Simultaneou

易经与风水-英语红宝书怎么用高效

2023年9月30日发(作者：皮德福)

Simultaneous chromatic dispersion and PMD

compensation by using coded-OFDM and

girth-10 LDPC codes

Ivan B. Djordjevic, Lei Xu*, and Ting Wang*

University of Arizona, Department of Electrical and Computer Engineering, Tucson, AZ 85721, USA

NEC Laboratories America, Princeton, NJ 08540, USA

ivan@

Abstract: Low-density parity-check (LDPC)-coded orthogonal frequency

division multiplexing (OFDM) is studied as an efficient coded modulation

scheme suitable for simultaneous chromatic dispersion and polarization

mode dispersion (PMD) compensation. We show that, for aggregate rate of

10 Gb/s, accumulated dispersion over 6500 km of SMF and differential

group delay of 100 ps can be simultaneously compensated with penalty

within 1.5 dB (with respect to the back-to-back configuration) when training

sequence based channel estimation and girth-10 LDPC codes of rate 0.8 are

employed.

OCIS codes: (060.4510) Optical communications; (999.9999) Polarization mode dispersion

(PMD); (999.9999) Chromatic dispersion; (060.4080) Modulation; (060.4230) Multiplexing;

(999.9999) Orthogonal frequency division multiplexing; (999.9999) Low-density parity-check

(LDPC) codes

References and Links

1. R. Prasad, OFDM for Wireless Communications Systems (Artech House, Boston 2004).

2. I. B. Djordjevic and B. Vasic, “Orthogonal frequency-division multiplexing for high-speed optical

transmission,” Opt. Express 14, 3767-3775 (2006).

3. W. Shieh and C. Athaudage, “Coherent optical frequency division multiplexing,” Electron. Lett. 42, 587-589

(2006).

4. A. J. Lowery, L. Du, and J. Armstrong, “Orthogonal frequency division multiplexing for adaptive dispersion

compensation in long haul WDM systems,” in Proc. OFC Postdeadline Papers, Paper no. PDP39, 2006.

5. I. B. Djordjevic and B. Vasic, “100 Gb/s transmission using orthogonal frequency-division multiplexing,”

IEEE Photon. Technol. Lett. 18, 1576-1578 (2006).

6. I. B. Djordjevic, “PMD compensation in fiber-optic communication systems with direct detection using

LDPC-coded OFDM,” Opt. Express 15, 3692-3701 (2007).

7. W. Shieh, “PMD-supported coherent optical OFDM systems,” IEEE Photon. Technol. Lett. 19, 134-136

(2006).

8. S. L. Jansen, I. Morita, N. Takeda, H. Tanaka, “20-Gb/s OFDM transmission over 4,160-km SSMF enabled

by RF-pilot tone phase compensation,” in Proc. OFC/ NFOEC 2007 Postdeadline Papers, Paper no. PDP15,

March 25-29, 2007, Anaheim, CA, USA.

9. B. J. Schmidt, A. J. Lawery, J. Amstrong, “Experimental demonstration of 20 Gbit/s direct-detection optical

OFDM and 12 Gbit/s with a colorless transmitter,” in Proc. OFC/ NFOEC 2007 Postdeadline Papers, Paper

no. PDP18, March 25-29, 2007, Anaheim, CA, USA.

10. A. Lowery, “Nonlinearity and its compensation in optical OFDM systems,” presented at ECOC 2007

Worksop 5 (Electronic signal processing for transmission impairment mitigation: future challenges).

11. I. B. Djordjevic, H. G. Batshon, M. Cvijetic, L. Xu, and T. Wang, “PMD compensation by LDPC-coded

turbo equalization,” IEEE Photon. Technol. Lett. 19, 1163 – 1165 (2007).

12. W. Shieh, X. Yi, Y. Ma, and Y. Tang, “Theoretical and experimental study on PMD-supported transmission

using polarization diversity in coherent optical OFDM systems,” Opt. Express 15, 9936-9947 (2007).

13. I. B. Djordjevic, S. Sankaranarayanan, S. K. Chilappagari, and B. Vasic, “Low-density parity-check codes

for 40 Gb/s optical transmission systems,” IEEE J. Sel. Top. Quantum Electron. 12, 555-562 (2006).

14. M. P. C. Fossorier, “Quasi-cyclic low-density parity-check codes from circulant permutation matricies,”

IEEE Trans. Inform. Theory 50, 1788-1794 (2004).

15. O. Milenkovic, I. B. Djordjevic, and B. Vasic, “Block-circulant low-density parity-check codes for optical

communication systems,” IEEE J. Sel. Top. Quantum Electron. 10, 294-299 (2004).

#92916 - $15.00 USDReceived 20 Feb 2008; revised 8 May 2008; accepted 16 Jun 2008; published 26 Jun 2008

16. J. L. Fan, “Array codes as low-density parity-check codes,” in Proc. 2nd Int. Symp. Turbo Codes and

Related Topics, Brest, France, pp. 543-546, Sept. 2000.

17. D. J. C. MacKay, “Good error correcting codes based on very sparse matrices,” IEEE Trans. Inform. Theory

45, 399-431 (1999).

18. R. M. Tanner, “A recursive approach to low complexity codes,” IEEE Trans. Inf. Theory IT-27, 533–547

(1981).

19. T. Mizuochi, Y. Miyata, T. Kobayashi, K. Ouchi, K. Kuno, K. Kubo, K. Shimizu, H. Tagami, H. Yoshida,

H. Fujita, M. Akita, and K. Motoshima, “Forward error correction based on block turbo code with 3-bit soft

decision for 10-Gb/s optical communication systems,” IEEE J. Sel. Top. Quantum Electron. 10, 376–386

(2004).

20. N. Cvijetic, L. Xu, and T. Wang, “Adaptive PMD Compensation using OFDM in Long-Haul 10Gb/s

DWDM Systems,” in Optical Fiber Comm. Conf., Paper OTuA5, Anaheim, CA (2007).

1. Introduction

Orthogonal frequency division multiplexing (OFDM) [1-10] represents a particular

configuration) being less than 1.5 dB, in an optical transmission system of aggregate rate

10 Gb/s.

The paper is organized as follows. The concept of LDPC-coded OFDM transmission is

introduced in Section 2. In Section 3 we describe a class of large-girth LDPC codes suitable

for use in coded-OFDM. In Section 4 we provide the numerical results to illustrate the

suitability of LDPC-coded OFDM in simultaneous chromatic dispersion and PMD

compensation. Finally, in Section 5 some important concluding remarks are given.

2. LDPC-coded optical OFDM Transmission

The transmitter and receiver configurations are shown in Figs. 1(a), and 1(b), respectively. On

the transmitter side the information-bearing streams at 10 Gb/s are encoded using identical

LDPC codes. The outputs of these LDPC encoders are demultiplexed and parsed into groups

of B bits corresponding to one OFDM frame. The B bits in each OFDM frame are

tottot

subdivided into N sub-channels with the i sub-carrier carrying b bits, .

QAMi

B=b

toti

∑

QAM

i=1

The b bits from the i sub-channel are mapped into a complex-valued signal from a -

point QAM signal constellation. For example, b=2 for QPSK and b=4 for 16-QAM. Notice

that different sub-carriers may carry different number of bits. The complex-valued signal

points from sub-channels are considered to be the values of the fast Fourier transform (FFT)

of a multi-carrier OFDM signal. The OFDM symbol is generated as follows: N

QAM

input

QAM symbols are zero-padded to obtain N

FFT G

input samples for inverse FFT (IFFT), N non-

zero samples are inserted to create the guard interval, and the OFDM symbol is multiplied by

the window function. The purpose of cyclic extension is to preserve the orthogonality among

sub-carriers when the neighboring OFDM symbols partially overlap due to chromatic

dispersion and PMD, and the role of windowing is to reduce the out-of band spectrum. For

efficient chromatic dispersion and PMD compensation, the length of cyclically extended

guard interval should be smaller than the total spread due to chromatic dispersion and DGD.

The cyclic extension is accomplished by repeating the last N

/2 samples of the effective

OFDM symbol part (N samples) as a prefix, and repeating the first N/2 samples as a

FFTG

suffix. After D/A conversion and RF up-conversion, the RF signal can be converted into the

optical domain using one of two possible options: (i) the OFDM signal can directly modulate

a distributed-feedback (DFB) laser, or (ii) the OFDM signal can be used as the RF input of a

Mach-Zehnder modulator (MZM). A DC bias component is added to the OFDM signal in

order to enable recovery of the QAM symbols using direct detection. Because bipolar signals

cannot be transmitted over an IM/DD link, the bias component should be sufficiently large so

that (when added to the OFDM signal) the resulting signal is non-negative. The main

disadvantage of this approach scheme is the poor power efficiency. To improve the OFDM

power efficiency two alternative schemes can be used: (i) the “clipped-OFDM” (C-OFDM)

scheme, which is based on single-side band (SSB) transmission and clipping of the OFDM

signal after the bias addition, and (ii) the “unclipped-OFDM” (U-OFDM) scheme, which is

based on SSB transmission using a LiNbO

MZM. To avoid distortion due to clipping at the

transmitter in the U-OFDM scheme, the information can be imposed by modulating the

electrical field of the optical carrier. In this way both positive and negative portions of the

electrical OFDM signal can be transmitted up to the photodetector. Distortion introduced by

the photodetector, caused by squaring, can be successfully eliminated by proper filtering, as

shown later in this Section. It is important to note, however, that the U-OFDM scheme is less

power efficient than the C-OFDM scheme. The SSB modulation can be achieved either by

appropriate optical filtering the double-side band signal at MZM output [see Fig. 1(a)] or by

using the Hilbert transformation of in-phase component of OFDM RF signal. The first version

requires the use of only in-phase component of RF OFDM signal, providing that zero-padding

is done in the middle of OFDM symbol rather than at the edges. The transmitted OFDM

signal is real and can be written as

st=s (1)t+D,

()()

OFDM

#92916 - $15.00 USDReceived 20 Feb 2008; revised 8 May 2008; accepted 16 Jun 2008; published 26 Jun 2008

where

⎧⎫

⋅t−kTj2

()

N/2−1

FFT

∞

⎪⎪

st=ReX⋅ee

OFDMi,k

()

⎨⎬

∑∑

wt−kT

()

Tj2ft

FFTRF

iN/2k

=−=−∞

FFT

⎪⎪

⎩⎭

is defined for t∈[kT-T/2-T, kT+T+T/2+T]. In the above expression X denotes the

GwinFFTGwini,k

ith subcarrier of the kth OFDM symbol, w(t) is the window function, and f is the RF carrier

frequency. T denotes the duration of the OFDM symbol, T

FFT

denotes the FFT sequence

duration, T is the guard interval duration (the duration of cyclic extension), and T denotes

Gwin

the windowing interval duration. D denotes the DC bias component, which is introduced to

enable the OFDM demodulation using the direct detection.

The PIN photodiode output current can be written as

where s

OFDM

(t) denotes the transmitted OFDM signal in RF domain given by (1). D is

introduced above, while R

denotes the photodiode responsivity. The impulse response of

the optical channel is represented by h(t), with operator * being the convolution operator. The

N(t) represents the amplified spontaneous emission (ASE) noise. The signal after RF down-

conversion and appropriate filtering, can be written as

⎤⎡

rt=itkcost∗ht+nt, (3)

()()()()

RFeRF

⎥⎢

where h

(t) is the impulse response of the low-pass filter, n(t) is electronic noise in the

receiver, and k

denotes the RF down-conversion coefficient. Finally, after the A/D

conversion and cyclic extension removal, the signal is demodulated by using the FFT

algorithm. The soft outputs of the FFT demodulator are used to estimate the bit reliabilities

that are fed to identical LDPC iterative decoders implemented based on the sum-product

algorithm [13].

For the sake of illustration, let us consider the signal waveforms and power-spectral

densities (PSDs) at various points in the OFDM system given in Fig. 1. These examples are

generated using SSB transmission in a back-to-back configuration. The bandwidth of the

OFDM signal is set to B GHz, and the RF carrier to 0.75B. With B we denoted the total

symbol transmission rate. The number of OFDM sub-channels is set to 64, the OFDM

sequence is zero-padded, and the FFT is calculated using 128 points. The guard interval is

obtained by a cyclic extension of 2x16 samples. The average transmitted launch power, in this

back-to-back example, is set to 0dBm. The OFDM transmitter parameters are carefully chosen

such that RF driver amplifier and MZM, as shown in Figs. 2(a)-2(b), operate in the linear

regime. The PSDs of MZM output signal, and the photodetector output signal are shown in

Figs. 2(c) and 2(d), respectively. The OFDM term after beating in the photodetector (PD), the

low-pass term, and the squared OFDM terms can easily be identified.

DFB

10-Gb/s data

streams

LDPCE

DEMUX+

……

LDPCE

Constellation P/S D/ARF

Mapperconverterconverterupconverter

IFFTMZM

DC bias

Clipping

DSB->SSB

Optical filter

SMF

it=Rst+D∗ht+Nt,(2)

()()()()

PINOFDM

⎡⎤

⎣⎦

{}

⎣⎦

(

)

(a)

from

SMF

…

PDFFT

RFBit reliability

downconvertercalculation

Carrier suppression

+DEMUX

A/D converter

LDPCD

…

10-Gb/s data

streams

(b)

Fig. 1. LDPC-coded OFDM: (a) transmitter configuration, and (b) receiver configuration.

LDPCE-LDPC encoder, LDPCD-LDPC decoder, S/P-serial-to-parallel converter, MZM-Mach-

Zehnder modulator, SMF-single-mode optical fiber, PD-photodetector, DSB-double-sideband,

SSB-single-sideband.

#92916 - $15.00 USDReceived 20 Feb 2008; revised 8 May 2008; accepted 16 Jun 2008; published 26 Jun 2008

[

]

0.08

0.06

0.04

0.02

0.00

0.12

C-OFDM

(a)

U-OFDM

0.08

0.04

(b)

0.00

-0.04

8000160002400032000

-0.08

8000160002400032000

Time, t [ps]

MZM out PSD, U-OFDM

(c)

-20

-40

-60

PD out PSD, U-OFDM

[

]

(d)

-20

-40

-60

-80

-3-2-10123

Normalized frequency, (f-f)/B

-80

-3-2-10123

Normalized frequency, (f-f)/B

Fig. 2. Waveforms and PSDs of SSB QPSK-OFDM signal at different points during

transmission for electrical SNR (per bit) of 6dB. (f denotes the optical carrier frequency, LD

denotes the laser diode).

Notice that our proposal requires the use of only I-channel, while both the coherent

detection version due to Shieh [3] and the direct detection version due to Lowery [4] require

the use of I- and Q-channels. In our recent paper [6] we have shown that coded OFDM is an

excellent candidate to be used in PMD compensation. In the same paper we designed the

girth-6 LDPC codes suitable for use in PMD compensation by coded-OFDM, because they

have small number of cycles of length 6. Moreover, the bit-error rate (BER) performance was

evaluated observing the thermal noise dominated scenario. In next Section, we will describe

the new class of quasi-cyclic LDPC codes suitable for use in coded-OFDM optical

transmission, the girth-10 LDPC codes. Those codes, significantly outperform the girth-6

codes, and provide about 0.5 dB improvement in coding gain over girth-8 LDPC codes and

about 1 dB improvement over turbo-product codes. Later, in Section 4, we study the

efficiency of coded-OFDM based on girth-10 LDPC codes in simultaneous suppression of

chromatic dispersion and PMD, observing the ASE noise dominated scenario.

3. Large girth block-circulant (array) LDPC codes

Now we turn our attention to the design of LDPC codes of large girth. Based on Tanner’s

bound for the minimum distance of an LDPC code [18]

⎧

⎢⎥

(g−2)/4

⎦⎣

−1,1+(r−1)g/2=2m+1

⎪

−

d≥(4)

⎨

⎣⎦⎣⎦

(g−2)/4(g−2)/4

⎥⎢⎥⎢

−1+(r−1),g/2=2m1+(r−1)

⎪

−r

⎩

()

where g and r denote the girth of the code graph and the column weight, respectively, and

where d stands for the minimum distance of the code. It follows that large girth leads to an

exponential increase in the minimum distance, providing that the column weight is at least 3.

( denotes the largest integer less than or equal to the enclosed quantity.) For example, the

⎣⎦

minimum distance of girth-10 codes with column weight r=3 is at least 10.

The structured LDPC codes introduced in this Section belong to the class of quasi-cyclic

[14,15] or array [16] codes. Their parity-check matrix can be represented by

#92916 - $15.00 USDReceived 20 Feb 2008; revised 8 May 2008; accepted 16 Jun 2008; published 26 Jun 2008

⎡⎤

IPP...P

c2c

⎢⎥

IPP...P

c2c

H=, (5)

⎢⎥

...............

⎢⎥

c2c

⎢⎥

IPPP

r−1r−1

⎣⎦

()

q−1c

...

()

r−

()

q−1c

where c

∈{0,1,…,q-1} (i=0,1,…,r-1), I is the identity matrix of dimension q, and P denotes

the permutation matrix

⎡⎤

0100...0

⎢⎥

0010...0

⎢⎥

..................

⎢⎥

0000...1

⎢⎥

⎣⎦

1000...0

The integers c

are to be carefully chosen according to [14] in order to avoid the cycles of

length 2k (k=3 or 4). According to [14] (see also [16]) the cycle of length 2k exists if we can

find the closed path in (5), denoted by (i,j), (i,j), (i,j), (i,j),…, (i,j), (i,j), such that

11122223kkk1

cj+cj+...+cj=cj+cj+...+cjmodq, (6)

12k231

iiiiii

12k12k

where q is the dimension of the permutation matrix P, and must be a prime number. The pair

of indices above denote row-column indices of permutation-blocks in (5) such that l

mm+1

≠l,

≠l (m=1,2,..,k; l∈{i,j}). In order to avoid the cycles of length 2k, k=3 or 4, we have to find

the sequence of integers c∈{0,1,…,q-1} (i=0,1,…,r-1; r<q) not satisfying the Eq. (6), which

can be done either by computer search or in a combinatorial fashion. For example, to design

the LDPC codes in [15] we introduced the concept of the cyclic-invariant difference set

(CIDS). The CIDS-based codes come naturally as girth-6 codes, and to increase the girth we

had to selectively remove certain elements from the CIDS. The design of LDPC codes of rate

above 0.8, column weight 3 and girth-10 using the CIDS approach is a very challenging, and

still an open problem. Instead, in this paper we solve this problem by developing an efficient

computer search algorithm, which begins with an initial set S. We add an additional integer at

a time from the set Q={0,1,…,q-1} (not used before) to the initial set S and check if the Eq.

(6) is satisfied. If the Eq. (6) is satisfied we remove that integer from the set S, and continue

our search with another integer from set Q, until we exploit all the elements from Q. The code

rate R is lower-bounded by

nqrq

−

=−≥

1r/n, R(7)

and the code length is nq, where n denotes the number of elements from S being used. The

parameter n is determined by desired code rate R

by n=r/(1-R). If desired code rate is set to

=0.8, and column weight to r=3, the parameter n=5r.

Example: By setting q=1129, the set of integers to be used in (5) is obtained as

S={0,1,4,11,27,39,48,84,134,163,223, 284,333,397,927}. The corresponding LDPC code has

rate R=1-3/15=0.8, column weight 3, girth-10 and length nq=151129=16935, which is about

twice shorter than turbo-product code (TPC) proposed in [19]. In the example above, the

initial set of integers was S={0,1,4}. The use of a different initial set will result in a different

set from that obtained above. In addition to this code, we also designed the

LDPC(24015,19212) code of rate 0.8, girth-10 and column weight 3.

The results of simulations for an additive white Gaussian noise (AWGN) channel model are

given in Fig. 3, where we compare the proposed LDPC codes against RS, concatenated RS,

turbo-product, and girth-8 LDPC codes. The girth-10 LDPC(24015,19212) code of rate 0.8

outperforms the concatenation RS(255,239)+RS(255,223) (of rate 0.82) by 3.35 dB, and

RS(255,239) by 4.75 dB, both at BER of 10

-7-10

. At BER of 10 it outperforms lattice based

#92916 - $15.00 USDReceived 20 Feb 2008; revised 8 May 2008; accepted 16 Jun 2008; published 26 Jun 2008

LDPC(8547,6922) of rate 0.81 and girth-8 by 0.44 dB, and BCH(128,113)xBCH(256,239)

TPC of rate 0.82 by 0.95 dB. The net effective coding gain at BER of 10

-12

is 10.95 dB, which

represents the largest net effective coding gain, on an AWGN channel, ever reported in optical

communications.

Given this description of LDPC-coded OFDM, and the design of LDPC codes to be used

in coded-OFDM, in the following Section we describe our application of interest: the use of

LDPC-coded OFDM in simultaneous chromatic dispersion and PMD compensation.

Uncoded OOK

RS(255,239)+RS(255,223) (R=0.82)

RS(255,239) (R=0.937)

BCH(128,113)xBCH(256,239) (R=0.82)

LDPC(8547,6922) (R=0.81, lattice, g=8, r=4)

Girth-10 LDPC(24015,19212) code(R=0.8, r=3)

-2

AWGN

-3

-4

-5

-6

-7

-8

-9

-10

567891011

Q-factor, Q [dB] (per information bit)

Fig. 3. The girth-10 LDPC code against RS, concatenated RS, turbo-product, and girth-8 LDPC

codes on an AWGN channel model.

4. Simultaneous chromatic dispersion and PMD compensation via LDPC-coded OFDM

The receiver commonly employs the trans-impedance amplifier (TA) design, because it

provides a good compromise between noise characteristics and supported bandwidth. The

received electrical field, at the input of the TA, in the presence of chromatic dispersion and

first-order PMD, can be represented by

⎛⎞

βωβω

⎧⎫

⎡⎤

−

j−L

⎜⎟

tot

⎪⎪⎪⎪

⎜⎟

⎡⎤

()

(8)

−1

⎝⎠

⎡⎤

+=+

EtFTFTEstbe,

()()

⎨⎨⎬⎬⎢⎥

⎢⎥

OFDM

⎦⎣

()

⎢⎥

⎪⎪⎪⎪⎣⎦

⎣⎦

⎭⎩

⎩⎭

where and represent the group-velocity dispersion (GVD) and second order GVD

ββ

parameters, L

tot

is the total SMF length, k is the splitting ratio between two principle states of

polarization (PSPs),

is the phase difference between PSPs, E is transmitted laser electrical

field amplitude, and N

and N represent x- and y-polarization ASE noise components. With

FT and FT we denoted the Fourier transform and inverse Fourier transform, respectively.

-1

The TA output signal can be represented by v(t)=R

FPINPIN

R|E(t)|+n(t), where R is the

photodiode responsivity, R is the TA feedback resistor, and n(t) is TA thermal noise. For

complete elimination of ISI, the total delay spread due to chromatic dispersion and DGD

should be smaller than the guard time:

βω

2totmaxtFFTmaxG

LΔ+DGD=DNΔf+DGD≤T,(9)

where D is the accumulated dispersion, Δf is the sub-carrier spacing, c is the speed of the

#92916 - $15.00 USDReceived 20 Feb 2008; revised 8 May 2008; accepted 16 Jun 2008; published 26 Jun 2008

light, and f is the central frequency set to 193.1 THz. The number of subcarriers N, the

FFT

guard interval T, GVD and second-order GVD parameters were introduced earlier. The

received QAM symbol of i-th subcarrier in the k-th OFDM symbol is related to transmitted

QAM symbol X

i,k

Y=heeX+n, (10)

i,ki,ki,k

where h is channel distortion introduced by PMD and chromatic dispersion, and is the

phase shift of i-th sub-carrier due to chromatic dispersion.

represents the OFDM symbol

phase noise due to SPM and RF down-converter, and can be eliminated by pilot-aided channel

estimation. Notice that in direct detection case, the laser phase noise is completely cancelled

by photodetection. To estimate the channel distortion due to PMD, h

and phase shift due to

chromatic dispersion , we need to pre-transmit the training sequence. Because in ASE noise

dominated scenario (considered here) the channel estimates are sensitive to ASE noise, the

training sequence should be sufficiently long to average the noise. For DGDs up to 100 ps, the

training sequence composed of several OFDM symbols is sufficient. For larger DGDs longer

OFDM training sequence is required; alternatively, the channel coefficients can be chosen to

maximize the log-likelihood ratios (LLRs) or someone can use the polarization beam splitter

to separate the x- and y-polarization components, and consequently process them. The phase

shift of ith subcarrier due to chromatic dispersion can be determined from training sequence

as difference of transmitted and received phase averaged over different OFDM symbols. Once

the channel coefficients and phase shifts due to PMD and chromatic dispersion are

determined, in a decision-directed mode, the transmitted QAM symbols are estimated by

⎞⎛

−j

X=h/heeY. (11)

ikik

⎜⎟

⎝⎠

The symbol LLRs λ(q) (q=0,1,…,2

-1) can be determined by

ˆˆ

ReXReQAMmapqImXImQAMmapq

⎡⎤⎡⎤

()()

⎣⎦⎣⎦

()

q=−−;q=0,1,...,2−1

⎡⎤⎡⎤

i,ki,k

−−

⎣⎦⎣⎦

()()

(12)

where Re[] and Im[] denote the real and imaginary part of a complex number, QAM denotes

the QAM-constellation diagram, N

denotes the power-spectral density of an equivalent

Gaussian noise process obtained from training sequence, and map(q) denotes a corresponding

mapping rule (Gray mapping is applied here). (b denotes the number of bits per constellation

point.) Let us denote by v the jth bit in an observed symbol q binary representation

=(v,v,…,v). The bit LLRs needed for LDPC decoding are calculated from symbol LLRs by

12b

()

qexp

⎤⎡

∑

q:v=0

⎣⎦

Lvˆ=log, (13)

⎡⎤

()

⎦⎣

∑

q:v=1

exp

()

Therefore, the jth bit reliability is calculated as the logarithm of the ratio of a probability that

=0 and probability that v=1. In the nominator, the summation is done over all symbols q

having 0 at the position j, while in the denominator over all symbols q having 1 at the position

The results of simulation, for ASE noise dominated scenario and single wavelength

channel transmission, are shown in Figs. 4-6, for the LDPC-coded SSB OFDM system with

aggregate rate of 10 Gb/s, 512 sub-carriers, RF carrier frequency of 10 GHz, oversampling

factor of 2, and cyclic extension with 512 samples. The modulation format being applied is

QPSK. The LDPC(16935,13550) code of girt-10, code rate 0.8, and column-weight 3,

designed as explained in Section 3 is used. In Fig. 4 we show the BER performance for DGD

of 100 ps, without residual chromatic dispersion. We see that uncoded case faces significant

performance degradation at low BERs. On the other hand, the LDPC-coded case has

#92916 - $15.00 USDReceived 20 Feb 2008; revised 8 May 2008; accepted 16 Jun 2008; published 26 Jun 2008

degradation of 1.1 dB at BER of 10 (when compared to the back-to-back configuration). In

-9

Fig. 5 we show the BER performance after 6500 km of SMF (without optical dispersion

compensation), for a dispersion map composed of 65 sections of SMF with 100 km in length.

The noise figure of erbium-doped fiber amplifiers (EDFAs), deployed periodically after every

SMF section, was set to 5 dB. To achieve the desired OSNR, the ASE noise loading was

applied on receiver side, while the launch power was kept below 0 dBm. We see that LDPC-

coded OFDM is much less sensitive to chromatic dispersion compensation than PMD.

Therefore, even 6500 km can be reached without optical dispersion compensation with

penalty within 0.4 dB at BER of 10

-9

, when LDPC-coded OFDM is used.

-1

-2

Back-to-back:

Uncoded

LDPC-coded

DGD of 100 ps:

Uncoded

LDPC coded

dispersion and DGD. It can also be noticed that, from numerical results presented here, that

the major factor of performance degradation in LDPC-coded OFDM with direct detection, in

ASE noise dominated scenario, is PMD. The main reason is that some of the subcarriers

completely fade away (see [20] for the detailed explanation). To improve the tolerance to

PMD someone may use longer training sequences and redistribute the transmitted information

among the subcarriers less affected by DGD, or to use the polarization beam splitter and

separately process x- and y-PSPs, in a fashion similar to that proposed for OFDM with

coherent detection [7,12]; however, the complexity of such a scheme would be at least two

times higher. Notice that for this level of DGD, the redistribution of power among subcarriers

not being faded away is not needed. For larger values of DGDs, the penalty due to DGD

grows as DGD increases, if the redistribution of subcarriers is not performed.

Back-to-back:

Uncoded

LDPC-coded

6500 km of SMF + DGD of 100 ps:

Uncoded

LDPC-coded

-1

-2

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