Robust optical FFH-CDMA communications: coding in place of frequency and temperature controls

Robust Optical FFH-CDMA Communications:Coding Avoids Frequency/Temperature Controls

Habib Fathallah and Leslie A. Rusch

COPL, Department of Electrical and Computer EngineeringLaval University, Québec, Canada G1K 7P4fhabib@gel.ulaval.ca, rusch@gel.ulaval.ca

Abstract- In WDM (Wavelength Division Multiplexing), transmitters require strin-gent and complex frequency control loops to avoid wavelength drift due to tem-perature fluctuations [1,2]. This makes the transmitters complex and bulky, whichhas limited the success of WDM for local and short-haul communications networks,as well as for manufacturing locales and other open areas where temperature con-trol is not feasible. We propose and analyze a technique we call robust fast fre-quency hopped code division multiple access (FFH-CDMA), particularly suitablefor typically access and switching applications. We demonstrate that robust FFH-CDMA is an efficient access technique, which avoids all frequency and temperaturecontrol requirements of WDM.

1 INTRODUCTIONIn WDM (Wavelength Division Multiplexing), optical sources must transmit at

precise, distinct wavelengths. Temperature fluctuations at the transmitter, can leadto significant wavelength drift. Stringent frequency control loops and complextracking systems are required to avoid this drift [1,2], involving optic-to-electronicand electronic-to-optic conversion devices. The result is heavy, cumbersome trans-mitters that are unsuitable for local area networks (LANs), fiber-to-home accessnetworks, short-haul interconnecting and multiplexing.

To avoid frequency stabilization we employ coded communications, in par-ticular code division multiple access (CDMA). In frequency-hopped CDMA thecoding is done in two dimensions: time and frequency. Each bit interval is parsed intime (into chip intervals) and a certain frequency is transmitted during each chip.The decoding operation recombines the signal that during coding was distributedacross the time and frequency domains. During traditional FFH-CDMA decoding,the two-dimensional code is translated only in the time dimension to achieve syn-chronization. Typically an absolute frequency reference is assumed, i.e., frequencysynchronicity, so that translation over the frequency domain is not necessary. Inradio frequency (RF) frequency-hopped CDMA, Doppler shift can corrupt the fre-quency reference so that the decoder must seek the auto-correlation peak whileshifting over both the time and frequency domains.

In optical communications there is no Doppler shift, however, frequency driftof source lasers due to environmental affects, especially temperature fluctuations,has a similar effect. Robust frequency-hopped CDMA allows communications de-

spite frequency drift, therefore obviating frequency-stabilization routines. In thispaper, we propose such a system and address two major concerns: 1) an device forencoding/decoding which can easily perform the correlation while shifting the codeover both the time and frequency domain, and 2) codes which have the necessaryauto-correlation peak over two dimensions and that also conform to the physicalconstraints of the coding device.

2 FREQUENCY DRIFT IN WDMIn DWDM (Dense Wavelength Division Multiplexing) communications, each

wavelength is dedicated to one independent transmitter/receiver pair. The wave-lengths coincide with a predetermined grid, fixing their values and spacing. TheITU standard fixes the wavelengths to a spacing of 100 GHz [2,3]. Physical char-acteristics of the optical devices in the transmission end, especially source lasers,can lead to wavelength drift due to environmental fluctuations, especially changesin ambient temperature. One obvious solution is the absolute control of semicon-ductor laser wavelengths for WDM applications.

2.1 Frequency Control ProblemOne popular way to stabilize the frequency of a laser source is to lock it to a

stable optical reference. High precision is usually obtained using an atomic or mo-lecular transition as an optical reference in the transmitter. Many atomic and mo-lecular transitions are available at both 1.3 and 1.55 µm, which are usually observedusing optogalvanic detection or optical pumping techniques [1,2]. While a sur-mountable impediment for long-haul communications, the complexity and bulk offrequency stabilization has limited the success of WDM for LANs, fiber-to-homeaccess networks, short-haul communications, security and surveillance systems. Inthese applications, we must: 1) reduce the system bulk and weight; 2) avoid envi-ronmental conditioning; 3) reduce system complexity, especially the number ofactive devices.

2.2 Non-Frequency Selective Environmental EffectsAs an alternative to frequency control techniques we propose a coded commu-

nications system that is robust to environmental effects that can be modeled as non-frequency selective. Whenever the environmental effect has the same behavior ateach frequency within the communications band in use, i.e. non-frequency selec-tive, communications is possible despite frequency drift.

Bragg gratings operates as an optical band-pass (or band stop) filter centered atthe so-called Bragg wavelength given by the expression lB neff= 2 L, and can bemodeled as having non-frequency selective drift in their reflectivity. The wave-length shift due to temperature variation ∆T can be expressed as

∆ Λ Λ ∆λ λB eff eff Bdn dT n d dT T= +/ / / /d i b g (1)

where the first term arises from the temperature dependency of the refractive index,and the second term is due to the thermal expansion of the fiber. The former isdominant and accounts for about ∼95% of the total shift. Experimental observations

show that [4]

∆ ∆λ λB B T/ .= × −6 67 10 6 (2)for the communications band. This equation indicates that wavelength shift ∆λB isfrequency selective, i.e., the constant of proportionality between ∆λB and ∆T isfrequency dependent. However, in practice, the wavelength drift is much narrowerthan the wavelength itself, (∼30 nm in the 1550 nm band), and therefore we canapproximate ∆ ∆λB T= × × ×−6 67 10 15506. .

3 OPTICAL FFH-CDMThe proposed coded robust communications system employs techniques bor-

rowed from CDMA. In optical FFH-CDMA [5-7], the information bits are encodedwith two-dimensional matrix codes simultaneously exploiting the time and fre-quency domains. The matrix codes are chosen to be as transparent (or mathemati-cally orthogonal) as possible. We use codes adopted for RF FFH-CDMA as astarting point for our analysis. These codes have their transparency optimized tocombat two affects: time domain asynchroncity (i.e., user transmissions have a ran-dom relative time delay modeled with a uniform distribution) and frequency do-main Doppler shift (due to relative motion of transmitter and receiver). Tempera-ture-induced frequency drift in optical systems will have characteristics differentfrom those of Doppler-shifted RF communications, and therefore the codes usedwill be sub-optimal. In Section 4, we show how the technique efficiently extractdesired transmitter information even if the transmitter environment temperaturevaries over a very wide range. We further evaluate the system performance usingsignal-to interference ratio (SIR) and bit error rate.

3.1 Encoding and DecodingIn an optical FFH system, each data bit is multiplied by a waveform called an

FFH code or sequence. This waveform is a concatenation of the so-called chippulses, each of which occupies a distinct frequency slot and is transmitted in a dis-tinct chip time slot of duration Tc. The FFH sequence can be mathematically ex-pressed as

c t d t jTk k j j cj

Nb g b g= −=∑ , ψ

1

(3)

where N is the length of the code (or the number of chips per bit), dk,j∈0, 1, for1≤ ≤j N , is the jth chip value of the kth user’s code, Tc is the chip duration andT NTc= is the bit duration. The chip signaling waveform y j ta f , for1≤ ≤j N , isusually assumed to be rectangular with unit energy. In an FFH encoding techniquebased on multiple Bragg gratings (see Figure 1), y j ta f is the time-convolution ofthe incident data-modulated, wideband, non-coherent short pulse with the jth singlegrating impulse response. The impulse response of a single grating is defined in thispaper as the inverse Fourier transform of the grating complex reflectivity.

In [5], we assumed that the data pulse is much narrower than the duration ofthe grating impulse response. However in [8], an experimental demonstration of the

λ3 λ2 λ4λ5 λ1λ6λ11 λ9

decoded signal

receivedsignal

data

encodedsignal

a)

1 cm 8mm

λ3λ2 λ4 λ5λ1 λ8λ7λ6

time

EE

EE

EE

EE

D

D

D

D

D

D

D

D

λ11λ10λ9 λ12λ0

wavelength

b)

Figure 1: a) FFH-CDMA encoder/decoder based on multipleBragg gratings, b) frequency-hop pattern of the encoder (re-spectively the decoder): the mark E (respectively D) marksthe encoder (respectively the decoder) pulses.

standard FFH-CDMAsystem shows that theimpulse response of ac-tual gratings is muchnarrower than datamodulated 300 ps pulses.The gratings introducevery little expansion ofthe incident pulse. Suchgratings can be modeledas a train of Dirac deltafunctions when com-pared to data pulseslasting hundreds of pico-seconds.

Since we assume anon-coherent source isused and with on/off datamodulation, the system is entirely positive. The decoding operation at the receiverconsists of a power-summation, i.e., square law detection and comparison of therecombined pulses with a constant threshold. The phase spectra of the gratings,therefore, need not add coherently. Figures 2 a) and b) report real measured char-acteristics of one standard FFH-CDMA encoder/decoder pair [3], where an8-wavelength multiple Bragg grating is written in a fiber segment of length 8 cm.Each grating is of length 8 mm and spacing between cascaded gratings is 2 mm, asdescribed in Figure 1 a).

Let cka be the transmitted FFH sequence at the temperature T T Tα α= +0 ∆ ,

where T0 is an arbitrary nominal temperature, ∆T is the amount of temperaturevariation which can change the transmitted wavelength by one increment, B0, i.e.,one frequency slot, and α a coefficient which quantifies the temperature variation inthe system. In a robust FFH-CDM system each user k, for k=1,...,K, transmits a se-quence ck

a corresponding to its own temperature Tka . At the receiver, the received

signal is a sum of all the transmitted signals

r t b c tk k kk

Kkb g b g= −

=∑ α τ

1

(4)

where bk Œ 0 1,k p is the kth user information bit during the bit interval (0,T), τk is thetime delay of user k with 0≤τk≤Tc and αk is the temperature variation coefficient ofuser k.

The receiver applies a matched filter to the incoming signal to extract the de-sired user’s bit stream. For simplicity, we assume that the desired user’s signal isdenoted by k=1, τ1=0. The matched filter output for bit interval (0,T) is thus

y c t r t dt c t c t dt b c t c t dtT T

k k kT

k

Kk= = + −z z z∑

=10

0 10

10 10

02

1b g b g b g b g b g b gα α τ (5)

where we have neglected the effects of quantum noise and thermal noise. The first

term in (5) corresponds to the desired user, the second is multiple access interfer-ence (MAI). Ideally, the users’ codes should be mutually orthogonal (or transpar-ent) leading to MAI equal to 0. When the temperature of the desired user’s envi-ronment shifts from the nominal temperature T0, the coefficient α1 0≠ , i.e., c t1

0( ),the locally generated desired user’s code, is spectrally shifted with respect to thetransmitted code to match c t1

1a ( ). In an ideal system, the first term is zero when thelocally generated code c t1

0( ) is not matched toc t11a ( ), i.e., the code should be or-

thogonal to any frequency shifted version of itself.Assuming the tuning of the matched filter from c t1

0( ) to c t11a ( ) is accomplished

by exploiting an auto-correlation peak of the code (described in the next section),the receiver output becomes

y c t r t dt c t dt b c t c t dt b N MAIT T

k k kT

k

Kk= = + − = +z z z∑

=10 1

2

0 102

11 1 1α α α α τb g b g b ge j b g b g (6)

In most CDM systems, the MAI is the most important noise source. For a largenumber of multiplexed signals, the probability density function of the MAI is ap-proximated to be Gaussian, using central limit arguments. To reduce MAI, codeswith specific transparency characteristics are required, especially codes that remain

-25

-15

-5

0

1550 1552 1554 1556 1558

a)

Ref

lect

ivity

(dB

)

λ2λ1 λ3 λ4 λ5 λ6 λ11λ9

-3dBpowerlevel

1550 1552 1554 1556 1558

b)

Rel

ativ

e gr

oup

dela

y (p

s)

Wavelength

0

200

400

600

800 E E

D

DD

E

D

D

E

D

D

D

E

E

Figure 2: a) Reflected spectrum from the encoder (solid line) and the decoder (dotted line),b) the group delay for each frequency bin of the encoder (solid line) and of the decoder(dotted line). The marks E and D have the same meaning as in Figure 1.

orthogonal (or almost-orthogonal) even under frequency and time shift.

3.2 Auto & Cross CorrelationIn FFH-CDMA, each user selects a set of N frequencies from a sets of q avail-

able frequencies S f fq= 1 , ... ,m r , where N q≤ : The frequency-hop pattern is usuallyrepresented by a matrix (N×q) showing the time and frequency plane.

Most codes developed for radio FFH-CDMA assume N=q. Only a few codefamilies can be generalized to N<q; all are sub-optimal. In our system the number Nis determined by the complexity of the encoder. The number of available frequen-cies q is limited by the expected maximum variation of the environmental parame-ters (discussed further in Section 4). Codes used in other optical FFH-CDMA sys-tems [5,8] are not appropriate, as frequency shifts lead to high correlation amongcodes. In this system, as described in Section 4-4.2, we modify square (N=q) ex-tended hyperbolic congruence codes, already resistant to RF Doppler shift, for thisrobust FFH-CDMA system. The codes used are one-coincidence sequences, i.e., 1)all of the sequences are of the same length; 2) in each sequence, each frequency isused at most once and 3) the maximum number of hits between any pair of se-quences for any time shift equals one [9]. These parameters are usually measuredusing the so-called auto-and cross-correlation functions defined in the following.

Let ck(i,j) be the frequency-hop pattern of user k when the time slot is i and thefrequency slot is j. Therefore ck(i,j) is one if there is a transmission for user k intime slot 1≤i≤N and frequency slot 1≤j≤q; otherwise it is zero. Let

Ω Ω1 2= =min maxk

kk

kα α

Each code in the code family should be selected to maximize the peak andminimize the sidelobes of the auto-correlation function over all delays τ and fre-quency shifts s:

R s i j i j swhere N Nand sm m m

i

N

j

q

τ ττ

, , ,,b g b g b g= − −

− + ≤ ≤ −− + ≤ ≤ −==

∑∑ c c00 1 2

1 11 1Ω Ω

(7)

where ideal auto-correlation corresponds to R Nm 0 0,a f = , the number of ones in thecode (or the weight of the code) and R sm t ,a f = 0 otherwise. Secondly, the cross-correlation should be minimized for all pairs of codes cm and cn in the code familyfor all delays τ and frequency shifts s:

R s i j i j swhere N Nand sm n m n

i

N

j

q

, , , ,,

τ ττb g b g b g= − −

− + ≤ ≤ −− + ≤ ≤ −==

∑∑ c c00 1 2

1 11 1Ω Ω

(8)

where ideal cross-correlation corresponds to R sm n, ,ta f = 0 for all time and fre-quency shifts. Note that cross-correlation term is visible in the MAI portion of thematched filter output in (5). Ideal auto and cross-correlation functions are difficultto achieve in realistic CDMA systems. We exploit codes having good, though notideal, auto- and cross-correlation properties.

3.3 SIR & Probability of ErrorTo evaluate the signal to interference ratio at the receiver output we re-

quire the probability density function of the time-shift (or delay) as well as that ofthe frequency-shift between any users. In an asynchronous system, the probabilitydensity function (pdf) of the random time delay is well modeled as uniform over abit duration, since no coordination exists between users. A uniform density is nothowever, necessarily a good model for the pdf of the temperature-induced fre-quency shift. Let P(s) be the probability of a discrete frequency-shift s so that

P s frequency shift s P ss

b g b g= = == +

−

∑Prob and( )Ω

Ω

1

2

1

1

1 (9)

In a real system the pdf can be estimated using statistical analysis of the parametersinfluencing environmental fluctuations.

Let Rm n, be the cross-correlation averaged over delay and frequency-shiftbetween codes m and n. Since the time-shift and the frequency-shift are independ-ent variables we arrive at

RN

P s R sm n m nN

N

s, , ,=

− =− +

−

=− +

−

∑∑12 1 1

1

1

1

1

2 b g b gττΩ

Ω(10)

The variance of the cross-correlation between codes m and n is

σ ττ

2 2

1

1

1

112 1

1

2

m n m n m nN

N

sNP s R s R, , ,,=

−−

=− +

−

=− +

−

∑∑ b g b gd iΩ

Ω(11)

Since we do not know which codes will be active at any given time, we further av-erage over all code pairs to arrive at σ σ2 2= E m n, , where E refers to expectationover uniformly distributed indices. With these definitions, the mean value of theMAI for K active users can be expressed as µ MAI m nK R= − 1b g , . Assuming the in-terfering users are statistically independent, the MAI has variance that can be ap-proximated as σ σMAI K2 21= −b g . The signal to interference can then be easily de-

rived as SIR N K= −2 21/ b gσ .The output of the matched filter is compared to the threshold η µ= +N MAI/ 2

to decide if a bit was transmitted. Using the Gaussian assumption for the MAI, andassuming the system is MAI limited (i.e., neglecting other noise sources) the prob-ability of error for equiprobable data is given by

P Prob y b 0 Prob y b 1 SIRe 1 1= ≥ = + ≤ = =12

η ηc h c ho t e jQ (12)

where Q e duux

xb g= −+ ∞z1 22 2/ /π .

4 ROBUST FFH-CDMA: THE SYSTEM PERFORMANCEIn Figure 3, we show a robust FFH-CDM system based on multiple Bragg

gratings and composed of one programmable receiver and K transmitters. Abroadband source and a rapid external modulator (at the chip-rate or lower) can beshared among all the users or a sub-group of users to generate a stream of

broadband short pulses (chip-duration). Each transmitter uses its own low rate (bit-rate) external modulator to insert its data sequence into the incoming short pulses,and its fixed multiple Bragg gratings to insert its own code.

4.1 Processing GainThe processing gain (PG) is a critical integer parameter for every CDMA sys-

tem, as it quantifies the resources shared between users via the codes. The higherthe PG, the higher the number of available codes and the greater the ease of dis-criminating among users. In FFH-CDM, the PG is expressed asPG number of available frequency slots number of available time slots= ×b g b gwhere the number of available wavelengths (or frequencies) in the system is limitedby 1) the bandwidth of the source and 2) grating tuning bandwidth. Since the com-bination of fiber stretching (up to 7 nm) and compression (up to 30 nm) leads tovery wide range of tunability in the system, we consider that the source bandwidthis the limiting factor for determining the number of available wavelengths. In tem-perature stabilized optical FFH-CDMA, the entire source bandwidth (BW) can beused for encoding, however this robust FFH-CDMA system requires upper andlower guard bands (each with bandwidth GB) to accommodate frequency drifts dueto environmental effects. This results in a decrease in the number of frequency slotsq, hence a reduction of the processing gain and the reduced capacity alluded tostandard FFH-CDMA. The number of frequency slots is reduced per

q BW B vs q BW GB B0 0 02= = −/ /. b gwhere B0 is the bandwidth of the frequency slot + spacing between slots.

4.2 Derivation of CodesThe encoding space is rectangular (N time slots < q frequency slots), however

most RF FFH-codes assume N=q. In [9], the authors developed a new family ofcodes called extended hyperbolic codes (EHC) which have good auto and cross-correlation properties under simultaneous frequency and time shift. The EHCplacement operator can be expressed as

Transmitter 1

Transmitter K

Fixed Bragg gratings

1:K1:K

1:K

data 1

Tunable Bragg gratings

Programmable Receiver

Broadband source

chip rateFigure 3: Proposed Robust FFH-CDM communications system.

y kik m p ik m p

ik m pi m,/b g b g b g b g

b g≡+ ≠ −

= −RST1

0mod if mod

if mod(13)

where, p is a prime number (for our system p=q), k=1,...,q-1 is the number of chiptime slots, i is the user number and m=0,...,q-1 is the so called message number. Forthis application we can select m=0 and consequently reduce the cross-correlationbetween codes. While the EHC codes are not rectangular, they can be adapted toour needs. Since the q q× codes satisfy the required auto- and cross-correlationproperties, any truncated codes of size q N× will also have the same properties.

While the EHC codes in [9] offer good performance for frequency shifts from-q to q, our system has shifts constrained by the guard bands (by construction theguard bands represent maximal frequency variation). Therefore our codes need tobe efficient for shifts equal to

− − ≤ ≤ −q q s q q0 02 2b g b g/ /We therefore propose the following modified EHC codes

y kik q k q

k qi N

k Ni,/ mod mod

mod,...,

, ,...,01 0

0 01 1

01 1b g b g b g b g

b g≡≠=

RST= −

= −ifif

for (14)

This derives q-1 different modified (or truncated) extended hyperbolic codes(TEHC), each of which uses N frequencies among the available q. It should benoted that the TEHC have performance at least as good as the EHC codes, howeverthey remain sub-optimal.

4.3 Illustrative exampleA super-fluorescent source (SFS) can provide a signal of 30 nm bandwidth

centered at the wavelength 1550 nm. Using the same Bragg gratings physical pa-rameters and wavelength spacing as in [5], we achieve up to q0=135 available dis-joint frequency slices. To match the OC 12 standard (622 MB/sec), a 16 cm of fiberis used in simulation for 12 gratings without significant overlap between chippulses (physical spacing between gratings is 4 mm). With these parameters a tem-perature stabilized FFH-CDMA has PG= q0×N= 135×12=1620.

In order to determine the guard bands required we assume T0=0° C to be themaximal temperature and 4 nm to be the maximum wavelength increase and/ordecrease expected due to temperature fluctuations. In [10], the fiber DFB laserspresented high linearity in a temperature range as wide as ∼400° C in the interval(-196° to 200° C), which leads to a frequency shift of ∼4 nm. This means that re-maining bandwidth for encoding is (30-2×4)=22 nm corresponding to ∼99 wave-lengths. For ease of code generation we select the prime number q=97 as the effec-tive available number of wavelengths to be shared among the robust FFH-CDMAsystem. Hence, the PG of the robust FFH-CDMA is PG 97 12 1164= × = .

4.4 Simulation results: Auto-correlation functionIn this section we simulate two 12-wavelength multiple Bragg gratings using

the parameters of the previous illustrative example and we restricted q to 17. We

derived the TEHCs and we simulated ambient temperature variation of one selecteddesired encoder scanning the interval –100 to +100 ºC in 5 ºC steps. This includes,for example, the range of ambient winter temperature in Québec City, where tem-peratures can be lower than –60 ºC in open areas, and where it can exceed 30 ºC inenclosed areas. Figure 6 shows that the correlation between an encoder maintainedat nominal initial temperature of 0 ºC and a decoder searching through all possibletemperature and time delays. We se a high central peak (weight 12) when the de-coder temperature is 0 ºC and time synchronization is achieved. Otherwise, the cor-relation leads to side-lobes with maximum weight of 2.

4.5 Probability of ErrorTo evaluate the system performance in terms of probability of error we assume

a Gaussian pdf for the temperature, with mean T0 and standard deviation η. Thetemperature of each transmitter is assumed to be independent and identically dis-tributed. Simulations use parameters from the previous Section and TEHC codesdescribed above. In the following we evaluate the probability of error of the systemaveraged over all users, as well as the probability of error of a single user. We pres-ent the probability of error as a function of the number of active users, for variousvalues of η, with ∆T the temperature shift leading to frequency drift B0.4.5.1 Average Probability of Error vs. Capacity

In Figure 5, we present the average probability of error of the proposed robustcommunications system as a function of the number of users. The curves clearlyshow that the higher the number of active transmitters, the higher the probability oferror. This is obviously due to the increased MAI interference. It is important tonote that the environment stabilized FFH-CDMA system gives an upper bound(curve (a)) to the probability of error in the system.

02

-11

0-100-50

050

100

Maximum side lobe of weight=2

The autocorrelation peak of weight=12

Side lobes of weight=1

Time shift (ns)

Temperature shift ∞C

Figure 4: Two-dimensional (time and temperature shifts) auto-correlation function.

Stabilization of the transmitters keeps all users’ signals in the central frequencyband and out of the guard bands, which leads to greater cross-correlation betweenusers and higher MAI. On the other hand, a uniform distribution of the frequencyshift gives a lower bound to the probability of error (curve (d)), as drifted codes

tend to cover the entire available bandwidth. Curves (b) and (c) correspond respec-tively to a temperature shift pdf with η=2.55×∆T and η=7.65×∆T.

As can be expected, decreasing η leads to increasing MAI and a degradation ofthe average probability of error. Figure 5 shows that for probability of error of 10-9

the system can accommodate ≈45 simultaneous users in a temperature-stable sce-nario, and 74 simultaneous users when the temperature is uniformly distributed.

20 30 40 50 60 70 80 90 10010

-18

10-16

10-14

10-12

10-10

10-8

10-6

10-4

Number of Users

BER

(des

ired

user

)

No Temperature Shift:Upper bound

Temperature Shift=2.55∆T

Temperature Shift=7.65∆T

Maximum Temperature Shift:Lower bound

Figure 5: Average Probability of Error vs. Capacity.

20 30 40 50 60 70 80 90 10010

-18

10-16

10-14

10-12

10-10

10-8

10-6

10-4

Number of Users

BER

(des

ired

user

)

No Temperature Shift:Upper Bound

Temperature Shift=2.55∆T

Temperature Shift=7.65∆T

Maximum Temperature Shift:Lower bound

Figure 6: Single Probability of Error vs. Capacity.

4.5.2 Single Probability of Error vs. CapacitySuppose that one transmitter exhibits a frequency shift, therefore pulses from

its signal are present in a guard band. This reduces its coincidence probability withthe interferers, i.e., reduces the MAI, and consequently improves its probability oferror. Figure 6 shows the single user probability of error versus the number of usersdepending on the frequency shift of the desired user. An unchanged temperatureshift gives an upper bound to the probability of error since the variance of the MAIis the highest. When the transmitter exhibits the maximum frequency shift the sin-gle probability of error achieves its lower bound since the MAI is at its lowest.

5 CONCLUSIONWe propose a new optical multiple access communications technique which is

robust to environmental fluctuations and avoids any stabilization or conditioning ofthe transmitters. We develop a modified version of Extended Hyperbolic Congru-ence codes to achieve environment-resistant codes. Simulations show that robustFFH-CDMA system can successfully extract the desired transmitter informationand reject the interference signal. For probability of error of 10-9 the system canaccommodate 45 to 74 simultaneous users depending on probability density func-tion of the temperature shift.

Applications include networks where the complexity and bulk of frequencycontrol has limited the success of WDM; including LANs, fiber-to-home accessnetworks, short-haul communications, security and surveillance systems, on-boardnaval and avionics communications. In such applications, the complexity and bulkof frequency stabilization of the transmitters has limited the success of WDM.

REFERENCES[1] T. Ikegami, S. Sudo and Y. Sakai, Frequency Stabilization of Semiconductor Laser

Diodes. Boston, MA: Artech House, 1995.[2] M. Guy, B. Villeneuve, C. Latrasse and M. Têtu, “Simultaneous Absolute Frequency

Control of Laser Transmitters in both 1.3 and 1.55 µm Bands for MultiwavelengthCommunication Systems,” Jour. of Lightwave Tech, vol. 14, pp. 1136-43, June 1996.

[3] International Telecommunication Union, Telecommunication Standardization Sector,Study Group 15, Question 25, Recommendation G.mcs, 1996.

[4] A. D. Kersey, “A Review of Recent Developments in Fiber Optic Sensor technology,”Optical Fiber Technology, vol. 2, pp. 291-317, 1996.

[5] H. Fathallah, L. Rusch, S. LaRochelle “Passive Optical Fast Frequency-Hop CDMACommunications System,” Jour. of Lightwave Tech., vol. 17, pp. 397-405, March 1999.

[6] ____, “Fast Frequency Hopping Spread Spectrum for Code Division Multiple AccessCommunications Networks (FFH-CDMA),” US Patent Pending, June 1998.

[7] H. Fathallah and L. A. Rusch “Robust Optical FFH-CDMA Communications-Coding inplace of Frequency and Temperature,” IEEE Jour. of Lightwave Tech., to appear 1999.

[8] H. Fathallah, P.-Y. Cortès, L. A. Rusch, S. LaRochelle, and L. Pujol, “ExperimentalDemonstration of an Optical Fast Frequency Hopping-CDMA Communications Sys-tem,” submitted to ECOC '99, March 1999.

[9] L. D. Wronski, R. Hossain and A. Albicki, “Extended Hyperbolic Congruential Fre-quency Hop Code: Generation and Bounds for Cross- and Auto-Ambiguity Function,”IEEE Transactions on Communications, vol. 44, no.3, pp. 301-305, April 1996.

[10] P. Varming, J. Hübner, M. Kristensen, “Five wavelength DFB fiber laser source,”Technical Digest of Optical Fiber Communications, pp. 31-32, 1997.