Coded Ofdm in Hybrid

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Published in IET Optoelectronics

Received on 29th October 2008

Revised on 3rd March 2009

doi: 10.1049/iet-opt.2008.0059

ISSN 1751-8768

Coded-orthogonal frequency divisionmultiplexing in hybrid optical networksI.B. DjordjevicDepartment of Electrical and Computer Engineering, University of Arizona, Tucson, AZ 85721, USA

E-mail: [email protected]

Abstract: Future Internet should be able to support a wide range of services containing large amount of

multimedia over different network types at a high speed. The future optical networks will therefore be hybrid,

composed of different single-mode fibre (SMF), multi-mode fibre (MMF) and free-space optical (FSO) links. In

these networks, novel modulation and coding techniques are needed that are capable of dealing with

different channel impairments, be it in SMF, MMF or FSO links. The authors propose a coded-modulation

scheme suitable for use in hybrid FSO fibre-optics networks. The proposed scheme is based on polarisation-

multiplexing and coded orthogonal frequency division multiplexing (OFDM) with large girth quasi-cyclic low-

density parity-check (LDPC) codes as channel codes. The proposed scheme is able to simultaneously deal with

atmospheric turbulence, chromatic dispersion and polarisation mode dispersion (PMD). With a proper design

for 16-quadrature amplitude modulation (QAM)-based polarisation-multiplexed coded-OFDM, the aggregate

data rate of 100 Gb/s can be achieved for OFDM signal bandwidth of only 12.5 GHz, which represents ascheme compatible with 100 Gb/s per wavelength channel transmission and 100 Gb/s Ethernet.

1 Introduction

The transport capabilities of fibre-optic communicationsystems have increased tremendously, primarily because ofadvances in optical devices and technologies, and haveenabled the Internet as we know it today with all itsimpacts on the modern society. In particular, dense

wavelength division multiplexing (DWDM) became a

viable, flexible and cost-effective transport technology[19]. Optical communication systems are developingrapidly because of the increased demands on transmissioncapacity. For example, the network operators are alreadyconsidering 100 Gb/s per DWDM channel transmission[2]. The free-space optical (FSO) communication [1012]is the technology that can address any connectivity needs inoptical networks, be it in the core, edge or access. Inmetropolitan area networks (MANs), the FSO can be usedto extend the existing MAN rings; in enterprise, the FSOcan be used to enable local area network (LAN)-to-LANconnectivity and intercampus connectivity; and the FSO is

an excellent candidate for the last-mile connectivity [11]. The future optical networks will allow the integration offibre optics and FSO technologies, and will, therefore, have

different portions of network being composed of fibre[either single-mode fibre (SMF) or multi-mode fibre(MMF)] and FSO sections. These hybrid optical networksmight have a significant impact in both military andcommercial applications, when pulling the ground fibre isexpensive and takes a long time for deployment. Given thefact that hybrid optical networks will contain both FSOand fibre-optic sections, one has to study not only how to

deal with atmospheric turbulence present in the FSOportion of the network, but also the influence of fibre non-linearities, PMD and chromatic dispersion in the fibre-optic portion of the network.

In this paper, we propose a coded-modulation scheme thatis able to simultaneously deal with atmospheric turbulence,chromatic dispersion and polarisation-mode dispersion(PMD) in future hybrid optical networks. Moreover, theproposed scheme supports 100 Gb/s per DWDM channeltransmission and 100 Gb/s Ethernet, while employing themature 10 Gb/s fibre-optics technology. The proposed

hybrid optical network scheme employs the orthogonalfrequency division multiplexing (OFDM) as themultiplexing and modulation technique, and uses the

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low-density parity-check (LDPC) codes as channel codes. With a proper design for 16 quadrature amplitudemodulation (QAM)-based polarisation-multiplexed coded-OFDM, the aggregate data rate of 100 Gb/s can beachieved for the OFDM signal bandwidth of only12.5 GHz, which represents a scheme suitable for 100 Gb/

s Ethernet. Note that arbitrary forward error correction(FEC) scheme can be used in proposed hybrid opticalnetwork. However, the use of large girth LDPC codes [2]leads to the channel capacity achieving performance.

We consider two scenarios: (i) the FSO channelcharacteristics are known on the transmitter side and (ii)the FSO channel characteristics are not known on thetransmitter side. In both scenarios, we assume that fibre-optic channel properties are known on the receiver side,obtained by pilot-aided channel estimation. Given the factthat transmitter and receiver nodes might be connected

through several FSO and fibre-optic links, and that FSOlink properties can vary significantly during the day, it isreasonable to assume that FSO link channel conditions arenot known on the receiver side. In the presence of rain,snow and fog, we assume that an RF feedback channel isused to transmit the channel coefficients to the transmitter,

which adapts the transmitted power and data rate accordingto the channel conditions.

The proposed scheme has many unique advantages:(i) demodulation, equalisation and decoding are jointlyperformed; (ii) it is able to operate in the presence ofchannel impairments over different optical links, be it inSMF, MMF or FSO, (iii) it has high-bandwidth efficiency(even 10 bits/s/Hz), (iv) it is compatible with future100 Gb/s Ethernet technologies; and (v) the employedcoded modulation provides excellent coding gains. We alsodescribe about how to determine the symbol reliabilities inthe presence of laser phase noise, and describe a particularchannel inversion technique suitable for dealing with PMDeffects.

The paper is organised as follows. In Section 2 we describethe proposed hybrid optical network and the correspondingcoded-modulation scheme. In Section 3 we describe the

channel model, whereas in Section 4 we describe thereceiver architecture and transmission diversity principle. InSection 5 we report numerical results, and some importantconclusions are given in Section 6.

2 Description of the proposedhybrid optical network

An example of a hybrid FSO fibre-optic network is shownin Fig. 1a. This particular example includes inter-satellitelinks and connection to aircrafts. The fibre-optic portion of

the network could be a part of already installed MAN or wide area network (WAN). The FSO network portionshould be used whenever pulling the ground fibre is

expensive and/or takes too much time for deployment,such as urban and rural areas, where the optical fibre linksare not installed. The corresponding hybrid opticalnetworking architecture is shown in Fig. 1b. We canidentify three ellipses representing the core network, theedge network and the access network. The FSO links canbe used in both edge and access networks. The hybridoptical networks impose a big challenge to the engineers,because the novel signal processing techniques should bedeveloped, which would be able to simultaneously dealatmospheric turbulence in FSO links, and with chromaticdispersion, PMD and fibre non-linearities in fibre-opticlinks. One such coded-modulation technique is describedin the rest of this section. By using the retro-reflectors theFSO systems can be applied even when there is no line of

sight between the transmitter and the receiver.

The proposed coded-modulation scheme employs thecoded-OFDM scheme with coherent detection. Note thatthe coded-OFDM scheme with direct detection has alreadybeen proposed by the authors in [12], as a scheme that incombination with interleaving is able to operate understrong atmospheric turbulence. The use of coherentdetection offers the potential of even 24 dB improvementover uncoded direct detection counterpart. One portion ofimprovement (10 13 dB) is comes from the fact thatcoherent detection can approach quantum-detection limit

easier than direct detection. The second portion (about11 dB) comes from the use of large-girth LDPC codes[1, 2]. Let us now describe the operation principle of

Figure 1 Illustration of hybrid optical networking principle

a A hybrid FSO fibre-optic network exampleb Hybrid optical networking architecture

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coded-OFDM scheme with coherent detection employingboth polarisations. Given the fact that the signal fromFig. 1 is going to be transmitted over the FSO links andover the fibre-optic links, we use a particular polarisationmultiplexing capable of eliminating the influence of PMD.

The transmitter and the receiver shown in Fig. 2, to be

used in hybrid optical network from Fig. 1, are able to

simultaneously deal with atmospheric turbulence, residualchromatic dispersion and PMD. The bit streamsoriginating from m different information sources areencoded using different (n, ki) LDPC codes of code rateri ki/n. ki denotes the number of information bits of theith (i 1, 2, . . . , m) component LDPC code, and n

denotes the codeword length, which is the same for all

Figure 2 Transmitter and receiver configurations for LDPC-coded OFDM hybrid optical system with polarisation multiplexing

and coherent detection

a Transmitter architectureb OFDM transmitter architecture

c Hybrid optical link exampled Receiver architecturee Coherent detector configuration. PBS/PBC



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LDPC codes. The use of different LDPC codes allows us tooptimally allocate the code rates. If all component LDPCcodes are identical, then the corresponding scheme iscommonly referred to as the bit-interleaved codedmodulation (BICM). The outputs of m LDPC encodersare written row-wise into a block-interleaver block. The

mapper accepts m bits at time instance i from the (m n)interleaver column-wise and determines the correspondingM-ary (M 2m) signal constellation point (fI,i, fQ,i) in atwo-dimensional (2D) constellation diagram such as M-aryphase-shift keying (PSK) or M-ary QAM. (Thecoordinates correspond to in-phase and quadraturecomponents ofM-ary 2D constellation.)

The OFDM symbol is generated as described below.NQAM input QAM symbols are zero padded to obtainNFFT input samples for inverse fast Fourier transform(IFFT) (the zeros are inserted in the middle rather than at

the edges), and NG non-zero samples are inserted to createthe guard interval. For efficient chromatic dispersion andPMD compensation, the length of the cyclically extendedguard interval should be longer than the total spreadbecause of chromatic dispersion and maximum value ofdifferential group delay (DGD). The cyclic extension isobtained by repeating the last NG/2 samples of theeffective OFDM symbol part (NFFT samples) as a prefix,and repeating the first NG/2 samples as a suffix. AfterD/ A conversion (DAC), the OFDM signal is convertedinto the optical domain using the dual-drive MachZehnder modulators (MZMs). Two dual-drive MZMs areneeded, one for each polarisation. The outputs of MZMsare combined using the polarisation beam combiner (PBC).

The same distributed feedback (DFB) laser is used as aCW source, with x- and y-polarisations being separated bya polarisation beam splitter (PBS).

The operations of all other blocks in the transmitter aresimilar to those we reported in [9, 12], and for more detailson OFDM with coherent detection an interested reader isreferred to an excellent tutorial paper by Shieh et al. [4].

The key idea of this proposal is to use the OFDM with alarge number of subcarriers (in the order of thousands) so that

the OFDM symbol duration becomes in order ofms, and bymeans of interleavers in the order of thousands overcome theatmospheric turbulence with temporal correlation in theorder of 10 ms. For the OFDM scheme to be capable ofsimultaneously compensating for chromatic dispersion andPMD, in addition to the atmospheric turbulence, the cyclicextension guard interval should be longer than the totaldelay spread because of chromatic dispersion and DGD, asindicated above.

The OFDM is also an excellent candidate to be used formulti-user access [known as OFDM access (OFDMA)]. In

OFDMA, subsets of subcarriers are assigned to individualusers. OFDMA enables time and frequency domainresource partitioning. In time domain, it can accommodate

for the burst traffic (packet data) and enables multi-userdiversity. In frequency domain, it provides furthergranularity and channel-dependent scheduling. InOFDMA, different numbers of sub-carriers can beassigned to different users, in order to supportdifferentiated quality of service (QoS). Each subset of sub-

carriers can have different kinds of modulation formats, andcan carry different types of data. The differentiated QoScan be achieved by employing the LDPC codes of differenterror correction capabilities. The OFDMA, therefore,represents an excellent interface between wireless/wirelineand optical technologies.

Because for high-speed signals a longer sequence of bits isaffected by the deep fade in the ms range due to atmosphericturbulence, we propose to employ the polarisationmultiplexing and large QAM constellations in order toachieve the aggregate data rate of RD 100 Gb/s, while

keeping the OFDM signal bandwidth in the order of10 GHz. For example, by using the polarisation-multiplexing and 16-QAM, we can achieve RD 100 Gb/sfor a OFDM signal bandwidth of 12.5 GHz, resulting in abandwidth efficiency of 8 bits/s/Hz. Similarly, by usingthe polarisation multiplexing and 32-QAM we can achievethe same data rate (RD 100 Gb/s) for a OFDM signalbandwidth of 10 GHz, with a bandwidth efficiency of 10bits/s/Hz.

The receiver description requires certain knowledge of thechannel. In what follows, we assume that fibre-optic channelcharacteristics are known on the receiver side, because thefibre-optics channel coefficients can easily be determined bypilot-aided channel estimation. On the other hand, thehybrid optical network may contain different FSO andfibre-optic sections, while the channel characteristics of theFSO link can change rapidly even during the day, it isreasonable to assume that FSO channel characteristics arenot known on the receiver side. The FSO transmitter canuse a retro-reflector and a training sequence to sense theFSO channel. We will further describe two concepts: (i)the transmitter does not have any knowledge about theFSO link and (ii) the transmitter knows the FSO linkproperties. When the transmitter knows the FSO link

properties, we can employ the transmitter diversity concept.Before resuming our description of the coded-OFDMreceiver, in the next section we provide more details aboutFSO and fibre-optic channel models.

3 Description of the channelmodel

A commonly used turbulence model assumes that the variations of the medium can be understood as individualcells of air or eddies of different diameters and refractive

indices. In the context of geometrical optics, these eddiesmay be observed as lenses that randomly refract the opticalwave front, generating a distorted intensity profile at the



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receiver of a communication system. The amplitude andphase fluctuations, also known as scintillation, represent themost important factors that limit the performance of anatmospheric FSO communication link. The most widelyaccepted theory of turbulence is given by Kolmogorov [10].

This theory assumes that kinetic energy from large

turbulent eddies, characterised by the parameter known asouter scale L0, is transferred without loss to the eddies ofdecreasing size down to sizes of a few millimetrescharacterised by the inner scale parameter l0. The innerscale represents the cell size at which energy is dissipatedby viscosity. The refractive index varies randomly acrossdifferent turbulent eddies and causes phase and amplitude

variations to the wave front. To account for the strengthof the turbulence, we use the unitless Rytov variance, givenby [10]

s2R 1:23 C2n k7=6L11=6 (1)

where k 2p/l is the wave number, l is the wavelength, L isthe propagation distance and Cn

2 denotes the refractive indexstructure parameter, which is constant for horizontalpaths. Weak fluctuations are associated with s2R, 1, themoderate with s2R 1, the strong with s

2R. 1 and

the saturation regime is defined by s2R2.1 [10]. In the weak turbulence regime, the Rytov method is commonlyused to represent the field of electromagnetic wave asfollows [10] (using cylindrical coordinates R (r, L), withL being the transmission distance)

U(R) ; U(r, L)

U0(r, L) exp[c(r, L)] (2)

where U0(r, L) is the electromagnetic field in the absence ofturbulence, and c is the complex phase perturbation becauseof turbulence. The complex phase perturbation can beexpressed as follows

c(r, L) log U(r, L)U0(r, L)

! XjY (3)

where X is the log-amplitude fluctuation and Y is thecorresponding phase fluctuation. In the weak turbulenceregime it is reasonable to assume X and Y to be Gaussian

random processes. To deal with phase fluctuations someonemay use active modal compensation of wave-front phasedistortion [13]. The residual phase variance after modalcompensation can be described in terms of Zernike termsby [13]

s2Y ZJD

d0

5=3(4)

where Dis the aperture diameter, d0 is the correlation lengthand ZJ denotes the Jth Zenrike term not being compensated(commonlyJ 3, 6, 10, 20). Through the paper we assume

D,

d0 and that the Zenrike terms beyond 5 are notcompensated, leading to typical sY being between 0.01 and0.1. We further assume that the OFDM system is designed

such that TOFDM , t0, where TOFDM is the OFDMsymbol duration and t0 is the correlation time, typicallybetween 10 ms and 10 ms. Therefore the Gaussian processX(t) can be described by multivariate Gaussian distribution

fX(t1),...,X(tn)(x1,. . .

, xn) 1

(2p)n=2det(CX)

exp 12

(x m)TC1X (x m) !

(5)

where x [x1 . . . xn]T, m [EfX(t1)g . . . EfX(tn)g]T, with

E[.] being the expectation operator. CX is the covariancematrix with elements (CX)i,j s

2XbX(ji2jjTOFDM/t0),

where the covariance function bX(t) is found to beexponential for both plane and spherical waves [10, 12]

bX(t) exp tj jt0

5=

3" #

(6)

s2X denotes the variance of the log-normally distributedamplitude, which for plane wave can be approximated as [10]

s2X ffi 0:56 k7=6L

0C2n (x)(L x)5=6 dx (7)

where the wave number k, propagation length L and therefractive index structure parameter Cn were introducedearlier. Note that sX is different from Rytov standarddeviation sRused earlier and for horizontal paths sR 2sX.

Therefore for the weak turbulence regime sX, 0.5.

We turn our attention now to the fibre-optic channelmodel. For the first-order PMD study the Jones matrix,neglecting the polarisation-dependent loss anddepolarisation effects because of non-linearity, can berepresented in a manner similar to [14] by

H(v) hxx hxy

hyx hyy

" # R1P(v)R, P(v)

ejvt=2 0

0 ejvt=2

" #(8)

where tdenotes DGD, R R(u, 1) is the rotational matrixdefined by

R cos

u

2

ej1=2 sin

u

2

ej1=2

sin u2

ej1=2 cos

u

2

ej1=2

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udenotes the polar angle, 1 denotes the azimuth angle and vthe angular frequency. For coherent detection OFDM, thereceived symbol vector ri,k [rx,i,k ry,i,k]

T at the ith OFDM



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symbol and kth subcarrier can be represented by

ri,k ai(k)ejfYH(k)si,kejfCD(k)ejfPN ni,k (9)

where si,k [sx,i,k sy,i,k]T denotes the transmitted symbol

vector,ni,k

[nx,i,k ny,i,k]

T

denotes the noise vector becauseof the amplified spontaneous emission (ASE) and the Jones matrix H was introduced in (8). Here we use index kto denote the kth subcarrier frequency vk. fCD(k) denotesthe phase distortion of the kth subcarrier because ofchromatic dispersion. fPN denotes the phase noise processfPN fT2fLO because of the laser phase noiseprocesses of transmitting laserfT and local laserfLO thatare commonly modelled as the Wiener Le vy processes[15], which are a zero-mean Gaussian processes withcorresponding variances being 2pDnTjtj and 2pDnLOjtj,

where DnT and DnLO are the laser linewidths oftransmitting and receiving laser, respectively. ai(k) denotes

the log-amplitude attenuation coefficient because of theatmospheric turbulence channel and fY is the residualphase noise process that remained after modal-phasecompensation, as described above. From (9) we can createthe equivalent OFDM channel model as shown in Fig. 3.

4 Description of the receiver andtransmission diversity scheme

In this section we describe the operation of the receiver byobserving two different transmission scenarios. In the firstscenario we assume that the transmitter does not have any

knowledge about the FSO channel. In the second scenario we assume that the transmitter has knowledge about theFSO link, which is obtained by using the short trainingsequence transmitted towards the retro-reflector. In bothscenarios we assume that the receiver knows the propertiesof the fibre-optic portion of the network obtained by pilot-aided channel estimation. This can be achieved byorganising the OFDM symbols in OFDM packets with

several initial OFDM symbols being used for channelestimation. Note that this approach is also effective inestimating the FSO channel properties in the regime of

weak turbulence. The immunity to atmospheric turbulencecan be improved by employing the diversity approaches[16]. To maximise the receiver diversity, the multiple

receivers should be separated enough so that independencycondition is satisfied. Given the fact that the laser beam isgetting expanded, during propagation it might not bepossible always to separate the receivers sufficiently enoughso that the independence condition is satisfied. On theother hand, by using transmission diversity instead, theindependence condition is easier to satisfy. Moreover, it hasbeen shown in [17] that transmitter diversity performsbetter comparable to the maximum-ratio combiningreceiver diversity. In transmission diversity, the signal to betransmitted from the ith transmitter, characterised by pathgain riexp[2jui], is pre-multiplied by complex gain ai ai

exp[2jui] (0 ai 1). On the receiver side, the weight aithat maximises the signal-to-noise ratio is chosen by [17]

ai riffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPLi1 r

2i

q

where L is the number of transmitter branches. When thechannel is not known on the transmitter side, we have toset up ai to 1, and ui 0, and use the Alamouti-typescheme instead [16, 17]. Note, however, that the

Alamouti-type receiver requires the knowledge of the FSOchannel, and as such is not considered here.

The operations of all blocks of receiver, except the symboldetector shown in Fig. 2d, are similar to those we reported in[9, 12]; for more details on OFDM with coherent detectionand chromatic dispersion compensation an interested readeris referred to [4]. Here we describe the operation of thesymbol detector block, the calculation of symbol log-likelihood ratios (LLRs) and the calculation of bit LLRs,in the presence of laser phase noise. By re-writing (9) inscalar form, we obtain, by ignoring the laser phase noise atthe moment to keep the explanation simpler

rx,i,k ai(k)ai(k)ejfY

hxx(k)sx,i,k hxy(k)sy,i,kh i nx,i,k (10)ry,i,k ai(k)ai(k)ejfY hyx(k)sx,i,k hyy(k)sy,i,k

h i ny,i,k (11)

where we used the index k to denote the kth subcarrier, indexi to denote the ith OFDM symbol, hij(k) (i, j[ fx, yg) arethe channel coefficients because of PMD introduced by (8),sx,i,k and sy,i,k denote the transmitted symbols in x- andy-polarisation, respectively; whereas the correspondingreceived symbols are denoted by rx,i,k and ry,i,k. The weightai is chosen in such a way so as to maximise the SNR, asexplained above. In (10) and (11) nx,i,k and ny,i,k denote the

ASE noise processes in x- and y-polarisation. In theabsence of ASE noise, (10) and (11) represent the systemof linear equations with two unknowns sx,i,k and sy,i,k, and

Figure 3 Equivalent OFDM channel model for hybrid optical

networks



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upon solving we obtain

~sx,i,k hxx= hxx

2 rx,i,k hxyhyy= hyy 2 ry,i,k !

1

hxxhxy= hxx

2

hyxhyy= hyy 2

(12)

~sy,i,k hyy

hyy

2 ry,i,k hyxh

yy

hyy

2 ~sx,i,k (13)

where ~sx,i,k and ~sy,i,k denote the detector estimates of symbolssx,i,kand sy,i,k transmitted on the kth subcarrier ofith OFDMsymbol. Note that the OFDM scheme with polarisationdiversity [4], assuming that both polarisations are used on atransmitter side and equal-gain combining on a receiverside, is the special case of the symbol detector described by(12) and (13). By setting sx,i,k sy,i,k si,k and using the

symmetry of channel coefficients, the transmitted symbolcan be estimated by

~si,k hxxrx,i,k hxyry,i,k

hxx 2 hxy 2

In the presence of laser phase noise the symbol detectorestimates are function of the laser phase noise process

~sx,i,k

(hxx= hxx

2)ejfPN rx,i,k (hxyhyy= hyy

2)ry,i,k

!

1 (hxxhxy= hxx 2)(hyxhyy= hyy 2)(14)

~sy,i,k hyye

jfPN

hyy

2 ry,i,k hyxh

yy

hyy

2 ~sx,i,k (15)

The detector soft estimates of symbols carried by thekth subcarrier in the ith OFDM symbol, ~sx(y)i,k, areforwarded to the a posteriori probability (APP) demapper,

which determines the symbol LLRs lx(y)(s) of x- (y-)polarisation by

lx(y) sjfPN

Re ~si,k,x(y)(fPN)h i

Re QAM(map(s)) 2

2s2

Im ~si,k,x(y)(fPN)h i

Im QAM(map(s)) 2

2s2

s 0, 1, . . . , 2nb 1 (16)

where Re[] and Im[] denote the real and imaginary parts of a

complex number, QAM denotes the QAM-constellationdiagram, s2 denotes the variance of an equivalent Gaussian

noise process originating from ASE noise and map(s)denotes a corresponding mapping rule (Gray mapping ruleis applied here). (nb denotes the number of bits carried by asymbol.) Note that symbol LLRs in (16) are conditionedon the laser phase noise sample fPN fT2fLO, whichis a zero-mean Gaussian process (the WienerLe vy process

[15]) with variance s2PN 2p(DnT DnLO)jtj (DnT andDnLO are the corresponding laser linewidths introducedearlier). This comes from the fact that estimated symbols~sx(y)i,k are functions offPN. To remove the dependence onfPN, we have to average the likelihood function (not itslogarithm), overall possible values offPN

lx(y)(s) log1

1exp lx(y) sjfPN

h i 1sPN

ffiffiffiffiffiffi2p

p(

exp f2

PN

2s2PN

df

PN)

(17)

The calculation of LLRs in (17) can be performed bynumerical integration. For the laser linewidths consideredin this paper it is sufficient to use the trapezoidal rule, withsamples offPN obtained by pilot-aided channel estimationas explained in [4].

Let us denote bybj,x(y) the jth bit in an observed symbol sbinary representation b (b1, b2, . . . , bnb) for x- (y-)polarisation. The bit LLRs required for LDPC decoding

are calculated from the symbol LLRs by

L(bj,x(y)) logP

s:bj0 exp lx(y)(s)h i

Ps:bj1 exp lx(y)(s)

h i (18)

Therefore the jth bit LLR in (18) is calculated as thelogarithm of the ratio of a probability that bj 0 andbj 1. In the nominator, summation is performed over allsymbols s having 0 at the position j. Similarly, in the

denominator summation is performed over all symbols shaving 1 at the position j. The bit LLRs calculated by(18) are forwarded to the corresponding LDPC decoders.

The LDPC decoders from Fig. 2d employ the sum-product-with-correction term algorithm. The LDPC codeused in this paper belong to the class of quasi-cyclic(array) codes of large girth (g! 10) [2, 18], so that thecorresponding decoder complexity is low compared torandom LDPC codes, and do not exhibit the errorfloor phenomena in the region of interest in fibre-opticscommunications (10215).

The parity check-matrix H of quasi-cyclic (QC) (N, K)LDPC codes [18] considered in this paper can be



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represented by

H

I I I . . . II PS[1] PS[2] . . . PS[c1]

I P2S[1] P2S[2] . . . P2S[c1]

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

I P(r1)S[1] P(r1)S[2] . . . P(r1)S[c1]

266664

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where I is the p p(pis a prime number) identity matrix, Pis the p p permutation matrix (pi,i1 pp,1 1, i 1, 2,. . . , p2 1; other elements of P are zeros), whereas r and crepresent the number of rows and columns, respectively.

The set of integers S are to be carefully chosen from the setf0, 1, . . . , p2 1g so that the cycles of short length, incorresponding Tanner (bipartite) graph representation ofthe parity-check matrix, are avoided. A bipartite (Tanner)graph is a graph whose nodes may be separated into twoclasses (variable and check nodes), and where undirected

edges may only connect two nodes not residing in the sameclass. The Tanner graph of a code is drawn according tothe following rule: check (function) node c is connected to

variable (bit) node v whenever elementhcv in a parity-checkmatrix H is a 1. There are N-K check nodes and N variablenodes. As an illustrative example, consider the H-matrix ofthe following LDPC code

H 1 0 1 0 1 01 0 0 1 0 10 1 1 0 0 10 1 0 1 1 0

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For any valid codeword v [v0 v1. . .vN21], the checks usedto decode the codeword are written as

Equation (c0): v0 v2 v4 0 (mod 2)




The bipartite graph (Tanner graph) representation of this

code is given in Fig. 4a. The circles represent the bit(variable) nodes, whereas the squares represent the check(function) nodes. For example, the variable nodes v0, v2 andv4 are involved in equation (c0), and therefore connected tothe check node c0. A closed path in a bipartite graphcomprising l edges that closes back on itself is called a cycleof length l. The shortest cycle in the bipartite graph is calledthe girth. The girth influences the minimum distance ofLDPC codes, correlates the extrinsic LLRs and thereforeaffecting the decoding process. The use of large girthLDPC codes is preferable because the large girth increasesthe minimum distance, and de-correlates the extrinsic info

in the decoding process. To check for the existence of shortcycles, one has to search over H-matrix for the patternsshown in Figs. 4band c. The codeword length is determined

by N jSjp, where jSj denotes the cardinality of set S, andthe code rate is lower bounded by (1-r/jSj). For example, byselectingp 1123 and S f0, 2, 5, 13, 20, 37, 58, 91, 135,160, 220, 292, 354, 712, 830g an LDPC code of rate 0.8,girth g 10, column weight 3 and length N 16 845 isobtained. For more details on LDPC codes an interestedreader is referred to an excellent book by MacKay [19].

5 Evaluation of the proposedhybrid optical network

We are turning our attention to the evaluation of theproposed hybrid optical network. We first compare theBER performance of the employed girth-10 LDPC codesagainst RS codes, concatenated RS codes, turbo-productcodes (TPCs) and other classes of LDPC codes. Theresults of simulations for an additive white Gaussian noise(AWGN) channel model are given in Fig. 5. The girth-10LDPC(24015,19212) code of rate 0.8 outperforms the

concatenation RS(255,239) RS(255,223) (of rate 0.82) by3.35 dB and RS(255,239) by 4.75 dB, both at BER of1027. The same LDPC code outperforms projective

Figure 4 Identification of short cycles in bipartite graph

representation of a parity-check matrix

a Bipartite graph of LDPC(6,2) code described by Hmatrix above.Cycles in a Tanner graphb Cycle of length 4c Cycle of length 6



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geometry (PG) (2,26)-based LDPC(4161,3431) (of rate0.825) of girth-6 by 1.49 dB at a BER of 1027, andoutperforms girth-8 LDPC(4320,3242) of rate 0.75 by

0.25 dB. At a BER of 10210

it outperforms lattice-based

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 by0.95 dB. The net effective coding gain at aBER of 10212 is

10.95 dB.

Figure 5 Large girth QC LDPC codes against RS codes, concatenated RS codes, TPCs, and previously proposed LDPC codes

Figure 6 The constellation diagrams for polarisation-multiplexed 16-QAM (the aggregate data rate is 100 Gb/s) after 500 psof DGD forsX 0.1, sY 0.05 and OSNR 50 dB observing the worst case scenario (u p/2 and 1 0) withouttransmission diversity

a Before PMD compensation

b After PMD compensation. The corresponding constellation diagrams in the presence of PMD only (for DGD of 500 ps)c Before PMD compensationd After PMD compensation



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In the simulation results shown in Figs. 6 and 7 we assumethat PMD channel coefficients are known at the receiver,because they can easily be determined by pilot-aidedchannel estimation [3, 4]. On the other hand, the FSOchannel may change significantly during the day time, andas such is difficult to estimate. To illustrate the efficiency of

this scheme, in Figs. 6a and b we show the constellationdiagrams for an aggregate rate of 100 Gb/s, correspondingto the M 16 QAM and the OFDM signal bandwidth of12.5 GHz in the presence of atmospheric turbulence(sX 0.1 and sY 0.05), before (see Fig. 6a) and after(see Fig. 6b) PMD compensation, assuming the worst-casescenario (u p/2 and 1 0). The correspondingconstellation diagrams in the presence of PMD only areshown in Figs. 6c and d. The proposed coded-modulationscheme is able to compensate for the PMD with DGD ofeven 500 ps in the presence of atmospheric turbulencecharacterised bysX 0.1 and sY 0.01.

In Fig. 7 we show the BER performance of the proposedscheme for both uncoded case (Fig. 7a) and LDPC-codedcase (Fig. 7b). The OFDM system parameters were chosenas follows: the number of QAM symbols NQAM 4096,the oversampling is two times, the OFDM signalbandwidth is set to either 10 GHz (M 32) or 12.5 GHz

(M 16) and the number of samples used in cyclicextension NG 64. For the fair comparison of differentM-ary schemes the optical signal-to-noise ration (OSNR)on the x-axis is given per information bit, which is alsoconsistent with digital communication literature [20]. Thecode rate influence is included in Fig. 7 so that thecorresponding coding gains are net-effective coding gains.

The average launch power per OFDM symbol is set to-3 dBm (and similarly as in wireless communications [20]represents the power per information symbol), and theGray mapping rule is employed. To generate thetemporally correlated samples according to (6), we used the

Figure 7 BER performance of the proposed hybrid optical network scheme

a Uncoded BER curvesb LDPC-coded BERsRD Denotes the aggregate data rate and BOFDM is the OFDM signal bandwidth. TD i: transmission diversity of order i



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Levinson Durbin algorithm [21]. From Figs. 6 and 7 it canbe concluded that PMD can be successfully compensatedeven in the presence of atmospheric turbulence. The mostof degradation is coming from the FSO channel, as shownin Figs. 6 and 7. The 32-QAM case with aggregate datarate RD 100 Gb/s performs 1.9 dB (at BER 10

26)

worse than 16-QAM (with the same aggregate rate)although the occupied bandwidth is smaller.

The net coded gain improvement (at BER of 1026) ofLDPC-coded OFDM over uncoded OFDM is between11.05 dB (M 16, sX 0.01, sY 0.01, correspondingto the weak turbulence regime) and 11.19 dB (M 16,sX 0.1, sY 0.01, corresponding to the mediumturbulence regime). The additional coding gainimprovement because of transmission diversity with twolasers is 0.19 dB for 32-QAM-based OFDM (sX 0.01and sY 0.1) at a BER of 10

26. On the other hand, the

improvement due to transmission diversity for the uncodedcase (at the same BER) is 1.26 dB. Therefore in the regimeof weak atmospheric turbulence, the improvements due totransmission diversity are moderate. On the other hand, inthe moderate turbulence regime the use of transmissiondiversity is unavoidable. Otherwise, the uncoded BER errorfloor is so high (see sX 0.5, sY 0.1 curve in Fig. 7a)that even the best LDPC codes are not able to handle, ifthe complexity is to be kept reasonably low. Withtransmission diversity, in moderate turbulence regime, weobtain BER performance comparable to the case in theabsence of turbulence regime, as shown in Fig. 7. Thestrong turbulence regime is not considered here because ofthe lack of an appropriate temporal correlation model. (Theatmospheric turbulence model described in Section 3 is nota valid model in the strong turbulence regime.)

The laser linewidths of transmitting and local laser were setto 10 kHz, so that the atmospheric turbulence, PMD and

ASE noise are predominant effects. For the influence oflaser phase noise on coherent OFDM systems, aninterested reader is referred to [22]. Note that BERthreshold required to achieve BER 1026 at the output ofthe LDPC decoder is 1.96 1022, and in this regionBER values for different laser linewidths are comparable.

6 Conclusion

We proposed a particular polarisation-multiplexed coded-OFDM scheme suitable for use in hybrid FSO fibre-optics networks. This scheme is able to simultaneously deal

with atmospheric turbulence, chromatic dispersion andPMD. We show that PMD can be compensated even inthe presence of atmospheric turbulence. We have foundthat the most of the degradation comesfrom FSO channel.

The proposed coded-modulation scheme supports 100 Gb/s per wavelength transmission and 100 Gb/s Ethernet. We

compare the BER performance of two schemes with a fixedaggregate rate of 100 Gb/s. The first scheme employs 32-QAM and occupies 10 GHz (with a bandwidth efficiency

of 10 bits/s/Hz), whereas the second scheme employs 16-QAM and occupies 12.5 GHz (with bandwidth efficiencyof 8 bits/s/Hz). We have found that the 16-QAM schemeoutperforms 32-QAM by 1.9 dB at a BER of 1026,although it has a higher bandwidth, because largerconstellation schemes are more sensitive to the atmospheric

turbulence. The net coded gain improvement (defined at aBER of 1026) of LDPC-coded 16-QAM OFDM, forsX 0.1 and sY 0.01, over uncoded-OFDM 11.19 dB.

The improvements because of transmission diversity aremoderate in weak turbulence, and significant in moderateturbulence regime.

7 Acknowledgment

This work was supported in part by the National ScienceFoundation (NSF) under Grant IHCS-0725405.

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