978-1-4673-5952-8/13/$31.00 ©2013 IEEE

Study on FEC Schemes for Optical Communication

Systems

Prabhmandeep Kaur

Department of Electrical Engineering

I. I. T. Delhi

New Delhi, India, 110016

prabhmandhillon16@gmail.com

Divya Dhawan1, V. K. Jain, Subrat Kar

1Department of Electronics & Electrical Comm. Engg.,

PEC University of Technology

Chandigarh

India, 160012

Abstract—In this paper third generation forward-error-

correction (FEC) schemes based on Projective Geometry (PG)

and block-circulant Low Density Parity Check Codes (LDPC) are

compared with the first and second generation FEC schemes. The

low complexity of both encoder and decoder and better bit error

rate (BER) performance makes LDPC codes an excellent choice

for optical communication systems/networks.

Keywords- Block-Circulant; Forward Error Correction (FEC);

Low Density Parity Check (LDPC) codes; Projective Geometry

(PG); optical communication.

I. INTRODUCTION

High-speed FEC architectures are becoming important in

optical communications due to the high demands being placed

on optical systems with the rapid growth of internet and

media-centric services. This has led to significant research in

this area during the last few years.

The FEC schemes for fiber optic communication systems

are classified into three categories [1]. First generation FEC

schemes consist of the (255, 239) Reed-Solomon (RS) code,

with only 6.7% overhead, standardized by the ITU for long-

haul submarine transmissions [2]. Optical networking

interface devices were developed consisting of four parallel

RS codecs, each operating at 2.5 Gb/s [2, 3]. The demand for

powerful FEC codes with high code rates and large coding

gains was further augmented as a number of high speed-long

haul optical communication systems began to be developed

with bit rates exceeding 10 Gb/s. The second generation

schemes saw the introduction of concatenated codes with

higher coding gains such as RS (223, 207) and RS (255, 223),

with 23% overhead. The third generation FEC schemes are

now seen as a promising way to reduce costs in high-capacity

transport systems by relaxing the requirements on expensive

optical devices as no major shifts in the system hardware are

required. They are based on soft decision decoding and consist

of the iteratively decodable codes such as turbo codes [4] and

low density parity check (LDPC) codes. It is shown that

LDPC codes can surpass the error correcting capabilities of

turbo codes [5, 6]. Also their hardware complexity is

significantly lower than the corresponding turbo codes whose

decoding algorithm is not suitable because of its very high

complexity of the Bahl, Cocke, Jelinek and Raviv (BCJR)

algorithm [7]. Using the iterative decoding algorithms LDPC

codes can achieve a near optimum performance that is very

close to the Shannon limit [8]. Due to their advantages in

terms of BER performance and hardware complexity, LDPC

codes have found many applications some of which include

the 10G Base-T Ethernet (IEEE 802.3an), the G.9960 (ITU-T

Standard for networking over power lines, phone lines and

coaxial cable), DVB-S2 (Digital video broadcasting) for

satellites, WiMAX (IEEE 802.16e standard for microwave

communications) and IEEE 802.11n standard for Wireless

LANs, China Multimedia Mobile Broadcasting (CMMB) to

name a few [9, 10].

In this paper, we study the Projective Geometry (PG) and

block-circulant LDPC codes and compare their performance

with the state-of-the-art FEC schemes used for optical

communications.

II. LDPC CODES

LDPC codes are a class of linear block codes deriving their

name from the characteristic of the parity-check matrix H that

has fewer 1s in comparison to 0s. These codes are based on

graphs known as bipartite or tanner graphs [11], having two

nodes namely the check (function) nodes representing the

parity bits and the variable (bit) nodes representing the code

bits. The check node is connected to variable node whenever

element hcv in a parity-check matrix is a 1. In a m x n parity-

check matrix, there are m = n - k check nodes and n variable

nodes. As an illustrative example, consider the H-matrix of the

following LDPC code.

(1)

Figure 1. Bipartite graph of LDPC (6, 2) code described by H-matrix in (1)

First discovered by Gallager [7] in the early 1960’s, they have

recently been rediscovered and generalized by MacKay and

Neal [12, 13]. The iterative decoding algorithms used provide

significant system performance enhancement. In order for the

message-passing decoding algorithm to perform better on such

graphs, the bipartite graphs of LDPC codes are desired to have

a large girth which is the size of the shortest cycle in the

bipartite graph. This is because it takes more iterations for the

extrinsic information originating from different nodes in the

bipartite graph to become correlated [6]. The original LDPC

codes presented by Gallager were regular, meaning that the

number of 1s per row and per column are both constant. But

the use of structured codes based on finite geometries is

preferred for optical communications due to various reasons as

given below.

i) These codes can lend themselves to simple encoders

that can be realized by feedback shift registers,

ii) They can be decoded using reduced complexity belief-

propagation algorithm and

iii) These codes assure large minimum codeword

distances due to their combinatorial structure.

Thus the LDPC codes constructed using projective

geometries and block-circulant structures make an excellent

choice for long-haul high-speed optical systems.

III. PROJECTIVE GEOMETRY BASED LDPC CODE

CONSTRUCTION

The design of codes using projective geometry is not a

novel idea but these designs have gained importance due to

their significant performance through the iterative decoding

algorithms that have been recently developed. They are

constructed from the elements of a Galois field. The finite

projective geometries [PG (m, ps)] are constructed using (m +

1)-tuples, x = {x0, x1, …xm} of elements xi from GF (ps) (p-a

prime, s-positive integer), not all simultaneously equal to zero,

called points. We take p = 2 and s = 6. Two (m+1)-tuples x =

{x0, x1,… xm} and y = {y0, y1,… ym} represent the same point

if , y = λx where λ ϵ GF (ps) is a nonzero element. So each

point can be represented in ps

- 1 ways, referred to as the

equivalence class. The number of points in PG (m, ps ) is given

by v where v = [p(m + 1)s

- 1] /ps - 1. The row weight is p

s + 1

and column weight is (pms

- 1)/ps - 1 column with exactly one

‘1’ common in any two columns and any two rows having at

most one ‘1’ in common. The corresponding Tanner graph is

free of cycles of girth 4 [15]. An approximate version of the

sum-product iterative decoding algorithm called the min-sum

is used as it requires only simple addition and finding

minimum operations [2]. Although the sum-product algorithm

performs well in various types of channels, its min-sum

adaptation is preferred for high speed applications.

IV. BLOCK-CIRCULANT LDPC CODE CONSTRUCTION

A more desirable approach to design high speed optical

communication systems is to use fast and simple circuits for

performing encoding and decoding. Recently, block-circulant

LDPC codes constructed using permutation blocks have been

found that provide excellent error correction performance.

These are based on a new class of combinatorial objects that

provide for well structured decoder architectures. Another

advantage of these structures is that they allow for small

storage requirements [16].

An rT × nT parity check matrix H of the block-circulant

code defined by the Tanner graph (called a protograph) can be

constructed by concatenating r × n sparse circulants of size T

× T. A circulant is a square binary matrix where the next row

can be constructed by a by a single right cyclic shift of the

present row without having the need to meet the specification

of each row having a Hamming weight 1 [17]. Any LDPC

code defined by a block-circulant H matrix is quasi-cyclic. A

quasi-cyclic code is one for which every cyclic shift by no

symbols of a codeword yields a codeword (where in a (n, k)

linear block code no is coprime with n). Thus, block-circulant

LDPC codes can thus have better minimum distances and

girths than row-circulant codes.

The parity check matrix of a block-circulant quasi-cyclic

LDPC code is shown below.

(2)

Here Ip represents the p × p identity matrix and Ip (pi,j) the

circulant shift of the identity matrix by r + pi,j (mod p)

columns to the right which gives the matrix with the r-th row

having a one in the (r + pi,j mod p)-th column.

V. SYSTEM DESCRIPTION

Figure 2. Optical system with LDPC encoder and decoder

Figure 2 shows a generalized optical communication system with LDPC encoder and decoder. The data from the source in the form of binary digits (0 or 1) is encoded by the LDPC encoder. This coded data is then used along with a modulator (e.g., Mach-Zehnder) to modulate continuous wave laser signals which is Distributed Feedback (DFB) laser. The modulated optical output is coupled onto a transmission channel which consists of optical fibers where Erbium Doped

Fiber Amplifiers (EDFAs) are deployed periodically to compensate for the transmission loss. Each fiber span consists of 50 km length followed by an EDFA. At the receiver end the optical signal first comes across an optical filter and then a photodiode (pin or avalanche) that converts the optical signal to an electric signal. After passing though the electric filter, this signal is fed to the LDPC decoder through a sampler which decodes and provides the original data. We studied the RZ modulated signal at 10 Gb/s providing an additional overhead depending on the codes used and varying the number of fiber spans.

VI. PERFORMANCE ANALYSIS OF PG-LDPC CODES

We studied the performance of PG (2, 26); LDPC (4161,

3431) codes with a rate of 0.83 and an overhead of 21.3%. The

system margin was calculated at a BER requirement of 10-9

.

The difference of system performance (measured or

simulated) and system performance requirement is a measure

of system viability and is called system margin. Traditionally

system margin is being calculated in units of Q-factor in dB.

Typical design objective for system margin is to be between 3

and 6 dB. The system margins of PG (2, 26); LDPC (4161,

3431) are compared with those of uncoded systems, RS (255,

239) codes and concatenated RS codes RS (223, 207) and RS

(255, 223) in Figs. 3 and 4.

Figure 3 shows that the system margin provided by PG (2,

26) LDPC codes exceeds the two RS schemes as well as the

uncoded system at a BER requirement of 10-9

. The coding gain

is found out to be around 9.13 dB for PG (2, 26) over uncoded

system. Also they provide a performance improvement of 1.76

dB over concatenated RS codes and 4.01 dB over RS (225,

239) codes.

Figure 3. BER versus the system margin (in dB) for different FEC

schemes

Figure 4. System margin (in dB) versus the number of spans for different

FEC schemes

Figure 4 shows the variations in the system margin with

the length of the optical link. Here also it can be seen that PG

(2, 26) codes outperform the other codes considerably. Figure

5 graphically shows the level of outperformance (in dB)

(represented on the x-axis) of PG (2, 26); LDPC (4161, 3431)

codes over other coding schemes.

Figure 5. Level of outperformance (in dB) of PG (2, 26); LDPC (4161,

3431) codes over other coding schemes.

VII. PERFORMANCE ANALYSIS OF BLOCK-CIRCULANT LDPC

CODES

Block-circulant LDPC (8176, 6716) codes with a code rate

of r = 0.82 and an overhead of 29.8% are compared with the

PG (2, 26): LDPC (4161, 3431) codes, the first generation FEC

scheme (RS (255, 239)), the second generation FEC scheme

(concatenated RS codes, RS (223, 207) and RS (255, 223))

along with the uncoded system at a BER requirement of 10-7

.

The system margin thus obtained is shown in Figure 6. Figure

7 shows the plot of the system margin with varying length of

the optical link.

Figure 6. BER versus the system margin (in dB) for different FEC

schemes

Figure 7. System margin (in dB) versus the number of spans for different FEC

schemes

Block-circulant LDPC (8176, 6716) codes are seen to

provide the best system margin among all the codes. They

outperform PG (2, 26): LDPC (4161, 3431) codes by 0.29 dB,

RS (223, 207) and RS (255, 223) codes by 2.01 dB, RS (255,

239) by 3.99 dB codes and provide a coding gain of 8.32 dB

over the uncoded system. The PG (2, 26): LDPC (4161, 3431)

outperform the RS (223, 207) and RS (255, 223), the RS (255,

239) and the uncoded system by 1.72 dB, 3.70 dB and 8.03

dB, respectively. This is represented graphically in Figure 8.

Figure 8. Level of outperformance (in dB) of block-circulant (8176, 6176)

LDPC codes and PG (2, 26); LDPC (4161, 3431) codes over other coding

schemes.

VIII. CONCLUSION

The performance of Block-circulant LDPC and the PG-

LDPC coding schemes is assessed and a significant

outperformance is demonstrated over the state-of-the-art FEC

schemes employed in optical communication systems. PG-

LDPC codes are shown to have a coding gain of around 9 dB

at BER of 10-9

. It is also observed that LDPC codes give better

performance by 0.29 dB with block-circulant structures as

compared to the PG-LDPC codes. The high code rate, large

minimum distances, simple decoding algorithms and excellent

BER performance make these codes an attractive choice for

long-haul high speed optical transmission.

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