hybrid pulse position modulation/ultrashort light pulse code...

2018 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 50, NO. 12, DECEMBER 2002

Hybrid Pulse Position Modulation/UltrashortLight Pulse Code-Division Multiple-Access

Systems—Part I: Fundamental AnalysisKwang Soon Kim, Member, IEEE, Dan M. Marom, Member, IEEE, Laurence B. Milstein, Fellow, IEEE, and

Yeshaiahu Fainman, Senior Member, IEEE

Abstract—In this paper, we propose a hybrid pulse positionmodulation/ultrashort light pulse code-division multiple-access(PPM/ULP-CDMA) system for ultrafast optical communications.The proposed system employs spectral CDMA encoding/decodingand PPM with very short pulse separation. The statistical prop-erties of the encoded ULP are investigated with a general pulsemodel, and the performance of the proposed PPM/ULP-CDMAsystem is investigated. It is shown that we can improve upon theperformance of other ULP-CDMA systems, such as those usingon–off keying, by employing PPM. It is also shown that we canimprove the performance of the proposed system by increasingthe effective number of chips, by increasing the number of PPMsymbols, and by reducing the ULP duration. The performanceanalysis shows that the aggregate throughput of the proposedPPM/ULP-CDMA system could be over 1 Tb/s.

Index Terms—optical code-division multiple-access (CDMA),pulse position modulation (PPM), ultrashort light pulse.

I. INTRODUCTION

I N RECENT years, design and analysis of optical commu-nication networks has been an active research area, as it

is commonly felt that a fiber-based optical communicationnetwork is the only way to meet the drastically increasingdemand for multimedia services in the near future. Althoughwavelength-division multiplexing (WDM) is the currentfavorite multiplexing technology for optical communication[1], code-division multiple access (CDMA) can be a desirablealternative, since it would provide asynchronous access to anetwork, a degree of security against interception, no need foreither frequency (or time) allocation or guard bands because

Paper approved by I. Andonovic, the Editor for Optical Networks and De-vices of the IEEE Communications Society. Manuscript received May 17, 2001;revised March 25, 2002. This work was supported by National Science Foun-dation (NSF). The work of K. S. Kim and D. M. Marom was supported by theKorea Science and Engineering Foundation (KOSEF) and the Fannie and JohnHertz Foundation, respectively.

K. S. Kim was with the Department of Electrical and Computer Engineering,University of California, San Diego, La Jolla, CA 92093 USA. He is nowwith the Mobile Telecommunication Research Laboratory, Electronics andTelecommunication Research Institute, Daejeon 305-350, Korea (e-mail:[email protected]).

D. M. Marom was with the Department of Electrical and Computer Engi-neering, University of California, San Diego, La Jolla, CA 92093 USA. He isnow with Advanced Photonics Research, Bell Laboratories, Lucent Technolo-gies, Holmdel, NJ 07733 USA (e-mail: [email protected]).

L. B. Milstein and Y. Fainman are with the Department of Electrical andComputer Engineering, University of California, San Diego, La Jolla, CA92093-0407 (e-mail: [email protected]; [email protected]).

Digital Object Identifier 10.1109/TCOMM.2002.806501

of complete utilization of the time and frequency by each user,and simplified network control.

Additionally, techniques of high-resolution waveform syn-thesis by spectral filtering of ultrashort light pulses (ULPs) in anoptical processor have been developed, advancing the science ofultrafast phenomena. By employing an orthogonal phase codeset as the spectral filter, the resulting optical waveforms can beused as a basis for a multiuser communication system with aspread time property [2]. In the ULP-CDMA format suggestedby Weiner,et al.[3], [4], each user has a unique phase code fromthe orthogonal set for encoding a pulse before transmission on acommon optical fiber carrier in an on–off keying (OOK) mod-ulation format. The desired signal is restored to a short pulseform (i.e., despread) at the receiver by applying a spectral filterconsisting of the phase-conjugated code used at the transmitterof interest. The optical waveforms from other users remain asencoded pulses with long duration and low intensity. The de-coded signal needs to be detected by a nonlinear thresholdingoperation [5], as the temporal variation of the signal is too fastfor direct electronic detection schemes.

On the other hand, in many recent studies including [6]–[10],various kinds of pulse position modulation (PPM) have been in-vestigated as ways to improve the performance of optical com-munication systems. In [11] and [12], however, it was shownthat, for a fixed throughput and chip time, there is no advan-tage in employing conventional PPM in place of OOK in op-tical CDMA systems. Thus, some modified schemes, includingoverlapping PPM (OPPM) [11], [13] and bipolar PPM withspectral CDMA encoding [14], have been investigated to in-crease the bandwidth efficiency. In the ULP-CDMA systems,however, the situation is different. Since a large bandwidth (ul-trashort pulsewidth) is already a given, it is of importance todetermine how to utilize the bandwidth effectively. Recently, aULP-CDMA system with PPM signaling has been reported in[15]. In this scheme, however, PPM is employed as it has beendone in spread spectrum CDMA systems: the pulse separationof the PPM is greater than the encoded signal duration. Thus,the large bandwidth of the ULP is not fully exploited and themaximum number, , of PPM symbols is given by the ratioof the symbol period to the CDMA encoded signal duration.One possible approach for better utilizing the large bandwidthof the ULP is to introduce PPM with a very small pulse sepa-ration symbol encoding. Then, the bandwidth efficiency wouldincrease if we increase the number of PPM symbols roughlyup to the ratio of the symbol period to the ULP duration. In

0090-6778/02$17.00 © 2002 IEEE

KIM et al.: HYBRID PULSE POSITION MODULATION/ULTRASHORT LIGHT PULSE CDMA SYSTEMS—PART I 2019

this scheme, however, a detector with very high time resolutionis necessary to discriminate the pulse positions. A new tech-nique for converting ultrafast temporal information to a spatialimage [16] may assist in the implementation of a high-resolu-tion time filter at the receiver. The ability to observe the temporalsignal by a spatial detector array enables parallel processing ofreceived signals at all PPM signal locations. Thus, the proposedPPM scheme using the ULP with very short pulse separationcan be implemented with electrical circuitry that is compatiblewith the symbol rate (typically, much lower than 1 GHz).

In this paper, we propose a hybrid PPM/ULP-CDMAscheme for ultrafast optical communications. The analysisof the proposed scheme spans two papers: Part I investigatesthe statistical properties of an ULP-CDMA network with ageneral pulse model, and provides a theoretical analysis onthe performance of the proposed hybrid PPM/ULP-CDMAscheme; in Part II, we consider some issues on practical systemimplementation and their effects on the performance, as wellas modified modulation/detection schemes to enhance theproposed PPM/ULP-CDMA system. The contributions of PartI of this paper include the following:

• the statistical properties of encoded ULPs are investigatedbased on a general pulse model;

• the performance of the proposed PPM/ULP-CDMA systemis analyzed by assuming fully asynchronous transmission andtaking the time-varying characteristics of the encoded ULP intoaccount, which were not fully considered in [4] and [15].

II. STATISTICAL PROPERTIES OFENCODED ULPS

There is a need for re-evaluating the statistical properties ofcoded ULP because the previous analysis [4] only treated a sim-plified model where the spectrum had a rectangular shape. Morerepresentative pulse shapes of an actual ULP from mode-lockedlasers are the Gaussian and sech profiles [17]. The structure ofthis section closely follows that in [4], including the statisticalproperties of a single-coded pulse and the correlation functionsof the coded pulses. However, in this paper, we do not confineour consideration to a specific pulse shape.

We begin with a normalized and dimensionless (both in do-main and in range) baseband temporal pulse shape withthe following properties: i) is real; ii) the duration and thepeak value of are both one (for pulses not limited in thetime domain, the duration is defined in the full-width half-max-imum (FWHM) sense); iii) is bounded by a function thatis positive, monotonically decreasing with, and integrable;

iv) , . Note that is alsodimensionless because the domain and range of are di-mensionless. Also note thatexists from property iii), where its FWHM bandwidth will bedenoted as . From the above properties, we getfor , where . Then, an input ULPwith duration and its associated spectrum are given by

(1)

Fig. 1. Schematic of spectrally encoding an ULP. (a) Spectrum of GaussianULP. (b) Spectral binary encoding sequence. (c) Resultant spectrum of anencoded Gaussian ULP.

where is the peak power of the pulse, and is the opticalcarrier frequency.

In the ULP-CDMA communication format, each user en-codes his transmitted pulse with a unique spectral filter. Thespectral filter is comprised of contiguous rectangular frequencybands, each of bandwidth, where each band is multiplied byan element of a coding sequence. The encoding filter is realizedby a spatial mask placed in the central Fourier plane of a spec-tral processing device—a standard optical setup consisting of apair of gratings and lenses in a 4-configuration—where thespectral components of the input ULP are spatially dispersed.The spectral filter can be implemented by either an etchedmask or a spatial light modulator [18]. For the purpose of thisanalysis, we assume the elements of the code take on the valuesof 1 (i.e., a binary code) with equal probability. Note thatthe analysis can be generalized to a broader set of complexsignature codes [19]. By using these quantized phase codeswith alphabet size greater than two, the number of availablesignature sequences is greatly enlarged, thus supporting manyusers. Also, the optical implementation of the encoding filterusing quantized phase codes is not significantly more difficultthan implementing binary codes, as opposed to the radiofrequency (RF) case where more hardware is required.

Since the bandwidth of the ULP may not be finite for certainpulse envelopes, we model the code as having an infinite numberof elements. In practice, its truncated version is used, and one ofthe results of the following analysis is that we find the effectivenumber of code elements contained in the bandwidth of the ULP,dependent on the pulse shape we assume.

A graphical representation of the encoding process is dis-played in Fig. 1. The spectrum of an encoded pulse is given by

(2)


where is the spectrum of the encoded waveform of user,is the encoding filter of user, when

and , otherwise, is the spectral chip bandwidth (inHz), and is the th element in the sequence used by user.Then, the encoded output waveform is given by

FT

(3)

where FT is the inverse Fourier transform, is the con-

volution operator, , the second equalityfollows the fact that

FT

and

Equation (3) describes the temporal evolution of an encodedwaveform for a known encoding sequence. When , theenvelope of the encoded waveform is determined by thefunction, while the infinite sum results in a waveform that re-sembles a noise burst.

For the purpose of this analysis, we model the elements of thecode, , as uncorrelated binary random variables. In addition,we make the following definitions:

(4)

(5)

(6)

where , as well as

and

In (4)–(6), the Fourier series identityis used, where

is the Dirac-delta function. With these definitions, statis-tical properties of an encoded waveform can now be inferred.

Proposition 1: The first and second moments of the encodedpulse are and

.Proof: Since , it is straightforward that

from (3). Again, from (3), we find the secondmoment of the encoded pulse as

(7)

where the second equality follows from the uncorrelated prop-erty of the code elements.

Lemma 1: If , we have

and

Proof: Since , has a much wider enve-lope than does . Thus, we have

(8)


Similarly, we get

(9)

Corollary 1: If , the expected intensity pro-file of an encoded waveform can be approximated as

, where and

is the effective number of chips.Proof: Since , we have

from Lemma 1. Thus, fromProposition 1, we get

Since is the autocorrelation function of , its durationis (approximately) twice that of , which yieldsfor . Thus, since , , , can beignored. Therefore, we get .

Note that the field autocorrelation shown inProposition 1is not a stationary function of time, and that the peak powerof the expected intensity of an encoded waveform has droppedby a factor of , compared to the input ULP peak power.FromCorollary 1, the FWHM duration of the expected intensityof an encoded waveform is now inversely proportional to,while its power spectrum remains the same, since a phase-onlylinear filter was used. Since the duration of an ULP is, theresulting time-bandwidth product of the encoded waveform hasincreased proportional to , that is, proportional to .We conclude that the effective number of chips is analogous tothe processing gain in conventional spread spectrum systems,and determines the amount of time spreading. Additionally, theenergy in the encoded pulse is equal to the energy in the inputpulse, as can be shown by integrating the intensities of the inputand the encoded ULP over .

As a last step, we analyze the intensity autocorrelation of anencoded waveform as follows:

Proposition 2: If , the intensity autocorrelation isgiven by

Proof: See Appendix A.From Propositions 1and 2, we can see that the field and

intensity autocorrelations are not stationary functions of time.Physical measurements of the autocorrelations perform time in-tegration by the slow electronics that detect the optical signal.By performing time integration on the nonstationary results ofPropositions 1and2, we get the time-integrated field and inten-sity autocorrelations as follows:

Lemma 2:

Proof: By substituting , we can rewrite (4) as

(10)

Thus, we have

Proposition 3: The time-integrated field autocorrelation isgiven by

Proof: Since the bandwidth of is confined within, the bandwidth of

is confined within , where and are arbitraryconstants. Thus, for , we have

, which implies, unless . Therefore,

from (7) andLemma 2, we have

Note that, fromProposition 3, we also have

Lemma 3: If , we have: 1); 2)

; and 3) ,where denotes “order of.”

Proof: See Appendix B.


Fig. 2. Field and intensity of an encoded ULP. (a) Real part of the field (asample). (b) Real part of the field (average). (c) Intensity (a sample). (d) Intensity(average).

Proposition 4: If , the time-integrated intensity au-tocorrelation is approximated as

(11)

Proof: Since , from Lemma 1, we have

(12)

Thus, fromProposition 2andLemma 3, we have

(13)

The interpretation ofProposition 4is as follows: for most ofthe time, the time-averaged intensity autocorrelation has a func-tional form of the autocorrelation function , whichis symmetric about with a peak autocorrelation valueof , decreases monotoni-cally to zero, and its duration is approximately ; however,it is doubled periodically at due to the second term

Fig. 3. Field and intensity correlation of an encoded ULP. (a) Real part ofthe field correlation (a sample). (b) Real part of the field correlation (average).(c) Intensity correlation (a sample). (d) Intensity correlation (average).

in the parenthesis of the right-hand side of (13), where the in-tensity profile of the input ULP reappears. This follows fromthe fact that the intensity samples separated byare correlatedwhen is within the vicinity (roughly ) of integer multiplesof , in which the periodicity comes from the spectral chipbandwidth. Note that the time-averaged field and intensity auto-correlations of the encoded waveforms yield measurements ofthe input ULP autocorrelation function , which is con-sidered a difficult task.

Computer simulations were performed to corroborate thefindings of the statistical properties. We used a Gaussian pulseshape given by . The simulationparameters were conducted for a fs pulse and aspectral chip bandwidth of GHz. Expectation valueswere approximated by averaging over 100 trials. Figs. 2 and 3summarize the computer simulation results, which match quitewell to Proposition 1, Corollary 1, andPropositions 3and4.Using the simulation parameters, and theeffective number of spectral chips is .

III. SYSTEM MODEL

The transmitter and receiver scheme for the hybridPPM/ULP-CDMA system considered in this paper are shownin Fig. 4. An ULP from a laser source is shifted into one ofthe possible signal locations with separation accordingto the data symbol with symbol period , and is CDMAencoded with its unique spectral filter. A decoding filter atthe receiver despreads the encoded signal back to a ULP,and the pulse position is detected through a photo detectorarray at the output of the time–space processor, which linearlyconverts the finite-length (determined by the time window ofthe time–space processor) interval of the incoming temporalsignal, at the instant determined by a reference pulse, into thecorresponding spatial image. More detailed discussion on thetime–space processor will be given in Part II. Finally, decisioncircuitry selects the symbol with the largest intensity, and then


Fig. 4. Transmitter/receiver scheme of the PPM/ULP-CDMA system.

the transmitted information is extracted from the detected pulseposition.

In order to convert a CDMA-decoded ULP to a spatial signalon the detector array, a timing reference is required, as men-tioned above. Since femtosecond-scale time synchronizationbetween transmitter and receiver is difficult, it must be ex-tracted from the transmitted signals. Sending a CDMA-encodedreference pulse along with the data is one possible solution. Asecond solution is to use previous data as a reference for thecurrent pulse position, together with an initial setup period.However, in Part I of this paper, we assume that a perfect timereference at the receiver is available by some means. Practicalimplementations and their effects on the performance of theproposed system will be considered in Part II.

Now, consider the signal at the desired user’s receiver. Inthe following analysis, we will consider only interuser interfer-ence, as it is the dominant source of degradation except at lowsignal-to-noise ratio (SNR). Let be the time delay of thethinterferer, including the effect of the PPM. Then, the receivedsignal is

(14)

where is the desired user’s encoded signal andis thenumber of users. Here, we assume thatis uniform over asymbol period, while the time delay of the desired user is setto zero without loss of generality. After passing through theCDMA decoder, the desired user’s signal despreads into a ULP,while the interfering signals remain spread out in time. Since weassumed random binary spreading sequences on all users, thestatistical properties of a falsely decoded (interference) signalare the same as those of an encoded signal.

Proposition 5: As ,converges to a nonstationary complex Gaussian

process with mean 0 and time-varying variance .

In addition, if is not an integer multiple of , the variancesof its real and imaginary parts are equal.

Proof: See Appendix C.By Proposition 5, we can approximate the complex

envelope of an interference signal, for a given, as a nonsta-tionary complex Gaussian process with mean zero and equaltime-varying variances of its real and imaginary parts given by

, when is sufficiently large.Now consider the correlation characteristics. Since PPM with

pulse separation is employed, the time differences betweeninterference samples of interest are ,. Since , we can set sufficiently greater thanbut sufficiently smaller than . Then,

for and any integer .Thus, from Proposition 1andLemma 1. Therefore, the interference samples can be assumeduncorrelated, which implies mutual independence since they areGaussian.

IV. PERFORMANCEANALYSIS AND NUMERICAL EVALUATION

In this section, we will analyze the performance of the pro-posed hybrid PPM/ULP-CDMA system. To facilitate the anal-ysis, we will assume that the intensity samples are taken in-stantaneously at the possible pulse positions and the detectionprocess is to choose the largest one. Let us define

(15)

where . Then, for fixed and , is acomplex Gaussian random variable with mean zero and the vari-ances of its real and imaginary parts given by , where

and

Let and be the intensities sampled at the desired signallocation , and another signal location which isapart from the desired slit , respectively. Due to thesymmetry of the and the probability density function(pdf) of , we can assume that without loss of generality.Then, the conditional pdfs of and are [20]

(16)

and

(17)


Fig. 5. N -level approximation.

respectively, where is the zeroth-order modified Besselfunction of the first kind. Let

and

be the effective number of interferers (i.e., interference variancenormalized by the peak variance, , of a single in-

terferer) at the desired location and the one separated byseconds, respectively. Then the following proposition holds:

Proposition 6: The pairwise probability, conditioned on, is

(18)

Proof: See Appendix D.From the conditional pairwise error probability, we get the

unconditional pairwise error probability as

(19)

To evaluate (19), the pdf of must be known. Sinceand are functions of , in principle, it could be obtainedfrom the pdf of . However, there does not appear to be a wayof obtaining the pdf in closed form. Thus, we will use threemethods to evaluate (19). The first two are numerical evalua-tions based on approximations of the pdf of obtainedby truncating the function in the time-varying variance,and will be described later; the other is an-level approxima-tion shown in Fig. 5. Note that a two-valued approximation wassuccessfully used to approximate Gaussian interference with arandomly varying variance in [21]. The-level approximationis obtained by quantizing the function with an -levelfunction, and can be described as follows:

• Let

and

• Assume that unless

, where , which is the minimum in-terval that contains the mainlobes of both and (seeFig. 5). This approximation helps reduce the region ofof in-terest and will be justified later.

• Let , , be the interval

and . In other words,the are equal-length intervals that span the region of interest

and is the averagevalue of within the interval . Then, we approximate

as if .Then, it is seen that , , due to the

symmetry of the function. The -level approximationdescribed above is a simple one, but may not be an optimal wayto represent with values. However, it will be shownlater that it yields a good approximation to the upper bound ofthe bit error probability.

The above approximation implies that interferers whose de-lays are not in can be assumed to have no effecton the desired user. Let us consider the case where the number ofinterferers within the delay interval is . Then,the probability of this event is

(20)In this case, each delay of theusers is within one of the ,

, equiprobably. Thus, each of the userscan take one of the values with equal probability

, while the other users do not interfere with thedesired user at all. Let be the number of interferers whosetakes the value of . Then, it is easily seen that ,

, and the probability of this event for a givenis . Therefore, we can approximate

(19) as

(21)

where

(22)

and

(23)

Note that (21) is a function of. There are ways ofchoosing the positions of the desired and interfering user from

possible positions. Among them, the number of -apartsignal position pairs is . By assuming equiprobable


Fig. 6. Bit error probabilities when� = 200 fs andN = 94.

Fig. 7. Bit error probabilities when� = 200 fs andN = 188.

information symbols, we get the union bound on the symbolerror probability as

(24)

Let be the average number of bit errors per the numberof bits in a symbol, caused by falsely detecting a symbol thatis -apart from the desired symbol. Then, we get the unionbound on the bit error probability as

(25)

Note that depends on the mapping rule of binary digitsinto PPM symbols. In the following numerical evaluations, Grayencoding is assumed.

In Figs. 6 and 7, the union bound on the bit error probabili-ties of the proposed hybrid PPM/ULP-CDMA system are shownfor when ,fs, fs, , and GHz ( ps,

) and GHz ( ps, ).The results from numerical method 1 are also based on (21).For a given , however, instead of using (22), we generatewith a uniform distribution over and evaluate

Fig. 8. Bit error probabilities when� = 100 fs and� = 200 fs.

and . Then,, at a given and , is evaluated fromProposition 6.

Finally, we evaluate the pairwise bit error probability by aver-aging each result of 20 000 runs obtained from (21). Note thatthe truncated function used in numerical method 1 con-tains only the mainlobe. The numerical method 2 is identical tonumerical method 1, except that it uses the largest sidelobe aswell as the mainlobe. In other words, is generated with a uni-form distribution over and (20) is modified as

The performance of the ULP-CDMA system using OOK [4],with the same , , and , is also plotted for comparison. Weslightly modify the bit error probability given in [4], by takingthe sinc-square form of the time-varying variance of the encodedpulse, which was simplified to a rectangular one in [4], intoaccount.

From the numerical results, we see that the results of method1 are quite close to those of method 2. Thus, it appears to bejustified to ignore the sidelobes of the functions in theanalysis. It is observed that the results from the-level ap-proximation match those obtained from the numerical methodsquite well for the two cases. Indeed, is shown to besufficient to get a good approximation for the two cases. Fromthe results, it is clearly seen that the aggregate throughput ofthe PPM/ULP-CDMA system is much higher than that of theULP-CDMA system with OOK, and that the performance ofthe proposed system gets better as the effective number of chips,

, increases. It is also seen that the bandwidth efficiency ofthe proposed system gets better asincreases.

In Fig. 8, the union bound for the bit error probabilities ofthe proposed hybrid PPM/ULP-CDMA system are shown for

GHz ( ps, ), GHz (ps, ), and GHz ( ps, ),when , fs, fs,

, respectively. The -level approximation withis used for the evaluation. The results shown in Figs. 6 and 7,with , are replotted for comparison. We can see that weimprove the performance of the proposed system by reducing


the ULP duration while the spectral chip bandwidth remainsconstant (therefore, the effective number of chips increases).This is due to the enlarged bandwidth by reducing the ULP dura-tion. It is also seen that we can improve the performance of theproposed system by reducingwhile the effective number ofchips remains constant (therefore, the spectral chip bandwidthincreases). This is also primarily due to the enlarged bandwidth.It indicates that, for a fixed effective number of chips (i.e.,isa constant), smaller (and larger ) yields better performancebecause the duration of an interfering signal is reduced, but yethas the same peak power reduction.

Now, consider the effect when we increase the number ofspectral chips with a fixed bandwidth. As increases, the du-ration of an encoded pulse increases, while the power of eachinterferer decreases. Thus, the number of effective interferers(those that put significant energy in the time window mainlobeof the desired user), also increases. Then, the interference canbe approximated as a stationary complex Gaussian process withvariance , inpart by invoking the central limit theorem. In this case, the SNRper symbol is , where

is the aggregate throughput. Since the pdfof the actual interference has a more impulsive nature (i.e., aheavier tail) than the Gaussian process, the performance ob-tained by the above approximation provides the limiting achiev-able one (also plotted in Fig. 8) for the proposed system. Thus,the limiting performance is given by the following proposition.

Proposition 7: Let be the required bit error probability.Then, the limiting aggregate throughput and the bandwidth ef-ficiency are given by

and

respectively.Proof: Since the union bound for the bit error proba-

bility of -ary orthogonal modulation/noncoherent receptionin an additive white Gaussian noise (AWGN) channel is

[20], we have, where .

The remaining part is straightforward, since the FWHMbandwidth of is .

For example, , ,fs, and results in Tb/s and

bps/Hz. When , and are reducedto 1 Tb/s and 0.233 bps/Hz, respectively. In Fig. 9, the max-imum achievable throughput, , is plotted as a function of

, showing that we can improve the performance by increasing.

V. CONCLUDING REMARK

In this paper, a hybrid PPM/ULP-CDMA system was pro-posed to efficiently exploit the large bandwidth of the opticalfiber channels. The proposed hybrid PPM/ULP-CDMA systememployed the PPM with very short pulse separation by virtue ofthe time–space processing technique in [16], while conventional

Fig. 9. Limiting aggregate throughput� for P = 10 and10 when� = 100 fs and� = 200 fs.

PPM/ULP-CDMA systems used PPM pulse separations compa-rable to a CDMA encoded signal duration. Thus, the bandwidthefficiency of the proposed system could be much higher thanthose of the conventional PPM/ULP-CDMA systems.

The statistical properties of a CDMA-encoded ULP, pre-viously done for a simple rectangular pulse assumption in[4], were investigated based on a general ULP model. Thepseudonoise characteristics of an encoded ULP was againconfirmed with the general ULP model. A set of computersimulations was performed to corroborate the derived statisticalproperties in the case of an encoded Gaussian-shaped ULP. Itwas shown that, with the general ULP model, the peak powerof an encoded ULP was reduced not by the number of chips asin [4], but by the effective number of chips that was determinedby the ULP duration, the ULP pulse shape, and the spectralchip bandwidth. It was also shown that the encoded ULP couldbe well approximated to a nonstationary complex Gaussianprocess with time-varying variance.

Through the analysis of the proposed hybridPPM/ULP-CDMA system, the -level approximationmethod was used to evaluate the union bound on the bit errorprobability of the proposed system. It was shown that the

-level approximation with could provide results closeto those obtained from numerical methods. For comparison, theperformance of the proposed hybrid PPM/ULP-CDMA systemwas plotted against that of the ULP-CDMA system usingOOK. From our results, it was shown that we can improve theperformance of the ULP-CDMA system substantially byemploying PPM. It was also shown that increasing the effectivenumber of chips, the possible signal positions in PPM, or thebandwidth, could provide better performance.

In order to investigate the performance limit of the proposedhybrid PPM/ULP-CDMA system, the case of a very largenumber of effective chips and a very large number of users wasconsidered. In this case, the interference was approximatedas a stationary complex Gaussian random process, and thelimiting aggregate throughput and the bandwidth efficiencywere derived as functions of the number of possible signalpositions in PPM and the required bit error probability.


The limiting achievable aggregate throughput was up to 1.5Tb/s and 1 Tb/s with fs ULP and PPM at

and 10 , respectively. In addition, it can befurther improved by increasing the number of possible signalpositions in PPM, theoretically up to the ratio of the symbolperiod to the ULP duration.

APPENDIX APROOF OFPROPOSITION2

The intensity autocorrelation is given by

(26)

The expectation of the product of the four code elements isnonzero for the following cases: a) when ,then ; b) when , then

; c) when , then; d) when ,

then . In case a), we have. The double sum-

mation term in cases a) and b) can be combined to yield, where

the double summation is separable. Similarly, cases a) and c)are combined to yield

. Finally, cases a) and d) are combined to yield. In

addition, and are real as shown in (4)and (5). Therefore, we have

.

APPENDIX BPROOF OFLEMMA 3

1) Since , from Lemma 1, we have

(27)

where the last approximation follows from the finite duration of. Thus, we have

2) Since , we have

from Lemma 1. Thus, we have

(28)

where the last equality follows by approximating with, so that

(29)

3) We getby approximating with in (6).

APPENDIX CPROOF OFPROPOSITION5

Let us define

and

Then, we have

From now on, we omit the superscript for con-venience. First, consider the real part, . Let

be an integer, where is a positive con-stant. Define . Then, ,

are zero mean mutually independentrandom variables with . Letus define


and

where

(30)

By using the identity

we get

(31)

Note that

where is the Kronecker delta function. Thus, as, we have

(32)

where the last equality follows from the fact that as

and

justified as follows. First, as , gets concen-trated at , which implies that

converges to a delta function when , and vanishes other-wise. Since

converges to . Similarly, when

gets also concentrated at , and vanishes otherwise. Since

converges to .Now, define

and . As in Appendix A, only theand , or and , orand terms are nonzero. Since the three

cases yield the same result, we have

(33)


By squaring (31), we have a double integral and a double sum-mation, in which there are four terms. The first term,, maybe expressed as

(34)

By taking the summation on the first cosine term overto and dividing it by , we get

(35)

Similarly, the second cosine term yields the same result exceptthat the argument of the delta function is . Then,we have

(36)

Now, consider the function

As , similar to what was discussed earlier, the secondhalf of the function, namely

gets concentrated at only when andvanishes otherwise. If , makes the first half,

, vanish. In addition, we have

Thus, it converges toas . Then, by substituting the delta function, we have

(37)

where the last inequality follows from the fact that, from thedefinition of , there exists a positive function that ismonotonically decreasing with respect toand is integrable,such that . Similarly, we can seethat, as , the summation of the other remaining termsin the square of (31) over to , divided by ,is finite by taking similar steps as in (34)–(37). Thus, we get

. Since impliesthat is identically zero, we can assume that

without loss of generality. Also, from (31), we have

Since is bounded by a positive function that is mono-tonically decreasing and integrable, we have

which implies . Thus, we get . Finally,we have

(38)

which shows that, as , converges to aGaussian process with mean zero and variance


by the Lyapunov version of the central limit theorem [22]. Simi-larly, also converges to a Gaussian process with meanzero and variance

As the last step, it can be shown that

(39)

where the second equality follows the fact that ,for , and the last equality follows the fact that

is an odd function of . Thus,converges to a complex Gaussian process with mean zero

and variance . Finally, when is not an integermultiple of , , whichimplies that the variances of its real and imaginary parts areequal, and given by .

APPENDIX DPROOF OFPROPOSITION6

From (16) and (17), we have

(40)

for . Note that the conditional prob-ability depends on only the variances andfor a given . Thus, we can rewrite (40) as

(41)

Therefore, we get

where and is shown as follows.Without loss of generality, we can assume that the delays areuniform over a symbol period centered at . Let bethe set of satisfying .Then, for an element , we define a collection

or

. Note that, for any , we get .Let us also define as the cardinality of .

Since is an even function, ,which implies . Thus, we get

. Since it is apparent that , we have. Note that, for a fixed , there are

more elements of that generate the same collection .Thus, by taking one element from each distinct collection, wecan construct a subset of satisfying i)and ii) for any , , .

Let be the measure of . Then, due to the independentand identically distributed (i.i.d.) uniform distribution of eachelement of , we can see that the pdf of, conditioned on

, is . Thus, we have

(42)

On the other hand, we have

(43)

where the second equality follows from the fact thatelements of take while the other elements take

, as their th element.From the definition of and , it is easily seen that

. Therefore, we get

REFERENCES

[1] V. W. S. Chan, K. L. Hall, E. Modiano, and K. A. Rauschenbach, “Ar-chitectures and technologies for high-speed optical data networks,”J.Lightwave Technol., vol. 16, pp. 2146–2168, Dec. 1998.

[2] P. M. Crespo, M. L. Honig, and J. A. Salehi, “Spread-time code-divisionmultiple access,”IEEE Trans. Commun., vol. 43, pp. 2139–2148, June1995.

[3] A. M. Weiner, J. P. Heritage, and J. A. Salehi, “Encoding and decodingof femtosecond pulses,”Opt. Lett., vol. 13, pp. 300–302, Apr. 1988.

[4] J. A. Salehi, A. M. Weiner, and J. P. Heritage, “Coherent ultrashort lightpulse code-division multiple-access communication systems,”J. Light-wave Technol., vol. 8, pp. 478–491, Mar. 1990.

[5] H. P. Sardesai and A. M. Weiner, “Nonlinear fiber-optic receiver forultrashort pulse code-division multiple-access communications,”Elec-tron. Lett., vol. 33, pp. 610–611, Mar. 1997.

[6] G. M. Lee and G. W. Schroeder, “Optical PPM with multiple positionsper pulsewidth,”IEEE Trans. Commun., vol. COM–25, pp. 360–364,Mar. 1977.

[7] C. N. Georghiades, “Modulation and coding for throughput-efficient op-tical systems,”IEEE Trans. Inform. Theory, vol. 40, pp. 1313–1326,Sept. 1994.

[8] D. A. Fares, “Heterodyne detection of multipulse signaling in opticalcommunication channels,”Microwave Opt. Technol. Lett., vol. 9, pp.201–204, July 1995.


[9] H. M. H. Shalaby, “Maximum achievable throughputs for uncodedOPPM and MPPM in optical direct-detection channels,”J. LightwaveTechnol., vol. 13, pp. 2121–2128, Nov. 1995.

[10] D. Shiu and J. M. Kahn, “Differential pulse-position modulation forpower-efficient optical communication,”IEEE Trans. Commun., vol. 47,pp. 1201–1210, Aug. 1999.

[11] H. M. H. Shalaby, “Performance analysis of optical synchronous CDMAcommunication systems with PPM signaling,”IEEE Trans. Commun.,vol. 43, pp. 624–634, Feb.-Apr. 1995.

[12] M. R. Dale and R. M. Gagliardi, “Channel coding for asynchronousfiber-optic CDMA communications,”IEEE Trans. Commun., vol. 43,pp. 2485–2492, Sept. 1995.

[13] H. M. H. Shalaby, “A performance analysis of optical overlappingPPM-CDMA communication systems,”J. Lightwave Technol., pp.426–433, Mar. 1999.

[14] E. D. J. Smith, R. J. Blaikie, and D. P. Taylor, “Performance enhance-ment of spectral-amplitude-coding optical CDMA using pulse-positionmodulation,” IEEE Trans. Commun., vol. 46, pp. 1176–1185, Sept.1998.

[15] K. Kamakura, T. Ohtsuki, and I. Sasase, “Optical spread time CDMAcommunication systems with PPM signaling,”IEICE Trans. Commun.,vol. E82-B, pp. 1038–1047, July 1999.

[16] P. C. Sun, Y. T. Mazurenko, and Y. Fainman, “Femtoscale pulse imaging:Ultrafast optical oscilloscope,”J. Opt. Soc. Amer. A, Opt. Image Sci., vol.14, pp. 1159–1170, May 1997.

[17] J.-C. Diels and W. Rudolph,Ultrashort Laser Pulse Phenomena:Fundamentals, Techniques, and Applications on a Femtosecond TimeScale. San Diego, CA: Academic, 1996.

[18] A. M. Weiner, D. E. Leaird, J. S. Patel, and J. R. Wullert, “Programmablefemtosecond pulse shaping by use of a multielement liquid–crystal phasemodulator,”Opt. Lett., vol. 15, pp. 326–328, Mar. 1990.

[19] F. M. Ozluturk, S. Tantaratana, and A. W. Lam, “Performance ofDS/SSMA communications with MPSK signaling and complexsignature sequences,”IEEE Trans. Commun., vol. 43, pp. 1127–1133,Feb.-Apr. 1995.

[20] J. G. Proakis,Digital Communication, 3rd ed. New York: McGraw-Hill, 1995.

[21] K. S. Kim, I. Song, S. C. Bang, and T.-J. Kim, “Performance analysisof forward link beamforming techniques for DS/CDMA systems usingbase station antenna arrays,”IEEE Trans. Signal Processing, vol. 48, pp.862–865, Mar. 2000.

[22] J. L. Doob,Stochastic Processes. New York: Wiley, 1953.

Kwang Soon Kim (S’94–M’99) was born in Seoul,Korea, on September 20, 1972. He received the B.S.(summa cum laude), M.S.E., and Ph.D. degrees inelectrical engineering from the Korea Advanced In-stitute of Science and Technology (KAIST), Daejeon,Korea, in 1994, 1996, and 1999, respectively.

He was a Teaching and Research Assistant at theDepartment of Electrical Engineering, KAIST, fromMarch 1994 to February 1999. From March 1999 toMarch 2000, he was with the Department of Elec-trical and Computer Engineering, University of Cal-

ifornia at San Diego, La Jolla, as a Postdoctoral Researcher. In April 2000,he joined the Mobile Telecommunication Research Laboratory, Electronics andTelecommunication Research Institute, Daejeon, Korea, where he is currentlya Senior Member of Research Staff. His research interests include detectionand estimation theory, channel coding and iterative decoding, array signal pro-cessing, wireless/optical CDMA systems, wireless CDMA MODEM design anddevelopment, and wireless OFDM systems.

Dr. Kim received the Postdoctoral Fellowship from the Korea Science andEngineering Foundation (KOSEF) in 1999. He also received the Silver Prize atthe Humantech Paper Contest in 1998, the Silver Prize at the LG Information andCommunications Paper Contest in 1998, and the Best Researcher Award fromthe Electronics and Telecommunication Research Institute (ETRI) in 2002.

Dan M. Marom (S’98–M’01) was born in Detroit,MI, in 1967. He received the B.Sc. and M.Sc.degrees from Tel-Aviv University, Tel-Aviv, Israel,in 1989 and 1995, respectively, and the Ph.D.degree from the University of California, San Diego(UCSD), in 2000. His doctoral dissertation dealtwith femtosecond-rate optical signal processing withapplications in ultrafast communications.

From 1996 through 2000, he was a Fannie andJohn Hertz Foundation Graduate Fellow at UCSD.In 2000, he joined Bell Laboratories, Holmdel, NJ,

to pursue his interests in optical communications.Dr. Marom received the IEEE Lasers and Electro-Optics Society Best Student

Paper Award in 1999.

Laurence B. Milstein (S’66–M’68–SM’77–F’85)received the B.E.E. degree from the City Collegeof New York, New York, NY, in 1964, and the M.S.and Ph.D. degrees in electrical engineering from thePolytechnic Institute of Brooklyn, Brooklyn, NY, in1966 and 1968, respectively.

From 1968 to 1974, he was with the Spaceand Communications Group of Hughes AircraftCompany, and from 1974 to 1976, he was a memberof the Department of Electrical and SystemsEngineering, Rensselaer Polytechnic Institute,

Troy, NY. Since 1976, he has been with the Department of Electrical andComputer Engineering, University of California at San Diego, La Jolla, wherehe is a Professor and former Department Chairman, working in the area ofdigital communication theory with special emphasis on spread-spectrumcommunication systems. He has also been a consultant to both government andindustry in the areas of radar and communications.

Dr. Milstein was an Associate Editor for Communication Theory for the IEEETRANSACTIONS ONCOMMUNICATIONS, an Associate Editor for Book Reviewsfor the IEEE TRANSACTIONS ONINFORMATION THEORY, an Associate TechnicalEditor for theIEEE Communications Magazine, and the Editor-in-Chief of theIEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS. He was the VicePresident for Technical Affairs in 1990 and 1991 of the IEEE CommunicationsSociety, and has been a member of the Board of Governors of both the IEEECommunications Society and the IEEE Information Theory Society. He is aformer member of the IEEE Fellows Selection Committee, and a former Chairof ComSoc’s Strategic Planning Committee. He is a recipient of the 1998 Mil-itary Communications Conference Long Term Technical Achievement Award,an Academic Senate 1999 UCSD Distinguished Teaching Award, an IEEE ThirdMillenium Medal in 2000, the 2000 IEEE Communication Society ArmstrongTechnical Achievement Award, and the 2002 MILCOM Fred Ellersich Award.

Yeshaiahu Fainman (M’93–SM’01) received thePh.D. degree from Technion-Israel Institute ofTechnology, Haifa, Israel, in 1983.

He is a Professor of Electrical and ComputerEngineering at the University of California at SanDiego, La Jolla. His current research interestsare in nonlinear space–time processes using fem-tosecond laser pulses for optical communications,near-field phenomena in optical nanostructuresand nanophotonic devices, quantum cryptographyand communication, 3-D quantitative imaging,

programmable, multifunctional diffractive, and nonlinear optics. He con-tributed over 100 manuscripts in referred journals and over 170 conferencepresentations and conference proceedings. Between 1993–2001, he served asa Topical Editor of theJournal of the Optical Society of Americaon OpticalSignal Processing and Imaging Science.

Dr. Fainman is a Fellow of the Optical Society of America and a recipient ofthe Miriam and Aharon Gutvirt Prize. He served on several conferences’ pro-gram committees, and organized symposia and workshops.

hybrid pulse position modulation/ultrashort light pulse code...

Documents