Electronic Dispersion Compensation - University Of Illinoissinger/pub_files/Electronic... · 2011-06-10 · A variety of optical dispersion compensation (ODC) techniques have been

© COMSTOCK & PHOTODISC

[Andrew C. Singer, Naresh R. Shanbhag,

and Hyeon-Min Bae]

Dto the steady growth in demand for bandwidth, the days whenoptical fiber was viewed as having unlimited capacity have cometo an end. Dispersion, noise, and nonlinearities pose substantialchannel impairments that need to be overcome. As a result, overthe last decade, signal processing has emerged as a key technolo-

gy to the advancement of low-cost, high-data rate optical communication systems. The unre-lenting progress of semiconductor technology exemplified by Moore’s Law provides an efficient

platform for implementing signal processing techniques in the electrical domain leading to what isknown as electronic dispersion compensation (EDC). In this article, we provide an overview of some of

the driving factors that limit the performance of optical links and highlight some of the potential opportu-nities for the signal processing community to make substantial contributions.

INTRODUCTIONThe optical fiber has traditionally been thought to have unlimited bandwidth. As a result, it has been the transmission

media of choice in backbone networks and is rapidly encroaching customer premises, enterprise networks, as well as back-plane and storage area networks. As shown in Figure 1, the key components of an optical fiber communication link are common

to those of many digital communication links. These may be broken into their roles within the transmitter, channel, and receiver.Within the transmitter, an information sequence is used to modulate the intensity of a laser source. For long-haul (LH) links

ElectronicDispersionCompensation[An overview of optical communications systems]

Digital Object Identifier 10.1109/MSP.2008.929230

IEEE SIGNAL PROCESSING MAGAZINE [110] NOVEMBER 2008 1053-5888/08/$25.00©2008IEEE

Authorized licensed use limited to: University of Illinois. Downloaded on July 27,2010 at 06:33:24 UTC from IEEE Xplore. Restrictions apply.

(typically greater than 200 km), the optical signal will propagatealong multiple spans of fiber with repeated optical amplification toovercome attenuation. In many regional or metro links (40km–200 km), the optical signal may propagate along a single spanof fiber without additional amplification. In very short-reach appli-cations, such as a local area network (LAN), the optical signal mayonly propagate a few tens to a few hunded meters through inexpen-sive, yet pervasive multimode fiber.

While the data rates for optical links exceed those of all other dig-ital communications media, in some respects, they are among theleast sophisticated, using simple on-off keying (OOK) and basebandcomparators for symbol-by-symbol data recovery. In this respect, theoptical link is quite primitive in that it has not needed to leveragethe tremendous advances in statistical signal processing and com-munication techniques that have enabled a number of importantadvances in both wired and wireless communications. These includedigital voice-band modems, cellular technologies, [such as globalsystem for mobile communications (GSM) and code division multi-ple access (CDMA)], broadband enabling technologies, (such as cablemodems and digital subscriber line (DSL) modems), and orthogonalfrequency division multiplexing (OFDM) technology in use in a hostof wireless digital transmission standards.

This trend is likely to reverse itself, as a dramatic change in thenature of optical communications occurred earlier this decadewhen carriers began to migrate from 2.5 Gb/s to 10 Gb/s transmis-sion rates. As a result of this transition, the performance of fiber

optic links in LH, metro, and enterprise networks became limitedby dispersion, or intersymbol interference (ISI), rather than onlyby noise. Figure 1 shows eye diagrams illustrating the transmittedand received symbol patterns as they would appear on an oscillo-scope at various stages through the optical fiber. Note how an opentransmit eye from which correct decisions can easily be made,begins to close due to dispersion and noise, as the distancethrough the fiber increases.

This is largely because group-velocity dispersion (GVD), or so-called chromatic dispersion (CD), in optical fibers (i.e., that lightpropagates with a wavelength-dependent velocity in fiber) increaseswith the square of the data rate. Though GVD is manageable for dis-tances of interest at data rates at or below 2.5 Gb/s, it becomes aserious impairment at 10 Gb/s. LH links at 2.5 Gb/s also need tohave built-in dispersion compensation for GVD. Polarization-modedispersion (PMD), which arises from manufacturing defects, vibra-tion, or mechanical stresses in the fiber, leading to additional pulsespreading at the receiver, also becomes serious in LH transmissionat 10 Gb/s and at even shorter reaches for higher data rates. Thetime-varying nature of PMD gives rise to the additional need foradaptive compensation at the receiver. Modal dispersion, arisingfrom geometrical properties of multimode fiber, are also consider-ably more severe at 10 Gb/s than at lower rates.

A variety of optical dispersion compensation (ODC) techniqueshave been implemented and many are practical at lower data rates.However, for 10 Gb/s systems, ODC is expensive and lacks the

[FIG1] Block diagram of an optical fiber link. The link can be broken into three key components: the transmitter, the optical channel,and the receiver. The transmitter contains a laser source that is modulated by an information bearing sequence, the channel consists ofpotentially repeated spans of optically amplified single-mode fiber, and the detector contains optical filtering, a photo-detector,electrical filtering, and subsequent electrical processing for clock and data recovery. For shorter reach links, the fiber may beunamplified and for very short reach applications, multimode fiber may be used. Also shown are data eye diagrams, which illustratethe transmitted and received symbol patterns as they would appear on an oscilloscope at various stages through the optical fiber.

FiberOptical

Amplifier

Forward ErrorCorrection

(FEC) Encoder

Symbol Mappingand Modulation

(E/O)

Laser SourceTransmitter

Fiber

0 km 75 km 125 km

Optical Channel

Optical Filter Photo Detector(O/E)

ElectricalFilter

Clock andData

Recovery

FECDecoder

Receiver

Data In

Data Out

IEEE SIGNAL PROCESSING MAGAZINE [111] NOVEMBER 2008


needed performance and flexibility. As a result, EDC techniqueshave emerged as a technology of great promise for OC-192 (10Gb/s) data rates and beyond. In addition, the unrelenting progressof semiconductor technology exemplied by Moore’s Law has pro-vided an implementation platform that is cost-effective, low power,and high performance. The next several decades of optical commu-nications will likely be dominated by the signal processing tech-niques that are used both for signal transmission, modulation andcoding, as well as for equalization and decoding at the receiver.

The design of signal processing-enhanced optical communica-tion links presents unique challenges spanning algorithmic issuesin transmitter and receiver design, mixed-signal analog front-enddesign, and very large scale integration (VLSI) architectures forimplementing the digital signal processing back-end. Though theseare tethered applications, reducing power is important due to strin-gent power budgets imposed by the systems, e.g., transponders andline cards, in which these devices reside. A cost-effective solution,i.e., a solution that meets the system performance specificationswithin the power budget, requires joint optimization of the signalprocession algorithms, VLSI architectures, and analog and digitalintegrated circuit design.

This article provides an overview of some of the driving fac-tors governing the development of current and next generationoptical communications systems, with a particular emphasis onthe tools, models, and methods by which the signal processingcommunity can play an integral role. Though texts and articleson optical communications are abundant [1]–[6], our focus ishighlighting some of the key challenges in this area in an effortto raise awareness of the potential opportunities for the signalprocessing community in this field.

SYSTEM MODELSIn this section, we briefly discuss some of the salient characteristicsin optical links of practical signal processing interest. We pay partic-

ular attention to models that can be used for system design, per-formance simulation and experimentation. These are the essentialbuilding blocks from which a signal processing development canmake a substantive and practical impact on current and future opti-cal communications systems.

THE TRANSMITTERThe transmitter uses a binary data sequence to modulate the inten-sity of a laser source. The simple transmitter model in Figure 2describes the modulation process for a variety of applications. Forexample, in short-reach applications such as 10G-Ethernet, storagearea networks, or in other enterprise applications, the data sourcemay be used to directly modulate the intensity of the laser driver,providing a low-cost, simple OOK or nonreturn to zero (NRZ) mod-ulation. While OOK is a form of linear pulse-amplitide modulation,NRZ is actually a nonlinear modulation in which the laser ampli-tude does not return to zero, but rather remains on when succes-sive ones are transmitted. Figure 2 shows that the transmit signalspectrum for an NRZ transmitter has finite bandwidth and is cen-tered at the carrier frequency of the optical source, typically around193 THz for 1,550 nm wavelength. The center frequency is 228THz for 1,310 nm wavelength, which can be obtained using therelationship f λ = c. Return-to-zero (RZ) modulation is a linearform of pulse amplitude modulation, in which the pulse shape hasduration less than a symbol period and the laser output thereforereturns to the zero state when transmitting a one. As it is difficultto completely attenuate the optical signal at such high data rates,one measure of the quality of the laser is the extinction ratio (seeFigure 2), which is a measure in decibels of the ratio of the opticalpower in a transmitted one to that in a transmitted zero, i.e.,

Er = 10 log10P1

P0. (1)

Direct modulation, for example, may achieve an extinction ratioas low as 8 dB. In less cost-sensitiveapplications, such as LH telecommuni-cations, metro area networks, or in theback-bone of a data-communicationsnetwork, laser sources with better spec-tral characteristics are often externallymodulated. This is accomplished, forexample, with a Mach-Zehnder modula-tor [1], employing an electro-optic crys-tal whose refractive index is modulatedby an applied voltage. This in turn cre-ates an intensity modulation of theapplied laser source. Such externallymodulated systems achieve closer to 15dB of extinction ratio. Unlike traditionalsignal processing applications wherehigher signal to noise ratio (SNR) usual-ly implies a higher quality signal [and alower bit-error rate (BER)], for opticalcommunications, a higher optical signal-to-noise ratio (OSNR) may not provide a

[FIG2] A simple transmitter block diagram is shown in which an informationsequence d [k ] is converted through a waveform generator to a modulation signalm(t ) = ∑

k d [k ] p (t − kT ) which is then used to modulate the intensity of a lasersource. Also shown is the finite bandwidth of the laser about the carrier fc and thefinite extinction ratio of the modulated laser source.

Δf

fc

0

WaveformGenerator

cos(2πfct)(Laser/Modulator)

d [k] m(t)

(Driver)

x(t )

2P0

2P1

1



lower BER, if the extinction ratio is poor. The power in the sig-nal component of the OSNR includes that in both the ones andthe zeros, while the BER of an optical link will be a function ofthe difference in these optical powers.

THE FIBERAfter the data has been transmitted from the source and coupledinto the optical fiber, it will propagate as an electromagnetic waveaccording to the nonlinear Schrödinger equation (NLSE)—thenonlinear wave equation governing the propagation of light infiber [2]—and undergo a number of channel impairments. Asshown in Figure 3(a), the propagating wave is supported by twoorthogonal axes of polarization, or polarization states within thefiber. The differential group delay experienced by these two modesgives rise to PMD causing pulse spreading at the output of thephotodetector in the receiver. In multimode fiber (MMF) [seeFigure 3(b)], the transmitted signal will excite a number of propa-gating modes, which will propagate at mode-dependent velocities.This results in modal dispersion at the receiver, where a singletransmitted pulse may spread into a number of adjacent symbolperiods, depending on the data rate, distance traveled, and fiberproperties.

Single-mode fiber (SMF) permits only a single optical modeto propagate thereby enabling greater propagation distances

before pulse spreading occurs. Short-reach SMF applicationstypically employ 1,310 nm wavelengths, as this represents thewavelength of minimum dispersion for silica fiber [2]. Longerreach applications use 1,550 nm wavelengths as this representsthe wavelength of minimum loss due to absorption. Thoughthe 1,550 nm wavelength enables greater transmission distancewithout amplification, it does so at the cost of necessitating dis-persion management at sufficiently high data rates. Long-reachapplications may include multiple spans of optical fiber that areperiodically amplified using optical amplifiers. The noiseinduced by the optical amplifiers eventually becomes sufficient-ly large that the data integrity is in jeopardy, after the signalpasses through a sufficient number of amplified fiber sections.At that point, optical-electrical-optical (OEO) regeneration isrequired whereby the signals are detected, converted to electri-cal format, error-corrected, and retransmitted through subse-quent sections of optical fiber.

LH links have also been designed employing a variety ofdispersion management methods to mitigate the accumulatedeffects of chromatic dispersion. These include spools ofnegative dispersion, or dispersion compensated fiber (DCF),inserted into the link to effectively cancel (to first order) theaccumulation of chromatic dispersion. Unfortunately, suchdispersion compensation comes with additional attenuation,

[FIG3] Optical fiber. (a) Single-mode fiber in which a single optical mode is coupled into two axes of polarization. After LH propagation,PMD gives rise to a differential delay τ between these two axes of polarization. (b) Multimode fiber in which multiple optical modesare coupled into the fiber (two shown using a geometric ray model). After propagation of only a few hundred meters, the pulse isdispersed due to modal dispersion. The outputs shown are three sample impulse responses developed by the IEEE 802.3aq workinggroup to model modal dispersion.

Fiber

Y-Polarization

X-Polarization

z

Y-Polarization

X-Polarization

z

(1-ρ2)1/2

ρτ(t)

(a)

Mode 1Mode 2

Z 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.10.20.30.40.50.60.70.80.9

1

Time (UI)

Am

plitu

de

PrecursorSymmetricalPostcursorδ(t)

0 t

Cladding

Cladding

Core

(b)



requiring additional optical amplification, and hence a reduc-tion in OSNR. With sufficient dispersion management, LHoptical links have been designed to be OSNR-limited, ratherthan receive power-limited. Shorter reach links are typicallyreceive-power limited and include a single unamplified spanof up to 80 km of fiber.

CDThe dependence of the index of refraction on frequency givesrise to a difference in the velocities with which the frequen-cies within the transmitted signal spectrum propagate andhence a build up of CD that spreads optical pulses in time.For single-wavelength systems, when the transmitted opticalpower is below 0 dBm (1 mW) and the propagation distanceis modest, say, less than 80 km, then the nonlinear effects ofthe fiber can be ignored to obtain a linear model for the CDeffects of the optical channel. In this case, an equivalentbaseband linear time invariant model for channel can bewritten in terms of its frequency response

HCD( f ) = exp

{− jπD

λ2

cLf 2

}, (2)

where D is the dispersion parameter at wave-length λ, L is the fiber length, and f is the equiv-alent baseband frequency. Typical fiber parametervalues are listed in Table 1. The top plots inFigure 4(a) illustrate a close match between amore accurate numerical integration of the non-linear Schrödinger equation (to be describedlater) and the linear model in (2) for 1 dBmlaunch power. However, significant differences in

behavior can be observed for +13 dBm launch power [bot-tom plots in Figure 4(a)]. The linear model is also inappro-priate for dense wavelength division multiplexing (DWDM)systems unless the wavelength spacing is sufficiently broadto diminish the resulting nonlinear effects that couple theadjacent wavelengths.

As evidenced in Figure 4(a), for high launch powers, thereis an additional accumulation of the effects of the dominantform of nonlinearity, known as the Kerr effect, which iscaused by the dependence of the index of refraction on signalintensity. The salient characteristics of LH propagation arebest described in terms of the nonlinear Schrödinger equa-tion that governs electromagnetic wave propagation in anoptical fiber [2]

E (z, t)z = −α ln(10)

20E (z, t) − β2

2E (z, t)tt

+ β3

6E (z, t)ttt + i γ |E (z, t)|2 E (z, t), (3)

[FIG4] Fiber modeling results for a 72-km fiber span. (a) Simulated eye diagrams for NRZ transmission using numerical integration of (a)the NLSE and (b) the equivalent linear model. Results are shown for (top) +1 dBm and (bottom) +13 dBm launch power. (b) Simulated(using NLSE) and measured eye diagrams for NRZ transmission (left) and a form of ODB transmission (right). Results are shown for (top)+1 dBm, (middle) +13 dBm, and (bottom) +17 dBm launch powers.

Numerical Integrationof NLSE +1dBm

Linear EquivalentModel +1dBm

Numerical Integrationof NLSE +13 dBm

Linear EquivalentModel +13 dBm

NRZ Transmission

(a) (b)

Measured Simulated

ODB Transmission

Measured Simulated

+1dBm

+13dBm

+17dBm

PARAMETER SYMBOL VALUEWAVELENGTH λ 1550 nmPROPAGATION VELOCITY c 2.997925 × 108 m/sFIBER ATTENUATION α 0.2 × 10−3 dB/mDISPERSION D 17 ps/nm-kmSECOND ORDER DISPERSION β2 = −Dc(10−6)/(2πf2

c ) 20 × 10−27 s2/mDISPERSION SLOPE β3 1.1 × 10−40 s3/mCENTER FREQUENCY fc 193.1 × 1012 HzCORE REFRACTIVE INDEX n2 2.6 × 10−20 m2W−1

CORE AREA A 90 × 10−12 m2

NONLINEARITY γ = 2πn2/λA 5 × 10−3 m−1W−1

[TABLE 1] TYPICAL PARAMETER VALUES FOR SINGLE MODE FIBERUSED IN LONG-HAUL COMMUNICATIONS.



where E (z, t) describes the electromagnetic field intensity as afunction of distance z and time t, the subscript z denotes thefirst partial derivative with respect to z, the subscripts tt and ttt

denote the second and third partial derivatives with respect to t,respectively. Typical parameter values for SMF are shown inTable 1.

A simple method exists for generating simulations overoptical fiber that have been shown to accurately reproduce lab-oratory measurements and hence can be employed to designvery high-speed commercial signal processing circuits [7].Noting that the nonlinear Schrödinger (3) can be written inthe form

E(z, t)z = (L + N ) E (z, t) (4)

where L and N are the linear and nonlinear operators,

(L)E = −α ln(10)

20− β2

2∂2E∂ t2 + β3

6∂3E∂ t3 (5)

(N )E = i γ |E|2E (6)

respectively. The split-step Fourier transform technique [8] is awidely-used method for its solution that successively solvesE (z, t)z = (L) E (z, t) and E (z, t)z = (N ) E (z, t) as illustrat-ed in Table 2. The transmitted optical signal E (t) is assumed tobe periodic, such that periodic boundary conditions can beassumed and the fast Fourier transform can be applied.Denoting E (z, ω) as the Fourier transform of E (z, t) withrespect to t, the basic approach is to divide the length L fiberinto L/dz segments of length dz and solve (5) and (6) sequen-tially in each segment until z = L is reached, at which point theoutput signal E (L, t) is obtained.

To simulate a 10 Gb/s NRZ transmission over 100 km ofSMF, an oversampled segment of data would be createdassuming periodic boundary conditions for, say, a 128-bsequence. A 10 Gb/s NRZ transmission would have most of itsenergy within a 0.2 nm (25 GHz) bandwidth, and thereforethe 100 km of SMF would exhibit a channel memory ofapproximately 340 ps corresponding to 3.4 symbols. As aresult, all 4-b patterns could be captured in a periodicallyrepeated 16-b pattern, though computations using periodical-ly repeated 128-b sequences or longer are typical. Figure 4(b)shows simulated eye-diagrams for NRZ and ODB modulationused to construct E (0, t). The simulated data is shownnoise free, while the measured data was taken at OSNRsranging from 10 dB to 20 dB. Note that the simulationscapture the significant nonlinearity that is experienced inthe high-launch power (+17 dBm) data for the NRZ trans-mission, as well as the relatively unperturbed ODB trans-mission at the same launch power. Longer sequences ofdata can be created by periodic concatenation, due to theassumed periodic boundary conditions. Noise realizations

can then be added to these longer sequences to enable practi-cal BER measurements.

PMDRecall from Figure 3(b) that single mode fibers support two axesof propagation, with each distinguished by its polarization [9].Optical birefringence in the fiber then causes these two axes topropagate at slightly different velocities, leading to PMD.Random changes in this birefringence along the fiber results inboth a random coupling between the modes as well as a statisti-cally time-varying nature to PMD. Excellent treatments existoutlining the basic theory and fundamental concepts of PMD[9]. We will restrict our attention to first-order models for PMD,which describe the differential group delay (DGD) of the fiber.

As shown in Figure 3(a), when light is coupled into the opti-cal fiber, a fraction ρ2 of the light energy will couple into one ofthe orthogonal polarization states of the fiber while the remain-der (1 − ρ2) couples into the other polarization state. These twostates will experience different group velocities over the thecourse of propagation through the fiber resulting in the pulsearrival times being separated by τ(t). The net result, to firstorder, is that PMD gives rise to a splitting of the optical power ina transmitted pulse into two pulses which arrive at the receiverat different times. When the optical signal is directly detectedafter propagating a distance L through a fiber with attenuationα, the effective optical channel tends toward a linear, time-varying system with impulse response

h(t, τ ) = e− ln(10)αL20

(ρδ(t − τ0) +

√1 − ρ2δ(t − τ0 − τ(t))

),

(7)

where τ0 is the baseline delay of the fiber and τ(t) is the instan-taneous DGD of the fiber due to first-order PMD. The meanDGD of the fiber, expressed in ps/

√km, gives a measure of the

expected delay between the two polarization axes. Longer linksand higher data rates are more adversely affected because theDGD is cumulative with distance. Also, adaptive receivers arerequired to mitigate PMD since DGD varies with time. ABellcore study [10] of field data measured a large cross sectionof cable plants, domestic and international, and indicated thatroughly 35% of pre-1991 fiber have mean DGD in excess of 1ps/

√km while only around 5% of post-1990 fiber have mean

DGD in excess of 1 ps/√

km. Solutions are sought that can mit-igate (or avoid) PMD over links with high values of mean DGDbecause replacement of installed legacy fiber with new fiber is

GIVEN E(t, 0)

WHILE z < L DO:E(z + dz/2, ω) = E(z, ω) exp

(−dz((α′/2) + j(β2ω

2/2) + j(β3ω3/6)

))E(z + dz, t) = E(z + dz/2, t) exp (jdzγ |E(z, t)|2)z = z + dz

END

[TABLE 2] SPLIT-STEP FOURIER METHOD FOR NUMERICALINTEGRATION OF THE NLSE.



too expensive. A good model of the variability of PMD over timein a fiber is given by the Maxwellian distribution for the value ofinstantaneous DGD τ(t) as a function of the mean DGD of thefiber [11]. That is, the marginal distribution for the DGD τ , in agiven fiber with mean DGD of τm, is given by

fτ (τ ) = 32τ 2

π2τ 3m

exp

{− 4τ 2

πτ 2m

}.

From this marginal distribution, we see that the instantaneousDGD of a fiber will exceed three times its mean DGD with proba-bility 4.2 × 10−5.

MODAL DISPERSION IN MMFMost of the fiber installed in buildings, campus networks, andother short-reach local area networks contain MMF, whichderives its name from the existence of multiple propagating spa-tial modes through the core of the optical fiber [see Figure 3(b)].

In SMF, the indices of refraction in the cladding and the core arecarefully controlled to allow only a single propagating mode. InMMF, tens to hundreds of propagating modes, depending on thefiber geometry, may be allowed. MMF can often provide the nec-essary bandwidth for short-reach applications at a fraction of thecost of SMF-based systems, due to the ease of its manufacture,packaging, and optical alignment [12].

A simple model for signal propagation in MMF captures thecoupling of the modulated laser source onto a set of Nm propa-gating modes, each of which propagates a slightly different dis-tance through the fiber [see Figure 3(b)]. If the detector area issufficiently large, then the output of a square-law detector yieldsa sum of powers of the modal arrivals since the modes are spa-tially orthogonal. The following finite-length impulse responsemodel for MMF propagation can be obtained when modal noiseand mode coupling is ignored

h(t) =Nm − 1∑k= 0

e− ln(10)αL/20 hk δ(t − τ0 − τk), (8)

where ∑Nm−1

k=0 |hk|2 = 1, Nm is the number of propagatingmodes, an amount equal to |hk|2 of the incident power is cou-pled into the k th mode, τk is the differential modal delay withrespect to the overall baseline delay of τ0 induced by the wave-length of the source, α is the attenuation, and L is the fiberlength. In the presence of modal coupling, the terms hk wouldbecome time-varying and can be modeled as Rayleigh fadingunder suitable assumptions on mode orthogonality [13].

RECEIVER MODELSAt the receiver, the optical signal is typically optically filtered toremove out-of-band noise, or to perform wavelength divisiondemultiplexing, and then passed on to a detector for optical-to-electrical conversion. Typical detectors include the avalanche

photodiode (APD) and the p-i-n (PIN)diode, each of which produce an electri-cal current proportional to the incidentoptical power. A transimpedance ampli-fier (TIA) then provides a voltage of ade-quate signal level for subsequentprocessing. Synchronization of a localoscillator to the receive data signal is asevere challenge in the multi-GHzregime, and as such, the clock must berecovered from the data signal itself. Astandard electrical receiver, referred toas a clock-data recovery (CDR) device,includes a system of phase-locked loops(PLLs) to recover the sampling clock,followed by a comparator (slicer) tomake symbol-by-symbol decisions. TheCDR, which is the most commonlydeployed receiver in optical links today,employs little or no signal processing.

DETECTORSIntensity modulated optical signals are detected (converted toelectrical signals) using a photodetector that converts incidentoptical power to an electrical current followed by a TIA whichproduces an output voltage. Modeling the intensity modulationas E (t) = m(t) cos (2π fct) , with fc around 200 THz, as inFigure 5 a photodetector, in the absence of noise and dispersion,can be viewed as a bandlimited square-law device, approximatelyproducing an output current

i(t) = R∫ t

t−τ

|E(t)|2dt = R∫ t

t−τ

|m(t)|22

dt,

where R is the responsivity of the detector (typically between0.7 A/W and 0.85 A/W for APD and p-i-n receivers), owing to thefinite integration bandwidth of the detector, where 1/τ � fc istypically on the order of 10 GHz, for a 10 Gb/s link, for example.

[FIG5] A block diagram model for a photodetector receiver is shown as an ideal square-law detector followed by a low-pass filter. The receive optical field E(t ) is shown withoptical power spectrum E(f ) before detection. After detection, the spectral components atDC and at 2fc are shown prior to the bandwidth limitation of the photodetector.

fc

E(f )

−2fc −2fc400 THz

0

E(t) i(t)

i(f )

Photodetector

2

LPF

10 GHz




The double-frequency term induced by the quadratic detector istherefore filtered out, leaving only the baseband demodulatedintensity signal m2(t), in the absence of dispersion. Sincem(t) > 0 for intensity modulation, the data can be completelyrecovered from m2(t) by a simple CDR in the absence of noiseand dispersion.

In the presence of noise, the detector can be modeled as asimple bandlimited square-law device, i.e.,

i(t) = R((Ex(t) + no,x(t))2 + (Ey(t) + no,y(t))2) + ne(t),

where the x and y polarizations are squared separately, eachwith their own signal and ASE noise components, and ne(t) isthe electrical noise seen at the detector. For unamplified links,no,x = no,y = 0 and additive Gaussian noise ne(t) of appropri-ate variance can be added after square-law detection.

The variance of ne(t) is derived from the sensitivity of thedetector, where the sensitivity of a detector (in decibels) isthe receive optical power level at which the BER, due todetector noise (assumed Gaussian) alone (in the absence ofdispersion), will be better than 10−12. Typical APD sensitivi-ties are in the range of −26 dBm with an overload for receivepowers in excess of −5 dBm. Typically used for amplifiedspans, PIN receivers have sensitivities in the range of −20dBm but can operate with receive optical powers in excess of+0 dBm. Unamplified spans of fiber typically use APDsbecause of their increased sensitivity. Detector noise inamplified spans can be neglected as optical noise and disper-sion will dominate BER calculations.

ANALOG FRONT-END (AFE)The AFE refers to the linear signal processing blocks in thereceive path. These include the TIA and the variable gain ampli-fier (VGA). The TIA is a linear amplifier that takes an input cur-rent iin(t) and generates an output voltage vo(t) where

vo(t) = −iin(t)RfAv

1 + Av, (9)

where Rf (Av/1 + Av) is the transimpedence (in ohms) and Av

is the open loop gain of the TIA amplifier kernel. Typical valuesof Rf reside in the range 800 � to 2 k�. The 3 dB bandwidth ofthe TIA fc,tia is typically designed to be approximately 70% ofthe symbol rate in order to achieve a balance between ISI andSNR. The performance of the TIA is also specified in terms ofits sensitivity.

A VGA is typically used to boost the receive signal level ifsubsequent processing will include more than a simple CDR.Linearity of the detector-TIA-VGA combination is importantif an ADC is used, in order to preserve the information pres-ent in the amplitude of the received signal. The VGA can bemodeled as a transconductance stage with an output loadresistor RL and an output capacitance CL. The low-frequencygain of such a stage is given by

Av = −gm RL, (10)

where gm is the transconductance of the transistor, and its 3 dBbandwidth is given by

fc,vga = 12π RLCL

. (11)

ANALOG-TO-DIGITAL CONVERTER An analog-to-digital converter (ADC) is typically modeled as anideal sampler followed by a memoryless quantizer. Such mod-els do not account for the many nonidealities that exist inpractice especially in the multigigahertz regime. A practicalADC consists of a band-limited front-end, a sampler, and aquantizer and may exhibit hysteresis and other pattern-dependent behavior. Nonidealities of the ADC are often aggre-gated in terms of the effective number of bits (ENOB) of theADC, which attempts to quantify the resulting quantizationnoise as if it were additive and independent of the input. At 10GHz, ADCs with 3–4 bits of resolution have been producedusing SiGe bipolar processes [7], [14] and more recently incomplimentary oxide-metal-semiconductor (CMOS) [15].

CLOCK-RECOVERY UNIT The clock-recovery unit (CRU) generates symbol timing infor-mation and the receiver clock from the received analog signalin the presence of dispersion, noise, and input jitter. A PLLforms the basis of the operation of a CRU. While conventionalPLLs generate significant jitter and cycle slips for long fiberlengths and at low OSNR, in LH applications, the CRU needs tosatisfy synchronous optical network (SONET) specifications onjitter transfer, jitter tolerance, and jitter generation. Jittermanifests itself as a random spread in the zero-crossings of asignal. The input jitter Jin is a function of the input SNR(SNRin), for which an approximate relationship is given by

Jin = kTtr√

SNRin(12)

where Ttr is the transition period, and k is a dimensionlessconstant.

A typical CRU uses a wideband PLL for tracking the jitterin the input signal so that the correct sampling phase is usedfor the data. However, the resulting recovered clock at theoutput of the PLL will contain jitter. A second narrowbandPLL is used to clean-up the jitter in the clock at the output ofthe first PLL. This requires the use of a first-in-first-out(FIFO) buffer at the output of the sampler to align data andoutput clock.

NOISE MODELSStatistical signal processing for optical channels provides anumber of new challenges owing to the nonlinearities presentin the channel coupled with the non-Gaussian and signal-dependent noise statistics. In amplified links, the dominantsource of noise as observed at the receiver is amplified


spontaneous emission noise, or ASE noise (see Figure 6). At mod-est transmission powers, ASE noise can be modeled as additive,circularly symmetric, white complex Gaussian over the spectralbandwidth of the transmitted optical signal. The OSNR is com-puted as the ratio of the signal power in the transmitted opticalsignal to the ASE noise power in a reference bandwidth at a refer-ence frequency adjacent to the transmitted signal. The typical noise refer-ence bandwidth used is 0.1 nm, which is approximately 12.5 GHz for1550 nm transmission. This is computed by noting that f = c/λ andtherefore | f | = (c/λ2)| λ| = 12.478 GHz, for λ = 1,550 nmand | λ| = 0.1 nm. As such, to simulate a given OSNR level, the noisevariance used in a Ts-sampled discrete-time simulation should be

σ 2ase = E {|E(t)|2}10−OSNR/10 Wsamp

Wref,

where Wsamp = 1/ Ts, is the sampling rate of the numerical sim-ulation, Wref = 12.5 GHz for a 0.1 nm noise resolution band-width and OSNR is the target OSNR. Since the signal willpropagate along two orthogonal axes of polarization, to ade-quately model the ASE noise, a separate noise simulation can berun for each polarization axis with the same target OSNR andthe results added after detection.

Modeling ASE noise as additive and Gaussian in the opticalchannel enables simulation and numerical studies to be carriedout using familiar methods. However, for directly detected opticalsignals, noise that was additive and Gaussian in the opticaldomain, becomes signal-dependent and non-Gaussian in the elec-trical domain. In particlar, denoting by r(t) the output of thedetector at time t and by y(t) the noise-free output of the opticalchannel, it can be shown [16] that the conditional probability den-sity function for r(t) given the value of y(t) is a signal-dependentnoncentral chi-square distribution with 2M degrees of freedom,

pr(t)(r | y(t) = y) = 1N0

(ry

)(M−1)/2

exp{− r + y

N0

}

× IM−1

(2√

ryN0

), (13)

where N0 is related to the average optical noise power over thebandwidth of the received optical signal, M is the ratio of theoptical to the electrical bandwidth, and Ik(.) is the kth modifiedBessel function of the first kind. The signal-dependence of thedetector output statistics are clearly seen through its depend-ence on the noise-free signal y(t).

END-TO-END SYSTEM MODELFor modeling or numerical simulation, each of the componentmodels given in this section can be combined as shown inFigure 7 to create an end-to-end system model for an opticallink, from transmitted information sequence to output detect-ed symbol sequence. The process begins with the informationsequence, which is mapped onto a modulating signal used tomodulate the intensity of the laser source. The optical signal isthen coupled into the optical fiber, which is modeled usingeither the linear CD model (2), MMF model (8), PMD model(7), or the NLSE model (3). The effect of any optical amplifica-tion is to introduce optical (ASE) noise, no(t) into each of thepolarization states of the optical signal. The optical signal isoptically filtered and converted to an electrical signal, at whichpoint electrical noise ne(t) is introduced from the detector-TIA-VGA chain, which also acts as a bandlimiting filter. Theelectrical signal is then sampled using a comparator or amultibit ADC from which the information sequence can thenbe estimated.

SIGNAL PROCESSING METHODSRecently, a number of efforts have emerged to bring signal pro-cessing into optical communcation links and employ a varietytechniques for EDC. In multimode fiber, analog feed-forwardequalizers (FFEs) and decision feedback equalizers (DFEs) havebeen proposed to mitigate modal dispersion. In such cost-sensi-tive applications, only relatively simple, CMOS-based tappeddelayline structures can be employed, as the receivers arerestricted to consume substantially less than 1W of electricalpower. Longer reach applications have opened up the possibilityof employing an ADC and subsequent digital processing of thereceived signal, including adaptive maximum likelihood


[FIG6] Block diagram link model shows two primary sources of noise are indicated. The primary source of noise in an amplified link isASE noise, indicated as optical noise no(t ), at the input the optical amplifier. In unamplified links, the dominant source of noise is theelectrical receiver noise ne(t ), which encompasses all sources of noise in the receiver, including that of the detector, TIA, andsubsequent electronics.

Tx

no(t)

d [k]

FAFiber Fiber

ne(t)

TIA

PD

FA: Fiber Amplifier

PD: Photo Detector



sequence estimation (MLSE)-based receivers [4], [5], [7], [17],[18]. The latter are known to be the optimal [in terms of mini-mizing sequence error-rate (SER)] receivers for digital transmis-sion over channels with memory [19]. Forward error correctionhas been used in optical links and typically has consisted of largeblock codes, though recent investigations have included bothconvolutional codes as well as the prospect of joint equalizationand decoding, or so-called turbo equalization [17], [20].

In this section, we will explore some of the basic models andstructures that have been proposed and brought into practicethat rely on digital signal processing methods to improve theperformance of optical links.

MODULATION FORMATSA variety of approaches for improving the performance of digitalcommunications can take place in the transmitter, such as cod-ing, adaptive modulation, and precoding. Since the dominantsource of distortion for 10 Gb/s LH links is CD, whose severity isproportional to the spectral width of the transmitted signal,more spectrally efficient modulation formats than OOK haverecently been proposed. In particular, a form of optical duo-binary (ODB) modulation is a three-level modulation driven by astate-machine (precoder), in which the transmitted signalincludes ones and zeros as in NRZ, however the carrier is addi-tionally phase-modulated by either 0 or π radians, depending onthe state of the precoder. As a result, the transmitted powerspectrum of ODB-modulated signals is roughly half that of NRZmodulation. For example, the power spectrum assuming rectan-gular pulse shapes, is given by

SODB ( f ) = Tsin2(2π f T)

(2π f T)2 (14)

SNRZ ( f ) = Tsin2(π f T)

(2π f T)2

(1 + 1

Tδ( f )

), (15)

which translates into roughly twice the dispersion tolerance/transmitted distance for links employing ODB, over thoseemploying NRZ [21].

ADAPTIVE EQUALIZATIONRelatively simple adaptive filtering techniques are readily able toelectronically mitigate modal dispersion using adaptive equaliza-tion [22]–[24] since a linear, time invariant (or slowly time-vary-ing—on the order of 10s of Hz [23]) model is a good approximationto the channel response for multimode fiber. The IEEE 802.3aqworking group [23] has developed preliminary standards for testingsubsystems based on adaptive equalization of MMF links and anumber of companies have released products based on EDC forMMF. Figure 3 shows three different MMF impulse responses devel-oped by the IEEE 802.3aq working group for assessing the perform-ance of adaptive equalization receivers for operation at 10 Gb/s andup to 300 m of fiber. Amplitudes of the three test responses areshown as a function of the bit period unit interval (UI).

The effects of modal dispersion from the input to the modula-tor to the output of the photodetector can be resonably wellmodeled with the following linear and slowly time-varying model

r(t) =∫ ∞

−∞h(τ, t)m(t − τ)dτ + w(t),

where w(t) is white Gaussian detector noise, m(t) =∑n d[n]p(t − nT) is the modulation waveform with pulse

shape p(t) transmitted per bit interval T and h(τ, t) is the equiv-alent electrical impulse response for the MMF and can bederived from (8) assuming mode orthogonality. The output r(t)can be written in terms of the transmitted bit sequence d[n] as

r(t) =∞∑

k=−∞d [k ] g(t − kT) + w(t),

where g(t) = h(t) ∗ p(t) is the convolution of the transmitpulse with the channel impulse response assuming thath(τ, t) = h(τ), i.e., that it is time invariant. It is well known[25] that the optimal receive filter for such a linear modulationover a linear channel is a matched filter to the transmit pulseg(t) and that a set of sufficient statistics for the optimal detec-tion of the sequence d[n] (for any criteria of optimality) can beobtained by sampling the output of this matched filter at thesymbol-synchronous intervals, t = nT.

[FIG7] An end-to-end system model for a fiber link is shown. The input to the link model is a binary information source, d [n ], which isthen used to modulate the intensity of a laser source. An amplified span of optical fiber is shown with the amplifier replaced by an ASEnoise source, no(t ). The receiver model includes an optical filter and square-law detector followed by a low-pass filter to emulate theoperation of the PIN/TIA/VGA detector chain. The electrical signal is then sampled with an ADC and symbol decisions d [n ] are madebased on these samples. A standard receiver (CDR) is a 1-b ADC has only 1 b, while a more sophisticated EDC-based receiver needs amultibit ADC if subsequent equalization and symbol detection is done digitally.

(PhotoDetector)

2

(TIA/VGA)

WaveformGenerator

cos(2πfct)(Laser/Modulator)

d [n] m(t)

(Driver)

Fiber

no(t)(Fiber

Amplifier)

ne(t)

1/T

EQ

(ADC)

d[n]^

(DSP)

Transmitter Receiver

(OpticalFilter)


An attractive means of implementing such a matched filterwhen the channel response is not known precisely in advance, isthe use of a fractionally-spaced adaptive equalizer, whose outputcan be written

y[n] = y(nT) =N∑

k= 0

ckr(nT− kτ), (16)

where the the delay-line spacing of the N taps in the equalizer, τ ,is typically a fraction of the symbol period, i.e., τ = T/Mf , whereMf = 2 is typical. The coefficients of the tapped delay line, orFFE, can be adapted using a variety of methods such as the least-mean square (LMS) algorithm, followed by subsequent symbol-spaced sampling. In principle, y[n] can exactly reproduce the T-spaced outputs of the optimal continuous-time matched filterg(nT− t) when the highest frequency in r(t) is less than 1/2τ ,i.e., τ -spaced sampling satisfies the sampling theorem.Fractionally spaced equalizers have the added benefit of havingperformance that is relatively insensitive to the sampling phase ofthe receiver, which is an important feature in the optical commu-nications, where the recovered clock is often fraught with jitter.

For MMF applications, T/2-fractionally-spaced equalizershave been implemented with reasonable success using as manyas 14 coefficients in the FFE, covering a span of seven T-spacedsymbols [6], [23]. A block diagram of a τ -spaced FFE is shown inFigure 8(a). FFE structures are typically implemented with ana-log tapped delay-line circuits in CMOS for MMF applications,where the cost (and therefore the power consumption) of atransceiver need to be kept to a minimum. Some of the chal-

lenges that arise include the realization of accurate, low-lossdelay-lines, such that the spacing τ between adjacent taps areequal and equal to a fraction of the bit-period, and that the sig-nal does not decay too substantially as it is passed from theinput to the end of the delay line. Another challenge is accurateimplementation of the coefficient multiplication at each tap inthe filter. As a result, for symbol decisions d[n] = 1 for y[n] > η

and d[n] = 0 for y[n] ≤ η and error signal e[n] = (d[n] − y[n]),the coefficient update algorithms using LMS, for step-size μ,

ck[n] = ck[n − 1] + μr(nT− kτ)e[n], k = 0, 1, . . . , N − 1

need not be overly complex, since the resolution with which themultiplications (16) can be carried out is limited. In a numberof practical implementations targeting modest levels of disper-sion, the error signal e[n] is replaced with a measure of theequalized eye opening at the decision device and the coefficientupdates are implemented using continuous-time feedback con-trol loops [6].

While feed-forward linear equalizers can perform well for lin-ear channels with modest frequency selectivity, they are not par-ticularly well-suited to channels with deep spectral nulls. Forsuch channels, the DFE shown in Figure 8, is a natural exten-sion of the FFE. A DFE typically consists of an FFE with an addi-tional linear (or nonlinear) filter used to process past symboldecisions d[n] in order to cancel postcursor ISI. Assuming pastsymbol descisions are correct, then the FFE portion of a DFEconverts as much of the precursor ISI as possible, within itsdegrees of freedom, into postcursor ISI, the effects of which will

[FIG8] Linear equalization techniques: (a) block diagram of an FFE and (b) block diagram of a fractionally spaced DFE.

r(t)

c1c0

y [n]

c1 cN1-1

cN-1

T

T

b1

T

y[n]

Feed-Forward Equalizer

Feedback Equalizer

b2

bN2

c0

r(t) r(t-τ)

r(t-τ)

r(t-2τ)

d [n]

d [n]

d [n-1]

d [n-2]

τ τ

r(t-2τ)τ τ

(a) (b)



be subtracted off by the feedback filter. The output of the DFEwith a linear feedback filter is given by

y[n] =N1−1∑k= 0

ck[n]r(nT− kτ) −N2∑

j= 1

bj[n]d [n − j], (17)

whose LMS coefficient updates are given by

ck[n] = ck[n − 1]+μr(nT− kτ) e[n],

k =0, 2, . . . , N1 − 1 (18)

bj [n] = bj [n − 1]+μd [n − j ] e[n],

j =1, 2, . . . , N2. (19)

EQUALIZATION OF NONLINEAR CHANNELSUnlike short-reach MMF applications, the effects of CD andPMD in longer reach applications cannot be modeled as linearand slowly time varying. The effect of direct (square-law)detection on such optically linear impairments is to makethem nonlinear in the electrical domain. The reason thatmodal dispersion remained linear through direct detection hasto do with the spatial orthogonality of the optical modes.Unfortunately, no such properties hold when there is chromat-ic dispersion present in the optical field. As a result, there is nodiscrete-time equivalent baseband channel, and adaptiveequalization techniques, such as the FFE or DFE discussedpreviously do not provide much benefit for SMF applications.Some success for small amounts of CD has been achievedusing DFEs with state-dependent thresholds [4] or with non-linear feedback taps, i.e., where the feedback filter contains notonly prior detected bits, but also pairwise products [6].

Rather, the SMF channel is best described as a nonlinear con-tinuous-time channel with finite memory whose output at thedetector can be written

r(t) = S(t; d [n], d [n − 1], . . . , d [n − Lc]) + n(t), (20)

where n(t) comprises optical noise (see the section “NoiseModels”) in an amplified link or electrical noise (see the sec-tion “Receiver Models”) in an unamplified span of fiber, andS(t; d[n], d[n − 1], . . . , d[n − Lc]) describes the noise-free,state-dependent, component of the signal, which is a func-tion of the Lc adjacent transmitted symbols. Sufficient statis-tics for optimal sequence detection [26] for suchcontinuous-time channels with memory arise from a baud-sampled (with perfect symbol timing) bank of matched fil-ters—one matched to the conditional mean waveform thatwould be observed for each possible state of the channel, i.e.,the 2Lc signals S(t; 0, 0, . . . , 0) up to S(t; 1, 1, . . . , 1) . Forexample, for a channel with CD that spans five symbol peri-ods, with binary signaling, a bank of 32 continuous-timematched filters would be required followed by 32 ADCs, asshown in Figure 9(a).

However, since the detected electrical signal is bandlimited to(at most) twice the bandwidth of the optical waveform (owing tosquare-law detection), then an oversampled ADC satisfying thesampling theorem could therefore comprise a set of sufficient sta-tistics for subsequent data detection. Note that in order to performthe appropriate matched filter operations, knowledge of (and syn-chronization with) the transmitted symbol timing would berequired. Further data reduction to one sample per bit periodcould be achieved by digitally performing the state-dependentmatched filters, from which a set of 2Lc statistics per symbol peri-od could be passed to a subsequent MLSE algorithm as shown inFigure 9(b), assuming oversampling by a factor of two. For datarates at or above 10 Gb/s, processing 32 such matched filters atline rate, together with a 32-state MLSE engine poses significantimplementation challenges.

MLSEThe optimum receiver in terms of minimizing the BER for a non-linear channel with memory, such as the optical fiber channel, isgiven by the maximum a-posteriori probability (MAP) detector

[FIG9] Sufficient statistics. (a) A bank of 2Lc continuous-time matched filters, each sampled with perfect symbol synchronization at rateT are shown, and (b) an equivalent bank of matched filters, in which an overampling ADC at a rate of T/2 is used followed by bank of2Lc digital matched filters producing the 2Lc statistics for subsequent processing. Note that to obtain sufficient statistics, theoversampling ADC and the sample rate compressor need to be symbol synchronous.

1/T

1/T

2/T

1/T

r(t) r (t)

r0[n]

r1[n]

rN[n]

r0[n]

r1[n]

rN[n]

(a) (b)

2

2

2

s(–t;0,0,…,0)

s(–t;0,0,…,1)

s(–t;1,1,…,1)

s(–nT/2;0,0…,0)

s(–nT/2;0,0…,1)

s(–nT/2;1,1…,1)



and can be computed using the BCJR algorithm [27], which is atwo-pass algorithm, requiring a forward and backward recursionover the entire data sequence. The MLSE algorithm provides theoptimal detector in terms of minimizing sequence error rate(SER) rather than BER. The MLSE algorithm provides nearly thesame performance as MAP detection for uncoded transmissions,or when symbol detection is carried out separately from FECdecoding, using only a single forward-pass through the data.

The Viterbi algorithm [19] can be used to recursively solve forthe SER-optimal transmitted sequence consistent with the obser-vations given a channel model at the receiver that can describethe probability density governing the observations r[n].Specifically, we can define the set of possible states at time index kas the 2Lc vectors s[k] = [d [k − 1], . . . , d [k − Lc]] , whered[n] ∈ {0, 1}. In addition, we define the set of valid state transi-tions from time index k to time index k + 1 as the 2Lc+1 possiblevectors of the form b[k] = [d [k], d [k − 1], . . . , d [k − Lc]] .Using this notation, the noise-free outputs of our channel modelx[n] = f(d [n], s[n]) and channel observations r[n] =x[n] + w[n], where the noise-free outputs, the channel observa-tions, and the noise samples, w[n] can be either scalars (in the caseof baud-sampled statistics), or vectors (in the case of over-sampledchannel outputs or state-dependent matched filter outputs).

Given a sequence of channel observations r[k] ={r[k], r[k − 1], . . . , r[k − Lr + 1]} of length Lr, the channelwhen viewed as a 2Lc -state machine can exhibit 2Lc+Lr possi-ble state sequences s[k] = {s[k], s[k − 1], . . . , s[k − Lr + 1]}(owing to the shift structure of adjacent states). Each validstate sequence corresponds to a path in the trellis represen-tation of the channel (see Figure 10). The MLSE algorithmseeks a particular state sequence or a path s[k] for whichthe probability of the channel observations sequencer[k] given s[k] is maximum, i.e.,

s[k ] = arg maxs[k ]

P(r[k ]|s[k ]) = arg maxs[k ]

P(r[k ]|s[k ]) P(s[k ])

= arg maxs[k ]

P(r[k ], s[k ]) (21)

assuming that each state sequence is equally likely. Assumingfurther that the noise samples are conditionally independent, wecan write

P(r[k ], s[k ]) =Lr∏

k= 0

P(s[k+1]|s[k ])Lr∏

k= 0

P(r[k ]|s[k+1], s[k ]),

and define the branch metric

λ(b[k ]) = − ln P(s[k + 1]|s[k ]) − ln P(r[k ]|b[k ]

= {s[k + 1], s[k ]})

to quantify the likelihood that a given state transition occurred.Then, we obtain the log probability of any state sequence, orpath, as a sum of branch metrics along the state transitions inthat path

L(s[k ]) = − ln

(P(r[k ], s[k ]) =

Lr∑k= 0

λ(b[k ])

),

which is also referred to as the path metric. The label S(s[k]) isused to designate the shortest path (the path with the smallestpath metric) through the trellis ending in state s[k], which isknown as a survivor path since all longer paths also ending instate s[k] are discarded. The path metric of the survivor path end-ing in state s[k] is labeled M(s[k]), i.e., M(s[k]) = L(S(s[k])).

This is best visualized using a trellis, as shown in Figure 10,where all permissable/valid state transitions b[k] have equal prob-ability. The Viterbi algorithm is now a dynamic programming

approach to finding the path through thetrellis of smallest accumulated branchmetric and is summarized in Table 3.

A number of practical considerationsare required to implement such an MLSEalgorithm for an optical link. First, sincethe data is not packet based, but rathercontinuously transmitted, it will be neces-sary, for latency reasons, to output bitdecisions as the data is processed. This iscalled the sliding window version of theViterbi algorithm, which requires a finitelook-back window of length D. It is desir-able to make the look-back window aslong as possible, in order to reduce theprobability that more than one of the sur-vivor paths (see Figure 10) will exist attime k − D when processing at time k.However, the storage requirements andcomplexity of the algorithm per outputsymbol will increase linearly in D. Wehave found that setting D equal to a small

[FIG10] MLSE: A trellis for a four-state (Lc = 2) MLSE algorithm with Lr = 5 is shown. Thestates are labeled based on the state vectors s[k ] = {d [k − 1 ], d [k − 2 ]} with the statenumber determined as S [k ] = 2d [k − 1 ] + d [k − 2 ]. The trellis diagram illustrates validstate sequences with solid lines as paths through the trellis. Above each line connecting twostates is the corresponding symbol that would have been transmitted for that transition. Thethick line path through the trellis illustrates the path of minimum accumulated branch metric(survivor path), based on the observed sequence r[n ]. Once the most likely path is determined,the corresponding sequence of transmitted symbols can be determined.

r [k–4] r [k–3]

0 0 0

1 0 1

2 1 0

3 1 1

Most Likely Sequence

Received Sequence

1 = arg min {M(s [k])}

S (s[k] = 1) = {0,2,3,3,1} d = {1,1,1,0}Trellis

S[k] = {d [k–1],d [k–2]}

0

0

0

0

11

11

0

0

0

0

11

11

0

0

0

0

11

11

0

0

0

0

11

11

r [k]r [k–1]r [k–2]

s[k] {0,1,2,3}∈

→




multiple of the channel memory Lc works well in practice, andsetting D > 3Lc does not improve performance significantly. ForD in this range and when more than one survivor path exists atk − D, selecting the one with the smallest path metric at kappears to work well in practice.

As illustrated in Figure 10, the conventional Viterbi decoderwould employ the current received sample r[n] to compute thebranch metrics, and update the four path metrics that correspondto the likelihood of the best path (survivor path) ending at each ofthe four states in one symbol period. The survivor paths arestored in memory. The path metric update step is followed by thetrace-back step where one of the survivor paths is traced back to apredetermined depth, D,and a decision is made. The survivor pathmetric update is performed using an add-compare-select (ACS)operation (see Table 3), which is recursive and therefore also diffi-cult to implement at optical line rates.

To carry out the branch metric computations, a probabilisticmodel for the observations r[k] given the state transitions b[k],i.e., given the transmitted bits d[k], . . . , d[k − Lc], is required.In a practical application of EDC, this model must be deter-mined adaptively from the channel observations r[n] withoutthe aid of any training, i.e., without any knowledge of the truebit sequence d[n].

One such model for a baud-sampled receiver uses an adaptivesystem identification algorithm based on a Volterra series expan-sion of the channel impairments [7], [18], [28]

r[n ] = c0 +Lc∑

k= 0

akd [n − k ] +Lc∑

k= 0

Lc∑�= 0,� = k

× bk,�d [n − k ]d [n − �] + · · · , (22)

where r[n] = r(nT) , are baud-sampled receiver outputs,d[n] ∈ {0, 1} are assumed transmitted channel symbols, andwhose parameters c0, ak, and bk,� can be adaptively estimatedusing the LMS algorithm with step-size μ

c0[n] = c0[n − 1] + μe[n] − ν(c0[n] − c0[0]), (23)

ak[n] = ak[n − 1] + μd [n − k ]e[n]

− ν(ak[n] − ak[0]), 0 ≤ k ≤ L (24)

bk,�[n] = bk,�[n − 1] + μd [n − k]d [n − �]e[n]

− ν(bk,�[n] − bk,�[0] 0 ≤ k, � ≤ L (25)

where e[n] = (rADC[n] − r[n]), for ADC outputs rADC[n], d[n]are past decided bits from the MLSE algorithm, and ν is a step-size parameter for the regularization terms c0[0], ak[0], andbk,�[0], which are used to incorporate prior knowledge and tomitigate accumulation of finite-precision effects. In a practicalimplementation, LMS updates can be performed at a large frac-tion of the line rate to reduce power, which also enables the useof MLSE outputs, or perhaps FEC-corrected outputs, whenmaking LMS updates. Given the model coefficents, (22) can beused to precompute the noise-free outputs for each state transi-tion in the trellis. The branch metrics can then be computed

using either a parametric or nonparametric model for the noisestatistics within each state. We have found that a suitably cho-sen parametric model for the noise statistics [7], e.g., state-dependent Gaussian, works well in practice, requires relativelyfew parameters to be estimated, and enables the use ofEuclidean branch metrics, which simplifies the architecture.

For the finite-state channel model, when the analog front-end of the receiver has an ADC with q bits of resolution, the out-put statistics can be nonparametrically modeled as 2Lc

state-dependent probability mass functions, pr(k|s), wherek = 0, . . . 2q and s = 0, . . . , 2Lc − 1. By collecting histogramsof the observations in each state, then, from training data, usinga simple Bayesian estimator, a recursive estimate of the proba-bility mass functions pr(k|s) can be shown [18] to take the form

pr(k|s)(n) = (1 − λ)pr(k|s)(n−1) + λ pk,

where pk is the kth entry of the normalized sample histogramover the nth observation interval, i.e., pk = l(k)/Nh, wherel(k) is the number of observations of bin k over a duration ofNh observations. Similarly, pr(k|s)(n−1) is the estimate afterthe (n − 1)th (prior) observation interval. While this approachworks well with training data, and can be used to provide reli-able estimates of the observational distributions for comput-ing limits of performance, it relies on accurate knowledge ofthe true state s in the collection of state-dependent his-tograms pr(k|s). If run without training data, then when oper-ating in a decision directed mode, any bit errors will naturallylead to histogram errors, which will tend to reinforce errorsin the tails of the distributions.

However, we have observed that by employing a kernel-based approach, these error-propagation effects can be readilymitigated. We update not only bin k but also bins k − 1 andk + 1 when an observation falls in bin k.In doing so, the esti-mated histograms run in decision-directed mode, withouttraining, closely approximate their estimates obtained withtraining, i.e., the empirically correct histograms. Specifically,we have found that by substituting the update of the probabili-ty in the kth bin with pk = (l(k − 1) + 2l(k) + l(k + 1))/4Nh ,and handling the edge cases such that l(−1) = l(0) andl(2q) = l(2q − 1), the sensitivity of the histogram-base methodto its initial estimate is practically eliminated along with thedetrimental effects of error propagation.

SET:S(s[k]) = s[0], S(m) = m, m = s[0]M(s[0]) = 0, M(s[m]) = ∞, m = s[0]

FOR: k = 0, . . . , K DOM(s[k + 1]) = mins[k](M(s[k]) + λ(b[k] = (s[k + 1], s[k])))FOR EACH s[k + 1]

OUTPUT: S(s[K] = c) FOR c = arg mins[K] M(s[K]).

[TABLE 3] THE VITERBI ALGORITHM. SURVIVORS S(s[k]) ARETHE SHORTEST PATHS ENDING AT EACH NODE s[k].SURVIVOR PATH METRICS M(s[k]) ARE THE PATHMETRICS ALONG THE SURVIVOR PATHS,M(s[k]) = L(S(s[k])).


CASE STUDY OF AN OC-192 EDC-BASED RECEIVERDesigning EDC-based receivers at OC-192 rates, i.e.,9.953–12.5 Gb/s, poses numerous signal processing and mixed-signal circuit design challenges. In this section, we illustratesome of the issues that arise in implementing EDC usingmixed-signal ICs by describing the design of an MLSE receiver[7] with adaptive nonlinear channel estimation. Key issuesinclude the design of a baud-sampled ADC, clock recovery inthe presence of dispersion, nonlinear channel estimation inthe absence of a training sequence, and the design of a high-speed Viterbi equalizer. This is the first reported design of acomplete MLSE receiver design meeting the specifications ofOC-192/STM-64 LH, ultra-LH (ULH), and metro fiber links atrates up to 12.5 Gb/s.

MLSE RECEIVER ARCHITECTUREThe MLSE receiver was designed as a two-chip solution in twodifferent process technologies that were chosen to match thefunctionality to be implemented. The analog signal condition-ing, sampling, and clock recovery functions were implementedin an AFE IC in a 0.18 μm, 3.3 V, 75 GHz SiGe BiCMOS process.The digital MLSE equalizer and the nonlinear channel estimatorwere implemented in a digital equalizer (DE) IC in a 0.13 μm,1.2 V CMOS process.

The architecture of the MLSE receiver is shown in Figure 11.The received signal is at a line rate of between 9.953 Gb/s foruncoded links and up to 12.5 Gb/s for FEC-based links. Thereceived signal is amplified by a VGA then sampled by a 4-b flashADC. A dispersion-tolerant CRU recovers a line rate clock for theADC. The 4-b line rate ADC samples are demuxed by a factor of1:8 to generate a 32-b interface to the DE IC, which implementsa four-state MLSE algorithm, i.e., it assumes a channel memoryof three symbol periods. The digital equalizer includes a blind,adaptive channel estimator of the form in (22) and a data decod-ing unit on-chip that requires no training.

MLSE EQUALIZER ALGORITHM AND VLSI ARCHITECTUREDesigning an MLSE equalizer at OC-192 line rates (9.953–12.5Gb/s) is extremely challenging because of the high data ratesinvolved. Conventional high-speed Viterbi architectures oftenemploy parallel processing or higher radix processing [29].Parallel/block processing architectures tend to suffer from edge-effects while higher-radix architectures have been shown toachieve speed-ups of up to a factor of two, which is not sufficientfor this application.

Instead, the architecture in [7] and [30] (see Figure 12)incorporates a number of techniques that enable the MLSEcomputations to complete within the bounds imposed by thehigh OC-192 line rates. First, the channel trellis is traversed in atime-reversed manner [see Figure 12(a)]. Doing so eliminatesthe trace-back step and the survivor memory associated withconventional approaches thereby providing a considerablepower saving while enhancing throughput.

Second, the ACS bottleneck associated with conventionalapproaches is eliminated by restricting the survivor path memo-ry to a reasonably small number (6) of symbol periods. With afixed survivor depth D, and time-reversed traversal of the trellis,it becomes possible to implement the ACS computations in afully feed-forward architecture. This feed-forward ACS unit isreferred to as the path-finder block in Figure 12. A feed-forwardarchitecture [31] can be easily pipelined in order to meet anarbitrary speed requirement.

Third, in order to avoid edge effects, past decisions areemployed to select among the four possible survivor paths.Though this choice results in a recursive stage referred to as thepath selector (see Figure 11) consisting of a series of multiplex-ers it is not difficult to meet the speed requirements. This isbecause the path selector is small and localized part of the archi-tecture and hence easily amenable to circuit optimizations.

The three innovations described above provide a favorabletradeoff between the BER with ease of implementation. Though

[FIG11] Block diagram of MLSE-based receiver.

VGA

CLK-DIV

1:8ADC

CRU

8:16 PathFinder

PathSelector

ChannelEstimator

CLK-DIST

DataInput DATA[15:0]

16 b32 b

ADCData

CLK

4 b

Digital Equalizer (DE) ICAnalog Front-End (AFE) IC

CLK



[FIG12] High-speed MLSE: (a) time-reversed trellis, in which a cold-start at r [k ] leads to four possible reverse-survivor paths ending atstates s [k − D ], (b) VLSI architecture, and (c) path selector.

DC0-DC3

DC4-DC7MUXSEL

MUXSEL

MUXSEL

MUXSEL

MUXSEL

MUXSEL

MUXSEL

MUXSEL

DE

LAY

DE

LAY

D

D

[7:6]

[15:14]

8

8 BITSOUT[15:8]

BITSOUT[7:0]

r [k–4] r [k–3] r [k–2] r [k–1] r [k]

0 0 0

0 1

1 0

1 1

1

2

3

Received Sequence

(a)

S[k] = {d [k–1],d [k–2]}

0

0

0

0

1

1

1

1

0

0

0

0

1

1

1

1

0

0

0

0

1

1

1

1

0

0

0

0

1

1

1

1

(b)

(c)

ADC0

ADC1

ADC15

DC7

DC1

DC0

ES

T0

ES

T1

ES

T7

Rec

over

edB

its

Branch MetricComputation

Path Finder Path Selector

ERRCOMP MIN

ERRCOMP MIN

ERRCOMP MIN

MIN

MIN

MIN

++

++

++



IEEE SIGNAL PROCESSING MAGAZINE [126] NOVEMBER 2008IEEE SIGNAL PROCESSING MAGAZINE [126] NOVEMBER 2008

these do not reduce frequency at which the architecture needsto be clocked, which remains at up to 12.5 GHz, they do resultin an architecture that can be easily transformed into one thatcan operate at a lower clock-frequency as described next.

The fourth innovation is to make multiple (Nd) decisionsalong the chosen survivor path in each clock cycle resulting in aclock frequency that needs to be 1/Nd times the line rate. Fifth,the architecture is unrolled [31] in time by a factor of Nu result-ing a further reduction of clock frequency by a factor of Nu.Thus, the final clock frequency is given by

fclk = RNuNd

, (26)

where R is the line rate. Algorithmically, unfolding [31] does notease the throughput problem as it expands the number of com-putations in the critical path by the same factor as it lengthensthe clock period. This fact does not present a problem in thiscase because the critical path is localized to the path selector,which as mentioned earlier, is a simple cascade of multiplexersand hence can be custom designed to meet the speed require-ments. A side benefit of making multiple decisions and employ-ing unfolding is that the equalizer doubles up as a 1 : NdNu

demultiplexer as well. Most receivers need a 1:16 demux func-tionality which is usually implemented using a separate demuxchip. By choosing Nd = 2 and Nu = 8, the DE chip demultiplex-es and decodes the data simultaneously.

MEASURED RESULTSThe MLSE receiver was tested with up to 200 km of SMF. TheMLSE receiver achieves a BER of 10−4 at an OSNR of 14.2 dBwith 2,200 ps/nm of dispersion as shown in Figure 13(a). Areceived power of −14 dBm is used throughout the testing.The MLSE receiver reduces the OSNR penalty of a standardCDR by more than 2 dB at CD of 1,600 ps/nm and BER of10−4. The penalty for a standard CDR increases rapidly beyond1,600 ps/nm of CD. Figure 13(a) also shows that the MLSEreceiver does not have a penalty in the back-to-back (i.e., 0km) configuration.

In [14], another group reports results for an MLSE receiverfor rates up to 10.7 Gb/s, however that MLSE receiver appearsto have a 2 dB OSNR penalty at back to back, i.e., in theabsence of dispersion. As a result, the measured results in [7]are better than those in [14] by 2 dB over the range of disper-sion from back-to-back through 1,600 ps/nm, after which theresults coincide.

The receiver has been shown to provide an error-free(BER < 10−15) post-FEC output, with a pre-FEC BER of 10−3

at 10.71 Gb/s with 2,000 ps/nm of dispersion. It can compensatefor 60 ps of instantaneous differential group delay (DGD) with a2 dB OSNR penalty BER = 10−6). The channel estimator in theMLSE engine adapts at a rate of 30 MHz and thus is easily ableto track variations in DGD, which are typically less than 1 KHz.

The MLSE receiver satisfies the SONET jitter tolerance spec-ifications with 2,200 ps/nm of dispersion [see Figure 13(b)]. A

HP’s OmniBER OTN was used for this measurement. The meas-ured output clock jitter is 0.5 psrms. The output BER is kept at10−3 and the CD is varied up to 2,200 ps with a 231 − 1 pseudo-random binary sequence (PRBS). The fixed BER is achieved byadjusting the OSNR while maintaining the received opticalpower at −14 dBm. The PLL output clock jitter is less than 0.64psrms across test conditions. The PLL maintains locked withoutcycle slips with up to 1,300 consecutive identical digits (CID) ata BER of 10−2, with 125 km SMF-28 optical fiber link.

The MLSE receiver illustrates the benefits of jointly address-ing signal processing algorithm design, VLSI architectures,mixed-signal integrated circuit implementation, as well as elec-tromagnetic signal integrity issues in packaging and circuitboard design.

ADVANCED TECHNIQUESIn this section, we will describe a few of the directions in whichnew advances are being made in optical communications.These include a number of new optical devices and techniquestogether with many unanswered questions and potential oppor-tunities for the signal processing community.

In the section “The Transmitter,” we described a basic modelfor both directly modulated and externally modulated opticalsources for intensity modulations, such as NRZ or RZ signaling.We also described a more spectrally efficient method, based on aform of optical duo-binary signaling, in which the phase of thecarrier is additionally modulated by either 0 or π radians, depend-ing on the state of the precoder. Attempts to achieve even greaterspectral efficiency in optical transmitters invariably require inno-vations in both the optical devices involved as well as in the trans-mit and receive signal processing. Recently, a number ofadditional phase-shift keyed modulation formats have beendemonstrated, including the differential quadrature phase shiftkeying (QPSK) format shown in Figure 14(a) [32]. The transmit-ter requires a digital precoder, and an optical encoder with dualMach-Zehnder modulators. Receiver decoding is performed opti-cally, using an optical delay-and-add structure to avoid the needfor generating a coherent local oscillator.

However, there are benefits of using a coherent receiver whenadditional signal processing is used for EDC. While the receivedsignal after direct detection is proportional to the received opticalpower, if a coherent receiver were used, then the received electri-cal signal would be proportional the electromagnetic field in thefiber. As a result, CD and PMD might appear as linear distortions,enabling linear equalization techniques to substantially outper-form their counterparts when used after direct (square-law)detection. Four photo detectors are presently required to recoverthe in-phase and quadrature (I and Q) components of the electro-magnetic field from both polarizations of the received optical sig-nal. If the length of fiber to be compensated were known inadvance, techniques for coherent moduation combined with lin-ear pre-equaliation [33] can be used. One advantage of electronicpre-equalization is that direct detection can still be used at thereceiver, in exchange for substantial signal processing followed bydigital to analog conversion at the modulator.


[FIG13] MLSE receiver test results: (a) chromatic dispersion test and (b) SONET jitter tolerance.

OSNR Versus CD (Linear Regime) at Fixed BER

8.0

9.0

10.0

11.0

12.0

13.0

14.0

15.0

16.0

17.0

18.0

−1,600−1,400−1,200−1,000 −800 −600 −400 −200 0 200 400 600 800 1,000 1,200 1,400 1,600 1,800 2,000 2,200 2,400

CD [ps/nm]

OS

NR

(0.

1nm

) [d

B]

Standard Transponder Tx /cdr Rx (BER 1e3)



Standard Transponder Tx /MLSE Rx (BER 1e-3)



0.1

1

10

100

1.00E+03 1.00E+04 1.00E+05 1.00E+06 1.00E+07 1.00E+08

Frequency (Hz)

Jitte

r A

mpl

itude

(U

I)

Mask (UI) Loop Back Loop Back 50 km 100 km 125 km

(b)

(a)



Polarization mode dispersion in the section “System Models”was viewed specifically as a channel impairment and therefore animpediment to higher-throughput communication.Polarization diversity has been seen as an opportunity for effec-tively doubling the transmission rate within the same opticalbandwidth [35], much in the same manner that it has been seenas an opportunity for diversity in wireless systems [36].Polarization diversity is obtained by controlling the couplingbetween the polarization states and tracking and detecting themseparately, at the receiver. The use of polarization controllers toenable polarization transmission diversity, together with dualpolarization detection at the receiver enables EDC that bothequalizes the CD and high-order PMD within each polarization.Such EDC techniques mitigate cross-talk due to higher-ordercoupling and can open a wide array of potential adaptive signalprocessing approaches.

Some researchers have posited the potential for exploitingmodal diversity, rather than viewing it as a fundamental impair-ment to high data rate communications [12], [34]. In [34], theconnection between the linear growth in capacity with the (mini-mum of the) number of transmit and receive antennas of multi-ple-input, multiple-output (MIMO) wireless links and thepossibility of achieving similar gains in capacity for MIMO opticallinks was made. For a setup using coherent optical modulation

and demodulation, with x[n] the modulated output of an array ofN transmit lasers and r[n] the output of an array of M coherentphoto-detectors, a model for the MMF channel with modal multi-plexing gives rise to an input-output relation of the form

r[n] = H x[n],

where H represents the aggregation of the modal coupling intothe optical fiber from the input array combined with the spatialsampling of the modal profile incident on the detector array [seeFigure 14(a)]. The effect of modal dispersion on the coherentlymodulated input is that each modulated carrier at the input willappear at the detector as a sum of randomly phased carrier sig-nals. Under conditions of sufficient modal mixing, the elementsof H can be modeled as uncorrelated complex Gaussian randomvariables and this is analogous to the Rayleigh fading modelused in MIMO wireless channels. Measurements of MMF chan-nels with coherent modulation have confirmed these results[12], [34] for simple 2 × 2 transmit and receive arrays. Suchmodal coupling, together with coherent modulation gives riseto an information capacity in the MMF channel of the form

C = log{

det[

IM + ρ

MHH H

]}

[FIG14] (a) A block diagram for a differential QPSK transmitter and receiver and (b) the MIMO modal multiplexing scheme of [34].

MZM

MZM

+

Enc

oder

Fiber

1/z

+/–

a

b

1/z

+/–

c

d

a+b

a–b

c+d

c–d

b[k]

b [2k–1]

b [2k]

Transmitter

Receiver

π/4

–π/4

π/2

d1[k]

Z

Tx1Laser

TxMLaser

dN[k]

Det1

DetN

MIMO

d1[k]

dN[k]

(a)

(b)



bits per second per hertz, where IM is the M × M identitymatrix, ρ is the average SNR of the coherent RF carrier, and H isthe normalized N × M transmission matrix comprising Rayleighfading elements, for a symbol rate that is long compared with themodal delay spread of the MMF link. Such a setup exploits theinherent spatial diversity enabledby the modal structure of thepropagation. This setup and set ofexperiments was focused on anarray of coherently modulatedlasers and an array of coherentlydemodulated detectors. Whilethere is increasing interest in theresearch community of coherentor partially coherent modulationtechniques for very high data-rate applications, at present, nearlyall optical links are intensity modulated and noncoherentlydetected. The extent to which this framework can be applied to apractical MMF link, with intensity modulation and noncoherentdetection is an open-ended question.

Additionally, at data rates commensurate with modern LANlinks, the modal delay spread would give rise to impulseresponses that span several symbol periods (see Figure 3). Thiswould provide an input-output MIMO model of the form

r[n] =L− 1∑l = 0

Hl x[n − l ],

where H [l ] comprise the l th matrix tap of the MIMO channel,i.e., hl[i, j] describes the relation between the i th element ofthe receiver array at time n and the jth element of the transmit-ter array at time n − l. In this regime, rather than the flat-fading MIMO wireless analogy, perhaps a better model is akin tothat of delay-spread MIMO wireless links, as discussed in thecontext of OFDM-based spatial multiplexing [37]. In [37], it isshown that such delay spread can actually give rise to advan-tages both in terms of outage capacity as well as the ergodiccapacity. The approach taken in [37] employs OFDM in order toachieve the multiplexing gains. In principle, such modulationhas indeed been shown to be feasible [38], [39].

Concerns over the nonlinear coupling between carriers, whichare typically assumed orthogonal in OFDM systems, indeed ariseand a careful study and understanding of their effects on thepotential for realizing even a practical single transmitter and sin-gle receiver OFDM system is needed. Further, study of the result-ing statistics of the impulse response matrices Hl in the MIMOcontext would need to be undertaken as well. A step in this direc-tion is given by [40], in which it was recently shown that themultiuser capacity of wavelength division multiplexing (WDM)systems is in fact not adversely impacted by nonlinearity (or non-linear coupling between adjacent wavelengths) when the receiverjointly demodulates all of the carriers. However, if a receiver forone of the channels uses only the output of one wavelength in thechannel, then the information capacity of that wavelength is sig-nificantly reduced due to nonlinearity, and saturates as the inter-

ference power becomes appreciable [40]. As such, low-complexitymultiuser detection algorithms for WDM links that incorporateEDC also pose an interesting set of research problems for the sig-nal processing community. If the nonlinearities present for LHWDM links are appreciable over the distances and spectral spacing

applicable to a coherent MIMOtransmission over MMF, then all ofthe benefits of OFDM transmis-sion would be lost along with themethods of OFDM spatial mutli-plexing. As with the flat-fadingcase, the focus of this approachwas on coherently modulatedtransmitting lasers and coherentlydemodulated optical detectors.

Whether any aspects of such an approach could carry over to non-coherent modulation and demodulation techniques is unknown.

CONCLUDING REMARKSThe potential for electrical signal processing to enhance the per-formance of optical links has been discussed for decades, datingat least as far back as the work of Personick [41], who provided asystem theoretic analysis of receiver design for optical systems.Some early work [42] discussed optimal detection of optical sig-nals and even considered the potential design of receiver struc-tures and derived expressions for minimum mean-squared errorreceivers, taking the form of linear transversal filters or decisionfeedback filters [43]. Many of these investigations were speculat-ing about the possibility of digital signal processing eventuallyforaying into optical communications. With the tremendousadvances in semiconductor technology and EDC-based productsmaking their way into commercial systems, it appears that thistime has come. We hope that this brief overview of the generalconcepts and highlighting of some of the salient features enablesfurther exploration into this fascinating and challenging topic.

ACKNOWLEDGMENTSThe authors thank Jason Stark for his help with the numericalsimulation codes and measurements.

AUTHORSAndrew C. Singer ([email protected]) received the S.B., S.M.,and Ph.D. degrees in electrical engineering, all fromMassachusetts Institute of Technology (MIT). He was a postdoc-toral affiliate at MIT in 1996 and from 1996 to 1998 was aresearch scientist at Lockheed Martin. Since 1998, he has beenat the University of Illinois at Urbana-Champaign where he isprofessor and director of the Technology Entrepreneur Center.His research interests are signal processing, communications,and machine learning. He was a Hughes Aircraft Fellow, recipi-ent of the Harold L. Hazen Award for teaching excellence, theNational Science Foundation CAREER Award, and a XeroxFaculty Research Award. He was named Willett Faculty Scholarand received the IEEE Journal of Solid-State Circuits Best PaperAward. He also serves as senior scientist at Finisar Corporation.

UNLIKE SHORT-REACH MMFAPPLICATIONS, THE EFFECTS OF CD

AND PMD IN LONGER-REACHAPPLICATIONS CAN’T BE MODELED

AS LINEAR AND SLOWLY TIME-VARYING.



[SP]

IEEE SIGNAL PROCESSING MAGAZINE [130] NOVEMBER 2008IEEE SIGNAL PROCESSING MAGAZINE [130] NOVEMBER 2008

Naresh R. Shanbhag ([email protected]) received theB.Tech. from the Indian Institute of Technology (IIT), New Delhi,M.S. from Wright State University, Ohio, and Ph.D. from theUniversity of Minnesota, all in electrical engineering. Since 1995,he has been a professor in the Department of Electrical andComputer Engineering at the University of Illinois, Urbana-Champaign. His research interests are low-power, high-perform-ance, and reliable integrated circuits for broadbandcommunications and signal processing systems. He received theIEEE Transactions Best Paper Award, Xerox Faculty ResearchAward, IEEE Leon K. Kirchmayer Best Paper Award, DistinguishedLecturer of IEEE Circuit and Systems Society, National ScienceFoundation CAREER Award, and the Darlington Best Paper Award.He also serves as senior scientist at Finisar Corporation.

Hyeon-Min Bae ([email protected]) received theB.S. degree in electrical engineering from Seoul NationalUniversity, Korea, in 1998 and the M.S. and Ph.D. degrees in elec-trical and computer engineering from the University of Illinois atUrbana-Champaign in 2001 and 2004, respectively. From 2001 to2007, he was with Intersymbol Communications, Champaign,Illinois. His research interests are high-performance, low-powermixed mode IC designs for broadband communications. Hereceived the Silver Medal of the Samsung Human-Tech ThesisPrize in 1998 and the 2006 IEEE Journal of Solid-State CircuitsBest Paper Award. Since 2007, he has been with FinisarCorporation. He is also a Visiting Lecturer with the Department ofElectrical and Computer Engineering and the CoordinatedScience Laboratory at the University of Illinois at Urbana-Champaign.

REFERENCES[1] G.E. Keiser, Optical Fiber Communications. New York: McGraw-Hill, 2000.

[2] G.P. Agrawal, Fiber-Optic Communication Systems. Hoboken, NJ: Wiley-Interscience, 2002.

[3] E.Forestieri and M. Secondini, Optical Communications Theory and Techniques.New York: Springer-Verlag, 2005.

[4] J. Winters and R. Gitlin, “Electrical signal processing techniques in long-haul fiberoptic systems,” IEEE Trans. Commun., vol. 38, no. 9, pp. 1439–1453, Sept. 1990.

[5] J. Winters, R. Gitlin, and S. Kasturia, “Reducing the effects of transmission impair-ments in digital fiber optic systems,” IEEE Commun. Mag., vol. 31, pp. 68–76, June1993.

[6] Q. Yu and A. Shanbhag, “Electronic data processing for error and dispersion com-pensation,” J. Lightwave Technol., vol. 24, pp. 4514–4525, Dec. 2006.

[7] H.M. Bae, J. Ashbrook, J. Park, N. Shanbhag, A.C. Singer, and S. Chopra, “An MLSEreceiver for electronic-dispersion compensation of OC-192 links,” IEEE J. Solid-StateCircuits, vol. 41, pp. 2541–2554, Nov. 2006.

[8] J. Weidman and B. Herbst, “Split-step methods for the solution of the nonlinearSchrodinger equation,” SIAM J. Numer. Anal., vol. 23, no. 3, pp. 485–507, 1986.

[9] J. Gordon and H. Kogelnik, “PMD fundamentals: Polarization mode dispersion inoptical fibers,” Nat. Acad. Sci., vol. 97, no. 9, pp. 4541–4550, Apr. 2000.

[10] J. Peters, A. Dori, and F. Kapron, “Bellcore’s fiber measurement audit of existingcable plant for use with high bandwidth systems,” in Proc. NFOEC’97, Sept. 1997, pp.19–30.

[11] M. Karlsson, “Probability density functions of the differential group delay in opticalfiber communication systems,” J. Lightwave Technol., vol. 19, pp. 324–332, 2001.

[12] A. Tarighat, R. Hsu, A. Shah, A. Sayed, and B. Jalali, “Fundamentals and challengesof optical multi-input multiple-output multimode fiber links,” IEEE Commun. Mag.,vol. 45, pp. 1–8, May 2007.

[13] G. Papen and R. Blahut, Lightwave Communication Systems. Course notes, 2005.

[14] A. Farbert, S. Langenbach, N. Stojanovic, C. Dorschky, T. Kupfer, C. Schulien, J.-P.Elbers, H. Wernz, H. Griesser, and C. Glingener, “Performance of a 10.7 Gb/s receiverwith digital equaliser using maximum likelihood sequence estimation,” in Proc.ECOC’2004, 2004, pp. PD–Th4.1.5.

[15] M. Harwood, N. Warke, R. Simpson, S. Batty, D. Colman, E. Carr, S. Hubbins, P.Hunt, A. Joy, P. Khandelwal, R. Killips, T. Krause, T. Leslie, S. Lytollis, A. Pickering, D.Sebastio, and A. Szczepanek, “A 12.5 Gb/s SerDes in 65 nm CMOS using a baud-rateADC with digital RX equalization and clock recovery,” in Proc. ISSCC 2007, 2007, pp.436–437.

[16] P. Humblet and M. Azizoglu, “On the bit error rate of lightwave systems with opti-cal amplifiers,” J. Lightwave Technol., vol. 9, pp. 1576–1582, Nov. 1981.

[17] H. Haunstein, T. Schorr, A. Zottman, W. Sauer-Greff, and R. Urbansky,“Performance comparison of MLSE and iterative equalization in FEC systems for PMDchannels with respect to implementation complexity,” J. Lightwave Technol., vol. 24,pp. 4047–4054, Nov. 2006.

[18] O. Agazzi, M. Hueda, H. Carrer, and D. Crivelli, “Maximum likelihood sequenceestimation in dispersive optical channels,” J. Lightwave Technol., vol. 23, pp. 749–763,Feb. 2005.

[19] G. Forney, “Maximum-likelihood sequence estimation of digital sequences in thepresence of intersymbol interference,” IEEE Trans. Commun., vol. 18, pp. 363–378,May 1972.

[20] R. Koetter, A. Singer, and M. Tuechler, “Turbo equalization,” IEEE SignalProcessing Mag., vol. 21, pp. 67–80, Jan. 2004.

[21] K. Yonenaga and S. Kuwano, “Dispersion-tolerant optical transmission systemusing duobinary transmitter and binary receiver,” J. Lightwave Technol., vol. 15, pp.1530–1537, Aug. 1997.

[22] R. Khosla, K. Kumar, K. Patel, C. Pelard, and S. Ralph, “Equalization of 10 GbEmultimode fiber links,” in Proc. 16th Annu. Meeting IEEE Lasers and Electro-OpticsSociety 2003, vol. 1, pp. 169–170.

[23] “IEEE draft p802.3aq/d2.1,” June 2005 [Online]. Available: www.ieee802.org/3/aq/

[24] C. Xia, M. Ajgaonkar, and W. Rosenkranz, “On the performance of the electricalequalization technique in MMF links for 10-gigabit Ethernet,” J. Lightwave Technol.,vol. 23, pp. 2001–2011, June 2005.

[25] S. Qureshi, “Adaptive equalization,” Proc. IEEE, vol. 73, pp. 1349–1387, Sept.1985.

[26] S. Benedetto, E. Bigliere, and V. Castellani, Digital Transmission Theory.Englewood Cliffs, NJ: Prentice-Hall, 1987.

[27] L.R. Bahl, J. Cocke, F. Jelinek, and J. Raviv, “Optimal decoding of linear codes forminimizing symbol error rate,” IEEE Trans. Inform. Theory, vol. 20, pp. 284–287, Mar.1974.

[28] W. Sauer-Greff, M. Lorang, H. Haunstein, and R. Urbansky, “Modified Volterraseries and state model approach for nonlienar data channels,” in Proc. IEEE SignalProcessing’99, 1999, pp. 19–23.

[29] P.J. Black and T.H. Meng, “A 140-Mb/s, 32-state, radix-4 Viterbi decoder,” IEEE J.Solid-State Circuits, vol. 27, pp. 1877–1885, Dec. 1992.

[30] R. Hegde, A. Singer, and J. Janovetz, “Method and apparatus for delayed recursiondecoder,” U.S. Patent 7 206 363, Apr. 2007.

[31] K.K. Parhi, VLSI Digital Signal Processing Systems: Design and Implementation.New York: Wiley, 1999.

[32] R. Griffin, “Integrated DQPSK transmitters,” in Optical Fiber CommunicationConf., 2005. Tech. Dig., Mar. 2005, vol. 3, p. OWE3.

[33] M.E. Said, J. Stich, and M. Elmasry, “An electrically pre-equalized 10 Gb/s duobi-nary transmission system,” J. Lighwave Technol., vol. 23, no. 1, pp. 388–400, 2005.

[34] H. Stuart, “Dispersive multiplexing in multimode optical fiber,” Science, vol. 289,no. 5477, pp. 281–283, 2000.

[35] C. Laperle, B. Villeneuve, Z. Zhang, D. McGhan, H. Sun, and M. OSullivan,“Wavelength division multiplexing (WDM) and polarization mode dispersion (PMD)performance of a coherent 40 Gbit/s dual-polarization quadrature phase shift keying(DP-QPSK) transceiver,” Post Deadline Papers OFC/NFOEC 2007, no. PDP16, 2007.

[36] R. Nabar, H. Blcskei, V. Erceg, D. Gesbert, and A. Paulraj, “Performance of multi-antenna signaling techniques in the presence of polarization diversity,” IEEE Trans.Signal Processing, vol. 50, pp. 2553–2562, Oct. 2002.

[37] H. Bölcskei, D. Gesbert, and A. Paulraj, “On the capacity of OFDM-based spatialmultiplexing systems,” IEEE Trans. Commun., vol. 50, pp. 225–234, Feb. 2002.

[38] J. Tang, P. Lane, and K. Shore, “Transmission performance of adaptively modulat-ed optical OFDM signals in multimode fiber links,” IEEE Photonics Technol. Lett., vol.18, pp. 205–207, Jan. 2006.

[39] W. Shieh and C. Athaudage, “Coherent optical orthogonal frequency division mul-tiplexing,” Electron. Lett., vol. 42, pp. 587–589, May 2006.

[40] M. Taghavi, G. Papen, and P. Siegel, “On the multiuser capacity of WDM in a non-linear optical fiber: Coherent communication,” IEEE Trans. Inform. Theory, vol. 52,pp. 5008–5022, Nov. 2006.

[41] S. Personick, “Receiver design for digital fiber optic communication systems,” BellSyst. Tech. J., vol. 52, pp. 379–402, July/Aug. 1973.

[42] G. Foschini, R. Gitlin, and J. Salz, “Optimum direct detection for digital fiber opticcommunication systems,” Bell Syst. Tech. J., vol. 54, pp. 1389–1430, Oct. 1975.

[43] D. Messerschmitt, “Minimum MSE equalization of digital fiber optic systems,”IEEE Trans. Commun., vol. 26, pp. 1110–1118, July 1978.


Documents

Electronic Dispersion Compensation - University Of Illinoissinger/pub_files/Electronic... · 2011-06-10 · A variety of optical dispersion compensation (ODC) techniques have been