Various lightpath protection schemes for a survivable WDM grooming network with dynamic traffic were investigated in Chapter 12. The nodes in the WDM grooming network are assumed to include ADM (add–drop multiplexer)-constrained grooming nodes. This chapter deals with the static survivable WDM grooming network design with wavelength continuity constrained grooming nodes. For static traffic the problem of grooming subwavelength level requests in mesh-restorable WDM networks, the corresponding path selection and wavelength assignment problems are formulated as ILP optimization problems.

Design problem

To address the survivable grooming network design problem, a network with W wavelengths per fiber and K disjoint alternate paths for each s-d pair can be viewed as W × K networks, with each of them representing a single wavelength network. For K = 2, the first W networks contain the first alternate path for each s-d pair on each wavelength. We number the networks from 1 to W, according to the wavelengths associated with them. The second set of W networks contain the second alternate path for each s-d pair on each wavelength. These networks are numbered from W + 1 to 2W, where the (W + i)th network represents the same wavelength as the ith network, i = 1, 2, …,W. Figure 13.1 illustrates this layered model for a six-node network with three wavelengths and two link-disjoint alternate paths. For each node-pair, it also depicts routing of two alternate paths for two connections in the network.

The two most important objectives for network operation are:

1. (i) capacity minimization

2. (ii) revenue maximization.

For capacity minimization, there are three operational phases in survivable WDM network operation: (i) initial call setup, (ii) short-/medium-term reconfiguration, and (iii) long-term reconfiguration. All three optimization problems may be modeled using an ILP formulation separately. A single ILP formulation that can incorporate all three phases of the network operation is presented in this chapter. This common framework also takes service disruption into consideration. Typically, most of the design problems in optical networks have considered a static traffic demand and have tried to optimize the network cost assuming various cost models and survivability paradigms. Fast restoration is a key feature addressed in the designs. Once the network is provisioned, the critical issue is how to operate the network in such a way that the network performance is optimized under dynamic traffic.

The framework for revenue maximization is modified to include a service differentiation model based on lightpath protection. A multi-stage solution methodology is developed to solve individual service classes sequentially and to combine them to obtain a feasible solution. Different cost comparisons in terms of the increase in revenue obtained for various service classes with the base case of accepting demands without any protection show the gains of planning and operation efficiency.

Capacity minimization

Among the three phases of capacity minimization the initial call setup phase is a static optimization problem where the network capacity is optimized for the given topology and the traffic matrix to be provisioned on the network.

Optical components are devices that transmit, shape, amplify, switch, transport, or detect light signals. The improvements in optical component technologies over the past few decades have been the key enabler in the evolution and commercialization of optical networks. In this appendix, the basic principles behind the functioning of the various components are briefly reviewed. In general, there are three groups of optical components.

1. (i) Active components: devices that are electrically powered, such as lasers, wavelength shifters, and modulators.

2. (ii) Passive components: devices that are not electrically powered and that do not generate light of their own, such as fibers, multiplexers, demultiplexers, couplers, isolators, attenuators, and circulators.

3. (iii) Optical modules: devices that are a collection of active and/or passive optical elements used to perform specific tasks. This group includes transceivers, erbium-doped amplifiers, optical switches, and optical add/drop multiplexers.

Fiber optic cables

The backbone that connects all of the nodes and systems together is the optical fiber. The fiber allows signals of enormous frequency range (25 THz) to be transmitted over long distances without significant distortion in the information content. While there are losses in the fiber due to reflection, refraction, scattering, dispersion, and absorption, the bandwidth available in this medium is orders of magnitude more than that provided by other conventional mediums such as copper cables. As will be explained below, the bandwidth available in the fiber is limited only by the attenuation characteristics of the medium at low frequencies and its dispersion characteristics at high frequencies.

Optical technology involves research into components, such as couplers, amplifiers, switches, etc., that form the building blocks of the networks. Some of the main components used in optical networking are described in Appendix A1. With the help of these components, one designs a network and operates it. Issues in network design include minimizing the total network cost, the ability of the network to tolerate failures, the scalability of the network to meet future demands based on projected traffic volumes, etc. The operational part of the network involves monitoring the network for proper functionality, routing traffic, handling dynamic traffic in the network, reconfiguring the network in the case of failure, etc. In this chapter, these issues are introduced in brief, followed by a discussion of the two main issues in network operation, namely survivability and how traffic grooming relates to managing smaller traffic streams.

Network design

Network design involves assigning sufficient resources in the network to meet the projected traffic demand. Typically, network design problems consider a static traffic matrix and aim to design a network that is optimized based on certain performance metrics. Network design problems employing a static traffic matrix are typically formulated as optimization problems. If the traffic pattern in the network is dynamic, i.e. the specific traffic is not known a priori, the design problem involves assigning resources based on certain projected traffic distributions. In the case of dynamic traffic the network designer attempts to quantify certain network performance metrics based on the distribution of the traffic. The most commonly used metric in evaluating a network under dynamic traffic patterns is the blocking probability.

A network is represented by a graph G = (V, E), where V is a finite set of elements called nodes or vertices, and E is a set of unordered pairs of nodes called edges or arcs. This is an undirected graph. A directed graph is also defined similarly except that the arcs or edges are ordered pairs. For both directed and undirected graphs, an arc or an edge from a node i to a node j is represented using the notation (i, j). Examples of five-node directed and undirected graphs are shown in Fig. A3.1. In an undirected graph, an edge (i, j) can carry data traffic in both directions (i.e. from node i to node j and from node j to node i), whereas in a directed graph, the traffic is only carried from node i to node j.

Graph representations. A graph is stored either as an adjacency matrix or an incidence matrix, as shown in Fig. A3.2. For a graph with N nodes, an N × N 0−1 matrix stores the link information in the adjacency matrix. The element (i, j) is a 1 if node i has a link to node j. An incidence matrix, on the other hand, is an N × M matrix where M is the number of links numbered from 0 to M - 1. The element (i, j) stores the information on whether link j is incident on node i or not. Thus, the incidence matrix carries information about exactly what links are incident on a node.

One of the important performance metrics by which a wide area network is evaluated is based on the success ratio of the number of requests that are accepted in the network. This metric is usually posed in its alternate form as the blocking probability, which refers to the rejection ratio of the requests in the network. The smaller the rejection ratio is, the better the network performance. Although other performance metrics exist, such as the effective traffic carried in the network, the fairness of request rejections with respect to requests requiring different capacity requirements or different path lengths, the most meaningful way to measure the performance of a wide-area network is through the blocking performance. To some extent the other performance metrics described above can be obtained as functions of the blocking performance.

Analytical models that evaluate the blocking performance of wide-area circuit-switched networks are employed during the design phase of a network. In the design phase these models are typically employed as an elimination test, rather than as an acceptance test. In other words, the analytical models are employed as back of the envelope calculations to evaluate a network design, rejecting those designs that are below a certain threshold.

Blocking model

The following assumptions are made to develop an analytical model for evaluating the blocking performance of a TSN.

• The network has N nodes.

• The call arrival at every node follows a Poisson process with rate λn. The choice of Poisson traffic is to keep the analysis tractable.

• […]

Feedback shift register (FSR) sequences have been widely used as synchronization, masking, or scrambling codes and for white noise signals in communication systems, signal sets in CDMA communications, key stream generators in stream cipher cryptosystems, random number generators in many cryptographic primitive algorithms, and testing vectors in hardware design. Golomb's popular book Shift Register Sequences, first published in 1967 and revised in 1982 is a pioneering book that discusses this type of sequences. In this chapter, we introduce this topic and discuss the synthesis and the analysis of periodicity of linear feedback shift register sequences. We give different (though equivalent) definitions and representations for LFSR sequences and point out which are most suitable for either implementation or analysis. This chapter contains seven sections, which are organized as follows. In Section 4.1, we give a general description for feedback shift registers at the gate level for the binary case and as a finite field configuration for the q-ary case. In Sections 4.2–4.4, we introduce the definition of LFSR sequences from the point of view of polynomial rings and discuss their characteristic polynomials, minimal polynomials, and periods. Then, we show the decomposition of LFSR sequences. We provide the matrix representation of LFSR sequences in Section 4.5 as another historic approach and discuss their trace representation for the irreducible case in detail in Section 4.6, which is a more modern approach. (The general case will be treated in Chapter 6.) LFSRs with primitive minimal polynomials are basic building blocks for nonlinear generators.

Randomness of a sequence refers to the unpredictablity of the sequence. Any deterministically generated sequence used in practical applications is not truly random. The best that can be done here is to single out certain properties as being associated with randomness and to accept any sequence that has these properties as random or more properly, a pseudorandom sequence. In this chapter, we will discuss the randomness of sequences whose elements are taken from a finite field. In Section 5.1, we present Golomb's three randomness postulates for binary sequences, namely the balance property, the run property, and the (ideal) two-level autocorrelation property, and the extension of these randomness postulates to nonbinary sequences. M-sequences over a finite field possess many extraordinary randomness properties except for having the lowest possible linear span, which has stimulated researchers to seek nonlinear sequences with similarly such favorable properties for years. In Section 5.2, we show that m-sequences satisfy Golomb's three randomness postulates. In Section 5.3, we introduce the interleaved structures of m-sequences and the subfield decomposition of m-sequences. In Sections 5.4–5.6, we present the shift-and-add property, constant-on-cosets property, and 2-tuple balance property of m-sequences, respectively. The last section is devoted to the classification of binary sequences of period 2n − 1.

Golomb's randomness postulates and randomness criteria

We discussed some general properties of auto- and crosscorrelation in Chapter 1 for sequences whose elements are taken from the real number field or the complex number field.

This book is the product of a fruitful collaboration between one of the earliest developers of the theory and applications of binary sequences with favorable correlation properties and one of the currently most active younger contributors to research in this area. Each of us has taught university courses based on this material and benefited from the feedback obtained from the students in those courses. Our goal has been to produce a book that achieves a balance between the theoretical aspects of binary sequences with nearly ideal autocorrelation functions and the applications of these sequences to signal design for communications, radar, cryptography, and so on. This book is intended for use as a reference work for engineers and computer scientists in the applications areas just mentioned, as well as to serve as a textbook for a course in this important area of digital communications. Enough material has been included to enable an instructor to make some choices about what to cover in a one-semester course. However, we have referred the reader to the literature on those occasions when the inclusion of further detail would have resulted in a book of inordinate length.

We plan to maintain a Web site at http://calliope.uwaterloo.ca/∼ggong/book/book.htm for additions, corrections, and the continual updating of the material in this book.

Binary sequences of period N with 2-level autocorrelation have many important applications in communications and cryptology. From Section 7.1, 2-level autocorrelation sequences are in natural correspondence with cyclic Hadamard difference sets with ν = N, κ = (N − 1)/2, and λ = (N − 3)/4. For this reason, they are named cyclic Hadamard sequences. In this chapter, 2-level autocorrelation always means ideal 2-level autocorrelation. There are three classic constructions for binary 2-level autocorrelation sequences that were known before 1997 (including some generalizations along these lines after 1997). One is m-sequences, described in Chapter 5, with period N = 2n − 1. The second construction is based on a number theory approach, including three types of sequences in Chapter 2, which are the quadratic residue sequences, Hall sextic residue sequences, and twin prime sequences. The period of such a sequence is either a prime or a product of twin primes. The third construction is associated with intermediate subfields. The resulting sequences have subfield decompositions and period N = 2n − 1. They include GMW sequences, cascaded GMW sequences, and generalized GMW sequences. Although the resulting sequences are binary, this construction relies heavily on intermediate fields and compositions of functions. As a consequence, it involves sequences over intermediate fields that are not binary sequences. The content of this chapter is organized as follows.

Overview

In the first three of the applications mentioned in the title of this chapter, one of the objectives (often the major objective) is to determine a point in time with great accuracy. In radar and sonar, we want to determine the round-trip time from transmitter to target to receiver very accurately, because the one-way time (half of the round-trip time) is a measure of the distance to the target (called the range of the target).

The simplest approach would be to send out a pure impulse of energy and measure the time until it returns. The ideal impulse would be virtually instantaneous in duration, but with such high amplitude that the total energy contained in the pulse would be significant, much like a Dirac delta function. However, the Dirac delta function not only fails to exist as a mathematical function, but it is also unrealizable as a physical signal. Close approximations to it – very brief signals with very large amplitudes – may be valid mathematically, but are impractical to generate physically. Any actual transmitter will have an upper limit on peak power output, and hence a short pulse will have a very restricted amount of total energy: at most, the peak power times the pulse duration. More total energy can be transmitted if we extend the duration; but if we transmit at uniform power over an extended duration, we do not get a sharp determination of the round-trip time. This dilemma is illustrated in Figure 12.1.