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A New Simple Method for Calculating the Bit Error Rate of OCDMA Systems

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A New Simple Method for Calculating the Bit Error Rate of OCDMA Systems
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  A New Simple Method for Calculating the Bit ErrorRate of OCDMA Systems Tamer Khattab, Maged Elkashlan, and Hussein Alnuweiri Department of Electrical and Computer Engineering University of British Columbia, Vancouver, British Columbia, Canada { tkhattab,magede,hussein} @ece.ubc.ca Abstract In this paper; we proposea novel simplified mathemati- cal analysis technique for modeling and calculating the ef- fect of multiple access interference on bit error rate in op- tical code division multiple access  OCDMA) systems. Our technique applies to OCDMA systems using optical orthog- onal codes  OOC) with optical time-domain spreading. The proposed analysisuses combinatorial methodson the com- bined signal at the output of the optical correlator decoder to derive amathematical expression for the bit error rate. 1 Introduction An optical code division multiple access  OCDMA) sys- tem uses a set of spreading codes calledoptical spreading codes  OSC) with certain autocorrelation and crosscorrela- tion properties to multiplex several users at the same timeon the same wavelength over a fiber-optic communication channel. Code multiplexing in fiber-optic communications can be performed using either time domain spreading [1] or spec- tral (frequency/wavelength) domain spreading [2]. In ad- dition hybrid schemes that use time domain spreading and frequency  wavelength) domain hoppingwas also discussed in the literature[3,4]. In this paper we focus on systems us- ing time domain spreading of optical signals. In timespreading methods, light sources generate pulses that representthe transmitteddata bits. Using on-off keying  OOK) for digital modulation, a source will produce a sin- gle pulse to represent a logic  1 bit, while it will produce no pulse for the case of logic  0 bit. The pulse duration, Tc, is usually very small compared to the data bit duration,Tb. Replicas of the pulse with reduced power are repeated at pseudo randomly selected chip locations (time-slots) within the bit duration causing a spread of the pulse power. The receiver recombines the pulse replicas allowing their pow- ers to add at a certain time-slot (chip location). This chip location is called the detection chip location. A detection of thesrcinal data bit is then performed. This recombina- tion can be performed using eitheractive devices such as a light source at thereceiver that correlates only withtime slots corresponding to the pseudo random code or passive devices such as an optical delay line correlator  ODLC). ODLCs [5] are devices based on optical fiber tapped de-lay lines that operate as optical correlation filters. ODLCs are utilized by optical transmitters and receivers to encode (spread) anddecode (de-spread) the optical signal. The time delays in the ODLC devices are determined ac- cording to the design of spreading codes. New families of spreading codes thatare suitable for optical signals have been investigated in the literature. The most common two families of optical spreadingcodes are: optical prime codes  OPC) (see [6] and citations within) and optical orthogonal codes  OOC) (see [7] and citations within). We considersystems using OOC for spreading of optical signals. This is due to the lower multiple access interference  MAI) levels introduced by these codes when compared to OPCs. Classical biterror rate  BER) calculations are derived us- ing combinatorial methods on the combined signal at the in- putof the optical correlator decoder. In this paper, we pro- pose a novel analytical method that considers the combined signal at the output of thecorrelatorreceiver to deduce the BER. Ourmethod provides analytical expressions for the BER ofsystems that are not easy to analyze using classical methods. 2Optical Orthogonal Codes In OOC, the circular2 autocorrelation withtime shift T   0 for a codeword and thecircular crosscorrelation between anytwo code words are minimized (or limited) while the zero-shift circular autocorrelation for a codeword is maximized. Formally, OOCs can be defined as follows: Definition 1 A (n, k, Aa, A,) OOC, C with cardinality 2Circular here refers to the fact that the codewords are correlated us- ing a circularbit rotation instead of a shiftin one end direction with zero insertions from the other end 1-4244-1521-7/07/ 25.00 )2007 IEEE 673  ICI = N, is defined as afamilyof N {0,]}-sequences of length n and weight k with circular autocorrelation  RXX  T) for sequence X C C and circular crosscorrelation  Rxy T) betweenanytwo sequences X, Y C C satisfyingthe following properties: n-1  Rxx T) = E Xj jGT j=o {k < Aa T   O T O n-1  RXY T) =E zi YjeT - AC,  2) j=O where xj is element number j in code X, yj is element num- ber j in code Y and e represents addition modulo-n. In this paper, we consider the case  a = AC = 1. We call this a (n, k,1) OOC and willrefer to it as OOC for simplic- ity. This is the case with the lowestmultiuser interference in an OCDMA system based on OOCs, which enhances the system performance. To avoid trivial code constructions, we consider only the case where N > 1 and n > k > 1. OOCs (or(n, k, 1)) codes possess the unique property that there is a maximum of a single  F -chip (pulse) overlap between any two different codewords belonging to the same familywith arbitrarycircular time shifts and a maximum of a single  1  -chip overlap between a code and a circular time shifted versionof itself. Detailed properties and construc- tion methods for OOCs are provided in[8,9]. 3 MAI Induced Errors in ODLC Decoders The ODLC baseddecoder has the same structure as the encoder with the delay pattern chosen so that they can re- verse the encoder effect and restore thesrcinal intended signal from a mix of multiplexed signals using different or- thogonal codes. This is achieved using a matched filter de- sign for the delay taps of the ODLC at the decoder side [10]. The receiver has an associated threshold detector, which only passes a signal if its total power is above a certain value. If we assume that the power per spread chippulse is Q, then the total power of the de-spread pulse is k x Q. The threshold device will pass a signal only if its total power is greater than or equal to h x Q, where 0 < h < k is the power threshold factor. The use of the threshold detection device ensures that the low amplitude pulses that are un- correlated with the decoderdelay pattern are blocked from causing background noise to thecorrelated pulse. Assume that we have an OOK based OCDMA system that contains N users each is spreadusing a unique code- word Ci selected from an OOC C. LetYj be the received signal from user j C { 1, 2,   N}. If we consider an ODLC matched to user i, then the intended signal is Yi and the MAI signal Ii is given by: (3) li{= E: y j(EII,2....NJ j:Xi An error at thereceiver can happen if thecorrelation of the MAI signal Ii with an ODLC decoder matched to Yi pro- duces a pulse of total power greater than theoptical thresh- old power value h x Q at a time-slot corresponding to a detected chip location in Yi while a corresponding logic  1 is not present in Yi. Classically, calculating the MAI induced biterror rate  BER) of time-encoded OOC-spread OCDMA signals was basedon using a combinatorial model that considers the dif- ferent scenarios that causes an unintended signal Yj, where j t i, at the input of the ODLC decoder to contribute a pulse of magnitude Q at a detectedchip location at the out- put of the ODLC decoder [5, 10]. The collective contribu- tions from different non-intended users produce a MAI sig- nal that induces a bit detection error when its power exceeds the threshold value under the assumption that the intended user is not sending a logic  1 bit value. The success of this method to derive mathematically tractable closed form models for BER is basedon the factthat using an optical timed gate at the output of thereceiver eliminates a large number of detection error scenarios. Indeed, the useof op- tical timed gate is necessary for the work of correlator re- ceivers  it performs the sampling part of the sample andcompare function of correlator receivers), but it requires strict bit level synchronization between thetransmitter and the receiver. If a simpler bit asynchronous receiver is to be used, modeling the BER using the classical method would be difficult due to the large number of different possible sce- narios produced by the classical model in this case. The same argument is valid if we consider errors caused by loss of synchronization in ODLC based correlator receivers. 4 BER Calculation Based on ODLC Output Signal We propose a new method to calculatethe BER of time- encoded OOC-spread OCDMA systems using ODLC en- coders/decoders. The new method is a modificationof the combinatorial method, described in Section 3, which con- siders the signal Zji received from user j at the output of the ODLC decoder matched to user i, where i, j C {1, 2, ... N} (See Fig. 1). Before we proceed with thederivation of the BER, we introduce two important theorems. Theorem 1 If an unintendedtime-spread signal Yj en- coded using an OOC Cj with code length n, code weight k and bit value of logic  1 is input to an ideal k-taps ODLC that is matched to user i t j, the corresponding signal Zji 674  4.1 ODLC Correlator Receivers with Op- tical Timed Gate Figure 1. ODLC decoder output signals for in- tended and unintended input signals. at the output of the ODLC decoder will have k2 pulses with equal power levels. Theorem 2 If an intendedtime-spread signal Yi encoded using an OOC Ci with code length n, codeweight k and bit value of logic  1 is input to an ideal k-taps ODLC that is matched to user i, the corresponding signal Zii at the output of the ODLC decoder will have k k -1) pulses with equal power levels and a single pulse with k times the power level of the other k k -1) pulses.This pulse will occur at a time-slot corresponding to the location of the last chip of the bit duration. We call this chip location the correlation peak location. Theorems 1 and 2 are illustratedin Fig. 1. Using the two theorems, we can summarize the error scenarios that occur when detecting the chipsof the spread OCDMA signal in the following corollary: Corollary 1 A chip detection error in time-encoded QOC- spread OOK-modulated OCDMA system based on ODLC encoder/decoder happens if and only if one of the following occurs: 1. More than h users out of the N users in the system contribute a pulse at any detection chip location otherthan the correlation peak location. 2. More than h users out of the N   unintended users in the system contribute a pulse at the correlation peak lo- cation and the intended user doesn  t contribute a pulse with power level higher than the threshold value at the same location. In our analysis, we assume that the data bits arrive at sources continuously with bit values drawnfrom indepen- dent identical random variables with uniform distributions. A logic  1 has a probability Pi and a logic  0 has a proba- bility   p1. Foran ODLC correlator decoder that employs optical timed gating and optical threshold detection with perfect bit-level synchronization between a receiver and its in- tended transmitter, the detection chip location will be thecorrelation peak location for every bit duration. Accord- ingly, for every n chips there is only one detectionchip location. Hence, using Corollary 1, we only consider the second part, as the first part cannot happen in this system.Therefore, the probability of error Pe is the probability that h or more out of the N -1 unintended users contribute a pulse at the correlation peak location while the intended user is sending a  0 bit. However, the probability Pr that a user contributes a pulse at a single chip location is given by Pr = P1i n Hence, the probability of bit error is given by i=h  4) k2 AN-1l-i n ( (5) where a( ) = N -1) The probability of chip detection error in this case is given by Pe (chip level) =-P n  6) because for every n chips, n -1 are guaranteed to be de- tected correctly as idle, because they are ignored by the re- ceiver. Itis only the chip at the correlation peak location that can introduce chip errors. The probability of biterror (or BER) Pe derived in (5) is exactly the same as the expressionderived for the same system using the classical input signal combinatorial anal- ysis. However, using our method, the mathematical deriva-tion was more intuitive and straightforward. 4.2 ODLC CorrelatorReceivers without Optical Timed Gate For an ODLC correlator decoder that does not employ optical timed gating, but still uses opticalthreshold detec- tion solely for detection, the detection chip location will be every chip of the intended transmitter s bit duration. Hence, usingCorollary 1, the probability of error Pe is the proba- bility that h or more out of the N users contribute a pulse at 675  any of the n -1 chips of the bit duration or h or more out of the N -1 unintended users contribute a pulse at the last chip of the intended transmitter s bit duration while the intended transmitter is sending a  0 bit. This probability of error is calculated at the chip level.  n this case, bit level probability of error does not have a meaning, since bit boundaries are not known to the receiver. Higher level components of the system resolve the received chips into actual bits. Considering the first part of thescenario,the N -1 unin- tended users have similar pulse patterns according to Theo- rem 1, while the intended user has a different pulse pattern according to Theorem 2. The second part of the scenario is identical to the case of optical timed gate. The probability of chip error Pe in this case can be expressed as Pe   n)(i) 1 ) Pn  I1 i  i P1 nj (1 + (121 11) Pn)  I +   (I 1n( k2 AN-1l-i k 2 P-i - n k2A N-l1-i n (7) where Pn is the probability that the intended user is con- tributing a pulse to a chip location and is given by P. pi k k- 1) n The expression derived in (7) cannot be easily deduced using the classical input signal combinatorial method. This is due to the factthat one would have to consider each sin- gle chip shift of the input signal and deduce the chip error rate  CER) from all these different shifts. It is the use of Theorems 1 and 2 that allowed for considering all the dif- ferent shifts collectively, hence, deducing an expression for the CER. 5 Performance Analysis In order to verify the expressions derived in (5) and (7) a simulation model is used to produce the BER/CER for the two consideredsystems and the results of simulation are compared to numerical calculations of both equations. In our simulation, as well as numerical results, we considers systems that utilize noiseless and lossless components, we ignore any optical dispersion effects, and we assume the use of symmetrical optical splitters and couplers. Accordingly, errors in our system are solely caused by MAI noise. More- over, we always choose an opticalthreshold value for the threshold detector that equals the usedcode weight  i.e., h = k). This assumption guarantees that our system is working at the point of maximum MAI rejection  minimum MAI induced errors)[10]. It can be seen from Figures 2 and 3 that the simulation and theanalytical results for the two analyzed systems are very close. This verifies the accuracyof the deduced mathe- matical model and the correctness ofour modeling method. We can also see from the same figures that increasing the code length enhances the systemperformance.This is ex- pected due to the lower probability of pulse collisions when code length increases. On the other hand, increasing the number of users will increase the MAI level. Thus, in- creases the error probability in both systems. While in OCDMA systemsusing optical timed gating at thereceiver we can definethe BER, this is not true for sys- temswithout optical timed gating at the receiver. This is because the latter systems do not define bit boundaries. The only known time boundary for these systems is the chipdu- ration. Hence, for these systems we use CER as aperfor- mance measure. To compare the performance of systems with and without optical timed gating, we plot the CER of the former systems in Fig. 4. It can be seen from the fig- ure that using optical timed gating at thereceiver greatly enhances theoverall error performance of the system. 6 Conclusions In this paper, we haveproposed a new method to de- rivea mathematical expression for the BER of OCDMA systems using time-encoding, OOC-based spreading and ODLC based encoders/decoders. The proposed method is based on two important theorems that we have constructed. The twotheorems explain the main properties of OCDMA signals at the output of ODLC decoder. Using these theo- rems we were able to develop a simple method forderiv- ing the BER basedon the OCDMA signal at the output of the ODLC. Using our newly developed method, we were able to derivethe BER for OCDMA systemswith and with- out optical timed gating at the decoder side. The numerical results of our derived models were compared with simula- tionresults to test its validity and accuracy. The comparison confirms the accuracyof our modeling method. References [1] M. Azizoglu, J. A. Salehi, and Y. Li. Opical CDMA via temporal codes. IEEE Transactions on Communi- cations, 40 7):1162-1170, July 1992. [2] M. Kavehradand D.Zaccarh. Optical code-division- multiplexed systems based on spectral encoding of non- coherent sources. IEEE Journal of Lightwave Technol- ogy, 13(3):534-545, March 1995. 676 N-1 71 2 2 -Pi) I: a i) Pi I n i=h  [3] K.Yu, J. Shin, and N. Park. Wavelength-time spreading optical CDMA system using wavelength multiplexers and mirrored fiber delay lines. IEEE Photonics Tech- nology Letters, 12(9): 1278-1280, September 2000. [4] B. Ni. The performances of optical code-division mul- tiple access communication systems. ProQuest Thesis Online UMI Number: 3185704, August 2005. [5] J. A. Salehi. Code division multiple-access tech- niques in optical fiber networks part I: Fundamen- tal principles. IEEE Transactions onCommunications,37 8):824-833, August 1989. [6] J. G. Zhang, W. C. Kwongz, and A. B. Sharma. Effec- tive design of optical fiber code-division multiple ac- cess networks using the modified prime codes and opti-cal processing. In Proceedingsof International Confer- ence on CommunicationTechnology WCC-ICCT 2000, volume 1, pages 392-397, Beijin, China, 21-25 Jan- uary2000. [7]S. Martirosyan and A. J. H.Vinck. A construction for optical orthogonal codes with correlation 1. IEICE Transactions on Fundamentals of Electronics, E85- A 1):269-272, January2002. [8]T. Khattab and H.Alnuweiri. Optical orthogonalcode construction using rejected delays reuse for increas- ing sub-wavelength switching capacity. IEEE Journalof Lightwave Technology, 24 9):3280-3287, September 2006. [9] F. R. K. Chung, J. A. Salehi, and V. K. Wei. Opti- cal orthogonal code Design, analysis, and application. IEEE Transaction onInformation Theory, 35 3):595- 604, May 1989. Figure 2. Bit error rate for OCDMA with timed gate and thresholddetector 1010 lo-,I 10 lo0 10 l 610 10 lo-1 o k = 3, n = 65Simulation   k = 3, n = 65 Analytical O k = 4, n = 145 Simulation k = 4, n = 145 Analytical 10 Figure 3. Chip error rate for OCDMA withouttimed gate and using thresholddetector only [10] T. Khattab and H.Alnuweiri. Optical GMPLS networks with code switch capable layer for sub- wavelength switching. In Proceedings ofGlobal TelecommunicationsConference GLOBECOM 04, vol- ume 3, pages 1786-1792, Dallas, TX, USA, 29 November-3December 2004. Figure 4. Chip error rate for OCDMA with timed gate and thresholddetector 677 -a

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