Abstract—In this paper, we analyze the maximum likelihooddecoding performance of Raptor codes with a systematic low-density generator-matrix code as the pre-code. By investigatingthe rank of the product of two random coefficient matrices, wederive upper and lower bounds on the decoding failure probabil-ity. The accuracy of our analysis is validated through simulations.Results of extensive Monte Carlo simulations demonstrate that forRaptor codes with different degree distributions and pre-codes,the bounds obtained in this paper are of high accuracy. Thederived bounds can be used to design near-optimum Raptor codeswith short and moderate lengths.
R ATELESS codes have been increasingly used in manytelecommunication systems [1], [2], [3], [4], including
cellular networks and satellite communication systems. Recent
Manuscript received January 21, 2015; revised June 30, 2015, October8, 2015, December 1, 2015, and January 17, 2016; accepted January 18,2016. Date of publication January 27, 2016; date of current version March15, 2016. This research is supported by Australian Research Council (ARC)Discovery projects DP110100538 and DP120102030 and Chinese NationalScience Foundation project 61428102 and 61210002. Zihuai Lin’s researchis supported by the Australian Research Council (ARC) Discovery projectDP120100405. The associate editor coordinating the review of this paper andapproving it for publication was A. Graell i Amat. (Corresponding author:Xiaohu Ge.)
P. Wang is with the School of Electrical and Information Engineering,University of Sydney, Darlington, N.S.W. 2008, Australia, and also withNational ICT Australia (NICTA), Sydney, N.S.W. 2015, Australia (e-mail:[email protected]).
G. Mao is with the School of Computing and Communications, Universityof Technology Sydney, Sydney, N.S.W., Australia, also with Data61,Sydney, N.S.W., Australia, also with Beijing University of Posts andTelecommunications, Beijing, China, and also with Huazhong University ofScience and Technology, Wuhan, China (e-mail: [email protected]).
Z. Lin is with the School of Electrical and Information Engineering,University of Sydney, Darlington, N.S.W. 2008, Australia (e-mail:[email protected]).
M. Ding is with the National ICT Australia (NICTA), Sydney, N.S.W. 2015,Australia (e-mail: [email protected]).
W. Liang is with the Research School of Computer Science,Australian National University, Canberra, A.C.T. 0200, Australia (e-mail:[email protected]).
X. Ge is with the School of Electronics and Information Engineering,Huazhong University of Science and Technology, Wuhan 430074, China(e-mail: [email protected]).
Z. Lin is with the College of Electrical Engineering, Zhejiang University,Hangzhou 310027, China (e-mail: [email protected]).
Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TCOMM.2016.2522403
work has shown that, by employing rateless codes, wire-less transmission efficiency and reliability can be dramaticallyimproved [5], [6].
Rateless codes are a class of forward error correction (FEC)codes with special properties, which were initially designedfor the binary erasure channel (BEC). Compared with conven-tional FEC codes with a fixed code rate, rateless codes havea number of advantages. Firstly, similar as low-density parity-check (LDPC) codes, rateless codes can be implemented withfar less complex encoding and decoding algorithms, which areattractive for implementation. Secondly, as suggested by thename, rateless codes are suitable for any code rate. They canautomatically adapt to instantaneous channel states and do notrequire feedback channels [1], [3], [5]. This is because theycan generate a potentially limitless stream of coded symbols,and all source symbols can be correctly decoded when thereare a sufficient number of successfully received coded sym-bols. Hence, rateless codes are desirable for certain channels,such as erasure multicast or broadcast channels, whose real-time channel erasure probability is very difficult to capture orestimate. Furthermore, they have the potential to replace theconventional automatic repeat request (ARQ) mechanism as anew mechanism of transmission control protocol [7].
Among the well-known rateless codes, two codes stand out.One is the Luby transform (LT) codes [3], which are the firstclass of practical digital fountain codes with an average decod-ing cost in the order of O(k log(k)) where k is the numberof source symbols. The other is the Raptor codes [1], whichare the first class of fountain codes with linear time encod-ing and decoding complexities. Raptor codes are concatenatedcodes, which combine a traditional FEC code with an LT codeto relax the condition that all input (source) symbols need tobe recovered in an LT decoder. Note that Raptor codes havealready been standardized in the 3rd Generation PartnershipProject (3GPP) [4] to efficiently disseminate data over a broad-cast/multicast network to provide multimedia broadcast andmulticast services.
Despite the successful application of Raptor codes in 3GPP,our understanding of Raptor codes is still incomplete due toa lack of complete theoretical analysis on their decoding errorperformance. Without analytical results, the optimization of thedegree distribution and other parameters of Raptor codes wouldbe extremely difficult.
In this paper, we investigate the performance of Raptorcodes by theoretically analyzing their decoding failure proba-bility under maximum likelihood (ML) decoding. The decoding
WANG et al.: PERFORMANCE ANALYSIS OF RAPTOR CODES 907
failure probability is the probability that not all source sym-bols can be decoded by ML decoding from a given numberof successfully received coded symbols. We consider a Raptorcode ensemble with a systematic (n, k, η) low-density gener-ator matrix (LDGM) code as the pre-code. In the case of theerasure channel, ML decoding is equivalent to solving a con-sistent system of m linear equations in k unknowns by meansof Gaussian elimination (GE). In this paper, we investigate thedecoding failure probability of Raptor codes by theoreticallyanalyzing the rank of the product of two random coefficientmatrices and deriving tight analytical bounds. The tightnessof the bounds is confirmed by extensive Monte Carlo simu-lations. More specifically, the contributions of this paper aresummarized in the following:
• Firstly, this paper provides analytical results (i.e. an upperbound and a lower bound) on the decoding failure perfor-mance of Raptor codes, using a systematic LDGM codeas the pre-code and assuming ML decoding.
• Furthermore, simulations are conducted to validate theaccuracy of the proposed bounds. That is, Raptor codeswith different degree distributions and pre-codes are eval-uated to verify the claims on the accuracy of the derivedupper and lower bounds.
The rest of the paper is organized as follows. Section IIreviews the related work. In Section III, a brief review of theencoding and decoding process of Raptor codes is given. InSection IV, a performance analysis of Raptor code is conductedby deriving an upper bound and a lower bound on the probabil-ity that not all source symbols can be successfully decoded by areceiver with a given number of successfully received codedsymbols. Section V validates the analytical results throughsimulations, followed by concluding remarks in Section VI.
II. RELATED WORK
In this section, we review related work on the analysis of theperformance of Raptor codes.
In general, there are two inter-related metrics to measure theperformance of Raptor codes. One is the bit error probabilityand the other is the decoding failure probability. To analyze thebit error probability of Raptor codes, Rahnavard et al. [7] pro-posed a method to compute the upper and the lower bounds onthe bit error probability of Raptor codes under ML decodingover the binary erasure channels (BEC). Despite the advancesin [7], their work can be further improved in the followingaspects. Firstly, the authors in [7] used a stochastic parity-check codes, i.e. (n, k, η) LDPC code, as the pre-code of Raptorcodes. All entries of the parity check matrix are assumed to beindependent and identically distributed (i.i.d) Bernoulli randomvariables [7]. Contrary to this assumption, in 3GPP standard[4], the pre-code of the standardized Raptor codes is a system-atic LDGM code. The use of the systematic LDGM code asthe pre-code is to guarantee that the parity check matrix is afull-rank matrix. Secondly, Rahnavard et al. assumed that theerasures on intermediate bit level are independent. As explainedin [8, Ch. 6.2.1], this assumption would only hold if a very longinterleaver was used. Using an interleaver in this setup, how-ever, is not reasonable. In [9], the authors derived the upper and
the lower bounds on the bit error probability of Raptor codesover Rayleigh fading channels assuming ML decoding.
In [1], Shokrollahi analyzed the decoding failure probabilityof Raptor codes with a finite length assuming belief propa-gation (BP) decoding. The analysis relies on the computationof the failure probability of the LT codes under BP decoding,which was derived in [10]. ML decoding, on the other hand, ismore computationally demanding than BP decoding for codeswith a large length. The analysis of the decoding failure prob-ability assuming ML decoding is however both important andsignificant, because it provides a benchmark on the optimumsystem performance that can be used to gauge the performanceof other decoding schemes. Furthermore, in [8] a pseudo upperbound on the performance of Raptor codes under ML decodingwas derived, under the assumption that the number of erasurescorrectable by the pre-code is small. This approximation isaccurate only when the rate of the pre-code is sufficiently high.For the more general case, the decoding failure probability ofRaptor codes still remains an open problem. In [11] it is shownthat the rank profile of the constraint matrix of a Raptor codedepends on the rank profile of the pre-code parity check matrixand the generator matrix of the LT code. The rank profile of theRaptor code cannot be determined if the rank profile of an LTcode with a general degree distribution is unknown. In our pre-vious work [6], we analyzed the rank profile of an LT code witha general degree distribution.
In this paper, we present theoretical analysis on the decod-ing failure probability of Raptor codes under ML decoding. Weconsider a Raptor code ensemble with a systematic (n, k, η)
LDGM code as the pre-code to guarantee that the parity checkmatrix is a full-rank matrix. Furthermore, we take into accountthe fact that the residual erasure events after LT decoding arenot independent, thereby deriving tighter bounds.
III. BACKGROUND OF RAPTOR CODES
This section is provided to familiarize the readers withthe basic idea of Raptor codes, their encoding and decodingalgorithms.
The encoding process of a Raptor code [1] is carried out intwo phases: a) encode k source symbols with a (n, k) error cor-rection code, which is referred to as the pre-code C, to formn intermediate symbols; b) encode the n intermediate symbolswith an LT code. Each coded symbol is generated by the follow-ing encoding rules of LT codes [3]. Firstly, a positive integer d(often referred to as the “degree” of coded symbols) is drawnfrom the set of integers {1, . . . , n} according to a probabilitydistribution � = (�1, . . . , �n), where �d is the probabilitythat d is selected and
∑nd=1 �d = 1. Then, d distinct interme-
diate symbols are selected randomly and independently fromthe n intermediate symbols to form the coded symbol to betransmitted using the XOR operation, where each intermedi-ate symbol is selected with equal probability. A Raptor codewith parameters (k,C,�) is an LT code with distribution � =(�1, . . . , �n) on n symbols which are the output symbols ofthe pre-code C.
An illustration of a Raptor code is given in Fig. 1. In prac-tice, the parity check matrix of the pre-code of Raptor codes is
908 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 64, NO. 3, MARCH 2016
Fig. 1. Two-stage structure of a Raptor code with a systematic pre-code.
a deterministic matrix. For example, in 3GPP standard [4], theparity check matrix of the pre-code of the standardized Raptorcodes is a systematic deterministic matrix. Using a system-atic deterministic matrix as the pre-code ensures that the paritycheck matrix of the pre-code is a full-rank matrix. However, itis difficult to obtain tractable analytical results of decoding per-formance for such Raptor codes. Therefore, in this paper weadopt a Raptor code ensemble with a semi-random (n, k, η)
LDGM code as the pre-code for analytical tractability whileensuring that the parity check matrix of the pre-code is a full-rank matrix. The generator matrix of the pre-code, denoted byGpre
n×k , can be written as Gpren×k = [Ik |Pk×(n−k)]T , where Ik is
an identity matrix of size k, and Pk×(n−k) is a k by (n − k)
matrix whose entries are i.i.d. Bernoulli random variables withparameter η. Such a code is denoted as an (n, k, η) LDGMcode. Furthermore, we can obtain the parity check matrix ofthis LDGM code as H(n−k)×n = [P(n−k)×k |I(n−k)](n−k)×n .
Let m, (m ≥ k), be the number of coded symbols that havealready been successfully received by a receiver and γ = m
k ,(γ ≥ 1) be the overhead of reception. When a coded symbolis received by a receiver, we use a 1 × k binary row vectorgLT
i Gpre to represent the coding information contained in thecoded symbol, where GLT is a kγ × n binary matrix, gLT
i isthe i th row vector of GLT and Gpre is a n × k binary matrix.Let [G]i, j be the entry in the i th row and the j th column ofthe matrix G. Particularly,
[gLT
i
]1, j is 1 if the coded sym-
bol is a result of the XOR operation on the j th intermediatesymbol (and other intermediate symbols); otherwise
[gLT
i
]1, j
equals 0. For[Gpre
]i, j , it is 1 if the i th intermediate symbol is
a result of the XOR operation on the j th source symbol (andother source symbols); otherwise
[Gpre
]i, j equals 0. Therefore,
a random row vector in this paper refers to the row vector ofa randomly chosen coded symbol where the coded symbol isgenerated using the Raptor encoding process described above.Recall that s = (s1, s2, . . . , sk) represents the k source sym-bols to be transmitted. The coded symbol can be expressed as:yi = gLT
i GpresT , where “sT ” is the transpose of s.Raptor codes can be decoded using a variety of decoding
algorithms. A commonly used decoding algorithm for Raptor
codes is the so-called “LT process” [3], but it is well known thatthe LT process is unable to decode all source symbols whichcan be possibly recovered from the received coded symbols.For example, the LT process relies on the existence of at leastone degree-one coded symbol to be received in order to startthe decoding process. For Raptor codes with limited lengths,ML decoding algorithm [12] has been proposed to replace theLT process. The performance of ML decoding is the same asthe Gaussian elimination. One way to apply the Gaussian elim-ination on Raptor codes is to solve a system of linear equationsgiven in the following.
GLTkγ×nGpre
n×ksTk×1 = ykγ×1,
where ykγ×1 = (y1, y2, . . . , ykγ )T . Then we can obtain thefollowing Lemma.
Lemma 1: A receiver can recover all k source symbolsfrom the kγ coded symbols under ML decoding if and onlyif (GLT
kγ×nGpren×k)kγ×k is a full-rank matrix, i.e. its rank equals
k [1].Note that in this paper, all algebraic operations and the
associated analysis are conducted in a binary field G F(2).
IV. PERFORMANCE ANALYSIS OF RAPTOR CODES
Denote by Akkγ the event that a receiver can success-
fully decode all k source symbols conditioned on the eventthat the receiver has successfully received kγ coded sym-bols. Obviously the event that (GLT
kγ×nGpren×k)kγ×k is a full-rank
matrix is equivalent to the event Akkγ . Let Ak
γ k be the com-
plement of event Akkγ . The main results of this paper are
summarized in Theorems 2 and 3.In this section, we shall analyze the probability Pr
[Ak
γ k
].
The analysis of decoding failure probability P DFk,n,γ = Pr
[Ak
γ k
]is conducted by analyzing the probability that the rank of(GLT
kγ×nGpren×k)kγ×k is not k.
A. Upper Bound on the Decoding Failure Probability of RaptorCodes
In this subsection, we will derive an upper bound on thedecoding failure probability of Raptor codes with a system-atic (n, k, η) LDGM code as the pre-code. The upper boundis formally stated in the following theorem.
Theorem 2: When a receiver successfully receives kγ codedsymbols generated using the Raptor code (k,C,�(x)), whereC is an (n, k, η) LDGM code, and the coded symbols receivedat the receiver are decoded using ML decoding, the probabil-ity that not all k source symbols can be successfully decodedby a receiver with the kγ, (kγ ≥ k) , received coded symbols,denoted by P DF
k,n,γ , is upper bounded by
P DFk,n,γ ≤
k∑i=1
(ki
) n−k+i∑r=i
(J (r))kγ D (i, r) , (1)
WANG et al.: PERFORMANCE ANALYSIS OF RAPTOR CODES 909
where
J (r) =n∑
d=1
�d
∑s=0,2, ..., 2
⌊d2
⌋ (rs
) (n−rd−s
)(
nd
) (2)
and
D(i, r) =(
n−kr−i
) [1 + (1 − 2η)i
2
]n−k−r+i
×[
1 − (1 − 2η)i
2
]r−i
(3)
and �d is the degree distribution of LT codes.
Proof: See Appendix A. �
B. Lower Bound on the Decoding Failure Probability of RaptorCodes
In addition to the upper bound in the previous subsection,in the following paragraphs, we derive a lower bound on thedecoding failure probability of Raptor codes which is formallystated in the following theorem.
Theorem 3: When a receiver successfully receives kγ codedsymbols generated using the Raptor code (k,C,�(x)), whereC is an (n, k, η) LDGM code, and the coded symbols receivedat the receiver are decoded using ML decoding, the probabil-ity that not all k source symbols can be successfully decodedby a receiver with the kγ, (kγ ≥ k) , received coded symbols,denoted by P DF
where 1(x) is an indicator function, 1(x) = 0 if x = 0 and1(x) = 1 otherwise, J (·) = 1 − J (·), D(w0, r0) is defined inEq. (3) and J (r0) is defined in Eq. (2).
Proof: See Appendix B. �
C. A Special Case of the Derived Bounds
When we apply a special degree distribution - a binomial
degree distribution [13] with �d = (nd)
(2n−1), 1 ≤ d ≤ n, Eq. (1)
can be further simplified into a much less (computationally)complex expression, for which Theorem 2 can be restated asthe following Corollary.
Corollary 4: When a receiver successfully receives kγ
coded symbols generated using the Raptor code (k,C,�(x))
where C is an (n, k, η) LDGM code, �(x) = ∑nd=1
(nd)xd
(2n−1), and
the coded symbols received at the receiver are decoded usingML decoding, the probability that not all k source symbols canbe successfully decoded by a receiver with the kγ, (kγ ≥ k) ,
received coded symbols, denoted by P DFk,n,γ , satisfies
P DFk,n,γ ≤
(2k − 1
)((2n−1 − 1)
(2n − 1)
)kγ
. (5)
Proof: See Appendix C. �For Theorem 3, we can simplify the lower bound into a
less (computationally) complex expression as well. This issummarized in the following Corollary.
Corollary 5: When a receiver successfully receives kγ
coded symbols generated using the Raptor code (k,C,�(x))
where C is an (n, k, η) LDGM code, �(x) = ∑nd=1
(nd)xd
(2n−1), and
the coded symbols received at the receiver are decoded usingML decoding, the probability that not all k source symbols canbe successfully decoded by a receiver with the kγ, (kγ ≥ k) ,
received coded symbols, denoted by P DFk,n,γ , satisfies
P DFk,n,γ
≥(
2k − 1) [ (2n−1 − 1)
(2n − 1)
]kγ
−(
2k − 1) (
2k−1 − 1)
×{[
(2n−1 − 1)
(2n − 1)
]3
+[
1 − (2n−1 − 1)
(2n − 1)
]3}kγ
. (6)
Proof: See Appendix D. �Compared with the general expressions in Theorems 2 and
3, the simplified expressions in Corollaries 4 and 5 allow usto easily observe the relationship between the decoding fail-ure probability and the parameters of the encoding rules, i.e.,k, n and γ . Additionally, the computation complexity of thederived upper bound can be reduced from O( 1
2 n2k(n − k)) toO(1). As for the lower bound, the computation complexity canbe reduced from O( 1
8 n6k3(n − k)3) to O(1).
V. SIMULATION RESULTS
In this section, we shall validate the accuracy of the ana-lytical results and the tightness of the proposed bounds, usingMATLAB simulations. Each point shown in the figures is theaverage result obtained from 106 simulations. For clarity, thesimulation parameters adopted in this section are summarizedin Table I.
A. Verification of the Derived Bounds
In this subsection, the number of source symbols is set tobe k = 20 and the degree distribution of Raptor codes followsthe widely used ideal soliton degree distribution [3]. Besides,the pre-code C is assumed to be (21, 20, 0.3) and (21, 20, 0.7)
LDGM codes respectively.
910 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 64, NO. 3, MARCH 2016
TABLE ISIMULATION PARAMETERS
Fig. 2. The decoding failure probabilities of Raptor codes with ideal solitondegree distribution and (n, k, η) LDGM codes as the pre-code versus overheadγ . Parameter for Bernoulli random variables η is set as 0.3 and 0.7.
In Fig. 2(a) and 2(b), both analytical and simulation resultsare presented on P DF
k,n,γ , the probability that not all k = 20source symbols can be successfully decoded by a receiver, fordifferent values of the reception overhead γ = m/k. As shownin Fig. 2(a) and 2(b), our analytical results, i.e., the upper boundand the lower bound, match the simulation results very well.This validates the accuracy of the analysis. However, whenthe overhead γ is small, there is still a gap between the upper(lower) bound and simulation results in Fig. 2(a) and 2(b). Thegap between the exact value and the upper bound is caused by
Fig. 3. The decoding failure probabilities of Raptor codes with (n, k, 0.7)
LDGM codes as the pre-code and different degree distributions versus overheadγ . The degree distributions of Raptor codes are chosen as ideal soliton degreedistribution [3], the standardized degree distribution in 3GPP [4, Annex B] andbinomial degree distribution [13].
the approximation used in Eq. (1), and the gap between theexact value and the lower bound is caused by Eq. (4).
B. Investigation of the Impact of Degree Distribution on theDecoding Failure Probability
In this subsection, we investigate the performance for differ-ent distributions of LT codes when we fix the pre-code C to be(21, 20, 0.7). The investigated degree distributions are dividedinto three cases.
• Case 1 uses the binomial degree distribution [13].• Case 2 investigates the widely used ideal soliton degree
distribution [3].• Case 3 is the standardized degree distribution in 3GPP
[4, Annex B]:
�3G P P (x) = 0.0099x + 0.4663x2 + 0.2144x3
+ 0.1152x4 + 0.1131x10 + 0.0811x11.
As shown in Fig. 3(a) and 3(b), for different degree distribu-tions, our analytical bounds agree very well with the simulationresults. The performance of Raptor codes with the binomial
WANG et al.: PERFORMANCE ANALYSIS OF RAPTOR CODES 911
Fig. 4. The decoding failure probabilities of Raptor codes with the binomialdegree distribution and (n, k, 0.7) LDGM codes as the pre-code at differentvalues of the overhead γ . The number of source symbols k is set to be 20, 40,70 and 100 respectively.
degree distribution outperforms those obtained with the otherthree degree distributions. Furthermore, the decoding failureprobability of Raptor codes with the binomial degree distribu-tion in Corollaries 4 and 5 are less computationally demandingcompared with those in Theorems 2 and 3. Therefore, we willuse Raptor codes with the binomial degree distribution in thefollowing simulations.
C. Investigation of the Impact of k on the Decoding FailureProbability of Raptor Codes
When the number of source symbols k varies from 20 to100, our analytical results still match the simulation resultsvery well. As shown in Fig. 4(a) and 4(b), at a larger valueof the source symbols, a less reception overhead γ = m/kis required to achieve the same performance on the decodingfailure probability.
VI. CONCLUSION
In this paper we studied the performance of finite-lengthRaptor codes with a systematic LDGM code as the pre-code,
and derived an upper bound and a lower bound on the decodingfailure probability of Raptor codes under ML decoding. Dueto the concatenated coding structure of Raptor codes, we ana-lyzed the rank behavior of the product of two random matricesto obtain the decoding failure probability. Furthermore, by con-sidering a special degree distribution, i.e. the binomial degreedistribution, we derived the simplified upper and lower bounds.On the basis of the results presented in the paper, we shallexplore the optimum degree distribution and optimal parame-ter setting of Raptor codes in different channels as our futurework.
APPENDIX APROOF OF THEOREM 2
In this appendix, we prove Theorem 2.According to the property of the matrix product [14,
Eq. (4.5.1)], we have
rank(GLTkγ×nGpre
n×k)
= rank(Gpren×k) − dim{N (GLT
kγ×n) ∩ R(Gpren×k)}, (7)
where N (•) is the right-hand null space of a matrix, R(•) is thecolumn vector space generated by a matrix and dim{V} repre-sents the number of vectors in any basis for a vector space V. Itfollows from the definition of Gpre
n×k given earlier that the rankof Gpre
n×k is k. It can then be readily obtained that
P DFk,n,γ = Pr[rank(GLT
kγ×nGpren×k) �= k]
= Pr[dim{N (GLTkγ×n) ∩ R(Gpre
n×k)} �= 0]. (8)
For convenience, let Wkγ,n,k be the event that dim{N (GLTkγ×n) ∩
R(Gpren×k)} �= 0. Now we need to analyze P DF
k,n,γ = Pr[Wkγ,n,k].
Provided that Gpren×k is the generator matrix of a sys-
R(Gpren×k)} �= 0, denoted by Wkγ,n,k , is equivalent to the event
that at least one column vector from R(Gpren×k) is in N (GLT
kγ×n),
i.e., ∪x∈R(Gpren×k)
GLTkγ×nx = 0, where x is a column vector of
R(Gpren×k). It can be readily shown that
Pr[Wkγ,n,k] = Pr[∪x∈R(Gpre
n×k)GLT
kγ×nx = 0]
≤∑
x∈R(Gpren×k)
Pr[GLT
kγ×nx = 0]. (9)
The column vector space R(Gpren×k) is partitioned into k sub-
space (V1,V2, . . . , Vk) and Vi is the subspace that contains allthe column vectors which are summation of i column vectors ofGpre
n×k . We denote �i as the set of indices of the column vectorsin Vi and there are (k
i ) elements in �i . Let xia be the ath, a ∈ �i
column vector in Vi . It can be shown that
∑x∈R(Gpre
n×k)
Pr[GLTkγ×nx = 0] =
k∑i=1
∑a∈�i
Pr[GLTkγ×nxi
a = 0].
(10)
912 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 64, NO. 3, MARCH 2016
Observe that xia = Ga
n×i 1i where Gan×i is the matrix formed
by i column vectors selected from k column vectors of Gpren×k
and 1i represent a i × 1 all one column vector. Let∣∣xi
a
∣∣ be theweight of column vector xi
a , using the law of total probability,we have
Pr[GLTkγ×nxi
a = 0]
=n∑
r=0
Pr[GLT
kγ×nxia = 0
∣∣∣ ∣∣∣xia
∣∣∣ = r]
Pr[∣∣∣xi
a
∣∣∣ = r]. (11)
Firstly, we need to calculate Pr[∣∣xi
a
∣∣ = r]. Provided Gpre
n×k =[Ik |Pk×(n−k)]T , in the first k entries of Ga
n×i 1i there are i ones.If∣∣xi
a
∣∣ = r , then there are r − i ones in the last n − k entries ofGa
n×i 1i , .i.e, Pa(n−k)×i 1i . Hence we can obtain that
Pr[∣∣∣xi
a
∣∣∣ = r]
= Pr[∣∣∣Pa
(n−k)×i 1i
∣∣∣ = (r − i)], (12)
and i ≤ r ≤ n − k + i . The rows of Pa(n−k)×i , i.e., p j , 1 ≤
j ≤ (n − k), are random binary row vectors, which are gen-erated independently. Each entry of Pa
(n−k)×i is i.i.d. Bernoullirandom variable with parameter η. Therefore, Pr[p j 1i = 0] =Pr[pk,k �= j 1i = 0]. The event that the j th entry in xi
a is zero isequivalent to the event that there are even number of ones inrow vector p j . Thus we have
Pr[p j 1i = 0] = Pr[∣∣p j
∣∣ is even]
=∑
s=0,2, ..., 2⌊
i2
⌋(is)η
s(1 − η)(i−s)
= [(η + (1 − η))i + (−η + (1 − η))i ]
2
= 1 + (1 − 2η)i
2. (13)
There are (n−kr−i ) possible combinations for r − i ones in the last
n − k entries. It follows that
Pr[∣∣∣Pa
(n−k)×i 1i
∣∣∣ = (r − i)]
= (n−kr−i ){Pr[p j 1i = 0]}n−k−r+i
× {1 − Pr[p j 1i = 0]}r−i . (14)
Combining Eq. (12), (13) and (14), we can obtain that
D(i, r) = Pr[∣∣∣xi
a
∣∣∣ = r]
=(
n−kr−i
) [1 + (1 − 2η)i
2
]n−k−r+i
×[
1 − (1 − 2η)i
2
]r−i
. (15)
For xia, xi
b,b �=a ∈ Vi , Pa(n−k)×i and Pb
(n−k)×i have the same prob-ability to form the same matrix formation. So we can obtain that
Pr[∣∣∣Pa
(n−k)×i 1i
∣∣∣ = (r − i)]
= Pr[∣∣∣Pb
(n−k)×i 1i
∣∣∣ = (r − i)],
which in turn leads to the conclusion that Pr[∣∣xi
a
∣∣ = r] =
Pr[∣∣xi
b
∣∣ = r]. Now, we calculate Pr
[GLT
kγ×nxia = 0| ∣∣xi
a
∣∣ = r].
The rows of GLTγ k×n , i.e., gLT
j , 1 ≤ j ≤ kγ , are random binaryrow vectors, which are generated independently. We have
Pr[GLT
kγ×nxia = 0
∣∣∣ ∣∣∣xia
∣∣∣ = r]
={
Pr[gLT
j xia = 0
∣∣∣ ∣∣∣xia
∣∣∣ = r]}kγ
. (16)
The degree of gLTj , i.e. the number of non-zero elements of
gLTj , is chosen according to the pre-defined degree distribution
� = (�1, . . . , �n) and each non-zero element is then placedrandomly and uniformly into gLT
j . It can be readily obtain that
Pr[gLT
j xia = 0
∣∣∣ ∣∣∣xia
∣∣∣ = r]
=n∑
d=1
�d Pr[gLT
j xia = 0
∣∣∣ ∣∣∣xia
∣∣∣ = r,∣∣∣gLT
j
∣∣∣ = d]. (17)
Let rij = (gLT
j1 xia1, gLT
j2 xia2, . . . , gLT
jn xian), where gLT
jk is[gLT
j
]1,k
and xiak is
[xi
a
]k,1. Then, we can obtain that
Pr[gLT
j xia = 0
∣∣∣ ∣∣∣xia
∣∣∣ = r,∣∣∣gLT
j
∣∣∣ = d]
= Pr[∣∣∣ri
j
∣∣∣ is even∣∣∣ ∣∣∣xi
a
∣∣∣ = r,∣∣∣gLT
j
∣∣∣ = d]
=∑
s=0,2, ..., 2⌊
d2
⌋(rs)(
n−rd−s)
(nd)
. (18)
Combining Eq. (17) and (18), we can obtain that
J (r) = Pr[gLT
j xia = 0
∣∣∣ ∣∣∣xia
∣∣∣ = r]
=n∑
d=1
�d
∑s=0,2, ..., 2
⌊d2
⌋(rs)(
n−rd−s)
(nd)
. (19)
Incorporating Eq. (16) into (19), it can be established that
Pr[GLT
kγ×nxia = 0
∣∣∣ ∣∣∣xia
∣∣∣ = r]
= [J (r)]kγ . (20)
We can obtain that Pr[GLTkγ×nxi
a = 0| ∣∣xia
∣∣ = r ] is only deter-
mined by the weight of xia rather than which i column vectors is
chosen from Gpren×k to obtain the summation xi
a . So we can con-clude that Pr[GLT
kγ×nxia = 0] = Pr[GLT
kγ×nxib = 0]. Recall that
there are (ki ) indices in �i . Combining Eq. (15), (20), (11) and
Eq. (10), yields the following results
P DFk,n,γ = Pr[Wkγ,n,k]
≤k∑
i=1
∑a∈�i
Pr[GLT
kγ×nxia = 0
]
=k∑
i=1
(ki
) n−k+i∑r=i
(n−kr−i
)⎡⎢⎣ n∑d=1
�d
∑s=0,2, ..., 2
⌊d2
⌋ (rs
) (n−rd−s
)(
nd
)⎤⎥⎦
kγ
×[
1 + (1 − 2η)i
2
]n−k−r+i [1 − (1 − 2η)i
2
]r−i
, (21)
which proves the theorem.
WANG et al.: PERFORMANCE ANALYSIS OF RAPTOR CODES 913
APPENDIX BPROOF OF THEOREM 3
Similar as that in [7, Lemma 10], by using the Bonferroniinequality [15], we can obtain a lower bound of Pr[Wkγ,n,k] as
P DFk,n,γ = Pr[Wkγ,n,k]
= Pr[∪x∈R(Gpren×k)
GLTkγ×nx = 0]
(a)≥∑
x∈R(Gpren×k)
Pr[GLTkγ×nx = 0]
− 1
2
∑x,y∈R(Gpre
n×k),x �=y
Pr[GLTkγ×nx = 0 ∩ GLT
kγ×ny = 0],
(22)
where x=Gpren×ka, a∈G F(2)k and y = Gpre
n×kb, b ∈ G F(2)k\a.The first term can be calculated by using Theorem 2. Recall thatVi is a subspace that contain all the column vectors which aresummation of i column vectors of Gpre
n×k , �i is the set of indicesof the column vectors in Vi and xi
a represents the ath, a ∈ �i
column vectors in Vi . It can be readily shown that∑x,y∈R(Gpre
n×k),x �=y
Pr[GLTkγ×nx = 0 ∩ GLT
kγ×ny = 0]
=∑
x∈R(Gpren×k)
∑y∈R(Gpre
n×k)\x
Pr[GLTkγ×nx = 0 ∩ GLT
kγ×ny = 0]
=k∑
i=1
∑a∈�i
∑y∈R(Gpre
n×k)\xia
Pr[GLTkγ×nxi
a = 0 ∩ GLTkγ×ny = 0],
(23)
where xia = Gpre
n×ka, |a| = i . Recall that y = Gpren×kb, b ∈
G F(2)k . We define three binary vectors z0, z1, and z2 ∈G F(2)k such that for t = 1, . . . , k, z0(t) = 1 if and only ifa(t) = 1 and b(t) = 1, z1(t) = 1 if and only if a(t) = 1 andb(t) = 0, and z2(t) = 1 if and only if a(t) = 0 and b(t) = 1.Let w0, w1 and w2 be the weights of vectors z0, z1, and z2,respectively. For xi
a , we have z0 + z1 = a and z0 + z2 = b.Hence we can obtain
Pr[GLT
kγ×nxia = 0 ∩ GLT
kγ×ny = 0]
= Pr[GLT
kγ×nGpren×kz0 = GLT
kγ×nGpren×kz1
∩ GLTkγ×nGpre
n×kz1 = GLTkγ×nGpre
n×kz2∣∣∣ |z0| = w0 ∩ |z1| = w1 ∩ |z2| = w2
]. (24)
Let Iz = {iz1, iz2, . . . , izτ } be the set of indices such that t ∈ Izfor z(t) = 1, we can obtain the sets of indices of vectors z0,z1, and z2 as Iz0 , Iz1 and Iz2 . Corresponding to the three setsIz0 , Iz1 and Iz2 , each column of the matrix Gpre
n×k , gprei , can
be divided into four mutually exclusive parts, gz0 , gz1 , gz2 and∪1≤i≤kgpre
i \(gz0 ∪ gz1 ∪ gz2), i.e., gz0 ∩ gz1 = {0}. Let gz0 bethe subset of ∪1≤i≤kgpre
i such that all the elements of this sub-set are selected from ∪1≤i≤kgpre
i according to the indices in set
Iz0 and Gprez0 be the matrix whose columns are elements of gz0 .
The length of gz0 is w0. The same operation is applied to theformation of gz1 and gz2 , in which the elements are selectedaccording to the indices in set Iz1 and Iz2 , and have lengths w1and w2, respectively. Let xw0 = Gpre
z0 1w0 , xw1 = Gprez1 1w1 and
xw2 = Gprez2 1w2 . Equivalently, Eq. (30) can be rewritten as,
Pr[GLT
kγ×nGpren×kz0 = GLT
kγ×nGpren×kz1
∩ GLTkγ×nGpre
n×kz1 = GLTkγ×nGpre
n×kz2∣∣∣ |z0| = w0 ∩ |z1| = w1 ∩ |z2| = w2
]= Pr
[GLT
kγ×nxw0 = GLTkγ×nxw1
∩GLTkγ×nxw1 = GLT
kγ×nxw2]. (25)
Recall that the rows of GLTkγ×n , i.e., gLT
j , 1 ≤ j ≤ kγ , are ran-dom binary row vectors, which are generated independently.We have
Pr[GLT
kγ×nxw0 = GLTkγ×nxw1
∩GLTkγ×nxw1 = GLT
kγ×nxw2]
={
Pr[gLT
j xw0 = gLTj xw1
∩gLTj xw1 = gLT
j xw2]}kγ
. (26)
According to the law of total probability, we have
Pr[gLT
j xw0 = gLTj xw1
∩gLTj xw1 = gLT
j xw2]
=n−k+w0∑r0=w0
n−k+w1∑r1=w1
n−k+w2∑r0=w2
Pr[∣∣xw0
∣∣ = r0]
× Pr[∣∣xw1
∣∣ = r1] Pr[∣∣xw2
∣∣ = r2]
× Pr[gLT
j xw0 = gLTj xw1
∩ gLTj xw1 = gLT
j xw2∣∣∣ ∣∣xw0∣∣ = r0 ∩ ∣∣xw1
∣∣ = r1 ∩ ∣∣xw2∣∣ = r2
]. (27)
For Pr[|xw0 | = r0], this can be calculated by using Eq. (15).Because all algebraic operations are conducted in a binary field,gLT
j xw0 can only be 1 or 0. Eq. (27) can be further written as :
Pr[gLT
j xw0 = gLTj xw1 ∩ gLT
j xw1 = gLTj xw2∣∣∣ ∣∣xw0
∣∣ = r0 ∩ ∣∣xw1∣∣ = r1 ∩ ∣∣xw2
∣∣ = r2
]= Pr
[gLT
j xw0 = 0 ∩ gLTj xw1 = 0 ∩ gLT
j xw2 = 0∣∣∣ ∣∣xw0∣∣ = r0 ∩ ∣∣xw1
∣∣ = r1 ∩ ∣∣xw2∣∣ = r2
]+ Pr
[gLT
j xw0 = 1 ∩ gLTj xw1 = 1 ∩ gLT
j xw2 = 1∣∣∣ ∣∣xw0∣∣ = r0 ∩ ∣∣xw1
∣∣ = r1 ∩ ∣∣xw2∣∣ = r2
]. (28)
914 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 64, NO. 3, MARCH 2016
Recall that xw0 = Gprez0 1w0 , xw1 = Gpre
z1 1w1 , xw2 = Gprez2 1w2 and
the columns of Gprez0 , Gpre
z1 , Gprez2 are mutually exclusive to each
other. So event that |xw0 | = r0 is independent of event that|xw1 | = r1 or |xw2 | = r2 and the event that gLT
j xw0 = 1 is inde-
pendent of event that gLTj xw1 = 1 or gLT
j xw2 = 1. Conditionedon |xw0 | = r0, |xw1 | = r1, |xw2 | = r2, the first part in Eq. (28)can be expressed as:
Pr[gLT
j xw0 = 0 ∩ gLTj xw1 = 0 ∩ gLT
j xw2 = 0
∣∣∣ ∣∣xw0∣∣ = r0 ∩ ∣∣xw1
∣∣ = r1 ∩ ∣∣xw2∣∣ = r2
]
= Pr[gLT
j xw0 = 0∣∣∣ ∣∣xw0
∣∣ = r0
]
× Pr[gLT
j xw1 = 0∣∣∣ ∣∣xw1
∣∣ = r1
]
× Pr[gLT
j xw2 = 0∣∣∣ ∣∣xw2
∣∣ = r2
]. (29)
Based on the previous analysis, we know that Pr[gLTj xw0 =
0∣∣∣ |xw0 | = r0] only relates to parameter r0. Let D(w0, r0) =
Pr[|xw0 | = r0] and J (r0) = Pr[gLTj xw0 = 0| |xw0 | = r0]. For
J (r0), it can be calculated by using Eq. (17) and (18). Basedon the previous analysis„ we know that J (r0) only relates toparameter r0 and D(w0, r0) is affected by parameter r0 and w0.Hence for the same parameters w0, w1 and w2, Eq. (25) has thesame result. Because xi
a �= y, we can obtain that w1 + w2 �= 0and w0 + w2 �= 0. For xi
a , when |z0| = w0, we have w1 = i −w0 and there are (i
w0) possible combinations of z0. For z2, there
are (k−iw2
) possible combination of z2 when |z2| = w2. InsertingEq. (25), (27), (27), (28) and (29) into (24), we can obtain:
When the binomial degree distribution (the expurgated stan-
dard random ensemble) [8], [13], i.e., �d = (nd)
(2n−1), 1 ≤ d ≤ n,
is inserted into Eq. (17), we can obtain that
Pr[gLTj xi
a = 0|∣∣∣xi
a
∣∣∣ = r ]
= (2n − 1)−1n∑
d=1
∑s=0,2, ..., 2
⌊d2
⌋(rs)(
n−rd−s). (32)
Similar to [13, Lemma 2], when the upper limit of the innersummation is changed from 2
⌊ d2
⌋to 2
⌊ n2
⌋, it will not affect
the result of Eq. (32). This is because (n−rd−s) with s > 2
⌊ d2
⌋equals 0.
Pr[gLTj xi
a = 0|∣∣∣xi
a
∣∣∣ = r ]
= (2n − 1)−1n∑
d=1
∑s=0,2, ..., 2� n
2 (rs)(
n−rd−s)
a= (2n − 1)−1∑
s=0,2, ..., 2� n2
(rs)
n∑d=1
(n−rd−s). (33)
The reason why the order of the two summations in Eq. (33)can be exchanged is because the inner summation variable s isnow independent of the outer summation variable d. Note that1 ≤ d ≤ n. Now we want to change the lower limit of the innersummation of Eq. (33) from 1 to 0 without affecting its result.
Pr[gLT
j xia = 0|
∣∣∣xia
∣∣∣ = r]
= (2n − 1)−1{∑
s=0,2, ..., 2� n2
(rs)[
n∑d=0
(n−rd−s) − (n−r
d−s)d=0]}
b= (2n − 1)−1[∑
s=0,2, ..., 2� n2
(rs)
n∑d=0
(n−rd−s) − (r
s)(n−rd−s)s=d=0].
(34)
WANG et al.: PERFORMANCE ANALYSIS OF RAPTOR CODES 915
Step (b) is because the term (n−rd−s)d=0 equals 0 for s �= 0. Hence,
only the case s = 0 needs to be considered. The terms (n−rd−s)
restricts d to s ≤ d ≤ n − r + s, such that
n∑d=0
(n−rd−s) =
n−r+s∑d=s
(n−rd−s) =
n−r∑d=0
(n−rd ) = 2n−r . (35)
Combining this term with the last expression for Pr[gLTj xi
a =0| ∣∣xi
a
∣∣ = r ] yields
[gLT
j xia = 0|
∣∣∣xia
∣∣∣ = r]
= (2n − 1)−1
⎛⎝2n−r
∑s=0,2, ..., 2� n
2 (rs) − 1
⎞⎠
= (2n − 1)−1(2n−r 2r−1 − 1) (36)
= (2n−1 − 1)
(2n − 1), (37)
where we have used identity∑
s even(rs) = 2r−1. We can
observe that Pr[gLTj xi
a = 0| ∣∣xia
∣∣ = r ] is independent from
the weight of xia , hence Pr[GLT
kγ×nxia = 0| ∣∣xi
a
∣∣ = r ] =Pr[GLT
kγ×nxia = 0]. Combining Eq. (16), (37), (10) and (8), we
can obtain that
P DFk,n,γ = Pr[Wkγ,n,k]
= Pr[∪x∈R(Gpre
n×k)GLT
kγ×nx = 0]
≤∑
x∈R(Gpren×k)
Pr[GLT
kγ×nx = 0]
= (2k − 1) Pr[GLT
kγ×nx = 0| |x| = r]
= (2k − 1)((2n−1 − 1)
(2n − 1))kγ . (38)
The proof of Corollary 4 is completed.
APPENDIX DPROOF OF COROLLARY 5
When the binomial degree distribution is inserted intoEq. (13), by using the result of Eq. (37), we can obtain that
J (r0) = Pr[gLTj xw0 = 0| ∣∣xw0
∣∣ = r0]
= (2n−1 − 1)
(2n − 1). (39)
Insert Eq. (39) into Eq. (25), we can obtain that
Pr[GLT
kγ×nGpren×kz0 = GLT
kγ×nGpren×kz1
∩ GLTkγ×nGpre
n×kz1 = GLTkγ×nGpre
n×kz2
| |z0| = w0 ∩ |z1| = w1 ∩ |z2| = w2]
=n−k+w0∑r0=w0
n−k+w1∑r1=w1
n−k+w2∑r0=w2
D(w0, r0)D(w1, r1)D(w2, r2)
× {[ (2n−1 − 1)
(2n − 1)]3 + [1 − (2n−1 − 1)
(2n − 1)]3}kγ
= {[ (2n−1 − 1)
(2n − 1)]3 + [1 − (2n−1 − 1)
(2n − 1)]3}kγ . (40)
Incorporating Eq. (40) into Eq. (30), we can obtain that∑xi
a �=y
Pr[GLTkγ×nxi
a = 0 ∩ GLTkγ×ny = 0]
=i∑
w0=0
∑w1=i−w0
k−i∑w2=0
1(w0 + w2)1(w1 + w2)(iw0
)(k−iw2
)
× {[ (2n−1 − 1)
(2n − 1)]3 + [1 − (2n−1 − 1)
(2n − 1)]3}kγ
= (2k − 2){[ (2n−1 − 1)
(2n − 1)]3 + [1 − (2n−1 − 1)
(2n − 1)]3}kγ . (41)
Combining Eq. (41), (23) and (22), we can obtain that
P DFk,n,γ = Pr[Wkγ,n,k]
≥∑
x∈R(Gpren×k)
Pr[GLTkγ×nx = 0]
− 1
2
∑x,y∈R(Gpre
n×k),x �=y
Pr[GLTkγ×nx = 0 ∩ GLT
kγ×ny = 0]
= (2k − 1)((2n−1 − 1)
(2n − 1))kγ − 1
2
k∑i=1
(ki )(2
k − 2)
× {[ (2n−1 − 1)
(2n − 1)]3 + [1 − (2n−1 − 1)
(2n − 1)]3}kγ
= (2k − 1)
[(2n−1 − 1)
(2n − 1)
]kγ
− (2k − 1)(2k−1 − 1)
×{[
(2n−1 − 1)
(2n − 1)
]3
+[
1 − (2n−1 − 1)
(2n − 1)
]3}kγ
. (42)
The proof of Corollary 5 is completed.
ACKNOWLEDGMENT
The authors would like to thank all the anonymous review-ers for their helpful comments and constructive suggestions toimprove early drafts of this paper.
916 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 64, NO. 3, MARCH 2016
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Peng Wang (S’12) received the B.Sc. degreein applied electronics from Beijing University ofAeronautics and Astronautics, Beijing, China, in2009, and the M.Eng. degree in telecommunicationsfrom the Australian National University, Canberra,A.C.T., Australia, in 2012. He is currently pursu-ing the Ph.D. degree in engineering at the Universityof Sydney, Sydney, N.S.W., Australia. He is alsowith the Sydney Research Laboratory, National ICTAustralia. His research interests include wirelessbroadcast networks, heterogeneous networks, graph
theory and its application in networking, channel/network coding, and 5Gcellular systems.
Guoqiang Mao (S’98–M’02–SM’08) received thePh.D. degree in telecommunications engineeringfrom Edith Cowan University, Mount Lawley, W.A.,Australia, in 2002. He was with the School ofElectrical and Information Engineering, Universityof Sydney, Sydney, N.S.W., Australia, between 2002and 2014. He joined the University of TechnologySydney, Sydney, N.S.W., Australia, in February 2014,as a Professor of wireless networking and the Directorof Center for Real-Time Information Networks. Thecenter is among the largest university research cen-
ters in Australia in the field of wireless communications and networking. He hasauthored more than 150 papers in international conferences and journals, whichhave been cited more than 3500 times. His research interests include intelligenttransport systems, applied graph theory and its applications in telecommunica-tions, wireless sensor networks, wireless localization techniques, and networkperformance analysis. He is an Editor of the IEEE TRANSACTIONS ON
WIRELESS COMMUNICATIONS (since 2014) and the IEEE TRANSACTIONS
ON VEHICULAR TECHNOLOGY (since 2010). He is a Co-Chair of the IEEEIntelligent Transport Systems Society Technical Committee on CommunicationNetworks. He has served as a Chair, a Co-Chair and a TPC member in alarge number of international conferences. He was the recipient of the TopEditor Award for outstanding contributions to the IEEE TRANSACTIONS ON
VEHICULAR TECHNOLOGY in 2011 and 2014.
Zihuai Lin (M’06–SM’10) received the Ph.D. degreein electrical engineering from Chalmers Universityof Technology, Gothenburg, Sweden, in 2006. Priorto that, he has held positions at Ericsson Research,Stockholm, Sweden. Following Ph.D. graduation,he worked as a Research Associate Professor withAalborg University, Aalborg, Denmark, and is cur-rently with the School of Electrical and InformationEngineering, University of Sydney, Sydney, N.S.W.,Australia. His research interests include graph the-ory, source/channel/network coding, coded modula-
tion, MIMO, OFDMA, SC-FDMA, radio resource management, cooperativecommunications, small-cell networks, and 5G cellular systems.
Ming Ding (M’12) received the B.S. and M.S.degrees (with first class Hons.) in electronics engi-neering, and the Doctor of Philosophy (Ph.D.) degreein signal and information processing from ShanghaiJiao Tong University (SJTU), Shanghai, China, in2004, 2007, and 2011, respectively. From September2007 to September 2011, he pursued the Ph.D.degree at SJTU while at the same time work-ing as a Researcher/Senior Researcher with SharpLaboratories of China (SLC). After achieving thePh.D. degree, he continued working with SLC as a
Senior Researcher/Principal Researcher until September 2014, when he joinedNational Information and Communications Technology Australia (NICTA).In September 2015, the Commonwealth Scientific and Industrial ResearchOrganization (CSIRO) and NICTA joined forces to create Data61, wherehe continued as a Researcher in this new R&D center located in Sydney,N.S.W., Australia. He has authored more than 30 papers in IEEE journalsand conferences, all in recognized venues, and about 20 3GPP standardiza-tion contributions, as well as a book Multipoint Cooperative CommunicationSystems: Theory and Applications (Springer). Also, as the first inventor, heholds 15 CN, seven JP, three U.S., two KR patents and has coauthored over100 patent applications on 4G/5G technologies. His research interests includeB3G, 4G, and 5G wireless communication networks, synchronization, MIMOtechnology, co-operative communications, heterogeneous networks, device-to-device communications, and modeling of wireless communication systems.He served as the Algorithm Design Director and Programming Director for asystem-level simulator of future telecommunication networks in SLC for morethan seven years. He has been Guest Editor/Co-Chair/TPC member of severalIEEE top-tier journals/conferences, e.g., the IEEE JOURNAL ON SELECTED
AREAS IN COMMUNICATIONS, the IEEE Communications Magazine, and theIEEE GLOBECOM Workshops. For his inventions and publications, he wasthe recipient of the President’s Award of SLC in 2012, and served as one of thekey members in the 4G/5G standardization team when it was awarded in 2014as Sharp Company Best Team: LTE 1014 Standardization Patent Portfolio.
Weifa Liang (M’99–SM’01) received the B.Sc.degree from Wuhan University, Wuhan, China, in1984, the M.E. degree from the University of Scienceand Technology of China, Hefei, China, in 1989, andthe Ph.D. degree from Australian National University,Canberra, A.C.T., Australia, in 1998, all in computerscience. He is currently a Full Professor with theResearch School of Computer Science, AustralianNational University. His research interests includedesign and analysis of routing protocols for software-defined networks (SDNs), wireless ad hoc and sensor
networks, cloud computing and mobile cloud computing, design and analysisof parallel and distributed algorithms, combinatorial optimization, and graphtheory.
WANG et al.: PERFORMANCE ANALYSIS OF RAPTOR CODES 917
Xiaohu Ge (M’09–SM’11) received the Ph.D. degreein communication and information engineering fromHuazhong University of Science and Technology(HUST), Wuhan, China, in 2003. He is a FullProfessor with the School of Electronic Informationand Communications, HUST. He is an AdjunctProfessor with the University of Technology Sydney(UTS), Sydney, N.S.W., Australia. He has worked atHUST since November 2005. Prior to that, he workedas a Researcher with Ajou University, Suwon, SouthKorea and Politecnico Di Torino, Turin, Italy, from
January 2004 to October 2005. He was a Visiting Researcher at Heriot-WattUniversity, Edinburgh, U.K., from June to August 2010. He has authoredabout 100 papers in refereed journals and conference proceedings and hasbeen granted about 15 patents in China. His research interests include mobilecommunications, traffic modeling in wireless networks, green communications,and interference modeling in wireless communications. He is leading severalprojects funded by NSFC, China MOST, and industries. He is taking partin several international joint projects, such as the EU FP7-PEOPLE-IRSES:project acronym S2EuNet (Grant no. 247083), project acronym WiNDOW(Grant no. 318992), and project acronym CROWN (Grant no. 610524). Heis a Senior Member of the China Institute of Communications, and a mem-ber of the National Natural Science Foundation of China and the ChineseMinistry of Science and Technology Peer Review College. He has been activelyinvolved in organizing more the ten international conferences since 2005.He served as the General Chair for the 2015 IEEE International Conferenceon Green Computing and Communications (IEEE GreenCom). He servesas an Associate Editor for the IEEE ACCESS, Wireless Communicationsand Mobile Computing Journal (Wiley), and the International Journal ofCommunication Systems (Wiley). Moreover, he served as the Guest Editorfor the IEEE Communications Magazine Special Issue on 5G WirelessCommunication Systems. He was the recipient of the Best Paper Award fromIEEE GLOBECOM 2010.
Zhiyun Lin (SM’10) received the bachelor’s degreein electrical engineering from Yanshan University,Qinhuangdao, China, in 1998, the master’s degreein electrical engineering from Zhejiang University,Hangzhou, China, in 2001, and the Ph.D. degreein electrical and computer engineering from theUniversity of Toronto, Toronto, ON, Canada, in 2005.He was a Postdoctoral Research Associate with theDepartment of Electrical and Computer Engineering,University of Toronto, from 2005 to 2007. He joinedthe College of Electrical Engineering, Zhejiang
University, in 2007. Currently, he is a Professor of systems control in thesame department. He is also affiliated with the State Key Laboratory ofIndustrial Control Technology, Zhejiang University. He held Visiting Professorpositions at several universities including Australian National University,Canberra, A.C.T., Australia, University of Cagliari, Cagliari, Italy, Universityof Newcastle, Callaghan, N.S.W., Australia, University of Technology Sydney,Sydney, Australia, and Yale University, New Haven, CT, USA. His researchinterests include distributed control, estimation and optimization, coordinatedand cooperative control of multiagent systems, hybrid and switched systemtheory, and locomotion control of biped robots. He is currently an AssociateEditor for Hybrid Systems: Nonlinear Analysis and the International Journal ofWireless and Mobile Networking.
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