On the Design of Network Coding for Downlink of Single-Relay Multi-Users Two-Way Network

Xiaoxi Guo

Beijing Key Laboratory of Network System Architecture and

Convergence, Beijing University of Posts and

Telecommunications Beijing, China

gxx880203@gmail.com

Jianjun Hao Beijing Key Laboratory of Network

System Architecture and Convergence,

Beijing University of Posts and Telecommunications

Beijing, China jjhao@bupt.edu.cn

Yijun Guo Beijing Key Laboratory of Network

System Architecture and Convergence,

Beijing University of Posts and Telecommunications

Beijing, China guo.yijun@hotmail.com

Abstract—In single-relay multi-users two-way network, the

aim is to transmit packets from each node to all nodes in order to implement information exchange between various nodes. Network coding has the benefit of high throughput efficiency compare to ARQ approach. In addition, random linear sparse network coding (RLSNC) can reduce the algorithm complexity based on general RLNC. In this paper, single-transmitting mixed network coding (STMNC) is proposed in this single-relay multi-users two-way network. The purpose is to increase the throughput efficiency with the least cost of latency and complexity.

Keywords-ARQ; Random linear Network coding; RLSNC; STMNC

I. INTRODUCTION Automatic Repeat-request (ARQ) is commonly used in

single-source single-receiver network as a reliable transport protocol. In that scenario, source node splits source information into several packets and sends them sequentially. In the meantime, the receiver feedbacks acknowledge frames to source node to inform whether the packet is received. If there is packet loss, the source node retransmits the packet after received NACK [1]. In a single-relay multi-users two-way network, relay collects all the data from each node and forwards them using ARQ method. But the throughput efficiency is low if the number of receiver is large enough. The performance is even worse as the channel is becoming worse.

In contrast, network coding approach can dramatically increase the throughput efficiency of a whole network [2]. Network coding was first proposed by Ahlswede, and widely studied in wireless network for its outstanding performance of throughput efficiency. In a network coding scheme, source packets are encoded using a function or algorithm into encoded packets. Then the encoded packets are sent to users. Network coding is maturely studied in two-way relay network [3] and the benefits and costs are analyzed and verified by many researchers.

Consider a multi-users two-way network, information from each node needs to be known to all nodes using a single relay. Random linear network coding (RLNC) is a solution other than ARQ. In RLNC approach, relay generates the encoding matrix

randomly from a finite field [4] and encodes all the source packets using this matrix. Even with better throughput efficiency[5], the encoding and decoding algorithm complexity of RLNC is relatively high [6].

In this paper, random linear sparse network coding (RLSNC)is proposed to simplify the encoding and decoding complexity of RLNC scheme in the downlink of single-relay multi-users two-way network. In addition, single-transmitting mixed network coding (STMNC) scheme is proposed and compared to ARQ, RLNC and RLPNC scheme in the aspects of throughput efficiency, latency and encoding/decoding complexity to verify its benefits.

The paper is organized as follows. Section II defines the single-relay multi-users two-way network system model, including the system scenario and transmission channel. Section III describes the transmission method of each scheme. Section IV analyzes the performance of RLPNC and STMNC theoretically. Section V demonstrates the practical simulation results of the throughput, latency and complexity. Analysis about the results is also raised in this section. Section VI concludes the whole paper.

II. SYSTEM MODEL

A. Scenario Consider a broadcast channel with a single relay R and K

receivers d1, d2,…,dK, as depicted in Figure2a. Channels between relay and receivers are all the same-erasure channel. And all the nodes, including the relay and receivers, are all work in half-duplex mode.

Figure 2a. Single-relay multi-users network

978-1-4673-5829-3/12/$26.00 ©2012 IEEE

In this paper, we assume that R has successfully received source packets from d1~dK. The discussion is focus on the transmission schemes in the downlink of the system. The aim is to exchange information efficiently among the large amount of nodes with the minimal overhead.

B. Binary erasure channel(BEC) In a binary erasure channel [7],as shown in Figure2b, the

sender X send 1 bit symbol to receiver Y each time slot. There are two situations when Y received the symbol:

1) Y received the data correctly, with the probability of (1-Pe), where Pe indicates the bite error rate.

2) Y cannot recognize the bit symbol, with the probability of Pe, then it just treats this bit as erased.

As a result, BEC can be an error-free channel, which is the channel model used in this scenario.

Figure 2b. Binary erasure channel

III. TRANSMISSION SCHEMES

A. ARQ scheme In ARQ scheme, relay R simply broadcasts K original

packets (s1~sK) one by one and retransmits packets if R receives NACK, until all the receivers get the correct packets. Obviously it needs two channel use one is for transmitting packets from relay to receivers. The other is for ACK/NACK feedback frame from receivers to relay [8].

ARQ scheme is always being a reliable scheme and the throughput efficiency is acceptable when there are only a few receivers. But as the number of receiver becomes larger, the throughput efficiency is dramatically decreasing. The possibility of i times transmission needed for a certain encoded packet is:

2 1( )i 1 1 1(1 )

i i i

ip p p p (1)

where 1 (1 )Kp is the possibility of once transmission and is the packet loss rate. Thus the needed number of

packets for a K-users network is 1

K ii

i p .

B. RLNC scheme Consider the scenario shown in Figure 2a above, relay R is

allowed to mix the incoming symbols and send out linear combinations following the basic rules of random linear

network coding (RLNC) [4]. More specifically, the encoder mixesS1,S2,S3,…,SK and outputs encoded packets t1,t2,…tN

using function1

t = G K

n k k nk

s , where G is the encode matrix

and the elements of G is randomly generated from GF(2n)[4].

At receiving ends, decoding function is 1

1

=K

k n k nn

s t G .

C. RLSNC scheme In RLNC scheme, encoding and decoding algorithm

complexity is really high (2×K2+K3). Some researchers are studying sparse encode matrix to simplify the complexity. As D.J.C. MacKay mentioned, binary matrix is used as encoding matrix GK×N[9].The transmission scheme is depicted in Fig.3a.

Figure 3a. RLSNC scheme

In RLSNC scheme, R randomly chooses some original packets to encode in every time slot. Receivers begin decoding process after received K non-related packets.

D. STMNC scheme In single-transmitting mixed network coding (STMNC)

scheme, original packets S1~SK is broadcasted in the first K time slots, but retransmission is not needed. Instead of retransmission, some redundant packets are added, following the single-transmitting packets, as depicted in Figure 4b.

Figure 3b. STMNC scheme

The redundant packets are generated using random linear network coding method, as described in the previous section. R stops to broadcast packets until every receiver has got all K linear non-related packets and can decode all the packets.

IV. THEORETICAL ANALYSIS

A. RLSNC scheme Figure 4a. shows the encoding process. Encoding matrix G

is a sparse matrix with binary elements.

Figure4a. RLSNC encodingprocess

The probability that a random K-by-K binary matrix is

invertible is

,for any K larger than 10. That means if we only transmit K packets, the possibility of successfully decoding is small, far from our expectation, which should be 0.999.

If some redundant packets are added, the result is totally different. The following equation is to describe the relationship between the number of redundant packet and probability that a receiver will not be able to decode the whole packets (E is the number of redundant packet).

(2)

In summary, the number of needed packets to have probability (1- )of success is 2

1logK [9]. Taking packet

loss rate into consideration, R needs to transmit

21( log )(1 )K packets theoretically. Obviously, it

cannot reach zero even =0, but as increases, the result is much better than ARQ scheme, on the condition that =0.0001.

Algorithm complexity is reduced to half of that in RLNC because there are only approximately K/2 source packets to be encoded into an encoded packet at one time slot.

B. STMNC scheme The encoding process of STMNC is depicted in Figure 4b.

Figure 4b. STMNC encodingprocess

When =0, it needs only K times transmission to achieve information exchange, the same as ARQ scheme and better

than RLNC/RLSNC. As increases, R needs to transmit more redundant packets. Thus the throughput efficiency of STMNC, RLNC and RLSNC are almost the same when is higher enough.

The latency in this scheme is shorten because most packets(about K(1- )) can be directly known by receivers. And other packets can be decoded when K linear non-related packets are received.

Complexity of STMNC is much smaller because each of the first K columns of encoding matrix only has one element. The multiplication complexity of encoding process is 2

KE ,

where E is the number of redundant packets.

V. SIMULATION RESULTS All the four schemes mentioned before are verified in the

aspects of throughput efficiency, time delay and computational complexity.

C. Throughput efficiency We use the number of transmitted packet to indicate the

throughput efficiency of all the schemes. In other words, if a scheme needs to transmit more packets at the same packet loss rate to achieve information exchange, that means the throughput efficiency is lower.

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1100

120

140

160

180

200

220

240

260

280

packet loss rate

num

ber o

f tra

nsm

itted

pac

ket

theoretical ARQpractical ARQpractical RLNCtheoretical RLSNCpractical RLSNCSTMNC

Figure 5a. Throughput efficiency of ARQ,RLNC,RLSNC and STMNC,

where K=100

As depicted in Figure.5a, the number of transmitted packet of ARQ scheme is much larger than RLNC, RLSNC and STMNC scheme. RLNC and RLSNC almost have the same performance. STMNC achieves the best when packet loss rate is low enough. Typically, ARQ scheme and STMNC only needs 100 packets to transmit when the packet loss rate is zero. However RLNC and RLSNC cannot reach zero at that point. That’s because the encoding matrix parameters of RLNC and RLSNC are selected randomly and the possibility of 100 packets that solvable is only 0.289.

D. LatencyWe first define the time of one packet transmission as a

time slot(t0). Figure5b. portrays the trend of decoded packet

amount at a certain time point on a receiver under the condition that K=100, =0.1.

0 50 100 150 200 250 3000

10

20

30

40

50

60

70

80

90

100

time slot

num

ber o

f cur

rent

dec

oded

pac

ket

ARQRLNCRLSNCSTMNC

Figure 5b. Latency of ARQ,RLNC,RLSNC and STMNC, where K=100

and =0.1

Among these four schemes, ARQ achieves better time delay in the first 100 time slots because receivers can get the original packets directly after correctly received, even though it may need more than one time slot to correctly receive a packet. It is for that reason that a receiver only gets no more than 40 original packets (S1~SK) while R have transmitted 100 already-there are more than 60 retransmissions.

The number of decoded packet of RLNC and RLSNC jumps to 100 at around 130 t0. It is at that time, the receiver gets all K linear non-related packets and decodes all the packets. RLNC and RLSNC spend less time to reach to 100 decoded packets than ARQ, even though in ARQ scheme, some certain packets can be quickly decoded.

STMNC combines the benefits of ARQ in the first 100 t0 and RLSNC in the whole process. A receiver can get about 90 packets at 100 t0 and decode all the packets at about 130 t0.

0 20 40 60 80 100 120 140 160 180 2000

1

2

3

4

5

6

7

8

9x 104

number of receiver(K)

com

plex

ity

RLNCRLSNCSTMNC

Figure 5c. Multiplication complexityof ARQ,RLNC,RLSNC

and STMNC, where =0.1

E. Computational complexity As is depicted in Figure 5c., we focus on the multiplication

complexity of encode and decode process of RLNC, RLSNC and STMNC, ignoring matrix reversion at the receivers, which is K3(the same in RLNC, RLSNC and STMNC). As for ARQ scheme, the multiplication complexity is always being zero because there is no packet combination.

VI. CONCLUSION In this paper, we discussed four schemes in single-relay

multi-users two-way network to study the performances of throughput efficiency, latency and algorithm complexity. Theoretical analysis and simulation results both prove that STMNC scheme achieves higher throughput efficiency and better latency with slightly complexity increase compare to ARQ scheme. In addition, STMNC scheme do not need feedback channel and it is much simple for R to just broadcast packets without receiving ACK/NACK.

However, STMNC is not really a secure transmission scheme compare to RLNC and RLSNC for its single-transmission process in the first 100 time slots. But it is definitely a good approach in a certificated system, which guarantees every receiver is a secure node.

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