Wireless Broadcasting With Network Coding

LU LU

Licentiate Thesis in Electrical EngineeringStockholm, Sweden 2011

Wireless Broadcasting With Network Coding

Copyright © 2011 by Lu Lu except whereotherwise stated. All rights reserved.

TRITA-EE 2011:051ISSN 978-91-7501-082-3

Communication TheorySchool of Electrical EngineeringKTH (Royal Institute of Technology)SE-100 44 Stockholm, Sweden

Printed by Universitetsservice US-AB.

Abstract

Wireless digital broadcasting applications such as digital audio broadcast (DAB)and digital video broadcast (DVB) are becoming increasingly popular since thedigital format allows for quality improvements as compared to traditional analoguebroadcast. The broadcasting is commonly based on packet transmission. In thisthesis, we consider broadcasting over packet erasure channels. To achieve reliabletransmission, error-control schemes are needed. By carefully designing the error-control schemes, transmission efficiency can be improved compared to traditionalautomatic repeat-request (ARQ) schemes and rateless codes. Here, we first studythe application of a novel binary deterministic rateless (BDR) code. Then, wefocus on the design of network coding for the wireless broadcasting system, whichcan significantly improve the system performance compared to traditional ARQ.Both the one-hop broadcasting system and a relay-aided broadcasting system areconsidered.

In the one-hop broadcasting system, we investigate the application of system-atic BDR (SBDR) codes and instantaneously decodable network coding (IDNC).For the SBDR codes, we determine the number of encoded redundancy packetsthat guarantees high broadcast transmission efficiencies and simultaneous low-complexity. Moreover, with limited feedback the efficiency performance can befurther improved. Then, we propose an improved network coding scheme thatcan asymptotically achieve the theoretical lower bound on transmission overheadfor a sufficiently large number of information packets.

In the relay-aided system, we consider a scenario where the relay node operatesin half duplex mode, and transmissions from the BS and the relay, respectively,are over orthogonal channels. Based on random network coding, a schedulingproblem for the transmissions of redundancy packets from the BS and the relay isformulated. Two scenarios; namely instantaneous feedback after each redundancypacket, and feedback after multiple redundancy packets are investigated. Wefurther extend the algorithms to multi-cell networks. Besides random networkcoding, IDNC based schemes are proposed as well. We show that significantimprovements in transmission efficiency are obtained as compared to previouslyproposed ARQ and network-coding-based schemes.

Keywords: wireless broadcasting, relay-aided system, systematic binary de-terministic rateless codes, random network coding, instantaneously decodable net-work coding.

i

Acknowledgments

This thesis is mainly the fruit of my hard work during the past two and ahalf years at KTH and it would not have been possible without the generouslove and support throughout my past life.

I owe my deepest gratitude to my advisors Prof. Ming Xiao and Prof.Lars K. Rasmussen for serving as friends and mentors during my study atKTH. Their deep insights and continuous encouragements have inspired mein the research and helped me overcome challenges on the way. I wouldalso thank Prof. Mikael Skoglund at KTH and Prof. Gang Wu at UESTCfor offering me the great opportunity of joining in the Asia-Link exchang-ing program, which expanded my vision and provided me the chance forstudying in Sweden.

I would like to express my thanks to Prof. Giuseppe Durisi for spendingthe time and efforts to act as my opponent.

Without the help and kindness of Dr. Chao Wang, Dr. Lei Bao, Zhong-wei Si, Jinghong Yang, Liping Wang, and Jinfeng Du, I would never haveleaded a happy life in Stockholm and never have achieved such fruitful re-sults in research. Thanks also go to everyone else who I have interactedwith at the Communication Theory Laboratory.

I would also like to thank all the finial supports including the Euro-pean Research Council under the European Community’s Seventh Frame-work Programme (FP7/2007-2013) / ERC grant agreement n◦ 228044, VIN-NOVA under the “Joint Sweden-China Strategic Cooperation Program,” theARC Grant DP0986089, and the VR grants 621-2008-4349 and 621-2009-4666.

Finally, I would like to dedicate my most special thanks to my dear momand my dear grandparents. All my academic successes can be traced backto them. Their unconditional love and huge support is always my source ofpower. I am so glad and lucky to have them as my family.

Lu LuStockholm, August 2011

iii

Contents

Abstract i

Acknowledgments iii

Contents v

Acronyms ix

I Introduction 1

Introduction 11 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1 Packet Erasure Channel . . . . . . . . . . . . . . . . 32.2 Error Control for Packet Erasure Channel . . . . . . 42.3 Network Coding . . . . . . . . . . . . . . . . . . . . 8

3 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . 133.1 One-hop Broadcasting System . . . . . . . . . . . . 143.2 Relay-Aided Broadcasting System . . . . . . . . . . 15

4 Problem statement and Research issues . . . . . . . . . . . 184.1 One-hop Broadcasting System . . . . . . . . . . . . 184.2 Relay-Aided Broadcasting System . . . . . . . . . . 20

5 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . 216 Conclusions and Further Works . . . . . . . . . . . . . . . . 23

6.1 Future works . . . . . . . . . . . . . . . . . . . . . . 24References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

II Included papers 31

A Efficient Wireless Broadcasting Based on Systematic Bi-nary Deterministic Rateless Codes A1

v

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . A22 System Description . . . . . . . . . . . . . . . . . . . . . . . A43 Complexity-Performance Trade-Off . . . . . . . . . . . . . . A54 Systematic BDR codes with limited feedback . . . . . . . . A85 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . A10

5.1 Systematic BDR codes Versus LT codes . . . . . . . A105.2 Systematic BDR Codes with and without Feedback A105.3 Systematic BDR Codes Versus ARQ . . . . . . . . . A12

6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . A12

B Efficient Network Coding for Wireless Broadcasting B11 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . B22 System description . . . . . . . . . . . . . . . . . . . . . . . B33 Proposed retransmission scheme . . . . . . . . . . . . . . . B44 Performance analysis . . . . . . . . . . . . . . . . . . . . . . B65 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . B10

5.1 Equal Erasure Probability . . . . . . . . . . . . . . . B105.2 Unequal Erasure Probability . . . . . . . . . . . . . B12

6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . B12

C Efficient Scheduling for Relay-Aided Broadcasting withRandom Network Codes C11 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . C22 System model . . . . . . . . . . . . . . . . . . . . . . . . . . C33 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . C4

3.1 States . . . . . . . . . . . . . . . . . . . . . . . . . . C63.2 Actions . . . . . . . . . . . . . . . . . . . . . . . . . C63.3 Probability Transition Function . . . . . . . . . . . . C63.4 Benefit Function . . . . . . . . . . . . . . . . . . . . C6

4 Scheduling Algorithms . . . . . . . . . . . . . . . . . . . . . C64.1 One Step Scheduling Algorithm . . . . . . . . . . . . C64.2 Multiple-Step Scheduling Algorithm . . . . . . . . . C8

5 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . C96 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . C11

D Design and Analysis of Relay-aided Broadcast using BinaryNetwork Codes D11 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . D22 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . D33 Retransmission Algorithms . . . . . . . . . . . . . . . . . . D5

3.1 Relay-aided ARQ . . . . . . . . . . . . . . . . . . . . D53.2 Relay-aided Network Coding . . . . . . . . . . . . . D5

4 Performance Analysis . . . . . . . . . . . . . . . . . . . . . D64.1 Relay-Aided ARQ . . . . . . . . . . . . . . . . . . . D6

vi

4.2 Relay-Aided Network Coding . . . . . . . . . . . . . D105 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . D146 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . D16

E Relay-Aided Multi-Cell Broadcasting with Random Net-work Coding E11 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . E22 System Description . . . . . . . . . . . . . . . . . . . . . . . E3

2.1 System Model . . . . . . . . . . . . . . . . . . . . . . E32.2 Network Coding Schemes . . . . . . . . . . . . . . . E42.3 Transmission Schemes . . . . . . . . . . . . . . . . . E6

3 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . E73.1 States . . . . . . . . . . . . . . . . . . . . . . . . . . E83.2 Actions . . . . . . . . . . . . . . . . . . . . . . . . . E83.3 Probability Transition Function . . . . . . . . . . . . E83.4 Benefit Function . . . . . . . . . . . . . . . . . . . . E8

4 Scheduling Algorithms . . . . . . . . . . . . . . . . . . . . . E94.1 One-Step Scheduling Algorithm . . . . . . . . . . . . E94.2 Multiple-Step Scheduling Algorithm . . . . . . . . . E9

5 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . E106 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . E13

vii

Acronyms

ACK: ACKnowledgement

ARQ: Automatic Repeat-reQuest

BEC: Binary Erasure Channel

BDR: Binary Deterministic Rateless

BS: Base Station

DAB: Digital Audio Broadcast

DVB: Digital Video Broadcast

GF: Galois Field

IDNC: Instantaneously Decodable Network Coding

LT: Luby Transform

MDS: Maximal Distance Separable

PEC: Packet Erasure Channel

P2P: Point To Point

SBDR: Systematic Binary Deterministic Rateless

TDD: Time Division Duplexing

UE: User Equipment

ix

Part I

Introduction

Introduction

1 Background

Wireless digital broadcasting applications such as digital audio broadcast(DAB) and digital video broadcast (DVB) are becoming increasingly popu-lar since the digital format allows for quality improvements as compared totraditional analogue broadcast. In a typical DAB/DVB scenario, a base sta-tion (BS) broadcasts information to a population of user terminals throughwireless broadcasting channels. In common terminology a user terminal isreferred to as user equipment (UE). With the digital format, error controlstrategies can be introduced to significantly improve the reliability of thebroadcast at both physical and at higher network layers (data link, networkand transport). The broadcast is based on packet transmission, where pack-ets are subject to channel noise, fading and interference at physical layer.Channel error correction may not be perfect; however, assuming perfecterror detection at higher layers a received packet at an UE is either error-free or discarded as erroneous. Consequently, the higher-layer broadcasttransmission from the BS to the set of UEs can be modelled as a broadcastpacket-erasure channel. To get reliable transmissions for the packet erasurechannel, error-control coding schemes are necessary.

Automatic repeat-request (ARQ) protocols [Dja99] have traditionallybeen used for error-control in digital broadcast systems. Though the pro-tocol is straightforward, ARQ becomes inefficient for broadcast in terms offeedback and retransmissions if the number of UEs is high.

Alternatively, digital fountain codes [Mac05] such as Luby Transform(LT) codes [Lub02] and Raptor codes [Sho06] have been proposed for wire-less broadcasting [MYZ06]. These codes are rateless in the sense that anunlimited number of encoded packets can be produced based on a finite num-ber of information packets. The BS continuously transmits coded packetsuntil a certain termination condition is satisfied. When fountain codes areused, the UEs attempt to decode after receiving a predetermined numberof coded packets. If the decoding is not successful, the UE collects furtherpackets and makes a new attempt to decode. This repetitive decoding pro-

2 Introduction

cess is costly in terms of computational resources at the UEs. Moreover,the redundancy of the fountain code is relatively high for a smaller numberof information packets.

As a type of rateless codes, binary deterministic rateless (BDR) codeswere proposed in [XMA07], [XAM08]. Unlike traditional rateless codes,BDR codes have the maximal distance separable (MDS) property with highprobability. By using this type of codes, the UEs can decode the original Nsource packets based on any N different received packets. With this prop-erty, the UEs only need to check the headers of each packets to distinguishthem and decode once when N different packets are received. The efficiencyincreases and the computational complexity decreases.

Recently, random network coding was proposed as another alternativefor error-control [LSM09b] to increase the transmission efficiency. Whenthe field size selected for the network code is sufficiently large, an UE isable to decode the original N data packets from any set of N successfullyreceived coded packets [LMKE08], [EOM06]. However, to ensure successfuldecoding in the random case, e.g., by matrix inversion, the UEs need to waitfor a sufficient amount of coded packets to arrive, leading to a potentiallarge decoding delay. Moreover, the matrix inversion at the UE side iscomputational complex [SV10].

To address the problems of using random network coding, binarynetwork coding schemes were considered [SST10, XLMWPS08, LXR+10,NTNB09]. We focus on considering a class of instantaneously-decodablenetwork coding (IDNC) schemes [NNB07,NTNB09,XLMWPS08,TNBG09,SST10,STK08,SV10,LWLed]. In such schemes, each network-coded retrans-mission packet contains at most one missing information packet for each UE.As compared to random linear network coding [THKE+06], IDNC schemesenjoy some benefits but also suffer some drawbacks. For example, an IDNCscheme may not be throughput-optimal. Conversely, when IDNC is usedthe decoding delay at the UE side is minimal [SV10], since a network-codedpacket can be decoded immediately by the users. In addition IDNC canbe easily implemented over GF(2), where only binary XOR operations isrequired in the encoding and decoding processes. It follows that the com-plexity is significantly reduced compared to codes operating over a largerfield.

Besides all these error-control methods, relay strategies [CG79] havebeen proposed to improve important features in wireless systems, such ascoverage, rates, and energy efficiency. In particular, fundamental resultson achievable rates for the broadcast relay channel promise significant im-provements in transmission efficiency [KGG05,LV07].

The main topics of this thesis is to design efficient error-control schemesfor reliable broadcasting. Both one-hop broadcasting systems and relay-aided broadcasting systems will be investigated.

The basic structure of the thesis is as follows. In Part I, we provide

2 Basic Concepts 3

1 bp−

1 bp−

bp

bp

Figure 1: Binary Erasure Channel, where ? represents the erasure sym-bol.

the basic concepts of the packet erasure channel and different error-controlschemes in Section 2. The one-hop and relay-aided broadcasting systemmodels are given in Section 3. In Section 4, our main research problems areformulated, while the contribution of the collected papers are summarizedin Section 5. Conclusion and future work are provided in Section 6. Part IIcontains all the related publications.

2 Basic Concepts

2.1 Packet Erasure Channel

Before we introduce the packet erasure channel (PEC), the binary erasurechannel (BEC) model is introduced. A bit transmitted over a BEC is re-ceived correctly with probability (1 − pb), and erased with probability pb.The symbol ? denotes an erasure. The BEC channel model is demonstratedin Fig. 1. The capacity of the BEC is

CBEC = maxp(x)

I(X ; Y )

= maxp(x)

(H(Y ) − H(Y |X))

= 1 − pb,

(1)

where X and Y are random variables representing the input bits and thechannel output symbols, respectively [Cov06].

The definition of the PEC is an extension of the BEC. In contrast to thebit-level transmission modelled by a BEC, the PEC models the packet-leveltransmission in a scalar way. In this case, a packet is received correctly

4 Introduction

with probability 1 − p, and erased with probability p. It is a commonchannel model of wireless communication systems. In practice, broadcast isbased on packet transmission, where packets are subject to channel noise,fading and interference at physical layer. Channel error correction maynot be perfect; however, assuming perfect error detection at the packet-layer, a received packet at the destination is either error-free or discardedas erroneous. Consequently, we can use the PEC model.

The PEC has been investigated in [Lap94, iF06, Joh09], where packet-wise maximum-distance separable (MDS) codes are shown to be optimalin terms of error probability. Considering a packet-wise MDS code over apacket-wise alphabet, the Singleton bound is satisfied with equality. There-fore, coding schemes applied across a sequence of broadcast informationpackets are typically considered for error control.

By using a similar definition as the BEC, the capacity of the PEC isdetermined as

CPEC = 1 − p, (2)

Theoretically, to successfully deliver N packets in a point-to-point (P2P)channel, the average number of transmissions X is

XP2P =N

1 − p, (3)

and the normalized overhead is

ηP 2P =1

1 − p. (4)

2.2 Error Control for Packet Erasure Channel

For reliable transmission, error control schemes are needed for the packeterasure channel. In this part, we will briefly introduce the common errorcontrol schemes. We focus on a system with only one transmitter and Mreceivers. Normally, the transmitter is the BS and the receivers are theUEs. Throughout this thesis, we use BS to denote the transmitter and theUEs to denote the receivers in our studied systems.

Automatic-Repeat-reQuest (ARQ)

The prevailing approach to packet-level coding is automatic-repeat-request(ARQ) error control protocol, e.g., [Dja99]. In a broadcasting scenario, aretransmission is requested by any UE with an erased packet; a strategythat becomes increasingly inefficient as the number of UEs in the broadcastincreases, both in terms of feedback and in terms of retransmissions. WhenARQ is used, the transmission process of each packet is independent and

2 Basic Concepts 5

has the same statistical performance. For comparison reasons, we give theperformance of the ARQ scheme in the one-hop broadcasting system andrelay-aided broadcasting system, respectively.

In the one-hop broadcast system with M UEs, if any of the UEs requeststhe packet the BS will retransmit it. The procedure is completed when allthe UEs get the packet. The normalized overhead is ( [NTNB09])

ηARQ =∑

i1,i2,...,iM

(−1)

M∑

j=1

ij −1

1 −M∏

j=1

pij

j

, (5)

where pj is the erasure probability of the BS-to-UEj link, ij ∈ {0, 1} and∃ij 6= 0, (j = 1, 2, ..., M).

In the relay-aided system, the requested packet is retransmitted eitherby the relay or the BS. We focus on a system with only one relay node andconsider a scenario where the relay node operates in half duplex mode. Also,transmissions from the BS and the relay, respectively, are over orthogonalchannels. If the relay gets the packet, it will be retransmitted by the relay.Otherwise, it is retransmitted by the BS. A model based on an absorbingMarkov chain is introduced to analyze the performance.

We define the state of the system as a vector of length M + 1. The BSis referred to as node 0, node i = 1, 2, ..., M refers to UE i, and node M + 1represents the relay. If node i receives the packet correctly, the ith entry ofthe state vector is equal to one. Otherwise, it is equal to zero. Thus, thereare 2(M+1) states, where state j is expressed as,

Sj =[

sj1, sj

2, ..., sjM+1

]

. (6)

We then define the probability state transition matrix T. The statetransition probability from state Si to Sj is denoted by ti,j , which can becomputed based on the corresponding channel erasure probabilities. pi,j

denotes the erasure probability of the link between node i and node j,(i ∈ 0, M + 1, j ∈ [1, . . . , M + 1]).

If siM+1 = 1, the relay retransmits the packet. We then have,

ti,j =

{

0 ∃sjk < si

k(k = 1, ..., M + 1)∏M

k=1 pek,0

M+1,k(1 − pM+1,k)ek,1 otherwise(7)

where ek,0 = I(sik, 1)[1 − (sj

k − sik)] and ek,1 = I(si

k, 1)(sjk − si

k). Here I(·)is a function to indicate whether the packet has been received by the nodek or not,

I(sik, 1) =

{

1 if sik = 0,

0 otherwise.(8)

6 Introduction

On the other hand, if siM+1 = 0, the BS retransmits the packet. Then,

ti,j =

{

0 ∃sjk < si

k(k = 1, ..., M + 1)∏M+1

k=1 pek,0

0,k (1 − p0,k)ek,1 otherwise(9)

Note that ti,j only depends on the current state, but not the previousstates. Thus, the whole transmission can be modelled as a Markov chain.Moreover, if the system enters the state Sj, where sj

k = 1 for k = 1, 2, ..., M ,the transmission stops and the system cannot leave it. These states arecalled absorbing states and there are r = 2 absorbing states. The othert = 2(M+1)−2 states are transient states. Thus, the transmission is modelledas an absorbing Markov chain.

We then use T to analyze this absorbing Markov chain. Renumber thestates so that the transient states come first. The transition matrix thencan have the following canonical form,

T =

transient absorbingtransientabsorbing

(

Q R0 I

)

. (10)

I is an 2×2 identity matrix, 0 is an 2×t zero matrix, R is a nonzero t×2matrix and Q is an t-by-t matrix. The entries of the submatrix Q give theprobabilities for being in each of the transient states after one transmissionfor each possible transient current state. Let nj be the expected number ofsteps before the chain is absorbed, given that the chain starts in state Sj ,and let n = [n1, n2, ..., n(N+1)M+1 ]T. Then,

n = (I − Q)−1c, (11)

where c is a column vector with all-one entries (theorem 11.4 and 11.5in [GS98]). Our system starts with the all-zero state, and we can get theexpected value of ηARQ,R by (11).

Rateless Codes

In this part, we consider two types of rateless codes, namely fountain codesand binary deterministic rateless codes.

Fountain CodesAlternatively, digital fountain codes [Mac05] such as LT codes [Lub02]and Raptor codes [Sho06] have been proposed for wireless broadcasting[MYZ06]. These codes are rateless in the sense that an unlimited number ofencoded packets can be produced based on a finite number of informationpackets. In this case the BS continuously transmits coded packets until acertain termination condition is satisfied. For example, the UEs can forward

2 Basic Concepts 7

positive acknowledgements (ACKs) to the BS as soon as successful decod-ing is achieved. Once ACKs are received from all UEs the BS starts thetransmission of the next sequence of encoded information packets. BesidesACK signals, the UEs will not feed back any other information to the BS.

When fountain codes are used, the UEs attempt to decode after receiv-ing a predetermined number of coded blocks. If decoding is not successful,the UE collects further packets and makes a new attempt to decode. Theprocess continues until decoding is successful. The repetitive decoding pro-cess is of course costly in terms of computational resources at the UEs.Furthermore, fountain codes are typically based on random sparse graphcoding techniques, and thus need redundant blocks in order to successfullydecode, i.e., the number of packets used for decoding is typically slightlyhigher than the number of information packets. The redundancy becomesnegligible as the number of information packets goes to infinity; however theredundancy is relatively high for a smaller number of information packets.Thus, fountain codes often require a large number of encoded blocks, whichmay not be suitable for delay-sensitive applications.

Deterministic Rateless CodesTo address the respective deficiencies of ARQ and fountain codes, we pro-pose the application of systematic binary deterministic rateless (BDR) codesas proposed in [XMA07], [XAM08], for error-control in wireless broadcast-ing systems. The systematic BDR (SBDR) coding scheme is proposedin [XAM08]. Non-systematic BDR codes are also possible, but require ad-ditional complexity [XMA07]. We therefore focus on systematic codes. Ifthere are N information packets, the sequence of packets for transmissionbased on SBDR consists of the systematic part of N − 1 information pack-ets, followed by N∗ BDR encoded packets generated based on a determin-istic generator matrix. The BDR codes are constructed to have maximum-distance separable (MDS) properties [XMA07], [XAM08] with high proba-bility, which means that the UEs are able to decode the source informationbased on receiving any distinct set of N packets.

The coding process is as follows. We assume the length of each infor-mation packet is L, where L + 1 is a prime [XMA07]. A coded block Cj ,j = 1, 2, ..., N∗, N∗ ≤ L + 1 is generated as

Cj = GjI, (12)

where I = [I1, I2, ..., IN ]T is an LN × 1 column vector denoting the sourceinformation bits, and Gj = [Gj,1, Gj,2, ..., Gj,N ] is an L × LN matrix de-noting the encoding matrix for Cj . The elements of Gj are determined

8 Introduction

as

Gj,i =

0 ... 0 ... 0 1 0 ... 00 ... 0 ... 0 0 1 ... 0

... ...0 ... 1 0 ... 0 0 ... 00 ... 0 1 ... 0 0 ... 0

L×L

. (13)

Let gk,t(k, t = 1, 2, ..., L) be the k-th row and t-th column element of Gj,i,

gk,t =

{

1,0,

t = ((i − 1) (j − 1) + k) mod (L + 1);otherwise.

(14)

By definition, Gj,i right-shifts a vector (i − 1)(j − 1) mod L bits. Thestructure of Gj,i ensures full-rank of the encoding matrix and meanwhileachieves low-complexity [XMA07]. Using this algorithm, an UE can recoversource packets from any of the N distinct received coded packets [XMA07].For the case of systematic BDR codes (SBDRCs), only N − 1 systematicsource packets can be used to guarantee this property (see [XAM08] fordetails). Therefore, there are N − 1 systematic information packets and N∗

encoded packets that will be used for transmission.

Network Coding

The use of network-coding-based error-control methods ( [ACLY00,LYC03,THKE+06,KM03]) is another alternative [XLMWPS08,NNB07,NTNB09].Here, a simple example is given to show the basic idea, as illustrated inFig. 2. There are two UEs in the system where both of them need packetsI1 and I2. After the BS broadcasts I1 and I2, UE1 and UE2 get I1 andI2, respectively. Other transmissions are erased. When ARQ is used, I1

and I2 will be retransmitted separately. By using network coding, I1 ⊕ I2

can be retransmitted. At UE1, the packet I2 can be retrieved by (I1 ⊕I2) ⊕ I1 while at UE2 the packet I1 can be retrieved by (I1 ⊕ I2) ⊕ I2.Assuming retransmission packets are correctly received at the two UEs, oneretransmission packet is sufficient using network coding, yet, two packetsare needed for ARQ. The transmission efficiency can thus be increased.

Error-control-based on network coding is the main topic of this thesis.In the follows, we will discuss network-coding-based broadcast schemes inmore detail.

2.3 Network Coding

Network coding theory is the basis for many bandwidth-efficient transmis-sion schemes in wireless networks. With network coding, the nodes in thenetworks is enabled to appropriately encode the incoming data before trans-mission to the next node. The ability to re-encode the data at the nodes

2 Basic Concepts 9

BS

1 2

1p

2p

1 2,I I

1I

2I

Figure 2: An example for the basic idea of Network coding.

results in a substantial bandwidth improvement over traditional store-and-forward networks [ACLY00,LYC03,THKE+06,KM03,LMKE08]. The ideaof network coding was first proposed in [ACLY00] for error-free transmis-sion. In [ACLY00], it is shown that the rate from the source to a set ofdestinations can reach the minimum of the min-cut capacities of the in-dividual destinations with network coding if the alphabet size (GF(q)) ofthe code goes to infinity, i.e., q → ∞. The classic example of networkcoding in a butterfly network is provided in Fig. 3. The capacity of eachlink is one bit. It can be easily seen that the min-cut between the sources and each destination (y or z) is two bits. Thus, by the max-flow min-cut theorem [ACLY00], the maximal flows are transmitting two bits froms to y and from s to z simultaneously. The objective of multicasting is toachieve the maximal flow capacity. Assume s has two bits, denoted as a1

and a2, to transmit to y and z. By using a traditional routing approach,the requirements cannot be achieved due to the bottleneck of the channelw → x. However, coding at the intermediate node w makes the capacity-achieving multicast feasible. At w, two incoming bits a1 and a2 are binaryadded (XOR) into one output bit a1 ⊕ a2. At y, a2 can be retrieved bya1 ⊕ (a1 ⊕a2) while a1 can be retrieved at z by the operation a2 ⊕ (a1 ⊕a2).Thus, the maximal capacity is achieved by coding in the intermediate nodew.

After network coding was proposed, a wealth of literature has been ap-peared. In [LYC03], linear network coding was shown to be sufficient toachieve the min-cut capacities of each destinations. To describe the frame-work based on network coding more efficiently, an algebraic framework isestablished in [KM03], where the network coding process can be describedusing transfer matrices. Thus, the problem of finding feasible coding schemeis converted into looking for a nonsingular transfer matrix for all the source-destination pairs. In [THKE+06], a random network coding scheme was

10 Introduction

z

u

y

t

w

x

s

1 2,a a

1a

2a

1a

2a

1 2a a⊕ 2

a1a

1 2,a a 1 2

,a a

Figure 3: A network coding example. a1 and a2 are information bitstransmitted from the source s to two destinations y and z simultane-ously. The modulo-2 addition operation is denoted by ⊕.

proposed for multicast, where each node independently and randomly en-code the incoming data to generate the outgoing data. The code achievescapacity with increasing code length. The main benefit of random networkcoding is the distributed implementation property, which avoids central con-trol and robust is against network changes.

Network Coding for Broadcasting

Recently, network-coding-based unicast, multicast and broacast schemeshave attracted a lot of attention. In this thesis, we focus on a one source(BS), multiple destinations (UEs) broadcasting system with packet-erasurechannels. Other networks, such as ad-hoc networks, are out of the scope ofour work. The reader is referred to [Wid05, FRWZ07, FOG08, FWB06] forinformation for such networks. For our network structure, the contributionsin the literature can roughly be divided into two types: network codingschemes based on large field size (q > 2) and binary network coding schemes.

If the coding coefficients are chosen from a finite filed (GF(q)) with a suf-ficiently large q, random network coding is throughput optimal [LMKE08].In this case, each coded packet will almost surely become linearly inde-pendent of all previously received coded packets and hence any UE isable to decode the original N data packets from any set of N success-fully received coded packets [LMKE08], [EOM06]. In [LSM09b, LMS09a,LSM09a, LMS09c, LMS09b], the transmission based on time division du-plexing (TDD) was studied. N packets are transmitted to the destination

2 Basic Concepts 11

using random linear network coding with the objective of minimizing themean time to complete transmission of that sequence of packets. Refer-ences [LSM09b, LSM09a] considered the one transmitter (BS), one desti-nation (UE) system and showed that there exits, an optimal number ofcoded packets to be transmitted back-to-back before stopping to wait foran acknowledgment (ACK), under the minimum time and minimum energycriteria, respectively. The system with one BS, multiple UEs is studiedin [LMS09a,LMS09c,LMS09b]. In [LMS09a,LMS09b], N packets are trans-mitted to M UEs. Mean time to complete the transmission of packets isstudied in [LMS09a]. In [LMS09b], the filed size consideration is discussed.It is shown that there is only a small degradation of throughput performanceby using random network coding with filed size 2, i.e., q = 2. On average,at most N + 2 coded packets are needed in order to decode if we performXORs of randomly selected packets from the N originals.

Random network coding can be throughput optimal; however, it hassome drawbacks. The throughput optimality comes at the cost of largedecoding delays at the UE side. Normally, a UE needs to collect a sufficientnumber of coded packets before being able to decode. Of course, there areapplications which are insensitive to such delays. For example, a softwaredownload or file download. The update starts to work when all the files arecompleted. Thus, the main desired objective is the mean completion timeas studied in [LMS09a,LMS09b]. However, we need to consider some otherdelay sensitive applications as well. There are two types of applications,namely order-sensitive applications and order-insensitive applications.

In the order-sensitive, such as applications in live modes, partial decod-ing of packets out of temporal order does not improve the system delayperformance. In [SSM08, SSM09, SSMed], they proposed a feedback-basedthroughput-optimal scheme to deal with the transmitter queue size, as wellas decoding and delivery delays at the destinations. A new concept nameddrop-when-seen is proposed, where the transmitter drops the packet if allthe destinations see the packet.

Another type of applications is for which partial decoding is beneficialand can result in lower delays irrespective of the order in which packets arebeing decoded. In this case, it is naturally to use multiple description codingtechniques [Goy01], in which every decoded packet brings new informationto the destination, irrespective of its order. In this thesis, our studied binarynetwork coding aims at this type of applications. Before we discuss relatedwork, some important definitions are given here.Definition 1: Instantaneously decodable: A transmitted packet is instanta-neously decodable at the i-th UE (i = 1, 2, . . . , M) if it is a linear combina-tion of source packets containing at most one source packet which has notyet been received correctly by the i-th UE. If the packet is instantaneouslydecodable for all the UEs, we call the code instantaneously decodable.Definition 2: Innovative packet: A transmitted packet is innovative for the

12 Introduction

i-th UE (i = 1, 2, . . . , M) if it contains source packet which has not yet beenreceived correctly by the i-th UE. Otherwise, it is a non-innovative packet.

For the network-coding-based schemes for the order-insensitive applica-tions, there are two sources of the delay. Firstly, delay can arise from thenon-innovative packets. If the packet is not innovative for the i-th UE, adelay is definitely incurred. Moreover, if the packet is not instantaneouslydecodable for the i-th UE, it cannot instantly retrieve novel information,which incurs a delay as well. Thus, a zero-delay scheme would require bothinstantaneously decodable and innovative properties. It has been shownin [KDF08] that for the case of M = 2 and M = 3 UEs, there exists anoffline algorithm that is delay-free. For M > 4, the authors prove that azero-delay algorithm does not exist. However, an offline algorithm is notpractical since it needs to know future channel realizations in advance. Foronline operation, a greedy algorithm achieves zero-delay for M = 2 whilefor M = 3, this is not always the case [KDF08]. In [BCMW09], an effectivesystematic online network coding algorithm was proposed targeting delaycontrol which does not enforce the instantaneously decodable property.

To limit the complexity of the transmission schemes, we considerthe class of instantaneously decodable network coding (IDNC) schemes[NNB07, NTNB09, XLMWPS08, TNBG09, SST10, STK08, SV10, LWLed].For this class of schemes, it is sufficient to consider linear network codingover (GF(2)) [SST10]. That is, the coded packets are formed using binaryXOR of the original source packets. Thus, the network coding scheme isperformed similarly as the one in [KRH+06]. Moreover, based on IDNC,the UEs do not need to wait for a number of packets before they can beginto decode or try to decode a coded packet. They decode the packet basedonly on the source packets received already.

For the IDNC schemes, the minimization of the completion time is amain issue. In [LWLed], a broadcast system with delay constraints is stud-ied. A Lyapunov analysis has been developed, which provides further in-sight into the dynamics of IDNC schemes. However, only the systems withM = 2 and M = 3 UEs are considered. In [NNB07, NTNB09], a IDNCscheme is proposed for the system with M = 2 UEs. A performance analy-sis is provided to show the obtained gain as compared to an ARQ scheme.Another IDNC scheme is studied in [XLMWPS08], where simulation resultsare provided. For a system with M UEs (M > 3), a hybrid IDNC schemeis studied for the one-hop broadcasting system. Theoretical analysis is pro-vided to quatify performance. An IDNC scheme based on graph theory wasinvestigated in [STK08, SV10]. A graph is formulated based on the feed-back information from all the UEs, where a maximal clique of the graphcan maximize the number of UEs which are capable of receiving an inno-vative packet. The maximal clique-searching algorithm is NP-hard. Thus,they also propose a maximum-weight vertex-search algorithm to encode thepacket which is lost by most of the UEs first. In general, there is a lower

3 System Model 13

bound for the completion time. If N packets are transmitted to M UEs,the number of transmissions X to be completed should not be less than thecase with M = 1. Thus, the normalized overhead is

η =X

N≥

1

1 − maxi pi

(15)

where pi is the erasure probability of the BS-to-UEi link, (i = 1, 2, . . . , M).It has been shown in [NNB07, NTNB09, XLMWPS08, TNBG09] that thelower bound can be achieved asymptotically with an increasing number ofpacket N .

In addition to IDNC schemes for independent erasure channels, schemesfor the Gilbert-Elliott channel were studied in [SST10, STK08], where aNP-hard set-packing integer-programming is formulated. To reduce thecomplexity, the UE which has the least probability with good channel willnot be considered if the innovative property cannot be guaranteed for allthe UEs.

All the above algorithms need feedback information after each packettransmission. Efficient IDNC scheme with reduced feedback information isstill an interesting topic. Moreover, how to generate coded packets with lowcomplexity is also of interest and is one of the motivations of our work.

Only limited work consider the applications of IDNC in a relay-aidedsystem. In [FZWL09], an algorithm was proposed based on IDNC for arelay-aided system. In this scheme, the BS and the relay only generatenetwork-coded packets once. Then, the network coded packets are retrans-mitted until all the UEs have retrieved all the source packets. Moreover, theBS retransmission phase is operated after the relay retransmission phase.They do not use feedback information to update successfully receive thegeneration of network-coded packets, which will degrade the efficiency ofthe scheme. The relay node can get more packets during the BS retrans-mission phase, which can be used for further retransmissions to increase theefficiency.

3 System Model

In this thesis, we consider reliable broadcasting based on network coding.Both one-hop system and relay-aided system are studied. In this section,the system models we will investigate are defined. Research will be posedin Section 4.

For a broadcast system with N packets to be transmitted, we assumethat only one packet can be transmitted during one time slot. Therefore,to measure system performance we define the overhead as

η = X/N, (16)

14 Introduction

BS

1 2 M...

1p

2p Mp

Figure 4: System model with one BS and M UEs.

where X is the total number of time slots until all M users successfully havereceived N packets. Our objective is to minimize the overhead η.

3.1 One-hop Broadcasting System

Assume there are N source packets Ii, i = 1, 2, ..., N to be transmitted fromthe BS to M UEs. The M BS-to-UE packet-erasure channels are assumedto be independent with packet-erasure probabilities pi, (i = 1, 2, . . . , M),respectively. The system model is shown in Fig. 4. For the one-hop system,we investigate broadcasting schemes based on SBDR codes (Section 2.2)and IDNC (Section 2.3).

Broadcasting based on SBDR codes

In Section 2.2, we briefly introduced SBDR codes. Here, we discuss theapplication of such codes for broadcasting. To simplify notation, let Ti

denote the output blocks from the BS, i.e., Ti = Ii for i = 1, 2, ..., N − 1and Ti = Ci−N+1 for i = N, ..., N + N∗ − 1. During transmission, an UEfeeds back an ACK to the BS once it has successfully received N distinctpackets. The BS, in turn, stops the transmission of a new packet once ithas received ACKs from all UEs. If ACKs are not received from all UEsafter all N + N∗ − 1 blocks are transmitted, the BS starts retransmittingthe packets from T1 as in [XAM08].

Broadcasting based on Binary Network Coding

As most of the existing works, we divide the transmission process into twophases: the information transmission phase and the redundancy transmis-sion phase. In the information transmission phase, the BS broadcasts N in-

3 System Model 15

formation packets, and during the transmission some packets are lost overthe respective BS-to-UE packet-erasure channels. Each UE subsequentlyfeedback a packet with indices of the erased packets, where we assume or-thogonal and error-free feedback channels.

In the redundancy transmission phase, the set of erased packets is di-vided into subsets such that at most one erased packet per UE is in anyparticular subset. This way, the instantaneously decodable property canbe guaranteed. The erased packets in a subset are then encoded with abinary network code modulo-2 addition (XOR) into one encoded block forretransmission.

3.2 Relay-Aided Broadcasting System

Broadcasting based on Random Network Coding

Again, we consider a BS with N information packets, I1, I2, ..., IN , for broad-cast to M UEs. Here, a relay node is included to assist the transmission,as shown in Fig. 5. We focus on a system with one relay to simplify thepresentation. The approach can readily be extended to systems with mul-tiple relays. The BS is referred to as node 0, node i = 1, 2, ..., M refers toUE i, and node M + 1 represents the relay. The links between nodes aremodelled as packet-erasure channels, where the erasure probability of thechannel between node i and node j is denoted by pi,j , with i = 0, M + 1,j = 1, 2, ..., M + 1, and i 6= j. We assume that the BS-to-relay channel isbetter than all the BS-to-UE channels, i.e., p0,M+1 < p0,i, i = 1, 2, ..., M ,and the relay-to-UE channels are better than the corresponding BS-to-UEchannels, i.e., p0,i > pM+1,i, i = 1, 2, ..., M . Moreover, we assume that theerasure probabilities are time-invariant and known at both the BS and therelay.

The overall transmission process consists of two phases: the informationtransmission phase and the redundancy transmission phase. In the informa-tion phase, the BS broadcasts N coded packets using random linear networkcoding [THKE+06]. Due to the fact that at least N packets are needed torecover the N information packets at the receiver side, no feedback infor-mation is used during this phase. The coded packet C0,k from the BS at

time slot k is generated as C0,k =∑N

j=1 γk,jIj , where γγγk = [γk,1, ..., γk,N ]is the coding vector of C0,k. Following the principles of network coding,the elements of γγγk are chosen independently and uniformly from the finitefield GF(q). During this phase, some packets are lost due to erasures in theBS-to-UE channels and the BS-to-relay channel. At the end of the infor-mation phase, each UE feeds back the rank of its own coding matrix to theBS and the relay, which will be defined later in this section. Likewise, therelay feeds back relevant information to the BS. For simplicity we assumeinstantaneous, error-free feedback channels.

16 Introduction

BS

1 2 M

Relay

...

0,1p

0,2p

0,Mp

0, 1Mp

1,M Mp

1,2Mp

1,1Mp

Figure 5: System model with one BS, one relay and M UEs.

During the redundancy transmission phase, the BS forms and transmitsredundancy network coded packets. On successfully receiving packets fromthe BS, the relay node does not seek to decode source information. Instead,the relay stores the received packets in memory for random network re-coding. Suppose that P packets are received by the relay after r − 1 > Ptime slots, denoted by V1, ..., VP . A redundancy packet CM+1,r from the

relay at time slot r is then generated as CM+1,r =∑P

p=1 βr,pVp, whereβββr = [βr,1, ..., βr,P ] is the coding vector of CM+1,r. Again, all elements ofβββr are chosen independently and uniformly from GF(q). Assuming that Vp

is the coded packet C0,spfrom the BS, we have

CM+1,r =

P∑

p=1

βr,pC0,sp =

N∑

j=1

(

P∑

p=1

βr,pγsp,j

)

Ij . (17)

Thus, the coding vector ΞΞΞu for the received packet at a receiver (any UEor the relay) may be represented as ΞΞΞu = [αu,1, ..., αu,N ], where αu,j = γk,j

if the received packet is a coded packet at time slot k from the BS, orαu,j =

∑P

p=1 βr,pγsp,k if the received packet is from the relay at time slot r.Assuming that a receiver has U successfully received packets, R1, ..., RU ,

at a given time slot, then the corresponding coding matrix G has αi,j onthe ith row and jth column, where i = 1, 2, ..., U ,j = 1, 2, ..., N . The N in-formation packets can be recovered if rank(G) = N [KM03]. Therefore, theranks of the coding matrices for the UEs and the relay become importantparameters. Gi refers to the coding matrix of node i. A received packet isreferred to as an innovative packet [LMKE08], [THKE+06] for the i-th re-ceiver if rank(Gi) increases by one. For the analysis in the following sectionswe consider the case that the field size q is sufficiently large [THKE+06].

3 System Model 17

1BS

2BS

R

Figure 6: System model of two BSs, one shared cell-edge relay and twoUEs in each cell.

In this case, a received coded packet from the BS is an innovative packetif rank(Gi) < N . Moreover, if rank(GM+1) > rank(Gi), then a receivedredundancy packet from the relay is an innovative packet for the i-th re-ceiver. If rank(GM+1) ≤ rank(Gi), then the probability that a receivedredundancy packet from the relay is an innovative packet for the receiver islow and it can be assumed as a non-innovative packet for the receiver.

The approach can be extended into multi-cell system. As shown in Fig.6, we consider a two-cell (cell one and cell two) system with two BSs, de-noted by BS1 and BS2, respectively, and with M1 and M2 cell-edge UEs incells one and two, respectively. BS1 has N1 information packets, denoted byI1

1, ..., I1N1

, to be broadcasted to the M1 UEs in cell one, and similarly, BS2

has N2 information packets, denoted by I21, ..., I2

N2, to be broadcasted to the

M2 UEs in cell 2. Without loss of generality, we assume N1 = N2 = N andM1 = M2 = M . Cell one and cell two share a relay [PPTJ09], denoted by R,to assist transmissions in both cells. In contrast to multiple cell-edge relays,the shared relay can coordinate transmissions and thus alleviate the needfor coordination among base stations. Furthermore, the costs of deployingmultiple relays is avoided. We consider the use of directional antennas atthe relay to increase coverage, and suppress interference [PPTJ09, PP97].By exploiting the directionality of the antenna, the relay can transmit tothe UEs in one cell without interfering with the transmission of the BS inanother cell. Hence, BS1 can transmit to respective cell-edge UEs simul-taneously with the relay transmitting to cell-edge UEs in cell 2; and viceversa. To constrain complexity and interference, however, the relay onlyassists one cell in a given time slot. Thus, the concurrent transmission ofthe relay to two cells is not allowed. Yet clearly, the simultaneous transmis-sions of the relay and one BS to different cells lead to higher transmissionefficiency.

Broadcasting based on Binary Network Coding

The system structure is the same as Fig. 5. We also assume that theBS-to-relay channel is better than the BS-to-UE channels, i.e., p0,M+1 <p0,i, i = 1, 2, ..., M , and the relay-to-UE channels are better than the BS-to-UE channels, i.e., p0,i > pM+1,i, i = 1, 2, ..., M . The relay node operates

18 Introduction

in half duplex mode. Moreover, the BS and the relay cannot transmitsimultaneously.

In our system, we divide the transmission phase into three phases: theinformation transmission phase, the BS retransmission phase and the relayretransmission phase. The information transmission phase is the same asthe one-hop system with IDNC. The difference comes from the retransmis-sion phases. The retransmission process exploits the benefits of the relayand of IDNC. In the BS retransmission phase, the BS conducts rounds ofretransmissions, which are each followed by feedback updates from UEs andthe relay. This process is repeated until all packets are received by all UEs,or until all packets still missing at the UEs are available at the relay. Therelay retransmits, again followed by feedback updates from the UEs nodes.This process is repeated until all the UE nodes have successfully receivedall the information packets.

Since the relay-to-UE channels are better than the BS-to-UE channels,the benefits of the relay are obvious. We note that though the relay ishalf-duplex, it does not increase the delay since it only receives when theBS transmits to the UEs. Although we consider IDNC, we do not requireinstantaneous decodability at the relay node, since individual informationpackets are only required at the UEs. This is beneficial for the systemefficiency. We use a simple example here to illustrate the reason. Let M = 2,N = 2, and UE1 gets I1, UE2 gets I2 while the relay gets nothing after theinformation transmission phase. The BS then retransmits I1 ⊕ I2, which isinnovative for both UEs. If the relay successfully receives the coded packet,while one or both UEs do not, then the relay can retransmit the receivedcoded packet as is, I1 ⊕ I2, without attempting decoding first. However, ifwe require that the packet is also instantaneously decodable for the relay,then the BS can only retransmit I1 and I2 separately since the relay haslost both packets. Obviously, this will affect the efficiency.

4 Problem statement and Research issues

In this section, we presents the problems we considered in our work.

4.1 One-hop Broadcasting System

Broadcasting based on SBDR codes

The code construction is subject to a series of constraints to ensure MDSproperties of the systematic BDR codes. Some theoretical insights into theeffects and impact of the code parameters have been obtained in [XMA07],[XAM08]; however, practical constraints for wireless broadcast applicationswere not considered. Here we consider a practical trade-off between com-plexity and efficiency in the design of the broadcast coding scheme.

4 Problem statement and Research issues 19

The systematic BDR codes of length N + N∗ − 1 are designed to haveMDS properties, and thus they can be decoded from any set of N receivedpackets. However, if a UE fails to receive N distinct packets, the BS hasno choice but to start retransmitting. With repetition, the resulting codehas no longer MDS properties, and thus, it is no longer guaranteed thatsuccessful decoding based on any N received blocks. The UE now has tomake sure that a set of N distinct packets have been received to warrantsuccessful decoding. To avoid complications in the decoding process, itis desirable for N∗ to be sufficiently large to ensure that N packets aresuccessfully received by all UEs within the transmission of N + N∗ − 1packets.

Conversely, due to delay and complexity issues it is desirable to restrictthe size of N∗. During the full transmission period, the BS is required to en-code N∗ coded packets and store N +N∗−1 packets, requiring resources andincurs a delay proportional to N∗. Furthermore, to allow for large N andstill ensure full rank of the generator matrix for any N packets, the lengthL of a packet must be larger than the maximum possible degree, Du

max, ofthe determinant of the respective generator matrix [XMA07], where

Dumax = (N − 1) N∗ + (N − 2) (N∗ − 1) + ...

+ (N − J + 2) (N∗ − J + 3) + (N − J + 1) (N∗ − J + 2) ,(18)

and J = min {N∗, N}. It is clear that L grows quickly with N∗, which inturn affects encoding and decoding complexity. The encoding complexityat the BS is of the order of O(LN∗), while the decoding complexity at anUE is of the order of O((LN∗)2).

Due to the constraints on the code construction, it is not possible tooperate BDR codes in an adaptive manner. Thus, a suitable compromisefor N∗ must be determined. This problem is considered in the thesis.

In addition, as a type of rateless code, SBDR codes can operate effi-ciently with only positive acknowledgement feedback. However, if limitedfeedback is made available at the BS, the transmission efficiency can besignificantly improved in terms of the overhead being substantially reduced.The improvements are particularly pronounced when N∗ is limited by e.g.complexity, delay or storage considerations. In fact, with limited feedbackit is possible to obtain a high transmission efficiency even when N∗ is small.

Broadcasting based on IDNC

We assume that ni packets are erased on the BS-to-UEi link, (i =1, 2, ..., M), during the information transmission phase. Let n = maxi ni

and i = arg maxi ni. It follows that if n > 0 then a new round of retrans-missions is required. Since UEi must receive at least n packets in order torecover all N information packets, n is a lower bound on the total numberof required retransmissions before the transmission is completed. Thus, the

20 Introduction

system can retransmit n without waiting for feedback information. Thisway, the amount of feedback can be reduced. The problem of designing theIDNC scheme in this scenario is another topic of this thesis.

4.2 Relay-Aided Broadcasting System

Broadcasting based on Random Network Coding

In the redundancy phase rank(Gi) may increase by one (unless rank(Gi) =N) if a packet is received successfully from the BS. Yet this may notbe true for a packet received successfully from the relay. Since ran-dom packet-erasures occur on the BS-to-relay channel, it is possible thatrank(GM+1) ≤ rank(Gi). Therefore, with some probability it is not pos-sible for UEi to receive an innovative packet from the relay. On the otherhand, p0,i > pM+1,i, i = 1, 2, ..., M . It follows that there is an inherentscheduling problem of whether the BS or the relay node should be allowedto transmit within a certain horizon of time slots in the redundancy phase.

During the redundancy phase, the scheduling problem will be investi-gated for two different scenarios. In the first case feedback information isreturned from all UEs (and the relay) for each redundancy packet. Werefer to this case as the one-step scheduling since we schedule redundancypackets for only one time slot ahead. Each instance of feedback comes ata cost in terms of resources and delay; hence, the one-step schedule is onlysuitable for small-scale systems in which the accumulated delay is negligibleand the feedback channels can be used frequently without causing undulystrain on system resources. If the number of UEs in the system is large, theamount of resources required to accommodate feedback, as well as the la-tency of the feedback channels, may be prohibitive. In the second scenariowe consider multiple-step scheduling, which is more suitable for systemswith a large number of UEs or where the feedback resources are limited.In this case feedback is provided after multiple redundant packets, thus re-ducing feedback requirements significantly. Redundancy transmissions arethen scheduled multiple steps ahead using dynamic programming (DP).

The situations in the multi-cell system is more complicated than the one-cell case where there are both intra-cell and inter-cell scheduling problems.The intra-cell scheduling problem is the same as the one-cell case. More-over, there is a inter-cell scheduling problem between two cells. Firstly, thetwo BSs cannot transmit at the same time, which leads to a BS selectionproblem. Also, when anyone of the BSs transmits, the relay can assist thetransmission in the other cell. However, when the half-duplex relay trans-mits, it cannot concurrently receive innovative packet from any BS. Yet therelay needs packets from the BSs to be able to assist users. Likewise, whenthe relay receives packets from a BS, it cannot transmit to the other cell.Hence, the shared relay scheduling is another important issue. We investi-

5 Contributions 21

gate the scheduling problem of both intra- and inter-cell scheduling. As thesame as the one-cell case, two scheduling problems will be studied based ondifferent requirements on the feedback.

Broadcasting based on IDNC

For this scenario, a related scheme was considered in [FZWL09]; however,the scheme in [FZWL09] can be modified to work more efficiently. Thescheme also divides the transmission into three phases. However, the re-lay retransmission phase is conducted before the BS retransmission phase.Since packets are lost at the relay due to packet-erasures in the BS-relaychannel, the relay node can also get innovative packets from the BS duringthe corresponding retransmission phase. Thus, in contrast to the protocolin [FZWL09], we allow the BS to retransmit first before the relay gets toretransmit. Compared to the scheme in [FZWL09], we might need largermemory at the relay node. The corresponding memory calculation is outof the scope of our current investigations and thus is considered as futurework. Moreover, we allow the use of feedback information to update theencoding process during the retransmission phases, which in turn improvesthe transmission efficiency.

5 Contributions

Paper A : Efficient Wireless Broadcasting Based on SystematicBinary Deterministic Rateless Codes [LXS+10]

We investigate the design and use of systematic binary deterministic rate-less (BDR) codes for information transmission over block-erasure broadcastchannels. BDR codes are designed to obtain a level of maximal distanceseparable (MDS) properties, making these codes ideal for the consideredbroadcast scenario. For a certain number of encoded redundancy blocks,we derive an expression for the probability that the MDS properties aremaintained. Moreover, if limited feedback is available, we extend the BDRcoding protocol to further improve the system performance. Numerical re-sults show that for a finite number of source blocks and as the numberof users grows the proposed systematic BDR codes performs significantlybetter than LT codes. The proposed schemes with feedback have betterperformance than traditional ARQ schemes.

Paper B : Efficient Network Coding for Wireless Broadcast-ing [LXR+10]

It has been shown in the literature that network coding can improve thetransmission efficiency of wireless broadcasting as compared to traditional

22 Introduction

ARQ schemes. In this paper, we propose an improved network codingscheme that can asymptotically achieve the theoretical lower bound ontransmission overhead for a sufficiently large number of information blocks.The proposed scheme makes use of an index allocation algorithm that dis-tributes information blocks that have been erased during transmission intoa minimum number of encoding sets , where each set represents the erasedblocks to be jointly network encoded and retransmitted. Numerical resultsshow that the proposed scheme enables higher transmission efficiencies thantraditional ARQ, and previously proposed networks coding schemes for wire-less broadcasting.

Paper C : Efficient Scheduling for Relay-Aided Broadcasting withRandom Network Codes [LXRS11]

We investigate efficient scheduling algorithms for a relay-aided broadcastingsystem using random network codes, where our objective is to maximize thetransmission efficiency. The broadcast from a base-station (BS) is dividedinto an information phase and a redundancy phase, where the half-duplexrelay assists in the redundancy phase. Time-division transmission is usedover packet-erasure channels, where the erasure probabilities of the BS-to-relay and relay-to-user links are lower than the BS-to-user links. Followingthe information phase, each user provides feedback on the status of receivedpackets to the BS and the relay, which in turn both generate redundancypackets for the redundancy phase. To improve efficiency, we formulate ascheduling problem for the transmissions of redundancy packets from theBS and the relay. We consider two scenarios; namely instantaneous feed-back after each redundancy packet, and feedback after multiple redundancypackets. In the first case the schedule is determined using a greedy algo-rithm, while in the second case the schedule is determined using dynamicprogramming. To determine the performance with instantaneous feedback,we develop an analytic approach based on a Markov chain. Numerical re-sults show that the transmission efficiency of the dynamic programmingalgorithm is close to the performance of the greedy algorithm, but requiressignificantly less feedback.

Paper D : Relay-Aided Broadcasting with instantaneously Decod-able Binary Network Codes [LXR11]

We consider a base-station broadcasting a set of order-insensitive packetsto a user population over packet-erasure channels. To improve efficiency wepropose a relay-aided transmission scheme using instantaneously-decodablebinary network coding. Our proposed scheme ensures that a coded packetcan be immediately decoded at the user side without delay. Moreover,only binary operations are required in the encoding and decoding processes,

6 Conclusions and Further Works 23

which decrease the computational complex. We further analyze the perfor-mance of the resulting broadcast scheme, and show that significant improve-ments in transmission efficiency are obtained as compared to previouslyproposed ARQ and network-coding-based schemes.

Paper E : Relay-Aided Multi-Cell Broadcasting with RandomNetwork Coding [LSXR10]

We investigate a relay-aided multi-cell broadcasting system using randomnetwork codes, where the focus is on devising efficient scheduling algorithmsbetween relay and base stations. Two scheduling algorithms are proposedbased on different feedback strategies; namely, a one-step scheduling al-gorithm with instantaneous feedback for each redundancy packet; and amulti-step scheduling algorithm with feedback only after multiple redun-dancy packets. For the latter case, dynamic programming is applied todetermine optimal scheduling. Numerical results show that the transmis-sion efficiency of the multi-step algorithm approaches that of the one-stepalgorithm, but requires significantly less feedback. They both significantlyoutperform corresponding ARQ and random scheduling approaches.

6 Conclusions and Further Works

In this thesis, we investigated wireless broadcasting systems with networkcoding. Both one-hop broadcasting systems and relay-aided broadcast-ing systems are studied. Based on the assumption of perfect error-code-detection in the physical layer, we consider the transmission on the packetlevel through packet-erasure channels.

In a one-hop broadcasting system, we investigated the application ofSBDR codes and IDNC. For SBDR-code-based broadcasting, systems withlarge code redundancy N∗ have theoretically higher transmission efficiency.Yet, the code redundancy N∗ is subject to practical constraints in terms ofcomplexity, delay, and storage considerations. We provided a framework forfinding a proper choice of the code redundancy N∗, based on formulatingthe probability of maintaining MDS properties of the code as a function ofthe code redundancy N∗. For systems constraint to small code redundancyN∗ and limited feedback, we developed a modified broadcast transmissionprotocol based on BDR codes and selective-repeat retransmission princi-ples. For IDNC based broadcasting, we proposed an efficient network codingscheme with reduced feedback information. Instead of using instantaneousfeedback following each redundancy transmission, our proposed scheme re-quires feedback after a certain number of transmissions. Theoretical analy-sis showed that our scheme can asymptotically achieve the theoretical lowerbound when the number of information packets are sufficiently large.

24 Introduction

Then, we analyzed broadcasting schemes in a relay-aided system. Westarted with a scheduling problem based on random network coding, whichdetermined whether the BS or the relay should conduct a redundancy trans-mission, where the objective was to maximize the transmission efficiency.Based on instantaneous feedback information after each time slot, we pro-posed a greedy algorithm to determine the schedule for the subsequent timeslot. Then, we used analytic approaches based on Markov chains to esti-mate the performance of one-step scheduling. Furthermore, to provide com-petitive efficiency with significantly less feedback we proposed a dynamicprogramming (DP)-based scheduling algorithm receiving feedback only af-ter multiple redundant packets have been transmitted. The schemes wereextended into the multi-cell systems with a shared relay. Then, we proposedIDNC schemes for the one-cell relay-aided system. Our proposed schemeswere divided into three phases, namely the information transmission phase,the BS retransmission phase and the relay retransmission phase. The pro-posed IDNC schemes have the merits of low-delay and low-complexity. Wealso provided an analytical framework for deriving the expected overhead.

6.1 Future works

Based on the methods and ideas in this thesis, we present some directionswhich are interesting for future works.

Optimality analysis based on Instantaneously decodable networkcoding

In the existing works, it has been shown that the maximal efficiency of theone-hop broadcasting system with network coding is 1/(1 − maxi pi), wherepi is the erasure probability of the BS-to-UEi link, (i = 1, 2, . . . , M).

However, the achievable efficiency of the relay-aided system with IDNCis an open problem. Thus, it is interesting to find out what is the optimalperformance based on IDNC. By comparing the resulted performance withthe random network coding, the impact of the field size constraint and theinstantaneously decodable constraint can be shown.

Channel with memory or correlation

The IDNC for a one-hop system with Gilbert-Elliott channel was studied in[SST10,STK08]. However, only simulation results of the proposed schemeswere provided without theoretical analysis. Thus, it is interesting to findout the optimal performance of the IDNC in the system with Gilbert-Elliottchannel, including both one-hop and relay-aided systems.

Besides IDNC, the performance of random network coding in the systemwith Gilbert-Elliott channel is interesting as well. The impact of correlation

References 25

or memory of the channel will give insight for system design.

System with feedback error

Till now, most efforts have been focused on the systems with perfect feed-back information. In practical system, this is not always the case. Thus,how to design network coding schemes with feedback error is another direc-tion of future work.

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