5 K \\ \\ \\ N \\ ' ~ \u2022 - \u2022 -J \" -' \u2014 : k = 6 i 300 400 500 600 700 800 900 1000 n Figure 5.8: Effect of changing code length on BER. u Figure 5.9: Effect of increasing number of users on BER. k Figure 5.10: Effect of changing code weight on BER. 10-3 1 0 \" 4 lO\"5 \u00a9 CL-IO-6 10\"7 1 0 - 8 h Figure 5.11: Effect of changing the optical threshold on BER (N u = 9). \\ \\ \\ n = 611 n = 686 n = 774 \" -\\ v. v. -V \\ S s \\ s X v. \\ X . ' X ' ' - C^O v x \" \u2022 k = 6 \\ N - \"V . ^ \\ i \\ N \u00ae in\"11 Q. 1 0 P A = 0.5 BPPM P A = 0.75 BPPM P A = 1 BPPM - e - P A = 0.5 OOK - o - P A = 0.75 OOK \u2014 e \u2014 P A ~ 1 OOK 1400 Figure 5.12: Effect of increasing average user activity PA on BER (k = 7) Figure 5.13: BER for a system using OOC with optimum code length. Figure 5.14: Optimum code length as a function of code weight and number of users. 10000 8000 6000 g. 4000 2000 0 20 capacity condition given by (5.31) and minimum BER condition given by (5.32). It can be seen from Figure 5.13 that the system BER has two regions; one with zero MAI errors, which is the region defined by the condition N u < hopt and the other is a region with MAI errors. In the region with MAI errors the BER of the system increases with N u and decreases with k due to the associated increase in h. It must be noted, however, that the effect of the coupled change in k and h on the BER is more prominent than the effect of N u on the BER. This is demonstrated by the rapid non-linear change in Pe with k and h as compared to the slower close to linear change in Pe with N u shown in the figure. Figure 5.14 shows the value of nopt associated with each value of Pe in Figure 5.13. 5.6 Concluding Remarks We have proposed a modulation method for optical fiber CDMA transmission based on encoding the bit level value at the chip level of the spread signal using binary pulse position modulation, hence, the name Chip-Level Modulated BPPM (CLM-BPPM). Our modulation method maintains the advantages of PPM modulation over OOK methods by allowing for easier clock recovery at the receiver side as well as an ability to detect source activity. Moreover, our method surpasses the previously proposed PPM methods based on bit level modulation because it preserves the autocorrelation and crosscorrelation properties of the optical spreading codes across the entire bit duration. An architecture for an optical communication system that employs the proposed scheme is provided. We have provide several variations of the main scheme based on applying different methods for detection and bit conflict resolution. In order to evaluate the performance of our proposed scheme based on bit error rate (BER) due to multiple access interference (MAI), we started by developing a general mathematical framework to describe the BER in terms of different system parameters. Our mathematical model incorporated for the first time the average source activity statis-tics into the BER calculations. We have applied the developed framework on one of the variations of our proposed scheme to get an expression for the BER. Using our developed mathematical model we have numerically evaluated the BER of CLM-BPPM with absolutely timed detection and '1'-value first chip conflict resolution and compared the results with simulation results to verify the mathematical model. The comparison shows that our mathematical model accurately represents the system performance under the given assumptions. From the results, we have seen that increasing the number of users increases the BER due to the higher level of interference present in the system. Our results also show that if the optimality conditions are maintained, increasing the code weight enhances the system performance due to the fact that it requires increasing the optical detection threshold, which means higher ability to suppress MAI. Separating the changes of code weight and optical detection threshold, we notice that in fact increasing the code weight alone while fixing the threshold results in increasing the BER. We have also examined the effect of the average source activity on the BER and have shown that increasing the average source activity increases the BER of the system. Compared to an OOK system, our CLM-BPPM system has a slightly higher BER, but as the source activity increases (load is increased), both systems show closer BER values. Asymptotically, a fully loaded system (P A \u2014 1) produces the same BER for OOK and CLM-BPPM. The results also demonstrate that the BER of a CLM-BPPM system operating at the optimum BER point (h = k) is highly sensitive to changes in the code weight k, which suggests that network designers should focus more on choosing the optimum value for the spreading code weight. In this chapter, we have developed the CLM-BPPM modulation scheme for OCDMA systems. We have also developed a general mathematical framework for calculating the BER for time-spread OCDMA systems. The framework was used to deduce the BER of our CLM-BPPM modulation technique. In the next chapter, we focus on developing efficient construction methods for OOC spreading codes. Our aim is to develop a low complexity algorithm that can construct bandwidth-efficient spreading codes. OOC Construction Using Rejected Delays Reuse In Chapter 5, we have proposed CLM-BPPM as a modulation method for OCDMA systems. We have also developed a general mathematical framework for the BER of time-spread OCDMA systems and used our developed model to deduce the BER for CLM-BPPM. In this chapter, we shift our focus to the problem of optical spreading code design. In particular, we focus on the design of OOC using greedy algorithms. We introduce a modified element-by-element greedy algorithm for constructing OOC with small code length given a certain code weight and number of codewords. Our modified algorithm is called Greedy Algorithm with Rejected Delays Reuse (RDR). The rest of this chapter is organized as follows. Section 6.1 is a brief introduction. In Section 6.2, we provide mathematical definitions and representations for OOCs and their main properties. Following that, in Section 6.3, an overview of construction methods for OOCs as well as a detailed description of a computationally efficient element-by-element greedy algorithm for constructing OOCs are provided. In Section 6.4, we provide a theo-rem that shows how the greedy algorithm provided in Section 6.3 can be enhanced then we introduce our new RDR construction method. In Section 6.5, we evaluate the effi-ciency of OOCs generated using RDR as compared to those generated using the existing element-by-element greedy algorithm by introducing an efficiency factor that provides a quantitative measure of the code's ability to expand sub-wavelength switching capacity. We call this factor the expansion efficiency factor. In Section 6.6, we provide numer-ical results that show how our newly designed OOCs outperform those designed using the original element-by-element greedy construction. Finally, we conclude the chapter in Section 6.7. 6.1 Introduction The use of OCDMA as a multiplexing (switching) layer at the sub-wavelength level in Generalized Multi-Protocol Label Switching (GMPLS) networks was introduced in Chap-ter 3 by proposing the OC-GMPLS architecture. In order to use OCDMA, the optical signal is spread either in the time domain (temporal spreading) using Fiber Tapped Delay Lines or in the wavelength domain (spectral spreading) using Arrayed Waveguide Grating (AWG) or Fiber Bragg Grating (FBG). In both cases a spreading code is required. Opti-cal Orthogonal Codes (OOC) [66] are widely used as optical spreading codes because they achieve the optimum criteria for unipolar {0,1} codes in terms of their autocorrelation and crosscorrelation properties. However, efficient OOCs are difficult to construct. . In [85], Argon et al presented an algorithm for constructing OOCs based on the extended set representation. Being a member of the class of greedy algorithms, the algorithm did not reconsider previously rejected delay elements in future construction steps. In this chapter, we show that rejected elements at one step can in fact be accepted in later steps. Based on this fact, we propose a new algorithm called Greedy Algorithm with Rejected Delays Reuse (RDR) to construct OOCs. We show through our numerical results that OOCs designed using RDR are generally shorter than those constructed using the classical element-by-element greedy algorithm. In order to study the effect of using shorter codes on switching capacity we define a switching capacity expansion efficiency factor and show that shorter codes lead to higher expansion efficiency factors, which means higher switching capacity. 6.2 Definitions and Mathematical Preliminaries In the following, we provide some mathematical definitions of Optical Orthogonal Codes and their set theoretic representations. Then, we provide concrete mathematical formulas that can be used to deduce the interdelay set theoretic representation of an OOC from its direct classical definition. Using our mathematical representation, as well as the mathematical properties of OOCs, we provide new proofs for two well known theorems on OOCs. In Chapter 3, we have stated the conditions on the autocorrelation and crosscorre-lation properties of an OOC. In the following, we provide a more mathematically solid definition of optical orthogonal codes and their different properties. Definition 6.1. A (n,k,X a,X c) OOC, C with cardinality \\C\\ = N, is defined as a family of N {0,1}-sequences of length n and weight k with circular autocorrelation TZXX(t) for sequence X \u20ac C and circular crosscorrelation 1ZXY{t) between any two sequences X,Y \u00a3 C satisfying the following properties: where Xj is element number j in code X, yj is element number j in code Y and \u00a9 represents addition modulo-n. In this chapter, we consider the case Aa = Ac = 1. We call this a (n, k, 1) OOC and will refer to it as OOC for simplicity. This is the case with the lowest multiuser 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 constructed using the above definitions will constitute sparse matrices. A (n, k, 1) OOC consists of an N x n matrix with N x k T s (k in each row) and the remaining elements being 'O's, where k \u00ab n. Thus, a direct representation of an OOC is not convenient. It is more compact and mathematically tractable to represent an OOC by its set of interdelays T between the locations of the '1' bits in each codeword. Another representation of OOC called the set theoretic representation uses sets that contain the k T = 0 (6.1) < A a r ^ O (6.2) column numbers of the 'l 's in C. In the rest of this chapter, we will use the set of interdelays T to represent an OOC. Each row of T represents a codeword from the OOC. The number of rows is the number of codewords and the number of columns is the code weight. Definition 6.2. The set of interdelays T for an OOC C is the set that contains the differences in column numbers (relative positions) between any two adjacent 'l's in each codeword (row) of C, treating each codeword as a circularly connected subset. Formally, where Uj is the jth '1' bit in row (codeword) number i, Uj is element in row i and column j of T, and col(cij) \u2014 j is the column number of element Cij \u20ac C. col(lij+i) - col(lij) , 1 < j < k - 1 T \u2014 ^ \" , 1 < i < N , col(l u) + n- col(li,fe) , j = k Definition 6.3. The extended set of interdelays T ext for an OOC C is the adjoint of the set T with the set that contains the summations of adjacent interdelays, up to (k - 1) adjacent interdelays, in each row of T, treating each row as a circularly connected subset. Formally, qj+Uj IText \u2022 ^ ti)V p, P~Qj Qj = 1 * 1 and weight k > 1, we have 1 < eitj < n \u2014 1. Accordingly, we have a pool of n \u2014 1 elements (integer numbers) out of which we consume k(k \u2014 1) elements per constructed codeword. This means that the maximum number of constructed codewords N is less than or equal to . Since this number must be an integer number, we have \u2022 6.3 Classical Element-by-Element Greedy Method for Constructing OOCs Designing an OOC can be formulated as an optimization problem. An OOC design is the construction of a set of N {0, l}-sequences with equal lengths n, and equal weights k that satisfy the autocorrelation and crosscorrelation properties stated in (6.1) and (6.2) with Aa = Ac = 1, subject to minimizing n for a given k and N. The methods of combinatorial designs are the roots of many code design methods for OOC [86,87]. An OOC can be constructed using known combinatorial methods (designs) by appropriately mapping the OOC design problem into one of the solved combinatorial design problems. Then the solution can be used to design the OOC. In [16], Chung et al proposed the greedy algorithm for constructing the code set C as well as several other techniques. The proposed greedy algorithm incorporates an exhaustive search through all the possible (\u00a3) codewords in order to find the codewords that satisfy (6.1) and (6.2). These codewords constitute C. Later in the same paper, the authors proposed an accelerated greedy algorithm, which constructs C using an iterative method over its set theoretic representation. In [85], Argon et al presented a variation of the accelerated greedy algorithm (Algo-rithm 6.1). The algorithm reduces the number of construction steps by using an iterative element-by-element greedy construction method for building the interdelays set represen-tation T of an OOC. The algorithm starts with an empty set and uses two nested loops Algorithm 6.1 OOCGreedy Require: N > 2, k > 2 Ensure: TZ Xx < \\,Uxy < lVX.y \u20ac C such that X ^Y 1: e \u00ab- 1 2: for all i such that 1 < i < N do 3: T ( i , l ) * - e 4: e e + 1 5: end for 6: for j \u2014 2 to k - 1 do 7: for i = 1 to iV do 8: 0 <- FALSE 9: while ->0 do 10: T ( i , j ) < - e 11: O <\u2014 ChecklntermediateOrthogonal(T) 12: e <\u2014 e + 1 13: end while 14: end for 15: end for 16: O <- FALSE 17: for all i such that 1 < i < N do 18: m ^ ^ E T f u l 19: end for 20: M x <\u2014 max{mx(i)} i 21: while ->0 do 22: for all i such that 1 < i < N do 23: T(i,k) *-e + (M x-mx(i)) 24: end for 25: O <- CheckCompleteOrthogonal(T) 26: e <- e + 1 27: end while to construct the set of interdelays representation of the designed OOC. At step {i,j), the outer loop points at codeword number i. All the codewords at rows {1, \u2014 1} are of length j, while the remaining codewords are of length j \u2014 1. The inner loop tries to find the smallest possible untested interdelay element dj at column number j to add to codeword (row) number i such that orthogonality conditions are satisfied. The functions ChecklntermediateOrthogonal(S) and CheckCompeleteOrthogonal(S) perform the orthogonality check (i.e., ensure that Theorem 6.4 is satisfied) of the OOC under construction. These functions are implemented by Algorithms 6.2 and 6.3 respec-tively. The major two steps in these functions are: 1. construct the extended set T ext 2. ensure that there are no repeated elements in T e 2 ; t The difference between the first function (Algorithm 6.2) and the second function (Algorithm 6.3) is in the first step (i.e., the construction of T e x t ) . The check for repeated elements in a set can be easily achieved using a quick sort followed by a linear check of every element and its immediate following neighbor. If there are no matches, then there are no repeated elements. Algorithm 6.2 ChecklntermediateOrthogonal(S') Require: T ^ *2,k> 2 Ensure: K Xx < 1,RXY < 1VX, Y \u20ac C such that X ^Y e <\u2014 1 , RD <\u2014 (j) for all i such that 1 < i < N do T(i, 1) <\u2014 e , e e + 1 end for for j = 2 to k \u2014 1 do for i = 1 to N do O <- FALSE for I = 1 to |RD| do T (i,j) RD (\/) , O f - ChecklntermediateOrthogonal (T) if O then , BREAK end if end for while ~>0 do T( i , j ) <\u2014 e , O ChecklntermediateOrthogonal(T) if \u2014>0 then RD RD + {e} end if e*-e + l end while end for end for 0 <- FALSE for all i such that 1 < i < N do k-1 o=i end for M x max{m x(i)} for I = I to |RD| do for all i such that 1 < i < N do T(i, k) *- RD(0 + (M x - mx{i)) end for 0 <\u2014 CheckCompleteOrthogonal(T) if 0 then BREAK end if end for while -i0 do for all % such that 1 < i < N do T{i,k) i AND m = j) OR m > j, before trying a new element, we retest the elements in the rejected list, starting with the smallest in ascending order. If one of these elements can be used, we use it and remove it from the rejected list. Otherwise, we test new elements. This process continues until the last element of T has been found. There are some extra modifications that can be added to the algorithm in order to increase its efficiency. One such modification is to restrict the list of rejected elements to elements that correspond to the second scenario in the proof of Theorem 6.6. This would save both space and computation time, although it would complicate the algorithm description making it more difficult to understand. i( Since the RDR algorithm is typically used off-line to calculate the OOC once and then use the same code for OCDMA communications, calculation efficiency is not a major factor at this point and we will sacrifice the efficiency for the sake \"of clarity. The RDR algorithm can be implemented asymptotically in 0(n 2 ) computation time, as proven in Appendix C. This computation time can be achieved in a fraction of a second on a GHz processor for codes of lengths in the order of n ~ 1000. This is considerably faster than the classical element-by-element greedy method given in [85] taking into consideration the fact that RDR codes produce smaller values for n while performing approximately the same number of steps. We observed while running the algorithm that the terminating elements of codewords (i.e., elements at column number k) are always much larger than all the other interde-lay elements of the codeword. This suggests that testing the rejected elements at the finalization step can be omitted to save computation time without affecting the resulting codes. The larger values of the terminating elements are due to the fact that orthog-onality conditions applied to these elements are more strict since they include circular adjacency into the calculation as shown in Algorithm 6.3. Another restriction on the values of these elements is that they have to be selected such that all the codewords have the same length n as shown in Algorithms 6.1 and 6.4. If this condition is relaxed (i.e., variable codewords are allowed) it would greatly reduce the magnitude of the terminating interdelays. OOCs designed using Algorithm 6.4 with their corresponding (i.e., having the same parameters) codes derived using Algorithm 6.1 are shown in Figure 6.1. One can notice that once an element is reused by the RDR algorithm (e.g., the value 27 at the 4th location in the last codeword of the {7,4} code), the elements after that are generally smaller than those used for the classical greedy algorithm, which results in a code with smaller length. {k,N} T TRDR {7,4} {1,5,12,24,31,50,144} {2,7,13,25,32,58,130} {3,8,15,28,37,60,116} {4,10,19,30,44,62,98} {1,5,12,24,16,48,139} {2,7,13,25,30,49,119} {3,8,15,28,31,53,107} {4,10,19,27,34,35,116} {6,6} {1,7,17,28,50,174} {2,9,18,31,51,166} {3,10,20,35,57,152} {4,12,22,39,59,141} {5,14,23,43,62,130} {6,15,26,46,67,117} {1,7,17,28,44,150} {2,9,18,31,46,141} {3,10,20,35,48,131} {4,12,22,32,39,138} {5,14,23,36,51,118} {6,15,26,43,56,101} Figure 6.1: Examples of OOC designs using RDR. 6.5 Sub-Wavelength Switching Capacity and Code Efficiency Analysis For an optical multiplexing system based on Code Division Multiple Access (OCDMA) at the sub-wavelength level [25], it is desired to be able to multiplex as many users as possible per wavelength channel without affecting the system error rate under ideal conditions. It was shown in [25] that the condition for an OCDMA system to operate as an error free transmission system under ideal conditions is to be able to completely suppress the Multiple Access Interference (MAI) caused by unwanted users on the intended received signal using an optical threshold (hard limiter) device at the output of the correlator receiver. This is possible if and only if N < k. In order to maximize the system capacity (maximize the number of multiplexed users per wavelength) we take N = k. (6.3) It was also shown in [25] that the label space expansion ratio p, which measures the increase in the system capacity per wavelength of a sub-wavelength OCDMA switched system as compared to a pure Wavelength Division Multiplexing (WDM) system, is given by: p = N. (6.4) Using (6.4), p for an error-free system is given by: p = k. (6.5) From Theorem 6.5, we get N < n - 1 k(k- 1)_ (6.6) which yields a maximum number of sub-wavelength multiplexed users (maximum system capacity per wavelength) given by: N = n \u2014 1 (6.7) which can be rewritten as N \u2022 k(k \u2014 1) = n \u2014 1 \u2014'(n \u2014 1 mod k{k - 1)). (6.8) In order to make the maximum use of the available system bandwidth, it is desired to have the smallest n that satisfies (6.8). Inspecting (6.8), the minimum value for n for given values of k and N is achieved when (n \u2014 1 mod k(k \u2014 1)) = 0. This happens when k(k - 1) | n - 1, (6.9) which reads k(k \u2014 1) divides n \u2014 1. Under this condition, (6.8) can be simplified to Umin = N \u2022 k{k - 1) + 1, (6.10) where nmin is the minimum code length. In order to analyze the optimum error-free system, we substitute for N and k from (6.4) and (6.5) respectively into (6.7) to get n - l IP(P- i). (6.11) which can be rewritten as p2(p \u2014 1)\u2014 n \u2014 1 \u2014 (n \u2014 1 mod p(p \u2014 1)). (6.12) Following the same reasoning used to deduce (6.10), we can write the minimum error-free code length n*min as n*mm = p2{p-l) + l. (6.13) Since the RDR algorithm is a greedy algorithm, it can only search for a local optima of the solution. The optimality in this case is localized to the current construction step only. Accordingly, the RDR does not guarantee that the achieved solution is the global optimum across all the different construction steps. This means that the codes designed using the RDR greedy algorithm (Algorithm 6.4) are generally suboptimal codes. In order to compare the efficiency of OOCs constructed using RDR with the efficiency of OOCs constructed using Algorithm 6.1 [85], we define an expansion efficiency factor rj as the ratio between the minimum code length nmin and the code length n of the OOC generated using suboptimal algorithms. IT'min , V = , (6-14) n which means that for an optimum code we have 77 = 1, while for a suboptimal code we have 0 < r] < 1. It must be noted, however, that there are no guarantees that for a general set of parameters {n, k, N}, an optimal construction method for a (n, w, A) OOC, which satisfies the equality condition of the Johnson's bound on N as defined in (6.7), exists. The existence of optimal constructions have been demonstrated for only a very small subset of the set of different combinations of {n, k, N} [88]. 6.6 Numerical Results Figure 6.2 shows the code efficiency for the RDR algorithm r j R D R as well as the code efficiency for the greedy algorithm rj as a function of code weight k and number of codes N (i.e., code cardinality |C|). Observe from the figure that OOCs designed using the RDR algorithm (Algorithm 6.4) have significantly superior code expansion efficiency over OOCs designed using classical element-by-element greedy algorithm (Algorithm 6.1). Observe also from the figure that the code efficiency is robust against increases in the code cardinality as it does not affect the code efficiency significantly. On the other hand, increasing the code weight rapidly degrades the code efficiency with the RDR being more robust and maintains a superior efficiency across different code weights. Figures 6.3 and 6.4 are plotted for the case where the system is operating at the maximum capacity under the error free transmission constraints. Figure 6.3 results demonstrate that codes designed using the RDR algorithm have an asymptotic efficiency of about 0.4 as compared to codes designed using the classical element-by-element greedy algorithm, which have an asymptotic efficiency of about 0.2. This shows that RDR designed codes are expected to constantly outperform classical greedy constructed codes even at asymptotically large values of code weights k. Figure 6.4 shows the code length nRDR for OOCs designed using the RDR (Algo-rithm 6.4), the code length n for OOCs designed using Algorithm 6.1, and the minimum \u2022 ^RDR T] 0.8 0.75 0.7 'H 0.65 0.6 0.55 0.5 0.45 Figure 6.2: Code expansion efficiency factor for RDR greedy and greedy OOCs. p+1 Figure 6.3: Code expansion efficiency factor for RDR greedy and greedy OOCs under error free conditions. x 10 Figure 6.4: Code length for RDR greedy, greedy, and optimal OOCs under error free conditions. code length n*min given by (6.13) for the same design parameters under the error free maximum capacity condition (6.4). Clearly the RDR achieves around 30% reduction in the code length n over the classical element-by-element greedy algorithm. This difference becomes more significant as we increase the label space expansion factor p (i.e., increase the number of multiplexed users). 6.7 Concluding Remarks In this chapter, we have proposed a new algorithm for construction of OOCs called Greedy Algorithm with Rejected Delays Reuse (RDR). The algorithm belongs to the family of element-by-element greedy construction algorithms used to generate OOCs. The algorithm incorporates a reuse mechanism for previously rejected smaller delay elements in order to minimize the code length. The codes generated using the RDR algorithm were shown to be generally shorter (have smaller code length n) than those generated using the classical greedy algorithm. In order to compare the efficiency of the codes generated using the two algorithms we compare their code lengths to the optimum code length using a code expansion efficiency factor r]. Numerical results show that OOCs generated using RDR have significantly higher rj than those using the classical element-by-element greedy algorithm. It was also found through numerical results that rj is largely affected by the increase in the code weight k. However, increasing the number of codes N has negligible effects on rj. Conclusions and Future Research We conclude this thesis with a summary of our work, highlights of the major thesis contributions, and some suggestions for future work. 7.1 Summary The main objectives of this thesis were: \u2022 The increase of sub-wavelength switching granularity in all-optical networks using code division multiplexing. \u2022 The development of enhanced optical code division multiple access schemes. \u2022 The development of mathematical performance models for sub-wavelength code switched mechanisms. In the following, we present a summary of each of the work chapters. In Chapter 2, we presented an overview of the major transmission methods used in OCDMA communica-tion systems. The overview spanned the different modulation and spreading techniques. It also discussed the different detection methods and the different coding families used in OCDMA systems. In addition, a detailed description accompanied by a mathemat-ical design for a class of OCDMA encoder\/decoder devices called Optical Delay Line Correlators is provided. Results in this chapter also appear in [31]. Chapter 3 proposed an architecture for optical networks capable of sub-wavelength switching using OCDMA based on expanding the label space of the GMPLS architecture. The architecture is called Optical Code Labeled GMPLS and employs a code switch capable layer into the GMPLS network to expand the switching granularity of all-optical core switches into the sub-wavelength level. .We have evaluated the switching capabilities of the OC-GMPLS using mathematical modeling to derive a proposed performance factor called the label space expansion ratio. Further, we derived the optimum value for the label space expansion ratio under different system constraints. Results of this work have been published in [25,26]. In Chapter 4, we presented an analytical model for calculating the throughput of the OC-GMPLS core switches. The model incorporates both network and physical layer parameters into the throughput calculation. We have used results from our model to show that there is a set of optimal parameters that maximizes the throughput. The values of these parameters as well as the value of the maximum throughput depend on the provided network and physical layer configurations as well as the required operational constraints. This work also appears in [26,27]. Chapter 5 proposed a new modulation scheme for OCDMA transmission. The scheme is based oh using BPPM to modulate the spread signal chip pulses, hence, the name Chip-Level Modulated BPPM. The transceiver architecture for implementing the pro-posed scheme in an all-optical fashion is provided. The proposed scheme enables the receiver to distinguish between the idle and zero transmitting sources within one bit du-ration. This was not previously possible using OOK methods. The proposed scheme also provides for a better clock recovery over OOK due to the guaranteed transitions within each bit duration as long as the source is active. In order to analyze the proposed system, a general mathematical framework is derived for calculating the bit error rate of any time spread, OOC encoded and correlation detection based OCDMA system. The mathemat-ical model incorporates, for the first time, the average source traffic parameters in the BER calculations. The mathematical framework is applied to the CLM-BPPM to deduce an expression for its BER. The numerical results and comparison with simulations show the high accuracy of the developed model. The results reveal that CLM-BPPM has an error rate that is very close to OOK systems with asymptotically equal values at higher load conditions. Results from this work appear in [28]. Chapter 6 proposed a new method for constructing OOC based on greedy element-by-element recursive construction methods. The proposed algorithm employs a reuse mechanism for previously rejected smaller delay elements in order to produce shorter OOC code lengths. The algorithm is called Greedy Algorithm with Rejected Delays Reuse (RDR). The complexity of the proposed algorithm is analyzed and its resulting code label space expansion efficiency normalized to optimal codes are calculated. The results show that RDR has less complexity than the classical greedy element-by-element methods. Further, the codes constructed using RDR have an expansion efficiency factor that is asymptotically an order of magnitude higher than that of the classically constructed codes. This work has been published in [29,30]. 7.2 Major Contributions In conclusion, among the main contributions of this thesis, we proposed: \u2022 An architecture for OCDMA based all-optical sub-wavelength switching in core GMPLS networks. The proposed architecture achieves higher level of flow isola-tion while maintaining all-optical switching speed at the network core. In order to demonstrate the advantages of our architecture we have derived a switching gran-ularity measure called label space expansion ratio and used it to derive optimal operating parameters under MAI error free condition. \u2022 An analytical model for calculating the throughput of OCDMA-enabled core GM-PLS switches. The model incorporates both the physical layer properties through the MAI induced BER as well as network layer parameters such as number of users and average packet length. The model was used to show that there are two major operating regions for the system. Optimal operating parameters which maximize the throughput were numerically deduced under each operating region. \u2022 A new modulation method for OCDMA called CLM-BPPM. The method is based on using BPPM deduced from the bit value to modulate the chip pulses. The method allows for better clock recovery and ability to detect transmitter activity as compared to OOK. It also provides a BER that is very close to OOK with equal asymptotical values at higher user activity. \u2022 A generalized mathematical framework for calculating the multiple access BER in OCDMA networks. The framework can be used to deduce the BER for any time-spread, OOC encoded OCDMA transmission with correlation detection. The accuracy of the mathematical model was compared against simulation models and was shown to achieve very close results. \u2022 A new efficient greedy element-by-element algorithm for constructing shorter opti-cal orthogonal codes. The algorithm was shown to produce smaller code lengths under same design constraints as compared to classical element-by-element greedy methods. The asymptotic computational complexity of the algorithm was deduced and was shown to be smaller than previously proposed algorithms. 7.3 Future Work We present below a list of research problems that can be investigated as possible exten-sions for the research work reported in this thesis. 1. The work in Chapter 6 was based on a set theoretic representation of OOC code-words as sets of interdelay elements. The auto and cross correlation properties were mapped into uniqueness properties among one set and across the different sets. It is worth investigating if a different representation of OOC might lead to better construction methods. An example of a representation that can be investigated is to represent code families as matrices of binary values. Then applying the auto and cross correlation constraints on these matrices one can deduce a set of construction rules for these matrices, using this set of rules in conjunction with a clever matrix search algorithm new construction methods can be established. For this technique to work, it is necessary to find a matrix representation that is unique per family of OOC codes and at the same time can exploit the auto and cross correlation in-duced constraints to the maximum resulting in minimizing the number of iterations during the matrix search operation. 2. Several variations of the RDR algorithm can be examined in search for a closer to optimal method. In the RDR algorithm codewords at the top of the iteration stack are always assigned shorter inter-delays than those below it. This results in producing a densely occupied area at the start of the code. This in turn will result in higher rejection of inter-delay values at subsequent codewords resulting in increasing their length. This increase in length reflects on the codes at the top of the stack when we reach the terminating stage due to the constraint that all codewords must have the same length. If a more balanced approach like moving through the iterations in a zig-zag manner so that in one round the assignment of inter-delays starts from the top codeword moving down. In the next round it goes in the reverse direction. This way a more balanced assignment of inter-delay values is achieved, which might result in reducing the code lengths. 3. The throughput of OC-GMPLS derived in Chapter 4 was derived under the as-sumption of a uniform fiber-wavelength-code assignment combination. In other words, the switch fills up all the fibers and wavelengths with a single flow each. Then, it starts rotating across the different wavelengths over the different fibers assigning a new flow to them one by one. This means that the average number of OCDMA flows on each wavelength are equal. This is not necessarily optimal in terms of the overall throughput. It will be a fruitful point of research to study the different permutations of OCDMA assignment given a certain number of flows to come up with the permutation that maximizes the overall system throughput. The next natural step will be to devise online algorithms that can assign the labels to attain, op.timality under different loading and channel conditions. 4. The analysis provided in Chapter 5 was carried under ideal optical channel assump-tions. It will be interesting to see how the system will perform when optical channel noise and non-linearities are introduced to the model. In particular it will be of great interest to compare the performance of CLM-BPPM with OOK assuming non-ideal optical channel conditions. 5. In optical networks, the error events are very rare events with very low probabilities of values as small as 10 - 1 2 . In order to directly simulate errors in these networks, simulation time becomes prohibitively large. This problem can be overcome by deriving mathematical models for the network performance as was done in this thesis. However, mathematical models are not always possible to attain and they require many assumptions that affects their accuracy. Which means that simulation remains an essential tool for analyzing these networks, both to fill in when mathe-matical modeling is not feasible and to check the accuracy of derived mathematical models. Statistical techniques such as importance sampling promise a solution to simulation of rare events. Using such techniques, it would be very useful to try to establish simulation models that can efficiently model optical networks. Bibliography [1] A, Kaheel, T. Khattab, A. Mohamed, and H. Alnuweiri. Quality-of-service mecha-nisms in IP-over-WDM networks. IEEE Communications Magazine, 40(12):38-44, December 2002. [2] B. E. A. Saleh and M. C. Teich. 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Combinatorial constructions of optical orthogonal codes for OCDMA systems. IEEE Communications Letters, 8(6):391\u2014393, January 2004. Proof of pr < 1 in Equation 4.1 In this Appendix, we provide a mathematical proof that a non-trivial OOC, which satisfies the orthogonality condition (3.3) satisfies also the following condition: k2 Pr = \u2014 <1 \u2022 n Proof. Starting from the orthogonality condition, we have N < n - 1 k(k- 1)_ (A.l) Therefore, we can write N < n - 1 k(k- 1)_ < n - 1 k(k- 1) (A.2) which can be rearranged to give N-k(k-l) 1. However, for a given n, the largest value of k in (A.3) is achieved by choosing the smallest possible value of N, which is achieved by choosing N = 2. Hence, 2k(k - 1) < n - 1 , (A.4) which can be rewritten as k2 + (k-l) 2 0 , (A.5) can be rewritten as k2 < n , (A.6) Substituting (A.6) in (4.1) we get k2 Pr = \u2014 < 1 n \u2022 Derivation of Equation 5.28 In Chapter 5, we gave a simplified mathematical representation of the BER for a CLM-BPPM using absolutely timed detection and 'l'-value first bit conflict resolution by (5.28). In the following, we derive this equation in details. Starting from the right hand side (RHS) of (5.27), we have RI Nu-lNu-l-l h-\\ N u-\\-l RHS = (1 - P a P x ) Y ^ Y , P ( l , m ) + ( 1 - P A ) Y , E P^rn)d>(l,m) . (B.l) l=h m=0 1=0 m=h R2 The first term (Rl) can be written as * = E V 1 - * ~ E _ i ( Nu ~ 1 ~ >om(i - * - \u00ab.)\u2022 l=h ^ ' m = 0 ^ , , \\ 1 J n \\ m J \\l-q1-q0 \u2022 l=h x m=0 x \u2022 \\ N u-1 \/ar i\\ \/ \\ Nu-l-l E ( \" y V 1 - ^ \" - 1 - ' . i=h A 1 J p(D which reduces to h-l RI = 1-]TP(1) , 1=0 where P(l) is the binomial distribution with parameters q\\, N u \u2014 1 and I. Considering the second term (R2), we can rewrite it as (B.2) where and 1=0 I tiil-qi-qo^-'-'SiW) , (B.3) N u-l-l E m=h N u - 1 - I m Qo 1 - q i I- qi- qo l-qi-qo N u-l-l h-l - E m=0 N u-l-l m Qo 1 - {l) -1 l< K,-l-h 0 otherwise. is cj)(l,m) rewritten as a function that depends on I only. Substituting (B.4) in (B.3), we get * = E ( N u , V 1 - n)\"-1-1*\u00ae - E - ft -1=0 \\ ' m=0 ^ ' ' h-l h-l h-l 1=0 1=0 m=0 Substituting (B.2) and (B.5) in (B.l) results in h-l \\ \/h-l h-l h-l RHS - (i - P A P X ) i - ^ p{i) + (i - p A ) E p v w i ) - E E p \\ 1=0 J \\l=0 (=0 m=0 \/ h-l (h-l h-l N = 1-PaP1 + J 2 P(l) [(1 - PAW) - (1 - PAPI)) - (1 - PA) E E ^ 1=0 \\ 1=0 m=0 Asymptotic Computation Time for the RDR Algorithm In this appendix, we provide an asymptotic computational complexity analysis for the RDR element-by-element greedy construction algorithm described in details in Chapter 6. The RDR algorithm performs its operations on the set T ext, which is composed of unique elements 1 < e^ < n. Hence, the algorithm -tests at most n elements (running at most n iterations). For each iteration, a new interdelay element is tested for addition to code number i at delay element number j in T at step (i, j), where 1 < i < N and 1 < j < k. Assuming that the algorithm will perform equal number of iterations in each step (i, j) to reach the accepted interdelay element, the total number of iterations per step will be n N xk ' In each iteration, the algorithm performs linear time operations (comparisons, additions, etc.) on, at most, all the elements of where T ^ is the partial set of T ext composed at step (i, j) of the algorithm. Taking the worst case scenario when i = N (an element is tested for addition to code number N), the total number of operations performed on all the elements of T ^ per iteration per step is given by: c0 x N x j(j - 1) , where c0 is an arbitrary constant. Accordingly, the maximum total number of operations N 0 performed by the algorithm is given by: N k 1=1 j=1 nxN k j=i nxN (k - l)k(k + 1) = \u00b0o\u2014; ^ = ^ x n x N{k-l){k + l) . o This means that the algorithm has 0(n x N x k2) computation complexity. However, we know from Johnson's inequality (proven in Theorem 6.5), that n-l N < k{k - 1) Therefore, O(N) < 0(\u00a3) Consequently, the computation complexity of the algorithm is i 71 0(nx^-x k2) = 0(n 2) . ","attrs":{"lang":"en","ns":"http:\/\/www.w3.org\/2009\/08\/skos-reference\/skos.html#note","classmap":"oc:AnnotationContainer"},"iri":"http:\/\/www.w3.org\/2009\/08\/skos-reference\/skos.html#note","explain":"Simple Knowledge Organisation System; Notes are used to provide information relating to SKOS concepts. 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Additional conditions apply, see Terms of Use https:\/\/open.library.ubc.ca\/terms_of_use.","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/rights","classmap":"edm:WebResource","property":"dcterms:rights"},"iri":"http:\/\/purl.org\/dc\/terms\/rights","explain":"A Dublin Core Terms Property; Information about rights held in and over the resource.; Typically, rights information includes a statement about various property rights associated with the resource, including intellectual property rights."}],"ScholarlyLevel":[{"label":"ScholarlyLevel","value":"Graduate","attrs":{"lang":"en","ns":"https:\/\/open.library.ubc.ca\/terms#scholarLevel","classmap":"oc:PublicationDescription","property":"oc:scholarLevel"},"iri":"https:\/\/open.library.ubc.ca\/terms#scholarLevel","explain":"UBC Open Collections Metadata Components; Local Field; Identifies the scholarly level of the author(s)\/creator(s)."}],"Title":[{"label":"Title","value":"Optical Code Division Multiplexing for sub-wavelength switching systems","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/title","classmap":"dpla:SourceResource","property":"dcterms:title"},"iri":"http:\/\/purl.org\/dc\/terms\/title","explain":"A Dublin Core Terms Property; The name given to the resource."}],"Type":[{"label":"Type","value":"Text","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/type","classmap":"dpla:SourceResource","property":"dcterms:type"},"iri":"http:\/\/purl.org\/dc\/terms\/type","explain":"A Dublin Core Terms Property; The nature or genre of the resource.; Recommended best practice is to use a controlled vocabulary such as the DCMI Type Vocabulary [DCMITYPE]. To describe the file format, physical medium, or dimensions of the resource, use the Format element."}],"URI":[{"label":"URI","value":"http:\/\/hdl.handle.net\/2429\/31083","attrs":{"lang":"en","ns":"https:\/\/open.library.ubc.ca\/terms#identifierURI","classmap":"oc:PublicationDescription","property":"oc:identifierURI"},"iri":"https:\/\/open.library.ubc.ca\/terms#identifierURI","explain":"UBC Open Collections Metadata Components; Local Field; Indicates the handle for item record."}],"SortDate":[{"label":"Sort Date","value":"2007-12-31 AD","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/date","classmap":"oc:InternalResource","property":"dcterms:date"},"iri":"http:\/\/purl.org\/dc\/terms\/date","explain":"A Dublin Core Elements Property; A point or period of time associated with an event in the lifecycle of the resource.; Date may be used to express temporal information at any level of granularity. Recommended best practice is to use an encoding scheme, such as the W3CDTF profile of ISO 8601 [W3CDTF].; A point or period of time associated with an event in the lifecycle of the resource.; Date may be used to express temporal information at any level of granularity. Recommended best practice is to use an encoding scheme, such as the W3CDTF profile of ISO 8601 [W3CDTF]."}]}