Hash tables are a dictionary structure of great practical importance and can be very efficient. The underlying idea is quite simple: we have a universe U and want to store a set of objects with keys from U. We also have s buckets and a function h from U to S = {0, …, s − 1}. Then we store the object with key u in the h(u)th bucket. If several objects that we want to store are mapped to the same bucket, we have a collision between these objects. If there are no collisions, then we can realize the buckets just as an array, each array entry having space for one object. The theory of hash tables mainly deals with the questions of what to do about the collisions and how to choose the function h in such a way that the number of collisions is small.

The idea of hash tables is quite old, apparently starting in several groups at IBM in 1953 (Knott 1972). For a long time the main reason for the popularity of hash tables was the simple implementation; the hash funcions h were chosen ad hoc as some unintelligible way to map the large universe to the small array allocated for the table. It was the practical programmer's dictionary structure of choice, easily written and conceptually understood, with no performance guarantees, and it still exists in this style in many texts aimed at that group.

In this chapter, we investigate structures and algorithms of cactus representations, which were introduced in Section 1.5.4 to represent all minimum cuts in an edgeweighted graph G. Throughout this chapter, we assume that λ(G) > 0 for a given graph G, which implies that G is connected. Let C(G) denote the set of all minimum cuts in G. In Section 5.1, we define a canonical form of cactus representations. In Section 5.2, we show that a subset of C(G) that consists of minimum cuts separating two given vertices, s and t, can be represented by a simple cactus structure. In Section 5.3, we design an O(mn + n2 log n) time algorithm for constructing a cactus representation R of C(G).

Canonical Forms of Cactus Representations

In this section, we discuss cactus representations for a subset of minimum cuts, and we prove the existence of two canonical forms, which we call the cycle-type and junction-type normal cactus representations. Such a canonical representation is useful in designing an efficient algorithm that constructs a cactus representation for all the minimum cuts of a given graph [244]. It also helps to efficiently test whether two given graphs have the same “structure” with respect to their minimum cuts, which is based on a planar isomorphism algorithm due to Hopcroft and Tarjan [126].

A cactus representation for a given subset C ⊆ C(G), if one exists, may not be unique unless we impose further structural restrictions.

In this chapter, some of the recent interesting research work related to UWB ranging and positioning are briefly reviewed. The purpose of the chapter is not to describe these studies in detail, but rather to point out specific recent references that may yield further research.

Development of accurate ranging/positioning algorithms

As discussed in the previous chapters, ranging and localization via UWB radios have been investigated extensively in the literature. While CRLB and ZZLB provide lower bounds on the ranging/localization accuracy, low-complexity and efficient estimators that approach these bounds in practical scenarios are still needed.

There are numerous recent research studies that aim at improving UWB ranging/localization accuracy. One research direction is joint estimation of range and location. In [170], it is shown that a two-step approach that uses independent decisions in ranging and localization steps is asymptotically optimal at high SNRs. However, it requires perfect estimates of delays, attenuations, and pulse shapes related to the received multipath components (MPCs) in order to construct an optimum correlation template at the receiver, which is very difficult to achieve in practice. Without perfect a-priori information of the channel parameters, such a two-step method returns unreliable TOA estimates during the ranging step. Since the measurements are separately performed at each reference node, without a constraint that all the measurements correspond to the location of the same mobile terminal, such approaches are suboptimal [137]. A better approach would be to make least commitment, where intermediate information is preserved and propagated till the end [393]. In other words, the received channel responses should not be discarded until a final decision regarding the target node location is made.

Wireless channel models carry significant importance for gaining insight into designing physical layer systems and selecting certain system parameters. For instance, in an IRUWB system, a design engineer might need to know how much apart to transmit two sequential pulses in order to avoid inter-frame interference at the receiver, or how likely the first arriving signal component contains the highest energy among all signal components for accurate ranging. Answers to such questions can be obtained either directly from channel measurements conducted in an environment of interest, or from statistical models derived from channel measurement campaigns.

There are various channel modeling techniques (e.g. ray tracing and statistical modeling) [93–96] and channel sounding methods (e.g. time-domain vs. frequency domain) [97, 98], which have been studied extensively in the literature. The focus of this chapter is not those well-known channel modeling techniques, but mainly the UWB channel models recently proposed and their interpretations for positioning applications.

Many UWB channel modeling campaigns have been performed within the past few years, mainly due to emerging UWB standards (e.g. multiband OFDM-UWB, IEEE 802.15.4a, and IEEE 802.15.3c) [96, 97, 99–101]. Although channel statistics and models of various frequency bands are publicly available, many of those do not explicitly include ranging-related statistics. Therefore, one of the aims of this chapter is to investigate UWB channel models from a range estimation perspective.

Designing a wireless system typically involves the steps illustrated in Fig. 3.1. First, application requirements need to be explored. Low attenuation at low frequencies makes through-the-wall communications and tracking applications attractive, but it is difficult to adopt sub-GHz UWB systems due to coexistence issues with existing narrowband systems.

As discussed in the previous chapter, the position of a mobile node in a wireless network can be estimated based on AOA, RSS, TOA, and/or TDOA of received signals. Due to their large bandwidths, UWB signals have very high time resolution, hence individual multipath components (MPCs) can be resolved at the receiver. While AOA, TOA, and TDOA approaches all benefit from this high resolution, AOA-based implementations have high complexity. Therefore, positioning based on TOA (or TDOA) estimation is the method of choice in UWB-based positioning systems [196] as opposed to AOA or RSS (which has low ranging accuracy)-based approaches. Therefore, the emphasis of this chapter is on time-based UWB ranging techniques.

The chapter is organized as follows. In Section 5.1, the time-based positioning problem is briefly re-visited and importance of accurate ranging for precise positioning is emphasized. The error sources in time-based ranging are discussed in Section 5.2. In Section 5.3, the time-based ranging problem is formulated and models for various transceiver types are studied. Section 5.4 reviews fundamental limits on the accuracy of time-based ranging, Section 5.5 investigates maximum likelihood (ML)-based techniques, and Section 5.6 presents alternative low-complexity ranging algorithms for UWB systems.

Time-based positioning

Consider a wireless network in which there are Nm reference nodes (RNs). The ith RN is located at (xi, yi), and a target node (TN) at position (x, y).

Commonly, an ultra-wideband (UWB) signal is defined to be a signal with a fractional bandwidth of larger than 20% or an absolute bandwidth of at least 500 MHz. The main feature of UWB signals is that they occupy a much wider frequency band than conventional signals; hence, they need to share the existing spectrum with incumbent systems. Therefore, certain regulations are imposed on systems transmitting UWB signals. In this chapter, after a detailed description of UWB signals, various regulatory rules on UWB systems in different parts of the world are investigated. Then, emerging UWB standards for wireless personal area network (WPAN) applications are studied.

Definition of UWB

Although Guglielmo Marconi's spark gap radio transmitters were sending UWB signals across the Atlantic Ocean in 1901, the rigorous investigation of UWB systems was stimulated by the studies on impulse response characterization of microwave networks in the 1960s [63, 64]. Instead of the conventional swept-frequency response characterization, a linear-time-invariant (LTI) system was characterized by its response to an impulse in the time domain. After employing impulses to characterize behavior of various systems, it was also realized that such impulses could also be used in radar and communications systems [65]. The first UWB communications patent was issued in 1973 to Gerald F. Ross on transmission and reception of baseband pulse signals [66].

Early names for UWB technology include baseband, carrier-free, non-sinusoidal and impulse. The term UWB was coined by the US Department of Defense in the late 1980s. A UWB signal is characterized by its very large bandwidth compared to the conventional narrowband systems.

Wireless communications are becoming an integral part of our daily lives. Satellite communications, cellular networks, wireless local area networks (WLANs), and wireless sensor networks (WSNs) are only a few of the wireless technologies that we use every day. They make our daily lives easier by keeping us connected anywhere, anytime.

Since more and more devices are going wireless every day, it is essential that future wireless technologies can coexist with each other. Ultra-wideband (UWB) is a promising solution to this problem which became popular after the Federal Communications Commission (FCC) in the USA allowed the unlicensed use of UWB devices in February 2002 subject to emission constraints. Due to its unlicensed operation and low-power transmission, UWB can coexist with other wireless devices, and its low-cost, low-power transceiver circuitry makes it a good candidate for short- to medium-range wireless systems such as WSNs and wireless personal area networks (WPANs).

One of the most promising aspects of UWB radios are their potential for high-precision localization. Due to their large bandwidths, UWB receivers can resolve individual multipath components (MPCs); therefore, they are capable of accurately estimating the arrival time of the first signal path. This implies that the distance between a wireless transmitter and a receiver can be accurately determined, yielding high localization accuracy.

Such unique aspects of UWB make it an attractive technology for diverse communications, ranging, and radar applications such as robotics, emergency support, intelligent ambient sensing, health-care, asset tracking, and medical imaging (see Fig. 1.1).

This chapter discusses three special topics related to ranging. First, techniques to mitigate various types of interference are presented. Second, carrier sensing methods that can be used to improve ranging performance for IEEE 802.15.4a networks are briefly reviewed. Finally, an overview of mechanisms that provide privacy and security for ranging signals and range information is given.

In this chapter, it is assumed that ranging is performed via frames that consist of preamble, start of frame delimiter (SFD), physical layer header (PHR) and payload, and also that the preamble is used for ranging (similar to the IEEE 802.15.4a systems studied in the previous chapter). Frames with longer preambles provide a higher processing gain for ranging due to improved SNR and lead to better ranging accuracy. This is because at high SNRs, detection of the direct path signal is easier. On the other hand, employing a longer preamble induces a drawback that the preamble becomes more vulnerable to interference and jamming attacks. In case that acquisition of a frame fails, the frame needs to be retransmitted.

Interference can be detrimental to ranging accuracy, even if it does not cause acquisition failure. At times the leading signal path gets buried under interference, so that it may be quite difficult to determine its arrival time. Remember from Chapter 6 that performance of ranging protocols is very sensitive to timing. This mandates rapid handling of all ranging related transmissions. If retransmissions of ranging frames were scheduled with high priority, regular data traffic would be penalized, and throughput and latency for the data traffic would degrade. Furthermore, each retransmission may potentially interfere with transmission of peer devices in the same network.

Ability to locate assets and people will be driving not only emerging location-based services, but also mobile advertising, and safety and security applications. Cellular subscribers are increasingly using their handsets already as mapping and navigation tools. Location-aware vehicle-to-vehicle communication networks are being researched widely to increase traffic safety and efficiency. Asset management in warehouses, and equipment and personnel localization/tracking in hospitals are among other location-based applications that address vast markets. It is a fact that application space for localization technologies is very diverse, and performance requirements of such applications vary to a great extent.

The Global Positioning System (GPS) requires communication with at least four GPS satellites, and offers location accuracy of several meters. It is used mainly for outdoor location-based applications, because its accuracy can degrade significantly in indoor scenarios. Wireless local area network (WLAN) technology has recently become a candidate technology for indoor localization, but the location accuracy it offers is poor, and also high power consumption of WLAN terminals is an issue for power-sensitive mobile applications. Ultra-wideband technologies (UWB) promise to overcome power consumption and accuracy limitations of both GPS and WLAN, and are more suitable for indoor location-based applications.

The Federal Communications Commission (FCC) and European Commission (EC) regulate certain frequency bands for UWB systems. These have prompted worldwide research and development efforts on UWB. Another consequence was development of international wireless communication standards that adopt UWB technology such as IEEE 802.15.4a WPAN and IEEE 802.15.3c WPAN.

The writing of this book was prompted by the fact that UWB is the most promising technology for indoor localization and tracking.

The previous chapter deals with detecting time of arrival of first signal path. Even though it is an essential and the very first step in TOA-based ranging and positioning systems, more need to be done to obtain range and position estimates. The TOA information makes sense only if the signal's time of transmission is known. Then, time of flight (TOF) of the signal can be easily computed. The TOF is directly proportional to the distance between a device that transmits the signal and the device that receives it. There are various protocols to transform any TOA information to a TOF and range estimate. Ranging protocols require actions to be taken at devices that are involved in ranging and positioning. The focus of this chapter is to study these protocols in detail.

The chapter consists of three parts. The first part provides an overview of communication protocol layers and explains functionalities of the service and management interfaces between adjacent layers. It is important to know what intra- and inter-device events take place for obtaining ranging information and how ranging-related information is passed from the physical layer to the upper layers for application's use. Interfaces play a key role in achieving this. A good example of an intra-device event is management of ranging signal parameters at the MAC sub-layer and then generation of the signal accordingly at the PHY. What constitutes an inter-device event is transmission and reception of ranging signals and ranging-related messages (e.g., time-stamps).

The second part takes a detailed look into well-known time-based ranging protocols and analyzes their advantages and drawbacks. There are numerous ranging protocols.