We now consider the performance of the digital modulation techniques discussed in the previous chapter when used over AWGN channels and channels with flat fading. There are two performance criteria of interest: the probability of error, defined relative to either symbol or bit errors; and the outage probability, defined as the probability that the instantaneous signal-to-noise ratio falls below a given threshold. Flat fading can cause a dramatic increase in either the average bit error probability or the signal outage probability. Wireless channels may also exhibit frequency-selective fading and Doppler shift. Frequency-selective fading gives rise to intersymbol interference (ISI), which causes an irreducible error floor in the received signal. Doppler causes spectral broadening, which leads to adjacent channel interference (small at typical user velocities) and also to an irreducible error floor in signals with differential phase encoding (e.g. DPSK), since the phase reference of the previous symbol partially decorrelates over a symbol time. This chapter describes the impact on digital modulation performance of noise, flat fading, frequency-selective fading, and Doppler.

AWGN Channels

In this section we define the signal-to-noise power ratio (SNR) and its relation to energy per bit (Eb) and energy per symbol (Es). We then examine the error probability on AWGN channels for different modulation techniques as parameterized by these energy metrics. Our analysis uses the signal space concepts of Section 5.1.

1. Introduction Let X be a set of points in Euclidean d-space Ed. Then conv X and aff X denote, respectively, the convex hull and the affine hull of X. Two points x and y are called antipodal points of X if there are distinct parallel supporting hyperplanes of conv X, one of which contains x and the other contains y. We say that X is an antipodal set if any two points of X are antipodal points of X. In the case that X is a convex d-polytope P, a related notion was recently introduced in [Talata 1999]. P is an edge-antipodal d-polytope if any two vertices of P, that lie on an edge of P, are antipodal points of P.

According to a well-known result of Danzer and Grünbaum [1962], conjectured independently by Erdős [1957] and Klee [1960], the cardinality of any antipodal set in Ed is at most 2d. Talata [1999] conjectured that there exists a smallest positive integer m such that the cardinality of the vertex set of any edge-antipodal 3-polytope is at most m. In an elegant paper, Csikos [2003] showed that m ≤ 12. In this paper, we prove that m = 8. THEOREM. The number of vertices of an edge-antipodal 3-polytope P is at most eight, with equality only if P is an affine cube.

Coding allows bit errors introduced by transmission of a modulated signal through a wireless channel to be either detected or corrected by a decoder in the receiver. Coding can be considered as the embedding of signal constellation points in a higher-dimensional signaling space than is needed for communications. By going to a higher-dimensional space, the distance between points can be increased, which provides for better error correction and detection.

In this chapter we describe codes designed for additive white Gaussian noise channels and for fading channels. Codes designed for AWGN channels typically do not work well on fading channels because they cannot correct for long error bursts that occur in deep fading. Codes for fading channels are mainly based on using an AWGN channel code combined with interleaving, but the criterion for the code design changes to provide fading diversity. Other coding techniques to combat performance degradation due to fading include unequal error protection codes and joint source and channel coding.

We first provide an overview of code design in both fading and AWGN, along with basic design parameters such as minimum distance, coding gain, bandwidth expansion, and diversity order. Sections 8.2 and 8.3 provide a basic overview of block and convolutional code designs for AWGN channels. Although these designs are not directly applicable to fading channels, codes for fading channels and other codes used in wireless systems (e.g., spreading codes in CDMA) require background in these fundamental techniques.

In Chapter 6 we saw that both Rayleigh fading and log-normal shadowing exact a large power penalty on the performance of modulation over wireless channels. One of the best techniques to mitigate the effects of fading is diversity combining of independently fading signal paths. Diversity combining exploits the fact that independent signal paths have a low probability of experiencing deep fades simultaneously. Thus, the idea behind diversity is to send the same data over independent fading paths. These independent paths are combined in such a way that the fading of the resultant signal is reduced. For example, consider a system with two antennas at either the transmitter or receiver that experience independent fading. If the antennas are spaced sufficiently far apart, it is unlikely that they both experience deep fades at the same time. By selecting the antenna with the strongest signal, a technique known as selection combining, we obtain a much better signal than if we had just one antenna. This chapter focuses on common methods used at the transmitter and receiver to achieve diversity. Other diversity techniques that have potential benefits beyond these schemes in terms of performance or complexity are discussed in [1, Chap. 9.10].

Diversity techniques that mitigate the effect of multipath fading are called microdiversity, and that is the focus of this chapter. Diversity to mitigate the effects of shadowing from buildings and objects is called macrodiversity.

1. The Background

Lennes [1911] was the first to present a simple three-dimensional nonconvex polyhedron whose interior cannot be triangulated without new vertices. The more famous example, however, was given by Schönhardt [1927]: he observed that in the nonconvex twisted triangular prism (subsequently called “Schönhardt's polyhedron“) every diagonal that is not a boundary edge lies completely in the exterior. This implies that there can be no triangulation of it without new vertices because there is simply no interior tetrahedron: all possible tetrahedra spanned by four of its six vertices would introduce new edges. Moreover, he proved that every simple polyhedron with the same properties must have at least six vertices. Later, further such nonconvex, nontriangulable polyhedra with an arbitrary number of points have been presented. Among them, Bagemihl's polyhedron [1948] also has the feature that every nonfacial diagonal is in the exterior.

The nonconvex twisted prism over an arbitrary n-gon would arguably be the most natural generalization of Schönhardt's polyhedron. Surprisingly enough, there has been no proof so far that it cannot be triangulated without new vertices. One of the reasons seems to be that — in contrast to Schönhardt's and Bagemihl's polyhedra — not every nonfacial diagonal lies completely outside the polygonal prism.

1. Introduction

The rank of a matrix M is one of the most important notions in linear algebra. This number can be defined in many different ways. In particular, the following three definitions are equivalent:

• • The rank of M is the smallest integer r for which M can be written as the sum of r rank one matrices. A matrix has rank 1 if it is the product of a column vector and a row vector.

• • The rank of M is the smallest dimension of any linear space containing the columns of M.

• • The rank of M is the largest integer r such that M has a nonsingular r x r minor.

Our objective is to examine these familiar definitions over an algebraic structure which has no additive inverses.

1. Introduction

Folding and unfolding problems have been implicit since Albrecht Dürer [1525], but have not been studied extensively in the mathematical literature until recently. Over the past few years, there has been a surge of interest in these problems in discrete and computational geometry. This paper gives a brief survey of most of the work in this area. Related, shorter surveys are [Connelly and Demaine 2004; Demaine 2001; Demaine and Demaine 2002; O'Rourke 2000]. We are currently preparing a monograph on the topic [Demaine and O'Rourke ≥2005].

In general, we are interested in how objects (such as linkages, pieces of paper, and polyhedra) can be moved or reconfigured (folded) subject to certain constraints depending on the type of object and the problem of interest. Typically the process of unfolding approaches a more basic shape, whereas folding complicates the shape. We define the configuration space as the set of all configurations or states of the object permitted by the folding constraints, with paths in the space corresponding to motions (foldings) of the object.

This survey is divided into three sections corresponding to the type of object being folded: linkages, paper, or polyhedra. Unavoidably, areas with which we are more familiar or for which there is a more extensive literature are covered in more detail. For example, more problems have been explored in linkage and paper folding than in polyhedron folding, and our corresponding sections reflect this imbalance. On the other hand, this survey cannot do justice to the wealth of research on protein folding, so only a partial survey appears in Section 2.5.

Wireless communications is, by any measure, the fastest growing segment of the communications industry. As such, it has captured the attention of the media and the imagination of the public. Cellular systems have experienced exponential growth over the last decade and there are currently about two billion users worldwide. Indeed, cellular phones have become a critical business tool and part of everyday life in most developed countries, and they are rapidly supplanting antiquated wireline systems in many developing countries. In addition, wireless local area networks currently supplement or replace wired networks in many homes, businesses, and campuses. Many new applications – including wireless sensor networks, automated highways and factories, smart homes and appliances, and remote telemedicine – are emerging from research ideas to concrete systems. The explosive growth of wireless systems coupled with the proliferation of laptop and palmtop computers suggests a bright future for wireless networks, both as stand-alone systems and as part of the larger networking infrastructure. However, many technical challenges remain in designing robust wireless networks that deliver the performance necessary to support emerging applications. In this introductory chapter we will briefly review the history of wireless networks from the smoke signals of the pre-industrial age to the cellular, satellite, and other wireless networks of today. We then discuss the wireless vision in more detail, including the technical challenges that must still be overcome.

The volume includes 32 papers on topics ranging from polytopes to complexity questions on geometric arrangements, from geometric algorithms to packing and covering, from visibility problems to geometric graph theory. There are points of contact with both mathematical and applied areas such as algebraic topology, geometric probability, algebraic geometry, combinatorics, differential geometry, mathematical programming, data structures, and biochemistry.

We hope the articles in this volume — surveys as well as research papers — will serve to give the interested reader a glimpse of the current state of discrete, combinatorial and computational geometry as we stand poised at the beginning of a new century.

1. Introduction

By a drawing of a graph G, we mean a drawing in the plane such that vertices are represented by distinct points and edges by arcs. The arcs are allowed to cross, but they may not pass through vertices (except for their endpoints) and no point is an internal point of three or more arcs. Two arcs may have only finitely many common points. A crossing is a common internal point of two arcs. A crossing pair is a pair of edges which cross each other at least once. A drawing is planar, if there are no crossings in it. A subdrawing of a drawing is defined analogously as a subgraph of a graph.

The crossing number cr(G) of a graph G is the minimum possible number of crossings in a drawing of G. The pair-crossing number pair-cr(G) of G is the minimum possible number of (unordered) crossing pairs in a drawing of G. In this paper we investigate the relation between the crossing number and the pair-crossing number. Clearly, pair-cr(G) ≤ cr(G) holds for any graph G. The problem of deciding whether cr(G) = pair-cr(G) holds for every G appears quite challenging. Let f(k) be the maximum cr(G), taken over all graphs G with pair-cr(G) = k. Obviously, f(k) ≥ k. Pach and Toth [2000] proved that f(k) ≤ 2k2. In fact, they proved this bound in a stronger version when the pair-crossing number is replaced by the so-called odd-crossing number, which is the minimum number of pairs of edges in a drawing that cross each other an odd number of times.

1. Introduction The systematic study of flag f-vectors of polytopes was initiated by Bayer and Billera [1985]. Billera then suggested the study of flag f-vectors of zonotopes; see the dissertation of his student Liu [1995]. The essential computational results of the field appeared in two papers by Billera, Ehrenborg and Readdy [Billera et al. 1997; 1998]. Here we present two classes of linear inequalities for the flag f-vectors of zonotopes. These classes are motivated by our recent results for polytopes [Ehrenborg 2005].

The flag f-vector of a convex polytope contains all the enumerative incidence information between the faces of the polytope. For an n-dimensional polytope the flag f-vector consists of 2n entries; in other words, the flag f-vector lies in the vector space ℝ2 . Bayer and Billera [1985] showed that the flag vectors of n-dimensional polytopes span a subspace of ℝ2n , called the generalized Dehn- Sommerville subspace and denoted by GDSSn. Bayer and Klapper [1991] proved that GDSSn is naturally isomorphic to the n-th homogeneous component of the noncommutative ring ℝ(c, d), where the grading is given by deg c = 1 and deg d = 2. Hence, the flag f-vector of a polytope P can be encoded by a noncommutative polynomial in the variables c and d, called the cd-index.

In this chapter we consider systems with multiple antennas at the transmitter and receiver, which are commonly referred to as multiple-input multiple-output (MIMO) systems. The multiple antennas can be used to increase data rates through multiplexing or to improve performance through diversity. We have already seen diversity in Chapter 7. In MIMO systems, the transmit and receive antennas can both be used for diversity gain. Multiplexing exploits the structure of the channel gain matrix to obtain independent signaling paths that can be used to send independent data. Indeed, the initial excitement about MIMO was sparked by the pioneering work of Winters, Foschini, Foschini and Gans, and Telatar predicting remarkable spectral efficiencies for wireless systems with multiple transmit and receive antennas. These spectral efficiency gains often require accurate knowledge of the channel at the receiver – and sometimes at the transmitter as well. In addition to spectral efficiency gains, ISI and interference from other users can be reduced using smart antenna techniques. The cost of the performance enhancements obtained through MIMO techniques is the added cost of deploying multiple antennas, the space and circuit power requirements of these extra antennas (especially on small handheld units), and the added complexity required for multidimensional signal processing. In this chapter we examine the different uses for multiple antennas and find their performance advantages. This chapter uses several key results from matrix theory: Appendix C provides a brief overview of these results.

1. Introduction and Preliminaries

Throughout this article, M denotes a compact subset of ℝn containing at least two points. By diamM we denote the maximum distance between points of M, but diameter of M also means the line segment connecting any pair of points of M that realize this distance (ambiguity is always avoided by the context). A Borsuk partition of M is a family of subsets of M, each of diameter smaller than diamM, whose union contains M. The Borsuk partition number of M, denoted by b(M), is the minimum number of sets needed in a Borsuk partition of M. It is obvious that b(M) is finite. It is also obvious that the maximum of b(M) over all bounded sets M in ℝn exists and is bounded above exponentially in n, since every set of diameter d is contained in a ball of radius d.

1. Introduction

The Delaunay tessellation is a useful canonical decomposition of the space around a given set of points in a Euclidean space E3, frequently used for surface reconstruction, molecular modelling and tessellating solid shapes [Delaunay 1934; Boissonnat and Yvinec 1998; Okabe et al. 1992]. The Delaunay tessellation is often used to compute its dual Voronoi diagram, which captures proximity. In its turn, it is often computed as a convex hull of points lifted to the paraboloid of revolution in one dimension higher [Brown 1979; Brown 1980]. As we sketch in this paper, there are a number of engineering decisions that must be made by implementors, including the type of arithmetic, degeneracy handling, data structure representation, and low-level algorithms.

We wanted to know what algorithm would be fastest for a particular application: computing the Delaunay tessellation of points that represent atoms coordinates in proteins, as represented in the PDB (Protein Data Bank) format [Berman et al. 2000]. Atoms in proteins are well-packed, so points from PDB files tend to be evenly distributed, with physically-enforced minimum separation distances. Coordinates in PDB files have a limit on precision: because they have an 8.3f field specification in units of angstroms, they may have three decimal digits before the decimal place (four if the number is positive), and three digits after.

Santalo [1940] showed, by an example, that Vincensini's proof [1935] of an extension of Helly's theorem was incorrect. Vincensini claimed to have proved that for any finite collection S of at least three compact convex sets in the plane, any three of which are met by a line, there must exist a line meeting all the sets. This would have constituted an extension of the planar Helly theorem [Helly 1923] to the effect that the same assertion holds if “line” is replaced by “point.” The Santalo example was later extended by Hadwiger and Debrunner [1964] to show that even if the convex sets are disjoint the conclusion still may not hold.

In 1957, however, Hadwiger showed that the conclusion of the theorem is valid if the hypothesis is strengthened by imposing a consistency condition on the order in which the triples of sets are met by transversals:

THEOREM [Hadwiger 1957]. If B1,…, Bn is a family of disjoint compact convex sets in the plane with the property that for any 1 ≤ I ≤ j ≤ k ≤ n there is a line meeting each of Bi,Bj,Bk in that order, then there is a line meeting all the sets Bi.

1. Introduction and Main Results

The Hadwiger number H(K) of a d-dimensional convex body K is the maximum number of mutually nonoverlapping translates of K that can be arranged so that all touch K. Often H(K) is called the translative kissing number of K as well. H*(K) is defined analogously with the restriction that all touching translates of K are pairwise disjoint. Trivially, H*(K) ≤ H(K). It is known that H(K) ≤ 3d — 1 [Hadwiger 1957], with equality attained only for parallelotopes [Groemer 1961].

An ad hoc wireless network is a collection of wireless mobile nodes that self-configure to form a network without the aid of any established infrastructure, as shown in Figure 16.1. Without an inherent infrastructure, the mobiles handle the necessary control and networking tasks by themselves, generally through the use of distributed control algorithms. Multihop routing, whereby intermediate nodes relay packets toward their final destination, can improve the throughput and power efficiency of the network. The Merriam-Webster dictionary lists two relevant definitions for ad hoc: “formed or used for specific or immediate problems”, and “fashioned from whatever is immediately available”. These definitions capture two of the main benefits of ad hoc wireless networks: they can be tailored to specific applications, and they can be formed from whatever network nodes are available. Ad hoc wireless networks have other appealing features as well. They avoid the cost, installation, and maintenance of network infrastructure. They can be rapidly deployed and reconfigured. They also exhibit great robustness owing to their distributed nature, node redundancy, and the lack of single points of failure. These characteristics are especially important for military applications, and much of the groundbreaking research in ad hoc wireless networking was supported by the (Defense) Advanced Research Projects Agency (DARPA) and the U.S. Navy. Many of the fundamental design principles for ad hoc wireless networks were identified and investigated in that early research.