Fábio Renan Durand and Taufik Abrão VOL. 3, NO. 9/SEPTEMBER 2011/J. OPT. COMMUN. NETW. 1

Distributed SNIR Optimization Based onthe Verhulst Model in Optical Code Path

Routed Networks With PhysicalConstraints

Fábio Renan Durand and Taufik Abrão

Abstract—In this work, the performance of a distributedpower control algorithm (DPCA), based on the Verhulst modelfor signal-to-noise plus interference ratio (SNIR) optimizationin optical code path (OCP) routed networks, was investigated.These networks rest on 2-D codes (time/wavelength) to estab-lish the OCP. The DPCA can be effectively implemented in eachnode, because it uses only local parameters. The SNIR modelconsiders multiple-access interference, amplified spontaneousemission at cascaded amplified spans, group velocity disper-sion, and polarization mode dispersion. Numerical resultshave shown SNIR convergence at power penalties of 7.94 and11.51 dB for 2.5 and 10 Gbps, respectively. These results couldbe utilized for adjustment of either the transmitted powerto a transmitter node or the gain to dynamic intermediaryamplifiers.

Index Terms—Distributed power control algorithm; Opticalcode division multiple access; Optical code paths.

I. INTRODUCTION

T he rapidly increasing demand for bandwidth, driven byQ1

the Internet Protocol (IP), has motivated the developmentof next generation optical networking. IP integration withwavelength division multiplexing (WDM) technology could re-duce the complexity and overhead associated with other layerprotocols [1]. WDM is probably evolving toward all-opticalQ2

networks (AON), which eliminate the optical-to-electrical-to-optical (OEO) conversion and allow for unprecedented trans-mission rates. Even so, over recent years, WDM technologyhas advanced notably, yet its wavelength number is limitedfor accommodating the future traffic demand. Moreover, thecapacity granularity is limited in wavelength, and sometimesit may be too large to accommodate the traffic between nodepairs [2,3]. Recent progress in high bit rate optical code divisionmultiplexing (OCDM) has allowed the application of thistechnology in code/wavelength routed networks [4,5]. This kindof network is compatible with generalized multiprotocol labelswitching (GMPLS), and it improves the capacity granularity

Manuscript received February 28, 2011; revised July 12, 2011; accepted July15, 2011; published August 3, 2011 (Doc. ID 143382).

Fábio Renan Durand is with the Technological Federal University of Paraná,UTFPR, CP 6101, Campo Mourão-PR, Brazil.

Taufik Abrão (e-mail: taufik@uel.br) is with the Department of ElectricalEngineering, the State University of Londrina, UEL-PR, Londrina, Brazil.

Digital Object Identifier 10.1364/JOCN.3.000001

and scalability of optical networks [2,5]. The advantages, likeasynchronous operation, distributed control, and differentiatedservices with quality of service (QoS) at the physical layer, havemade OCDM rather attractive [6]. These networks use hybridWDM/OCDM to create optical code paths (OCPs) based onwavelength/code, and each different code identifies a distinctuser or logic channel [3]. In a common channel, the interferencethat may arise between different user codes is known asmultiple-access interference (MAI). Hence, the number of userssharing the channel simultaneously would be limited by theMAI [6].

In optical networks, for a dynamic traffic scenario, theobjective is to minimize the blocking probability of theconnections by routing and assigning channels, and tomaintain an acceptable level of optical power and adequateoptical signal-to-noise ratio (OSNR) all over the network.Furthermore, different channels can travel via different optical Q3

paths and also have different levels of QoS requirements.The QoS in WDM networks depends on OSNR, dispersion,and non-linear effects [7–10]. The predominant impairment inthe OSNR is given by amplified spontaneous emission (ASE)noise accumulation in chains of optical amplifiers and its effecton the OSNR [7]. Therefore, it is desirable to set networkparameters (optical power or amplifier gains) in an optimalway, based on on-line decentralized iterative algorithms toaccomplish such adjustment [9,10]. This dynamic optimizationwould result in increased network flexibility and capacity [9].

In the case of WDM/OCDM-based optical code routednetworks, the signal-to-noise plus interference ratio (SNIR)optimization problem appears to be a huge challenge, sincethe MAI introduces the near–far problem [11]. Furthermore,if the distances between the nodes are quite different, likein real optical networks, the signal powers received fromvarious nodes are significantly distinct. Thus, considering theWDM/OCDM cross-connect as the reference, the performanceof closer nodes is many orders of magnitude better thanthat of far ones. Thus, efficient power control is needed toovercome this problem and enhance the performance andthroughput of the network; this could be achieved throughSNIR optimization [11–13]. Analogously to a code divisionmultiple-access (CDMA) cellular system, the power control(centralized or distributed) is one of the most important issues,since unbalanced received powers have a significant negativeimpact on both performance and capacity. Power control is

1943-0620/11/090001-09/$15.00 © 2011 Optical Society of America

2 J. OPT. COMMUN. NETW./VOL. 3, NO. 9/SEPTEMBER 2011 Fábio Renan Durand and Taufik Abrão

the most effective way to avoid the near–far problem and toincrease the SNIR [14].

The optical power control problem has recently been investi-gated in the context of local area networks (LANs) with opticalcode division multiple-access (OCDMA) aiming to solve thenear–far problem [11,12,15] and establish the QoS at the phys-ical layer [13,16,17]. In [11], the effect of the near–far problemand MAI mitigation was studied using a centralized power con-trol mechanism. In [12], a detailed review of the power controlproblem was presented including the use of distributed algo-rithms. In [15], an iterative algorithm based on [12] was ap-plied to solve the near–far problem for 2-D (time/wavelength)OCDMA codes considering the receiver noises. However, thesenoises are discarded when compared to the level of ASE noisefrom the pre-amplifier [12]. On the other hand, in [13,16,17],the concept of transmitting at different power levels for usersof distinct types was analyzed. Different power levels wereobtained by combining power attenuators [13], adjustableencoders/decoders [16], and adjustable transmitters [17].Furthermore, the optimal selection of the system parameters,such as transmitted power and information rate, wouldimprove their performance [18–21]. For instance, in [18], theoptical power control and time-hopping method for multimediaapplications using a single wavelength was proposed. Thisapproach accommodates various data considering only onesequence by changing the time-hopping rate. However, in orderto realize such a system, an optical selector device consisting ofa number of optical hard-limiters is needed [18].

On the other hand, in [19], a multi-rate and multi-power-level scheme using an adaptive overlapping pulse-positionmodulator (OPPM) and an optical power controller wasintroduced. The bit rate varies depending on the number ofslots in the optical OPPM system, and it has the advantagethat no change in the code sequence is required despite userinformation rate distinctions. The power level can be achievedby accommodating users with various transmitted powers.The power controller requires only a power attenuator, andthe difference in the powers does not cause changes in theinformation rates. In [20], a joint hybrid power and non-linearprogramming rate control algorithm for overlapped opticalfast frequency-hopping (OFFH) systems was proposed. Themulti-rate transmission is achieved by overlapping consecutivebits while coded using fiber Bragg gratings (FBGs). Thetransmitted optical signal intensity is directly adjusted fromthe laser source with respect to the transmission datarate. The proposed algorithm provides a joint transmissionpower and overlapping coefficient allocation strategy obtainedvia solution of a constrained optimization problem, whichmaximizes the aggregate system throughput, subject to peaklaser transmission power constraints. Additionally, in [21],a control algorithm to solve the unfairness problem for theresource allocation strategy presented in [20] was analyzed.Besides, a unified framework for allocating and controlling thetransmission rate and power in such a way that it could beapplied for any expression of the system capacity was proposedand implemented.

The related works cited above have demonstrated theutilization of resource allocation aspects in order to regulatethe transmitted power, bit rate variation, and the numberof active users, aiming to maximize the aggregate through-

put of LAN–OCDMA systems. However, all these issueshave not been investigated considering routed OCP-based Q4

networks. Indeed, distributed iterative algorithms with highperformance-complexity tradeoffs must be taken into accountto achieve superior figures of merit in OCP networks oper-ating under physical layer imperfections. Hence, OCP-basednetwork optimization issues constitute open research topics,and have not been envisaged in the literature so far.

It is worth noting that routed OCP-based networks bringa new combination of challenges with the power control,like amplified spans, multiple links, accumulation, andself-generation of the optical ASE noise, as well as the MAIgenerated by the OCP. Besides, the dispersive effects, such aschromatic or group velocity dispersion (GVD) and polarizationmode dispersion (PMD), are signal degradation mechanismsthat significantly affect the overall performance of opticalcommunication systems [3,7,22]. In this context, the presentpaper proposes and analyzes for the first time, as far as theauthors are aware, the performance of a distributed powercontrol algorithm (DPCA) based on the Verhulst model forSNIR optimization in wavelength-hopping time-spreading coderouted networks. The SNIR model considers MAI between theOCP based on 2-D codes (time/wavelength), ASE at cascadedamplified spans, and GVD and PMD dispersion effects. TheVerhulst model tries to describe the temporal evolution ofthe number of individuals of some biological species. In thismodel, physical space and food limitations are assumed. Thelimitation of these resources prevents unlimited populationgrowth [23]. The analytical–iterative characteristics of theDPCA based on the Verhulst model are found to be attractivedue to the DPCA performance–complexity tradeoff comparedwith other optimization methods that use either matrix Q5

inversion or numerical methods [20,21].

The paper is organized as follows. Section II describes theOCP networks. Section III presents the SNIR optimization.Section IV describes the performance evaluation. In Section V,the numerical results are shown. Finally, Section VI containsthe conclusion of the paper.

II. NETWORK ARCHITECTURE

The OCPs are established by a code sequence thatis transmitted through the optical network and receivedby optical correlators at the receiver end. The networkconsidered is formed by nodes that have optical core routers,interconnected by WDM/OCDM links [24] with OCPs that aredefined by patterns of short-wavelength pulses, as shown inFig. 1.

The optical core router consists of parallel code converterrouters forming a two-dimensional router node [25], andeach group of parallel code converters is pre-connected to aspecific output, performing routing by selecting a specific codefrom the incoming broadcasting traffic. This kind of routerdoes not require light sources or optical–electrical–opticalconversion, and it can be scaled by adding new modules.The other network equipment, such as code-processing devices(encoders and decoders at the transmitter and receiver), could

Fábio Renan Durand and Taufik Abrão VOL. 3, NO. 9/SEPTEMBER 2011/J. OPT. COMMUN. NETW. 3

edge routercore router

OCP1OCP2

Optical corenetwork

Fig. 1. (Color online) Optical code path routed network architecture.

be made using robust, lightweight, and low-cost platforms withcommercial-off-the-shelf technologies [26].

The OCDM can be divided into non-coherent unipolarsystems, based only on optical power intensity modulation [2],and coherent bipolar OCDM systems, based on amplitudeand phase modulation [3]. The performance of coherent codesis higher than that of non-coherent ones when analyzingthe signal-to-interference ratio (SIR) [6]. This effect occursbecause the bipolar code is true-orthogonal and the unipolarcode is pseudo-orthogonal. However, the main drawbackto coherent OCDMA lies in the technical implementationdifficulties, concomitant with the utilization of phase-shiftedoptical signals [3,5]. The non-coherent codes can be classifiedinto one-dimensional (1-D) and two-dimensional (2-D) codes. Inthe 1-D codes, the bits are subdivided in time into many shortchips with a designated chip pattern representing a user code.On the other hand, in the 2-D codes, the bits are subdividedinto individual time chips, and each chip is assigned to anindependent wavelength out of a discrete set of wavelengths.The 2-D codes have better performance than the 1-D codes,and they can significantly enhance the number of active andpotential users [27]. However, 2-D codes have been applied onlyQ6

in local and access networks [5]. Recently, the utilization of 2-Dcodes to obtain OCP routed networks was proposed, and itsperformance was evaluated by simulation, considering coding,topology, load condition, and physical impairment [24–26,28].

2-D codes can be represented by Nλ × NT matrices, whereNλ is the number of rows, equal to the number of availablewavelengths, and NT is the number of columns, equal to thecode length. The code length is determined by the bit periodTB, which is subdivided into small units called chips, eachof duration Tc = TB/NT . In each code, there are w shortpulses of different wavelengths, where w is called the codeweight. The (Nλ×NT,w, λa, λc) code is the collection of binaryNλ × NT matrices, each of the code weight w; λa and λcare non-negative integers and represent the constraints onautocorrelation and cross-correlation [27].

III. SNIR OPTIMIZATION

In the present approach, the SNIR optimization is basedon the definition of the minimum power constraint (alsocalled sensitivity level) assuring that the optical signal canbe detected by all optical devices. The maximum power

constraint guarantees the minimization of non-linear physicalimpairments, because it makes the aggregate power on a linkbe limited to a maximum value.

The power control in optical networks appears to be an opti-mization problem. Denoting Γi as the carrier-to-interferenceratio (CIR) at the required decoder input, in order to get acertain maximum bit error rate (BER) tolerated by the i-thoptical node, and defining the K-dimensional column vector ofthe transmitted optical power p = [p1, p2, . . . , pK ]T , the opticalpower control problem consists in finding the optical powervector p that minimizes the cost function [12] J(p) = 1Tp =∑K

i=1 pi , subject to the constraints

Γi =G ii piGamp

GampK∑

j=1, j 6=iG i j p j +2N eq

sp

≥Γ∗,

Pmin ≤ pi ≤ Pmax ∀i = 1, . . . ,K ,

(1)

where 1T = [1, . . . ,1], Γ∗ is the minimum CIR to achieve adesired QoS, G ii is the attenuation of the OCP consideringthe power loss between the nodes, according to the networktopology, G i j corresponds to the attenuation factor for theinterfering OCP signal at the same route, Gamp is the totalgain at the OCP, N eq

sp is the spontaneous noise power (ASE)for each polarization at cascaded amplified spans, pi is thetransmitted power for the i-OCP, and p j is the transmittedpower for the interfering OCP. Q7

Using matrix notations, Eq. (1) can be written as[I −Γ∗H]p ≥ u, where I is the identity matrix, H is thenormalized interference matrix, the elements of which can beevaluated by Hi j = G i j /G ii for i = j and zero for anothercase; thus ui = Γ∗N eq

sp /G ii , where there is a scaled version Q8

of the noise power. Substituting inequality by equality, theoptimized power vector solution through the matrix inversionp∗ = [I −Γ∗H]−1u could be obtained. The matrix inversion isequivalent to centralized power control, i.e., the existence ofa central node in power control. The central node stores allthe information about the physical network architecture, suchas fiber length between nodes, amplifier position and regularupdate for the OCP establishment, and traffic dynamics. Theseobservations justify the need for on-line SNIR optimizationalgorithms, which have provable convergence properties forgeneral network configurations [9,12]. A DPCA synthesisconsists in the development of a systematic procedure for thevector p’s evolution in order to reach the optimum value p∗, Q9

based on the target SNIR γ∗i , γi , and pi values. The optimumsolution for the power allocation problem satisfies the followingassociated iterative process [14]:

pi[n+1]= pi[n]−α(1−

γ∗iγ[n]

)pi[n], i = 1, . . . ,K , (2)

where n is the number of iterations and α is the numericalintegration step that converges to 0 < α < 1. This equationrepresents the classical DPCA proposed by Foschini andMiljanic for CDMA wireless networks, and each optical nodecan be effectively adapted for OCP networks. This DPCA couldbe implemented in each optical node, because all necessaryparameters (SNIR level given by γ∗i and the transmitted powerpi[n]) are known in the node. Thus, Eq. (2) depends on local

4 J. OPT. COMMUN. NETW./VOL. 3, NO. 9/SEPTEMBER 2011 Fábio Renan Durand and Taufik Abrão

parameters, just allowing the power control to work in adistributed manner.

In this paper, the Verhulst model, previously used for thepower control problem solution in cellular radio systems [29],is proposed for the adaptation to the OCP network scenario.Verhulst formulated a model trying to describe the temporalevolution of the number of individuals of some biologicalspecies. His model assumes physical space and food limita-tions. The limitation of these resources impedes unlimitedpopulation growth. The dynamic Verhulst model is defined bythe following differential equation [23]:

p = z(p)= p(1− p

p∗). (3)

The analytical integration of results for (3) givesQ10

p(t)= exp(t)p(0)p∗p∗+ p(0)[exp(t)−1]

, (4)

where the asymptotic behavior of this solution is given by

limt→+∞ p(t)= p∗ (5)

for any strictly positive initial condition p(0). When p(0) > p∗,the sign of dp/dt will be negative and, as a consequence, p(t)will monotonically decrease until p (+∞) = p∗. For the case ofpositive dp/dt sign, p(t) will increase until p(+∞)= p∗ [23,29].The implementable discrete version of the Verhulst model, interms of power update for each of K nodes in the network, isdescribed by [29]

pi[n+1]= (1+α)pi[n]−α(γ [n]γ∗i

)pi[n], i = 1, . . . ,K . (6)

The Verhulst DPCA has a remarkable characteristic: theestimated SNIR, γ[n], inside the brackets in Eq. (6), is in thenumerator. This property makes the algorithm’s sensitivityindependent of the SNIR real value in the estimation of errors.As a result, under estimated error conditions, the VerhulstDPCA exhibits a smaller discrepancy from the optimum powervector solution and better convergence than the classicalDPCA [29]. It is known that a function, like φ = κγ−1 (SNIRin the denominator is inherent to all known DPCAs proposedin the literature, the classical one is illustrated in Eq. (2) [14]),has a differential increment dφ=−κγ−2dγ, i.e., the amount ofφ variation for small γ deviations depends on the actual valueof γ. However, for a function like φ = κγ, as in the VerhulstDPCA, there is dφ=−κdγ, which is independent of the actualvalue of γ. Besides, 0 < α < 1 is the factor responsible for theconvergence speed: values close to 1 indicate fast convergenceand simultaneously determine the quality of the solution (afterconvergence) in terms of the normalized mean squared error(NMSE) related to p∗ and calculated by

NMSE[n]=E

[∥∥p[n]−p∗∥∥2

‖p∗‖2

], (7)

where ‖ · ‖ denotes the square Euclidean distance to theorigin and E[.] the expectation operator. Hence, the qualityof the solution achieved by the iterative Verhulst equation is

RX

G0

G1

G2

P2G2

P1 P0

Nsp–2 Nsp–1

G1 G0

Pre-amplifier

Nsp–0

RX

Fig. 2. Cascading amplifiers.

measured by how close to the optimum solution p[n] is at then th iteration.

The Foschini and the proposed DPCA algorithms weresimulated for positive and no greater than one valuesof α (numerical integration step), since both algorithmsdiverge outside this interval [29]. One should notice thatthe Verhulst equilibrium procedure, obtained by recursivelyapplying Eq. (6), characterizes a distributed power controlproblem, since the received interference power could beefficiently estimated. The proposed DPCA can be effectivelyimplemented in each node, because the Verhulst equilibriumin Eq. (6) depends on local parameters, just allowing thepower control to work in a distributed manner. In [29], elevenpropositions to prove the convergence of the Verhulst DPCAwere discussed. Thus, the theoretical validity of the Verhulstconvergence will not be demonstrated herein.

IV. PERFORMANCE EVALUATION

A. Signal-to-Noise Plus Interference and Bit Error Rate

The SNIR at each OCP, considering the 2-D codes, is givenby [13]

γi =N2

T (G ii piGamp)/σD

σ2GampK∑

j=1, j 6=iG i j p j +2N eq

sp

, i = 1, . . . ,K , (8)

where G ii is the attenuation of the OCP taking into accountthe power loss between the nodes, according to the networktopology, G i j corresponds to the attenuation factor for theinterfering OCP signal at the same route, Gamp is the totalgain at the OCP, N eq

sp is the ASE noise for each polarizationat cascaded amplified spans, pi is the transmitted power forthe i-OCP, and p j is the transmitted power for the interferingOCP.

The average variance of the Hamming aperiodic cross-correlation amplitude is represented by σ2 [27]. The parameterσD represents the reduction of the optical power peak,available at the receiver and caused by temporal spreadingof optical pulses resulting from GVD and PMD effects [30].σD is formally defined in Eq. (10). The SNIR and thecarrier-to-interference ratio in Eq. (1) are related to thefactor NT /σ, i.e., γi ≈ (NT /σ)2Γi . The bit error probability(BER) is given by Pb(i) = er f c

(pγi /2

), where the Gaussian

approximation is adopted.

The ASE at the cascaded amplified spans is given by themodel presented in Fig. 2 [10].

Fábio Renan Durand and Taufik Abrão VOL. 3, NO. 9/SEPTEMBER 2011/J. OPT. COMMUN. NETW. 5

This model considers that the receiver gets the signal froma link with cascading amplifiers, numbered as 1,2, . . . , startingfrom the receiver. The pre-amplifier can be contemplated asthe number 0 cascade amplifier. Let G i be the amplifiergain, i.e., Nsp − i will be its spontaneous emission factor.The span between the i-th and the (i − 1)-th amplifiers hasthe attenuation G ii . Let Pti be the mark power at the i-thamplifier input. The equivalent spontaneous emission factor isgiven by [10]

N eqsp = Nsp−1(G1 −1)G iiG0 +Nsp−0(G0 −1)

G1G iiG0 −1. (9)

Calculating the N eqsp factor recursively, one can find the

noise at the cascading amplifiers. Without loss of generality,all employed optical amplifiers provide a uniform gain, settingthe maximum obtainable erbium-doped fiber amplifier (EDFA)to 20 dB across the transmission window. This is a reasonableassumption for the reduced number of wavelengths in thecode transmission window (four wavelengths), considering theoptical amplifier gain profile, where the maximum difference ofthis gain is 0.4 dB for the wavelength, which is the most distantone from the central wavelength (1550 nm), with spectralspacing of 100 GHz [9,31].

The dispersive effects, such as chromatic or GVD and PMD,are signal degradation mechanisms that significantly affect theoverall performance of optical communication systems [7]. ThePMD effect appears to be the only major physical impairmentthat must be considered in high capacity optical networks,which can hardly be controlled due to its dynamic andstochastic nature [28,32]. On the other hand, the GVD causesthe temporal spreading of optical pulses that limits the productline rate and link length [7]. The pulse spreading due tothe combined effects of the GVD and the first-order PMD forGaussian pulses is given by [30,33]

σD =

(1+ Cpβ2di j

2τ20

)+

(β2di j

2τ20

)

+ x− 1

2(1+C2

p

) ×[√1+ 4

3

(1+C2

p

)x−1

]1/2

, (10)

where Cp is the chirp parameter, τ0 = TC /{2[2ln(2)]1/2

}is the

RMS pulse width, Tc is the chip period at half maximum, β2 =−Dλ2

0/2πc is the GVD factor, D is the dispersion parameter, c is

the speed of light in vacuum, x =∆τ2/4τ20 and ∆τ= DPMD

√dii ,

DPMD is the PMD parameter, and dii is the link length.

Although there is a difference in the GVD for each wave-length, resulting from time skewing between the wavelengths,the consideration of the same GVD value for the entiretransmission window is reasonable for a small number ofwavelengths, as for the present code [33,34]. On the otherhand, this approximation is utilized to obtain an analyticaltreatment of the GVD and the PMD in the same and lesscomplex formalism, rather than applying a formalism basedon numerical methods [35]. The utilization of compensationschemes, based on a combination of arrayed waveguide grat-ings (AWGs) and optical delay lines [36], optimum thresholddetection [34], and code pattern pre-skewing [34], will alleviate

1

2

3

4

5

6

7

89

10 11

12

183

133

871

2

3

4

5

6

7

89

10 11

12

73

183

133

1816

50

83

110

37

8762

84

44

28

74

34

54

69

20

Imatra

Lappenranta

Lahti

Hämeenlinna

Riihimäki

Vantaa

Espoo

Helsinki

KotkaPorvoo

Kouvola

Järvenpää

Fig. 3. (Color online) The high-speed southern Finland networktopology consisting of 12 nodes and 19 bi-directional links.

and almost eliminate the GVD effect. The compensationscheme, based on the AWGs and the delay lines, can worknot only as a compensation device, but also as a decoder [36].In this context, the present approximation demonstrates atradeoff between the complexity of the formalism and theaccuracy of the results [33].

B. NMSE Evaluation Considering SNIR Error Estima-tion

In a real scenario, the SNIR estimations at each node arenot perfect, in the sense that the values obtained from theseestimations present a random error characteristic. In order toincorporate this characteristic, a random error is added to thecalculated SNIR at each iteration. The ratio of the estimatedand the real SNIR values is given by (1+ ε), where ε will beconsidered as the random variable with a uniform distributionwithin the range of [−δ; +δ]; hence, the estimated SNIR at eachiteration is given by

γi = (1+ε)γi , ∀i and ε∼U [−δ;+δ] . (11)

V. NUMERICAL EXAMPLE

For the purpose of computational simulations, the southernFinland network topology, illustrated in Fig. 3, was chosen.This topology, consisting of 12 nodes and 19 bi-directional links,has been utilized in other studies [3,32], since its dimensionsare adequate to apply the WDM/OCDM technology consideringthe dispersive effects [28]. In the literature, WDM/OCDMviability has been observed in networks that have dimensionsa little bit greater than metro networks, where the wavelengthgranularity (WDM) is too high; thus, the granularity smallerthan a wavelength can be switched and forwarded to the opticalcode domain [3,37]. The OCPs were generated in each node,using a shortest path algorithm for all destination nodes. Inthis way, the maximum OCP (obtained by the shortest path

6 J. OPT. COMMUN. NETW./VOL. 3, NO. 9/SEPTEMBER 2011 Fábio Renan Durand and Taufik Abrão

0

5

10

15

20

25

30

0

5

10

15

20

25

30

0

5

10

15

20

25

30

SNIR

(dB

)

0 50 100 150 200 250 300 350 400

Iterations(a) = 0.1

0 20 40 60 80 100 120 140 160 180 200

Iterations

Without GVD and PMD effects

Without GVD and PMD effects

Without GVD and PMD effects

Power penalty7.94 dB

Power penalty7.94 dB

Power penalty7.94 dB

Power penalty11.51 dB

Power penalty11.51 dB

Power penalty11.51 dB

Bit rate = 2.5 Gbps

Bit rate = 2.5 Gbps

Bit rate = 2.5 Gbps

Bit rate = 10 Gbps

Bit rate = 10 Gbps

Bit rate = 10 Gbps

0 10 20 30 40 50 60 70 80 90 100

Iterations

(c) = 0.9

(b) = 0.5

SNIR

(dB

)SN

IR (

dB)

Fig. 4. (Color online) The SNIR evolution for the number of iterationsfor α of (a) 0.1, (b) 0.5, and (c) 0.9.

algorithm) presents a length of 290 km, and the mean OCPlength is 130 km. Furthermore, in this analysis, the minimumEDFA spacing is 60 km, and all receiver nodes utilize an opticalpre-amplifier that contributes to gain and noise power. As has

(c) 10 Gbps

0

0.05

0.1

0.15

0.2

Avg power Avg power Avg power

0

0.5

1

1.5

2 5

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Iterations Iterations Iterations

0 200 400 0 200 400 0 200 400

(a) No PMD and GVD (b) 2.5 Gbps

Pow

er a

lloca

tion

per

node

(m

W)

Fig. 5. (Color online) Transmitted power per node × number ofiterations, α= 0.1.

been shown in previous works (for instance, see [12]), the ASEnoise is larger than all other kinds of noise at the receiver side;this fact justifies the ASE noise study for the OCPs in networkswith a length of less than 60 km.

Herein, we considered an amplifier gain of 20 dB with aminimum spacing of 60 km, fiber attenuation of 0.2 dB/km,DPMD of 0.1 ps/

pkm, and D of 15 ps/nm/km. Losses for

encoder/decoder and router architecture of 6 dB and 20 dB,respectively, were included in the power losses, taking intoaccount the AWG structure [24]. The parameters are a codeweight of 4 and a code length of 101.

Figure 4 shows the SNIR evolution of the Verhulst DPCAfor the number of iterations for α of (a) 0.1, (b) 0.5, and (c)0.9. The node 10 (Porvoo) was considered as reference takinginto account two situations: i) without GVD and PMD effects,ii) with GVD and PMD effects of 2.5 and 10 Gbps, respectively.

It can be seen from Fig. 4 that the SNIR reaches the targetvalue when the iterations increase for each transmitted node,i.e., the node from which the OCPs were originated. Thetarget SNIR for all the nodes is equal, and if the perfectpower balancing is assumed, it could be demonstrated thatthe maximum SNIR is limited by the number of nodes [29].Therefore, the SNIR theoretical upper bound is determined bythe number of nodes and code parameters. However, when theGVD and PMD effects are considered, there is are penaltiesof 7.94 and 11.51 dB for 2.5 and 10 Gbps, respectively.These penalties represent the received power reduction due totemporal spreading.

When Figs. 4(a), 4(b), and 4(c) are compared, it canbe noticed that the convergence velocity depends on theparameter α. For α of 0.1, 0.5, and 0.9, the numbers of requirediterations for the total convergence of all nodes to the targetSNIR are 280, 60, and 40, respectively. The increase of α arisesfrom the increment of the SNIR convergence velocity, althoughthe quality of the answer found is affected [29].

The power penalty, considering the GVD and PMD effects,could be mitigated by transmitted power optimization in eachnode, as shown in Figs. 5 (α= 0.1), 6 (α= 0.5), and 7 (α= 0.9).

Fábio Renan Durand and Taufik Abrão VOL. 3, NO. 9/SEPTEMBER 2011/J. OPT. COMMUN. NETW. 7

(c) 10 Gbps

Avg power Avg power Avg power

0

0.05

0.1

0.15

0.2

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0 100 200

Iterations Iterations Iterations

0

0.5

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1.5

2

0 100 200 0 100 200

(a) No PMD and GVD (b) 2.5 Gbps

Pow

er a

lloca

tion

per

node

(m

W)

Fig. 6. (Color online) Transmitted power per node × number ofiterations, α= 0.5.

0 50 100 0 50 100

(c) 10 Gbps

Avg power Avg power Avg power

0

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0 50 100

Iterations Iterations Iterations

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(a) No PMD and GVD (b) 2.5 Gbps

Pow

er a

lloca

tion

per

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(m

W)

Fig. 7. (Color online) Transmitted power per node × number ofiterations, α= 0.9.

In Figs. 5, 6, and 7, we can observe the optimized powerallocated to each node for the same situation, as illustratedin Fig. 4, considering (a) no GVD and PMD effects, (b) and (c)the GVD and PMD effects for 2.5 and 10 Gbps, respectively.Figures 5, 6, and 7 show the optimum transmitted powerfor each network node obtained by taking into account thecentralized control plotted with the horizontal dotted lines, andthe transmitted power when the Verhulst DPCA is employed.We can observe the convergence values for the Verhulst DPCAwhich could be calculated at each node without the need for acentral node.

In Figs. 5, 6, and 7, the average transmitted power (AvgPower) is the same: (a) without dispersion considerations of0.133 mW, (b) the average transmitted power of 0.834, and (c)1.88 mW is necessary when the PMD and the GVD are includedin the present model for 2.5 and 10 Gbps, respectively. In allsituations, the maximum power, allocated to each node, is lessthan the limit of excited non-linear effects [1].

0100

200300

400

050

100150

200

00.1

0.20.3

0.40.5

00.1

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0.40.5

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mal

ized

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ared

err

or, N

SEN

orm

aliz

ed s

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ed e

rror

, NSE

Error estimate, δ

Error estimate, δ

Iteration, N

Iteration, N

(a) = 0.1

1

0.8

0.6

0.4

0.2

0

(b) = 0.5

00.1

0.20.3

0.40.5

Error estimate, δ

020

4060

80 100Iteration, N

Nor

mal

ized

squ

ared

err

or, N

SE

1

0.8

0.6

0.4

0.2

0

(c) = 0.9

1

0.8

0.6

0.4

0.2

0

Fig. 8. (Color online) NMSE × number of iterations × SNIR errorestimation (δ).

Figures 8(a), 8(b), and 8(c) show the NMSE for the numberof iterations, considering the SNIR error estimation on aniteration basis, for α = 0.1, 0.5, and 0.9, respectively. TheSNIR estimation error was modeled according to Eq. (11), withδ ∈ [0;0.5]. Q11

Furthermore, from Figs. 8(a), 8(b), and 8(c), one canconclude that even under a strong SNIR error estimationof 50% (δ = 0.5), the NMSE is reduced significantly aftera number of iterations that is inversely proportional tothe convergence speed factor, 0 < α < 1. Hence, for slowconvergence scenarios, i.e., α = 0.1 (Fig. 8(a)), the NMSE is

8 J. OPT. COMMUN. NETW./VOL. 3, NO. 9/SEPTEMBER 2011 Fábio Renan Durand and Taufik Abrão

not a concern after N = 400 iterations, whereas for rapidconvergence, α = 0.9, the convergence is achieved early, withN = 50 iterations (Fig. 8(c)), but at the cost of a higher NMSE.

VI. CONCLUSIONS

In this work, the SNIR optimization with the VerhulstDPCA was investigated in OCP routed networks consideringMAI, ASE at cascaded amplified spans, GVD, and PMD. Theresults obtained have shown SNIR convergence at powerpenalties of 7.94 and 11.51 dB for 2.5 and 10 Gbps, respectively.On the other hand, the optimized transmitted power, definedby the Verhulst DPCA, could mitigate the dispersion effectseven under strong SINR error estimation conditions, and itconverged to the optimized transmitted power calculated bythe centralized power control. The DPCA advantage lies in itscharacteristics of being effectively implemented in each node,since only local parameters are necessary.

ACKNOWLEDGMENTS

This work was supported in part by the National Council forScientific and Technological Development (CNPq) under Grant303426/2009-8 and the Araucaria Foundation, PR, Brazilunder Grant 045/2007.

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Fábio R. Durand received his M.S. degreein Electrical Engineering from the São CarlosEngineering School of São Paulo State, Brazil,in 2002, and his Ph.D. degree in ElectricalEngineering from the State University ofCampinas (UNICAMP), São Paulo, Brazil, in2007. He is currently a Professor at theTechnological Federal University of Paraná(UTFPR), Campo Mourão, Brazil. His researchinterest has been focused on photonic tech-

nology, WDM/OCDM networks, heuristic and optimization aspects ofOCDMA networks, and PMD impairments.

Taufik Abrão (M’97) received his B.S.,M.Sc. and Ph.D., all in Electrical Engineering,from the Polytechnic School of the University ofSão Paulo (EPUSP), Brazil, in 1992, 1996, and2001, respectively. He is currently an AssociateProfessor at the Electrical Engineering De-partment of the State University of Londrina(UEL), Brazil. In 2007 and 2008, he was aVisiting Professor at the Department of SignalTheory and Communications of the Polytechnic

University of Catalonia (TSC/UPC), Barcelona, Spain. His researchinterests include multi-user detection, MC-CDMA and MIMO systems,resource allocation, heuristic and convex optimization aspects ofMC-CDMA systems, and 4G systems. He is the author or co-authorof more than a hundred research papers published in specializedperiodicals and symposia.