access channels with correlated sources,” Bell Syst. Tech. J., vol. 52, no. 7, pp. 1037–1076, Sep. 1973. [39] T. M. Cover, A. El Gammal, and M. Salehi, “Multiple access channels with arbitrarily correlated sources,” IEEE Trans. Inf. Theory, vol. 26, no. 6, pp. 648–657, Nov. 1980. [40] N. J. Laneman, D. N. 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. [41] R. Comroe and D. Costello, “ARQ schemes for data transmission in mobile radio systems,” IEEE J. Sel. Areas Commun., vol. 2, no. 4, pp. 472–481, July 1984. [42] R. Wolff, Stochastic Modeling and the Theory of Queues. Upper Saddle River, NJ: Prentice-Hall, 1989. [43] G. Caire and D. Tuninetti, “The throughput of hybrid-ARQ protocols for the Gaussian collision channel,” IEEE Trans. Inf. Theory, vol. 47, no. 5, pp. 1971–1988, July 2001. [44] P. Gupta and P. R. Kumar, “Towards an information theory of large networks: an achievable rate region,” IEEE Trans. Inf. Theory, vol. 49, no. 8, pp. 1877–1894, Aug. 2003. [45] A. Papoulis and S. U. Pillai, Probability, Random Variables and Stochastic Processes, 4th edition. McGraw Hill, 2002. [46] Q. Li, S. H. Ting, and C. K. Ho, “Nonlinear network code for high throughput broadcasting with retransmissions,” in Proc. IEEE ISIT, June 2009, pp. 2853–2857. [47] Q. Li, S. H. Ting, and C. K. Ho, “Joint network and channel coding for wireless networks,” in Proc. SECON Workshops, June 2009. Qiang Li received his B.Eng degree in communication engineering in 2007 from the School of Communication and Information Engineering, University of Electronic Science and Technology of China. He is currently working towards his Ph.D. degree in Nanyang Technological University, Singapore. His research interests include cooperative communications, wireless network coding, and cognitive radios.

See Ho Ting received the B.Eng., M.Eng., and Ph.D. degrees in electrical and electronic engineering in 2002, 2004, and 2006, respectively, from the Tokyo Institute of Technology, Japan. From April 2006, he is an Assistant Professor in Nanyang Technological University, Singapore. He received the Young Researcher Encouragement Award from the IEEE VTS Japan Chapter in 2002. In 2005, he received the Ericsson Young Scientist Award and the IEICE Outstanding Paper Award. His current research interests include MIMO-OFDM systems, cooperative communications, and wireless network coding. Chin Keong Ho received the Ph.D. degree from Eindhoven University of Technology, The Netherland, and the M.Eng. and B.Eng. (first-class Honors) degrees from the National University of Singapore. Since August 2000, he has been with the Institute for Infocomm Research, A*STAR, Singapore. He conducted research work in Philips Research while working towards his Ph.D. degree. Prior to earning his Ph.D., Dr. Keong worked on the signal processing aspects of multi-carrier and multi-antenna communications. His current research interest lies in wireless cooperative and adaptive communications.