A class of generic decentralised distributed fountain coding schemes is introduced and the tools of analysis of the performance of such schemes are presented. It is demonstrated that the developed approach can be used to formulate a robust code design methodology in a number of instances. We show that two non-standard applications of fountain codes, fountain codes for distributed source coding and fountain codes for unequal error protection lie within this decentralised distributed fountain coding framework

Doufexi, A

Ismail, M

Piechocki, RJ

Sejdinovic, D

English

A class of generic decentralised distributed fountain coding schemes is introduced and the tools of analysis of the performance of such schemes are presented. It is demonstrated that the developed approach can be used to formulate a robust code design methodology in a number of instances. We show that two non-standard applications of fountain codes, fountain codes for distributed source coding and fountain codes for unequal error protection lie within this decentralised distributed fountain coding framework.A class of generic decentralised distributed fountain coding schemes is introduced and the tools of analysis of the performance of such schemes are presented. It is demonstrated that the developed approach can be used to formulate a robust code design methodology in a number of instances. We show that two non-standard applications of fountain codes, fountain codes for distributed source coding and fountain codes for unequal error protection lie within this decentralised distributed fountain coding framework

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                          Sejdinovic, D., Piechocki, R. J., Doufexi, A., & Ismail, M. (2010).Decentralised distributed fountain coding: asymptotic analysis and design.IEEE Communications Letters, 14(1), 42 - 44.10.1109/LCOMM.2010.01.091541Link to published version (if available):10.1109/LCOMM.2010.01.091541Link to publication record in Explore Bristol ResearchPDF-documentUniversity of Bristol - Explore Bristol ResearchGeneral rightsThis document is made available in accordance with publisher policies. Please cite only the publishedversion using the reference above. Full terms of use are available:http://www.bristol.ac.uk/pure/about/ebr-terms.htmlTake down policyExplore Bristol Research is a digital archive and the intention is that deposited content should not beremoved. However, if you believe that this version of the work breaches copyright law please contactopen-access@bristol.ac.uk and include the following information in your message:• Your contact details• Bibliographic details for the item, including a URL• An outline of the nature of the complaintOn receipt of your message the Open Access Team will immediately investigate your claim, make aninitial judgement of the validity of the claim and, where appropriate, withdraw the item in questionfrom public view.42 IEEE COMMUNICATIONS LETTERS, VOL. 14, NO. 1, JANUARY 2010Decentralised Distributed Fountain Coding:Asymptotic Analysis and DesignDino Sejdinovic, Robert Piechocki, Angela Doufexi, and Mohamed IsmailAbstract—A class of generic decentralised distributed fountaincoding schemes is introduced and the tools of analysis of theperformance of such schemes are presented. It is demonstratedthat the developed approach can be used to formulate a robustcode design methodology in a number of instances. We show thattwo non-standard applications of fountain codes, fountain codesfor distributed source coding and fountain codes for unequalerror protection lie within this decentralised distributed fountaincoding framework.Index Terms—Rateless codes, asymptotic analysis, decen-tralised code designI. INTRODUCTIONWE assume that the reader is acquainted with standardfountain code design and analysis, cf. [1], [2], [3] forthe overview of fountain codes.The aim of a decentralised distributed fountain coding(DDFC) scheme is to reliably recover the data possiblydistributed across a set of nodes in a network, called sourcenodes, at another set of nodes, called collector nodes, withminimal number of transmissions. In our setting, we assumethat each collector node seeks to recover a data sequencex = (𝑥1, 𝑥2, . . . , 𝑥𝑘), consisting of 𝑘 data packets 𝑥𝑖 ∈ 𝔽𝑏2,𝑖 = 1, 𝑘, which are vectors of length 𝑏 over 𝔽2, and thateach source node in a network has access to a subset of datasequence x. Furthermore, each source node is oblivious ofwhich data packets are available at other source nodes andthe sets of packets available at different source nodes are notnecessarily disjoint. Each source node uses a fountain code,i.e., LT or Raptor code, to produce encoding packets over itssubset of data packets and multicasts these encoding packetsto collector nodes. The challenge in designing an efficientand robust DDFC scheme lies within the fact that sourcenodes do not cooperate and thus are not able to producea resulting bitstream resembling that of a good fountaincode. Rather, they produce localised encoding packets, linearprojections restricted to their respective subsets of coordinatesin data sequence x. However, we will show that by usingan appropriate generalised version of standard techniques foranalysis of sparse graph codes and fountain codes, we canformalise a robust code design methodology for a numberof important instances of the DDFC framework. It is aninteresting insight that for some typical single source multicastfountain coding problems such as fountain codes for unequalerror protection (UEP), it may still be useful to study theseproblems as instances of DDFC, where a single source nodeManuscript received July 29, 2009. The associate editor coordinating thereview of this letter and approving it for publication was V. Stankovic.D. Sejdinovic´, R. Piechocki, and A. Doufexi are with the Centre forCommunications Research, Department of Electrical and Electronic Engineer-ing, University of Bristol, Bristol, UK (e-mail: {d.sejdinovic, r.j.piechocki,a.doufexi}@bristol.ac.uk).M. Ismail is with Toshiba Research Europe Ltd, TelecommunicationsResearch Laboratory, Bristol, UK (e-mail: mohamed@toshiba-trel.com).This work was supported by Toshiba TRL Ltd.Digital Object Identifier 10.1109/LCOMM.2010.01.091541produces various classes of localised encoding packets in orderto attain the UEP property.II. PROBLEM OVERVIEWLet 𝑘, 𝑏 ∈ ℕ, 𝜀 > 0, and 𝑛 = ⌈𝑘(1 + 𝜀)⌉. Let x =(𝑥1, 𝑥2, . . . , 𝑥𝑘) be a data sequence of 𝑘 data packets 𝑥𝑖 ∈ 𝔽𝑏2,𝑖 ∈ 𝑁𝑘 that needs to be communicated to the collectornodes.1 Assume that a collector node obtains a sequencey = (𝑦1, 𝑦2, . . . , 𝑦𝑛) of 𝑛 encoding packets produced in adecentralised fashion at a number of source nodes, i.e., thecode overhead available to the collector node is 𝜀. We willdescribe the decentralised generation of the encoding packetsby a weighted complete bipartite graph 𝒢 = (𝒜,ℬ,Θ), illus-trated in Fig. 1a. In 𝒢, nodes 𝒜 = {𝐴1, 𝐴2, . . . 𝐴𝑟} representa disjoint partition of 𝑁𝑘, such that ∀𝑖 ∈ 𝑁𝑟, ∣𝐴𝑖∣ = 𝜋𝑖𝑘, forsome 𝜋𝑖 ∈ [0, 1], and nodes ℬ = {𝐵1, 𝐵2, . . . 𝐵𝑠} represent adisjoint partition of 𝑁𝑛, such that ∀𝑗 ∈ 𝑁𝑠, ∣𝑌𝑗 ∣ = 𝛾𝑗𝑛, forsome 𝛾𝑗 ∈ [0, 1], and Θ = (𝜃𝑗𝑖 ) is an 𝑘×𝑛 matrix, such that 𝜃𝑗𝑖is the weight associated with the edge 𝐴𝑖𝐵𝑗 . The weights arenormalized such that ∀𝑗 ∈ 𝑁𝑠,∑𝑖∈𝑁𝑟 𝜃𝑗𝑖 = 1. Note that also∑𝑖∈𝑁𝑟 𝜋𝑖 = 1,∑𝑗∈𝑁𝑠 𝛾𝑗 = 1, by construction. We can thinkof 𝐴𝑖, 𝑖 ∈ 𝑁𝑟, and 𝐵𝑗 , 𝑗 ∈ 𝑁𝑠 as determining the divison ofraw data packets and encoding data packets, respectively, intoclasses: sequence 𝑥𝐴𝑖 is the 𝑖-th class of raw data packets andsequence 𝑦𝐵𝑗 is the 𝑗-th class of encoding packets. Each ofthe 𝑠 source nodes has direct access to a certain portion ofvector x, and the set of coordinates available to source node𝑗 ∈ 𝑁𝑠 is 𝑥𝐶𝑗 , where 𝐶𝑗 = ∪𝜃𝑗𝑖 ∕=0𝐴𝑖. Graph 𝒢 is a key partof our system model, as it characterises: (1) availability ofdata at source nodes: node 𝑗 has access to sequence 𝑥𝐴𝑖 if𝜃𝑗𝑖 ∕= 0 (2) rate of production of encoding symbols at eachof the source nodes: sequence 𝑦𝐵𝑗 was produced at node 𝑗,and (3) bias introduced towards certain portions of data information of encoding packets: during the generation of eachencoding packet, packets from sequence 𝑥𝐴𝑖 are used withprobability 𝜃𝑗𝑖 at node 𝑗. Similar code construction involvingbias for the case of a single source node is used in UEPfountain codes [5], which we refer to as weighted LT coding.In addition to graph 𝒢, overall encoding process is describedby a set of degree distributions {Ω𝑗(𝑥)}𝑠𝑗=1, where distributionΩ𝑗(𝑥) is used for LT encoding at user node 𝑗. Thus, with𝒢 and {Ω𝑗(𝑥)}𝑠𝑗=1, we have fully described an instance ofdata collection via independent decentralised distributed LTencodings, i.e., a particular code ensemble which we denote byDDLT(𝒢, {Ω𝑗(𝑥)}𝑠𝑗=1, 𝑘). We are interested in the decodingperformance of this ensemble as 𝑘 →∞.Example 1 (Uniform LT encodings): Source node 𝑗 ∈ 𝑁𝑠chooses uniformly from all the available data packets: valuesof 𝜃𝑗𝑖 are proportional to the sizes of 𝐴𝑖, whenever 𝜃𝑗𝑖 ∕= 0,1Throughout the paper, we adopt the following notation. For any naturalnumber 𝑚, denote by 𝑁𝑚 the set {1, 2, . . . , 𝑚}. For data sequence x =(𝑥1, 𝑥2, . . . , 𝑥𝑘) and 𝐴 ⊆ 𝑁𝑘 , we denote 𝑥𝐴 := (𝑥𝑖 : 𝑖 ∈ 𝐴).1089-7798/10$25.00 c⃝ 2010 IEEESEJDINOVIC et al.: DECENTRALISED DISTRIBUTED FOUNTAIN CODING: ASYMPTOTIC ANALYSIS AND DESIGN 43k1π1An1γ……11θclasses of source packets21θ12θ s1θs2θsrθ1rθ22θ2rθk2π2AkrπrAn2γ nsγclasses of 1B 2B sBencoding packets(a)kp21−1An211Bpk2A2Bkp21−3An21pp+−11pp+−11pp+12pp+12(b)k1π1Ak2π2A1θ 2θnB1(c)k1π1Ak2π2Anγ nγ11π 2π11B22B(d)Fig. 1. (a) Generic DDFC scheme, (b) Symmetric DSC problem in Example 2 as an instance of DDFC, (c) Weighted UEP LT codes [5] for two classesof importance as an instance of DDFC, (d) EWF codes [6] for two classes of importance as an instance of DDFC.i.e., 𝜃𝑗𝑖 =𝜋𝑖∑𝑙∈𝐶𝑗 𝜋𝑙. This means that the user 𝑗 is performinga simple LT encoding over vector 𝑥𝐶𝑗 .III. GENERALISED AND-OR LEMMAAssume that each user is producing encoding packets asdescribed and once a collector node successfully receives asufficient amount of these encoding packets, it attempts torecover x. We can capture the asymptotic performance of thebelief propagation decoder at the collector node as a functionof the code overhead 𝜀 by the generalisation of the originalAnd-Or tree analysis [4]. As in the standard And-Or treeanalysis and other density evolution arguments, we derivethe asymptotic performance of the decoder by looking at thestructure, and specifically at the degree distributions, on thedecoding factor graph. The proof of the lemma is omitted asit is a straightforward generalisation of the standard And-Orlemma. What distinguishes this generalised setting from theoriginal one is that the nodes in the decoding factor graphare divided into classes, each of the classes of nodes possiblyhaving a different degree distribution. Classes are introducednaturally - class 𝑖 of the input nodes corresponds to the rawdata packets 𝑥𝐴𝑖 and class 𝑗 of the output nodes correspondsto the encoding packets 𝑦𝐵𝑗 .Lemma 1 (Generalised And-Or lemma): For all 𝑖 ∈ 𝑁𝑟,the packet error rate within sequence 𝑥𝐴𝑖 of a belief prop-agation decoder for an ensemble DDLT(𝒢, {Ω𝑗(𝑥)}𝑠𝑗=1, 𝑘) ata collector node, as 𝑘 → ∞, is equal to 𝑦𝑖 = lim𝑙→∞ 𝑦𝑖,𝑙,where 𝑦𝑖,𝑙 is given by:𝑦𝑖,0 = 1, (1)𝑦𝑖,𝑙+1 = exp[−(1 + 𝜀)𝑠∑𝑗=1𝜃𝑗𝑖𝛾𝑗𝜋𝑖Ω′𝑗(1−𝑟∑𝑚=1𝜃𝑗𝑚𝑦𝑚,𝑙)].In some special cases of DDFC, derived asymptotic anal-ysis allows us to formulate a robust linear programmingoptimisation routine to calculate asymptotically good degreedistributions, similarly to, e.g., [3], and the remainder of thispaper illustrates two simple examples of such use of thegeneralised And-Or lemma.IV. SYMMETRIC DISTRIBUTED SOURCE CODING (DSC):TWO NODES WITH COMMON DATAExample 2: Assume that two source nodes 𝑆1 and 𝑆2contain more than a half of all the packets in the data sequencex = (𝑥1, 𝑥2, . . . , 𝑥𝑘) (but the number of packets at 𝑆1 and 𝑆2is the same). A certain portion 𝑝 < 1/2 of the packets iscommon to 𝑆1 and 𝑆2 but they have no knowledge of whichpackets are available at the other source node. A collectornode would ideally need just slightly more than 𝑘 encodingpackets to recover the entire information sequence.This Example outlines a simple DDFC problem, and itsgraphical representation is given in Fig. 1b. Note that there arethree different classes of packets: class 𝐴1 of1−𝑝2 𝑘 packetsavailable only at node 𝑆1, class 𝐴2 of 𝑝𝑘 packets availableat both nodes, and class 𝐴3 of1−𝑝2 𝑘 packets available onlyat node 𝑆2. Let each source node produce encoding packetswith an LT code with degree distribution Ω(𝑥) over (1−𝑝2 +𝑝)𝑘packets available at that node and let us assume collector nodeobtains an equal number of encoding packets from each sourcenode. The recursive equation describing the asymptotic packeterror rate in class 𝐴1 is thus given by:𝑦1,𝑙+1 = exp[−1 + 𝜀1 + 𝑝Ω′((1 − 𝑝)(1− 𝑦1,𝑙) + 2𝑝(1− 𝑦2,𝑙)1 + 𝑝)].(2)It is easily checked that 𝑦3,𝑙 = 𝑦1,𝑙 and that 𝑦2,𝑙 = 𝑦21,𝑙,∀𝑙 ≥ 0. Thus, one can trace the asymptotic behaviour of allthree packet error rates with a single parameter, which allowssimple transformation of the above recursive equations intothe linear program optimisation procedure.Now, let us for the sake of simplicity assume 𝑝 = 1/3,i.e., a third of all the packets are available at both sourcenodes. Simple transformations yield the following genericlinear program2:LP : min𝑑max∑𝑑𝜔𝑑𝑑(3)34𝜔(1− 𝑧𝑖2− 𝑧2𝑖2) ≥ − ln(𝑧𝑖), 𝑖 ∈ 𝑁𝑚,where 1 = 𝑧1 > 𝑧2 > ⋅ ⋅ ⋅ > 𝑧𝑚 = 𝛿 are 𝑚 equidistantpoints on [𝛿, 1]. The solution of this linear program is an edgeperspective degree distribution with maximum degree 𝑑maxwhich reaches the packet error rate of 𝛿 within classes 𝐴1and 𝐴3 (and 𝛿2 within class 𝐴2) at the minimum overhead.The degree distribution we obtained using this linear programwith 𝛿 = 0.01 and 𝑑𝑚𝑎𝑥 = 100 successfully takes advantageof the fact that two sources contain correlated information, andit is given by: Ω∗(𝑥) = 0.0020𝑥 + 0.4305𝑥2 + 0.2205𝑥3 +0.0793𝑥5+0.1097𝑥6+0.0508𝑥12+0.0409𝑥13+0.0343𝑥30+2In linear programs given in the paper, variables (to be optimised) arenon-negative, i.e., 𝜔𝑑 ≥ 0, 𝑑 ∈ 𝑁𝑑max ; 𝜔(𝑥) = Ω′(𝑥)/Ω′(1) is the edgeperspective output degree distribution.44 IEEE COMMUNICATIONS LETTERS, VOL. 14, NO. 1, JANUARY 2010−0.1 −0.05 0 0.05 0.1 0.15 0.210−510−410−310−210−1100overhead εpacket error rate  y1, y3(k=∞, Ω*)y2 (k=∞, Ω*)y1, y3 (k=6⋅104, Ω*)y2 (k=6⋅104, Ω*)y1, y3 (k=1.5⋅105, Ω*)y2 (k=1.5⋅105, Ω*)y1, y3 (k=6⋅104, Ωraptor)y2 (k=6⋅104, Ωraptor)Fig. 2. Symmetric DSC with LT codes.0.0106𝑥32 + 0.0215𝑥100. Simulation results for large block-lengths, 𝑘 = 6 ⋅ 104 and 𝑘 = 1.5 ⋅ 105 are consistentwith our asymptotic analysis, as demonstrated in Fig. 2. Forcomparison, we included results for a typical Soliton-likedegree distribution3 Ω𝑟𝑎𝑝𝑡𝑜𝑟(𝑥) proposed for Raptor codes [3],which is clearly penalised by higher error floors in this setting.V. FOUNTAIN CODES FOR UNEQUAL ERROR PROTECTIONWhereas LT and Raptor codes typically provide equal errorprotection for all source data, there are cases where certaindata parts are considered to be more important and requirehigher error protection, e.g., transmission of video compressedwith a layered coder, and unequal error protection (UEP)fountain coding schemes have recently been proposed [5], [6].Asymptotic analysis of these codes are special cases of Lemma1. Although these codes can be fully centralised, it is useful tothink of them as of instances of DDFC, as this allows rigorousasymptotic analysis and an insight into the design of codeparameters.Example 3 (UEP codes for two classes of importance):The data sequence x = (𝑥1, 𝑥2, . . . , 𝑥𝑘) consists of twoclasses of importance: class 𝐴1 consists of 10% of alldata packets which require higher error protection, whereasremaining 90% in class 𝐴2 are considered less important.The encoder can utilise a weighted UEP LT code [5], whichuses a nonuniform distribution across data packets, i.e.,𝜃1/𝜋1 > 𝜃2/𝜋2, which means that the data packet in class 𝐴𝑖is selected with probability 𝜃𝑖𝜋𝑖𝑘 , 𝑖 = 1, 2. Alternatively, theencoder can utilise an EWF code [6], by dividing encodingpackets into two classes - class 𝐵1 of packets encoding onlyclass 𝐴1 of source packets (the first window of data packets),and class 𝐵2 encoding the entire data sequence. Two kindsof UEP fountain codes with two classes of importance areillustrated in Fig. 1c and 1d as instances of DDFC. Byapplying Lemma 1, the recursive equation for asymptoticanalysis of weighted UEP LT codes is given by:𝑦𝑖,𝑙+1 = exp(−(1 + 𝜀) 𝜃𝑖𝜋𝑖Ω′(1− 𝜃1𝑦1,𝑙 − 𝜃2𝑦2,𝑙)),for 𝑖 = 1, 2, which is consistent with the analysis in [5],and, in an analogous manner, the asymptotic analysis of EWF3Ω𝑟𝑎𝑝𝑡𝑜𝑟(𝑥) = 0.0080𝑥 + 0.4936𝑥2 + 0.1662𝑥3 + 0.0726𝑥4 ++0.0826𝑥5+0.0561𝑥8+0.0372𝑥9+0.0556𝑥19+0.0250𝑥65+0.0031𝑥66−0.05 0 0.05 0.1 0.15 0.210−510−410−310−210−1100overhead εpacket error rate  y1, (k=∞, Ω*)y2, (k=∞, Ω*)y1, (k=105, Ω*)y2, (k=105, Ω*)y1, (k=∞, Ωraptor)y2, (k=∞, Ωraptor)Fig. 3. Weighted LT codes with an optimised degree distribution.codes in [6] is another special case of Lemma 1. Furthermore,with appropriate transformations, we can optimise degreedistribution for weighted LT codes using the following genericlinear program:LP : min𝑑max∑𝑑𝜔𝑑𝑑(4)𝜔(1− 𝜃1𝑧𝜃1/𝜋1𝑖 − 𝜃2𝑧𝜃2/𝜋2𝑖 ) ≥ − ln(𝑧𝑖), 𝑖 ∈ 𝑁𝑚,where 1 = 𝑧1 > 𝑧2 > ⋅ ⋅ ⋅ > 𝑧𝑚 = 𝛿 are 𝑚 equidistantpoints on [𝛿, 1]. The solution of this linear program is an edgeperspective degree distribution with maximum degree 𝑑maxwhich reaches the packet error rate of 𝛿𝜃1/𝜋1 within class 𝐴1and 𝛿𝜃2/𝜋2 within the class 𝐴2 at the minimum overhead. Aresult of this optimisation for 𝑑max = 100, 𝛿 = 0.0075 inthe weighted LT instance with 𝜃1 = 0.184, 𝜋1 = 0.1, weobtain degree distribution Ω∗(𝑥) = 0.0080𝑥1 + 0.4226𝑥2 +0.2769𝑥3+0.1515𝑥6+0.0214𝑥7+0.0524𝑥13+0.0123𝑥14+0.0338𝑥27+0.0212𝑥74. We chose parameter 𝜃1 = 0.184 in or-der to compare the asymptotic and large blocklength (𝑘 = 105)packet error rate of the calculated degree distribution with thatof Ω𝑟𝑎𝑝𝑡𝑜𝑟(𝑥), which was used in [5] with the bias parameterequivalent to 𝜃1 = 0.184 (this bias parameter minimisespacket error rate at the overhead 𝜀 = 0.03). The resultsare presented in Fig. 3 and clearly indicate how a designeddegree distribution outperforms Ω𝑟𝑎𝑝𝑡𝑜𝑟(𝑥) both in terms ofthe packet error rates and the code overhead. Nonetheless,we have noted a fragile behaviour of the designed degreedistributions at smaller blocklengths and the robust finitelength design with appropriately modified LP optimisationremains a problem for further investigations.REFERENCES[1] D. J. C. MacKay, “Fountain codes," IEE Proc. Commun., vol. 152, no.6, pp. 1062-1068, 2005.[2] M. Luby, “LT codes,” in Proc. IEEE Symp. Foundations of ComputerScience (FOCS), Vancouver, Canada, Nov. 2002.[3] A. Shokrollahi, “Raptor codes,” IEEE Trans. Inf. Theory, vol. 52, no. 6,pp. 2551-2567, June 2006.[4] M. Luby, M. Mitzenmacher, and A. Shokrollahi, “Analysis of randomprocesses via and-or tree evaluation,” in Proc. SIAM Symp. on DiscreteAlgorithms (SODA), San Francisco, USA, Jan. 1998.[5] N. Rahnavard, B. N. Vellambi, and F. Fekri, “Rateless codes with unequalerror protection property,” IEEE Trans. Inf. Theory, vol. 53, no. 4, pp.1521-1532, 2007.[6] D. Sejdinovic´, D. Vukobratovic´, A. Doufexi, V. Senk, and R. Piechocki,“Expanding window fountain codes for unequal error protection,” IEEETrans. Commun., vol. 57, no. 9, pp. 2510-2516, 2009.

Analysis of random processes via and-or tree evaluation,” in

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Rateless codes with unequal error protection property,”

Decentralised distributed fountain coding: asymptotic analysis and design

A class of generic decentralised distributed fountain coding schemes is introduced and the tools of analysis of the performance of such schemes are presented. It is demonstrated that the developed approach can be used to formulate a robust code design methodology in a number of instances. We show that two non-standard applications of fountain codes, fountain codes for distributed source coding and fountain codes for unequal error protection lie within this decentralised distributed fountain coding framework. © 2010 IEEE

Sejdinović, D

Piechocki, R

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Decentralised distributed fountain coding: Asymptotic analysis and design

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Decentralised distributed fountain coding: asymptotic analysis and design

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