Page 1

Protograph-based LDPC Convolutional Codes

for Correlated Erasure Channels

Aravind R. Iyengar∗, Marco Papaleo†, Gianluigi Liva‡,

Paul H. Siegel∗, Jack K. Wolf∗, Giovanni E. Corazza†

∗University of California, San Diego, La Jolla CA 92093, USA

email: {aravind,psiegel,jwolf}@ucsd.edu

†University of Bologna, DEIS-ARCES, Viale Risorgimento, 2 - 40136 Bologna, Italy

email: {mpapaleo,gecorazza}@arces.unibo.it

‡Institute of Communications and Navigation - DLR, Postfach 1116, 82230 Wessling, Germany

email: Gianluigi.Liva@dlr.de

Abstract—We consider terminated LDPC convolutional codes

(LDPC-CC) constructed from protographs and explore the per-

formance of these codes on correlated erasure channels including

a single-burst channel (SBC) and Gilbert-Elliott channel (GEC).

We consider code performance with a latency-constrained message

passing decoder and the belief propagation decoder. We give

theoretical bounds on the code efficiency over the SBC and describe

a construction that achieves this bound. We show that the designed

codes with belief propagation (BP) decoding perform as well as

the regular LDPC-CCs presented in the literature on the binary

erasure channel (BEC) and the GEC, while achieving significant

gains on the SBC. In the case of windowed decoding, our codes

perform much better than the best known regular LDPC-CCs over

the BEC and the GEC, with very low decoding latencies.

I. INTRODUCTION

LDPC ConvolutionalCodes (LDPC-CC) were first introduced

in [1]. In [2] protograph-based terminated LDPC-CCs which

can approach capacity over the binary erasure channel (BEC)

were introduced. In this paper we are interested in evaluating

the performance of this family of codes over correlated erasure

channels. The analysis is carried out considering both the

classical belief propagation (BP) decoder and the windowed

decoder (WD) introduced in [3]. The motivation behind the

adoption of a windowed decoder is the possibility to trade-off

erasure correcting capability for reduced decoding latency.

We devote our attention to a subclass of asymptotically

rate-1/2, regular LDPC-CCs and make some key observations

related to the protograph structure. We analyze the performance

on erasure channels with memory, specifically the single-burst

channel (SBC) and the Gilbert-Elliott channel (GEC). Our aim

is to achieve the best possible performance on the SBC and the

GEC, while maintaining good performance on the BEC. We

give bounds on the maximal size of a single burst of erasures

– the maximum tolerable burst length (MTBL) – that the BP

decoder can recover on the SBC, and we propose a construction

rule that allows us to design codes with MTBL value guaranteed

to be linearly increasing as a function of the memory msof the

convolutional code. We show that an (n,k) terminated LDPC-

CC belonging to the class of asymptotically regular (J,2J)

codes [2] can never reach the MTBL performance of an (n,k)

**A. R. Iyengar and M. Papaleo contributed equally to this work.

The work of A. R. Iyengar is supported by the National Science Foundation

under Grant CCF-0829865.

maximum distance separable (MDS) code. Finally we analyze

the performance of the proposed codes over the GEC through

numerical simulations.

This paper is organized as follows. In Section II we introduce

the LDPC-CC terminology. We will introduce the two ensem-

bles and codes used as continuing examples throughout the

paper. In Section III, we describe the two decoding algorithms

considered in the analysis. Section IV presents the erasure

channels under consideration and the relevant figures of merit

for each channel. We then present the results obtained through

numerical simulation in Section V. We finally summarize our

findings in Section VI.

II. PROTOGRAPH-BASED TERMINATED LDPC-CC

A protograph [4] is a relatively small bipartite graph from

which a larger graph can be obtained by a copy-and-permute

procedure – the protograph is copied M times, and then the

edges of the individual replicas are permuted among the M

replicas to obtain a single, large bipartite graph. Suppose the

protograph possesses NP variable nodes (VNs) and MP check

nodes (CNs), with degrees Jj,j = 1,...,Np, and Ki,i =

1,...,Mp, respectively. Then the derived graph will consist of

n = NPM VNs and m = MPM CNs. The nodes of the

protograph are labeled, so that if the VN Vj is connected to

the CN Ci in the protograph, then Vj in a replica can only

connect to one of the M replicated Ci’s.

Protographs can be represented by means of an MP× NP

bi-adjacency matrix B, called the base matrix where the entry

Bm,n represents the number of edges between CN Cm and

VN Vn (a non-negative integer, i.e., multiple parallel edges –

multiedges – are permitted.). The copy-and-permuteoperation is

realized by replacing each edge (multiedge)in the base matrix B

with a size-M permutation matrix (the sum of as many distinct

size-M permutation matrices as the number of multiedges,

respectively). For different values of M, different blocklengths

of the derived Tanner graph can be achieved keeping the original

graph structure imposed by the protograph. This means that

the density evolution analysis can be performed within the

protograph instead of the unstructured ensemble.Furthermore,

protographs impose a structure on the derived graph, which

facilitates the design of fast decoders and efficient encoders, as

well as a refined control on the derived graph edge connections.

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Analogous to LDPC block codes, LDPC-CC can also be

derived by a protograph expansion. The parity-check matrices of

these codes are composed of blocks of size-M square matrices.

Let a be the greatest common divisor (gcd) of J and K, the VN

and CN degrees respectively. Then there exist positive integers

J′and K′such that J = aJ′, K = aK′, and gcd(J′,K′) = 1.

Starting from these parameters, it is possible to define the

ensemble CP(J,K) of LDPC-CC. If we start the convolutional

code at time instant t = 1 and terminate it after L instants, we

obtain a block code, described by the base matrix1B[1,L]as

in (1). The protograph of the terminated code has Np= LK′

VNs and Mp= (L + ms)J′CNs,

B[1,L]=

B0(1)

B1(1)

...

B0(2)

B1(2)

...

...

...

...

...

Bms(1)

B0(L)

Bms(2)

B1(L)

...

Bms(L)

(1)

where ms = a − 1 is the memory of the LDPC-CC and

Bi(t),i = 0,..,ms are J′× K′identical component base

matrices with all entries equal to one. The rate of the LDPC-CC

is therefore

?J′

RL= 1 −

?L + ms

LK′= 1 −

?

1 +ms

L

?

(1 − R)

(2)

where R = 1 −

Note that RL → R and the LDPC-CC has a regular degree

distribution [2] when L → ∞. The construction above was

proposed in [5] and allows the construction of some (J,K)

regular LDPC-CCs. However, not all (J,K) regular LDPC-CC

can be constructed, e.g. with this construction ms becomes

zero if J and K are relatively prime and consequently the

resulting code is not convolutional. In [2], the authors addressed

this problem by proposing a construction rule based on edge

spreading.

In the following we analyze the case of (J,2J) regular

LDPC-CC. Our aim is to analyze this subclass of asymptotically

rate-1/2 codes and find protograph structures achieving good

performances in both memoryless and correlated erasure chan-

nels. Since in this case we always have J′= 1 and K′= 2, the

component base matrices Bi,i = 0,...,msare 1 × 2 matrices.

As a consequence, we can use two polynomials to represent

the entries of the terminated protograph B. The first column of

the protograph B would comprise the first elements of the Bi

matrices p0,p1,··· ,pms, with which we associate a polynomial

p(x) = p0+p1x+···+pmsxmsof degree at most ms. Similarly,

the second column of B comprises the second elements of

the Bimatrices, q0,q1,··· ,qmsand with these we associate a

polynomial q(x) = q0+q1+···+qmsxms, also of degree at most

ms. Then, the (2i+1)thcolumn of B can be associated with the

polynomial xip(x), and the (2i + 2)thcolumn the polynomial

xiq(x). We will use the polynomial of a column and its index

J′

K′ is the rate of the non-terminated code.

1For a time-invariant terminated LDPC-CC, Bi(t) = Bi∀ t = 1,2,··· ,L,

and i = 0,1,··· ,ms.

interchangeably, e.g. when we say “choosing the polynomial

xip(x),” we mean that we choose the (2i + 1)thcolumn of B.

Similarly, by “summations of polynomials p(x) and q(x),” we

mean the collection of the corresponding columns of B. Note

that for the exnsembles to be (J,2J) regular, we will further

have the constraint p(1) = q(1) = J.

The terminated ensemble can now be defined by specifying

the parameters L and Bi, ∀ i = 0,...,ms, or equivalently, by

giving L, p(x) and q(x). Note that our construction is similar

to the one in [5] except that we do not require that ms= a−1.

We further disregard the requirement that the Bi matrices are

identical and have only ones.

Ensembles and Codes

Throughout the paper, we consider the performance of two

code ensembles on a multitude of channels. The first code

ensemble is a regular LDPC-CC ensemble constructed in [2],

and the second is constructed to perform well with windowed

decoding [3]. For fair comparison between the two ensembles,

i.e., for comparing two ensembles of the same rate and same

asymptotic degree distribution, we keep the parameters ms, J

and L the same. Here we choose ms= 2, J = 3, K = 2J = 6

and L = 20, so that the rate RL = 0.45. Note that both the

code ensembles are (3,6) regular LDPC-CC ensembles, and we

refer to them henceforth as ensembles A and B. Ensemble A

is defined by

B0= [11], B1= [11], B2= [11]

(3)

and ensemble B by

B0= [22], B1= [0 1], B2= [1 0].

(4)

Equivalently, the ensembles A and B can be defined using the

polynomials pA(x) = qA(x) = 1 + x + x2and pB(x) = 2 +

x2,qB(x) = 2 + x respectively. Let us denote the protographs

defined by these submatrices (or polynomials) as BAand BB

respectively. While dealing with finite-length codes, we keep

the parameter M (and consequently, the blocklength n) also the

same for the two codes. In such finite length comparisons, the

code protographs have been expanded by means of circulant

permutation matrices2. For this purpose, a girth optimization

technique based on [6], [7] has been adopted [8]. The codes

constructed from ensembles A and B are denoted A and B

respectively.

III. DECODERS

We consider two particular iterative decoders described be-

low.

A. Belief Propagation (BP) Decoder

Since we are interested in terminated LDPC-CC, decoding

can be performed in a manner similar to the case of an LDPC

block code, i.e. each frame carrying a codeword obtained

through the termination can be decoded with the sum-product

algorithm (SPA). Note that the BP decoder has to wait for

the entire codeword to be received to start decoding, and this

implies a large latency and buffer requirement at the decoder.

2It follows that the resulting terminated LDPC-CCs can be seen also as quasi-

cyclic block codes.

Page 3

B. Windowed Decoder (WD)

It is desirable to avoid the large latency of the BP decoder de-

scribed above in many applications, e.g. [9]. The convolutional

structure of the code imposes a maximum distance constraint

on the VNs connected to the same parity-check equations –

two VNs that are at least K′(ms+ 1) columns apart (in the

protograph) cannot be involved in the same equation. This

characteristic can be exploited in order to perform continuous

decoding of the received stream through a window that slides

along the bit sequence.

Consider a terminated parity-check matrix H built from a

base protograph B and a sliding window that covers3WJ′

rows and WK′columns of the protograph B. At the first time

instant, t = 1, the decoder performs message passing iterations

over the equations within the sliding window with the aim of

decoding all of the first MK′symbols in the window, called the

targeted symbols. The sliding window shifts down J′rows and

right K′columns in B after a maximum number of message

passing iterations have been performed, or when all targeted

symbols within the window have been recovered, and continues

decoding at the new position at the second time instant, t = 2.

This is summarized in Figure 1. The latency of the WD is

Fig. 1.

regular LDPC-CC with L = 16 and M = 512 at time instant t = 2.

Windowed decoder for sliding window size W = 4, for a (3,6)

therefore a fraction w =W

the BP decoder. It is clear that the performance achievable by

the WD improves with increase in window size (latency) as

more equations are considered. In [3], this windowed decoder

was analyzed for the BEC.

Lof the latency in comparison with

IV. CHANNELS AND FIGURES OF MERIT

A. Binary erasure channel (BEC)

The BEC is one over which each symbol of the transmitted

codeword is erased independently with a probability ε. The

performance of LDPC-CCs on the BEC has been extensively

studied, e.g. [1], [2], [3]. In the asymptotic setting, the figures

of merit for the two decoders discussed in Section III are the BP

threshold ε∗

were studied in [3].

For the finite length analysis, we will compare the perfor-

mance with the Singleton bound, which represents the perfor-

mance achievable by an (n,k) MDS code. This bound for the

BPand the windowed threshold ε∗

W, both of which

3When the sliding window goes beyond the protograph, it is assumed to be

bounded by the boundaries of the protograph.

codeword error rate (CER) PCW for the BEC can be expressed

as

n

?

j=n−k+1

For ensemble4B, the trade-off between latency and perfor-

mance in terms of ε∗

PCW=

?n

j

?

εj(1 − ε)n−j

(5)

Wis shown in Figure 2. Also shown is

0.150.250.35 0.450.55 0.650.75

0.42

0.44

0.46

0.48

0.5

w = W/L (Latency)

ε∗

W

Capacity

Ensemble B

3456789101112131415

W (Window Size)

Fig. 2.Trade-off between latency and performance for ensemble B.

the Shannon limit for windowed decoding – we give a limit

for windowed decoding, rather than for the ensemble because

for window sizes ms < W ≤ L, the rate of the code within

the window is 0.5, although the rate for the entire code is

RL= 0.45. The reduced limit achievable with WD (ε∗= 0.5

rather than ε∗= 0.55) is perhaps the cost for reduced latency.

Finite length performance of codes constructed from ensembles

A and B over the BEC is evaluated in Section V-A.

B. Single-burst channel (SBC)

We now consider the performance of the ensembles over a

channel that inserts a single burst of erasures within a codeword.

In this case, we are interested in the MTBL [10], defined as

the maximal length of a single burst of erasures that can be

recovered by the BP decoder, and denoted ∆max. Let us define

a protograph stopping set to be a subset SBof the VNs of the

protograph B whose neighbouring CNs are connected at least

twice to SB. Let us denote the minimum number of consecutive

columns of the protograph that contains a protograph stopping

set by sB

simple bounds for the MTBL based on sB

later.

For ensemble A, the first two columns of BAform a stopping

set. This is clear from the highlighted columns below

min. This parameter is of interest because we can give

min, as will be shown

11

0

1

1

1

0

...

0

1

1

1

0

...

0

0

1

1

1

...

0

0

1

1

1

...

···

···

···

···

···

...

1

1

1

1

0

0

...

0

0

...

.

4Ensemble A has zero windowed threshold [3].

Page 4

Therefore, sBA

graph stopping set, sBA

For ensemble B, the highlighted columns of BB in the

following matrix form a stopping set.

min≤ 2. Since no single column forms a proto-

min≥ 2, implying sBA

min= 2.

2

2

1

0

0

0

...

0

2

0

1

0

...

0

0

0

2

0

1

...

0

0

2

1

0

...

···

···

···

···

···

...

0

1

2

1

00

0

...

0

...

.

Thus, sBB

constitute a protograph stopping set, it is clear that sBB

so that sBB

a protograph that achieves an sB

ms.

Proposition 1 (Achievability of sB

regular LDPC-CC with memory msand J > 2, we can achieve

sB

Proof: Consider the ensemble given by p(x) = (J − 1) +

xmsand q(x) = (J − 1) + x, and the polynomial r(x) =

p(x)a(x)+q(x)b(x) that represents all (22ms−1−1) non-trivial

subsets of the first (2ms− 1) columns of B, for all choices

of polynomials a(x) and b(x) with coefficients in {0,1} and

maximal degrees (ms− 1) and (ms− 2) respectively, except

the choice of all zero coefficients. It is not hard to see that

r(x) is a monic polynomial of degree (ms+ i1) where i1 is

the degree of a(x) when a(x) ?= 0, and a monic polynomial of

degree (1+i2) where i2is the degree of b(x) when a(x) = 0.

Thus, sB

min≤ 4. As no three consecutive columns of BB

min> 3,

min= 4. In fact, we can give a general construction of

minthat increases linearly with

min= 2ms): For a (J,2J)

min= 2ms.

min> 2ms− 1. Finally, notice that

p(x) + xms−1q(x) = (J − 1) + (J − 1)xms−1+ 2xms,

with all coefficients strictly larger than 1. p(x) corresponds to

the first column of the protographand xms−1q(x) to the (2ms)th

column. Thus, we have sB

Note that ensemble B in (4) is an example of the above

construction. We bring to the reader’s attention here that other

constructions that achieve sB

the construction in Proposition 1 is not unique in achieving the

mentioned sB

however, is that if msis increased to get larger sB

rate in (2) decreases linearly. We prove in the following that for

any protograph, we cannot have sB

construction achieves the maximal sB

Proposition 2 (Converse): For a (J,2J) regular LDPC-CC

with memory msand J > 2, sB

Proof: We will prove this by showing that in any pro-

tograph with K = 2J and memory ms, we can find 2ms

consecutive columns that will always contain a protograph

stopping set. We know that since the code has a memory ms,

at least one of the polynomials defining the protograph of the

ensemble – p(x) and q(x) – is of degree ms. Without loss of

generality, let us assume the degree of p(x) to be ms, and that of

q(x) be j, 1 ≤ j ≤ ms. Also, let the first non-zero coefficient5

in p(x) be the coefficient of xip, 0 ≤ ip≤ ms− 1 and that in

q(x) be the coefficient of xiq, 0 ≤ iq≤ j − 1. Then consider

min= 2ms.

min= 2ms are also possible, i.e.

min. A drawback in this protograph construction,

min, the code

min> 2msso that the above

min.

min≤ 2ms.

5We disallow ip = ms or iq = j because in this case a single column of

B is a protograph stopping set, i.e. sB

min= 1 < 2ms.

2ms consecutive columns of the protograph B starting with

a column corresponding to the polynomial p(x). Consider the

subset of columns of B corresponding to the polynomial r(x) =

p(x)(xiq+xiq+1+···+xj−1)+q(x)(xip+xip+1+···+xms−1).

We claim that this is a protograph stopping set. To see this,

consider ˆ p(x) = xip+ xmsand ˆ q(x) = xiq+ xj. Note that

p(x) and ˆ p(x) (resp. q(x) and ˆ q(x)) have the same minimum

and maximum degrees. Thus, p(x) differs from ˆ p(x) in that

its coefficients of xipthrough xmsare possibly larger than the

corresponding coefficients of ˆ p(x). Similarly for q(x) and ˆ q(x).

With ˆ p(x) and ˆ q(x) as the column polynomials defining B, the

columns corresponding to the above subset can be associated

with the polynomial ˆ r(x) = ˆ p(x)(xiq+ xiq+1+ ··· + xj−1) +

ˆ q(x)(xip+ xip+1+ ··· + xms−1), which can be written as

ˆ r(x) = (xip+ xms)(xiq+ xiq+1+ ··· + xj−1)+

(xiq+ xj)(xip+ xip+1+ ··· + xms−1)

= 2(xip+iq+ ··· + xip+j−1+ xip+j+ ··· + xms+j−1).

Note that ˆ r(x) has all coefficients more than 1 and that the

minimum and maximum degrees of ˆ r(x) are (ip+ iq) and

(ms+j−1) respectively. These degrees are the same as those of

r(x), and thus r(x) can only differ from ˆ r(x) in having larger

coefficients. Therefore, r(x) also has all coefficients greater

than 1. This shows that the chosen subset of columns form a

protograph stopping set. Since all these columns are contained

in a maximum of 2msconsecutive columns of the protograph

B, sB

We now show the relation between the parameters ∆maxand

sB

Lemma 1: ∆max≤ MsB

Proof: Clearly, all of the MsB

check matrix corresponding to the sB

of B that contain a protograph-stopping set must contain a

stopping set of the parity-check matrix. Therefore, if all symbols

corresponding to these columns are erased, they cannot be

retrieved, implying ∆max≤ MsB

Corollary 1: A (J,2J) regular LDPC-CC can never achieve

the MTBL of an MDS code.

Proof: From the Singleton bound, we have ∆max≤ n −

k = (L + ms)M, assuming that the parity-check matrix is full

min≤ 2ms.

min, assuming that sB

min≥ 2.

min− 1.

mincolumns of the parity-

minconsecutive columns

min− 1.

rank. From Lemma 1 we have ∆max ≤ MsB

2msM − 1. Since we require ms≤ L for a non-negative code

rate in (2), ∆max≤ 2msM − 1 ≤ (L + ms)M − 1 < (L +

ms)M, which shows that the MTBL of an MDS code can never

be achieved.

Despite the above discouraging result, we can guarantee an

MTBL that linearly increases with ms.

Lemma 2: ∆max≥ M(sB

Proof: From the definition of sB

of the two extreme columns is completely known, all other

symbols can be recovered, for otherwise the remaining columns

will have to contain a protograph stopping set, violating the

minimality of the stopping set distance sB

columns are pivots of the stopping set [11]). The largest solid

burst that is guaranteed to have at least one of the extreme

columns completely known is of length M(sB

Therefore, ∆max≥ M(sB

min− 1

Prop. 2

≤

min− 2) + 1.

min, it is clear that if one

min(The two extreme

min− 2) + 1.

min− 2) + 1.

Page 5

The actual values of the MTBL for the codes A and B are

given in Section V-B.

C. Gilbert-Elliott Channel (GEC)

Finally, we evaluate the performances of codes A and B on

the GEC [12], [13]. In this model, the channel is either in a

“good” state G, where we assume the erasure probability is 0,

or in a “bad” state B, in which we will assume the erasure

probability is 1. The state process of the channel is a first-order

Markov process with the transition probabilities P{B → G} =

g and P{G → B} = b. With these parameters, we can easily

deduce [14] that the average erasure rate ε = P{B} =

and the average burst length ∆ =

GEC to be parameterized6by the pair (ε,∆) and evaluate the

performance of codes A and B in Section V-C.

The figure of merit in this case is the CER in comparison with

that of an (n,k) MDS code, which is given by the Singleton

bound [13].

b

b+g

1

g. We will consider the

V. NUMERICAL RESULTS

In this section we analyze the performance achievable by

codes A and B over the channels described in Section IV.

The terminated protographs describing the two ensembles have

been expanded by a factor M = 512, resulting in codes with

blocklength n = LMK′= 20480 and rate 0.45. The codes are

constructed such that the girth is 12 for both codes A and B.

A. BEC

In Figures 3 and 4, the symbol error rate (SER) and the CER

performance are depicted for codes A and B.Also shown

0.10.20.30.40.50.6

10

−4

10

−3

10

−2

10

−1

10

0

ε

SER

A − W=3

B − W=3

A − W=5

B − W=5

A − W=10

B − W=10

A − BP

B − BP

Fig. 3.SER performance for BP and Windowed Decoding over BEC.

in Figure 3 (as ticks on the ε-axis) are the ε∗

the ensembles, which are 0.4882 and 0.4881 respectively for

ensembles A and B, and the ε∗

Figure 2). As can be observed, code B clearly outperforms code

A for small window sizes (W = 3,5), confirming the effective-

ness of the proposed design rule for windowed decoding. For

BPvalues for

Wvalues for ensemble B (see

6Note that there is a one-to-one correspondence between the two pairs (b,g)

and (ε,∆).

0.10.20.30.4 0.5 0.6

10

−4

10

−3

10

−2

10

−1

10

0

ε

CER

A − W=3

B − W=3

A − W=5

B − W=5

A − W=10

B − W=10

A − BP

B − BP

SB

Fig. 4.

Singleton bound (SB).

CER performance for BP and Windowed Decoding over BEC with

larger window sizes (W = 10), the two codes show almost the

same performance, with code A performing slightly better. Also

shown for comparison in Figure 4 is the CER for an MDS code

(marked SB for Singleton bound).

B. SBC

The MTBL for codes A and B was computed using an

exhaustive search algorithm, by feeding the decoder with a solid

burst of erasures and testing all the possible locations of the

burst. The MTBL for the codes we considered were 1023 and

1751 for codes A and B, respectively. Note that for code A, the

MTBL ∆max= 1023 = 2M−1, i.e., code A achieves the upper

bound from Lemma 1. More importantly, the maximum possible

∆maxwas achievable while maintaining good performance over

the BEC with the BP decoder. However, the MTBL for code

B, ∆max = 1751 < 2047 = 4M − 1, is much smaller

than the corresponding bound from Lemma 1. In this case,

whereas other code constructions with ∆maxup to 2045 were

possible, a trade-off between the BEC performance and MTBL

was observed, i.e. the code that achieved ∆max = 2045 was

found to be much worse over the BEC than both codes A and

B considered here. Such a trade-off has also been observed

by others, e.g. [15]. Nevertheless, our code design does give

a large increase in MTBL (> 70%) without increasing the

memory ms, i.e. without any sacrifice in terms of code rate.

The MTBL achieved as a fraction of the maximum possible

MTBL (achieved by MDS codes) ∆max/(n − k) was roughly

9.1% and 15.5% for codes A and B respectively.

C. GEC

In Figures 5, 6 and 7 we show the CER performance obtained

for codes A and B over GEC channels with ∆ = 10,50,100

respectively, and ε ∈ [0.1,0.6]. As can be seen from the figures,

for W = 3, code B always outperforms code A, while for

W = 5 there is no such gain when ∆ = 100. However, for

W = 10 and for BP decoding, code A slightly outperforms

code B, similar to the trend observed for the BEC.

Note that the code B outperforms A for small ε when the

average burst length ∆ = 100 for large window sizes and for

Page 6

0.10.20.3 0.4 0.50.6

10

−4

10

−3

10

−2

10

−1

10

0

ε

CER

A − W=3

B − W=3

A − W=5

B − W=5

A − W=10

B − W=10

A − FULL

B − FULL

SB

Fig. 5.CER Performance on GEC with ∆ = 10 with Singleton bound (SB).

0.10.20.30.40.50.6

10

−4

10

−3

10

−2

10

−1

10

0

ε

CER

A − W=3

B − W=3

A − W=5

B − W=5

A − W=10

B − W=10

A − FULL

B − FULL

SB

Fig. 6. CER Performance on GEC with ∆ = 50 with Singleton bound (SB).

0.10.20.30.4 0.50.6

10

−4

10

−3

10

−2

10

−1

10

0

ε

CER

A − W=3

B − W=3

A − W=5

B − W=5

A − W=10

B − W=10

A − FULL

B − FULL

SB

Fig. 7.CER Performance on GEC with ∆ = 100 with Singleton bound (SB).

BP decoding. This can be explained because in this regime, the

probability of a burst is small but the average burst length is

large. Therefore, when a burst occurs, it is likely to resemble

a single burst in a codeword, i.e. the channel in this regime is

similar to the SBC in which case we know that the code B is

stronger than A. Also note the significant gap between the BP

decoder performance and the Singleton bound, suggesting that

unlike some moderate length LDPC block codes [16], LDPC-

CCs are far from achieving MDS performance.

VI. CONCLUSION

In this paper we have analyzed the performance of

(J,2J) protograph-basedterminated LDPC convolutional codes

(LDPC-CC) on correlated erasure channels. We have derived

lower and upper bounds for the maximum tolerable burst length

(MTBL) and proposed a construction rule that allows us to

design protographs whose MTBL increases linearly with the

memory ms. These protographs also have good performance

under BP and windowed decoding over the BEC. We proved

that (J,2J) regular LDPC-CC cannot achieve the MTBL of

MDS codes. Finally, we evaluated the performance of the

proposed LDPC-CC on the GEC through numerical simulations,

showing that the convolutional structure leads to far-from-

optimal performance with respect to the Singleton bound.

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