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range-independent scheme, only needs the existence of beacon signals and does not require any measurements to be taken We look at each of these schemes in Sections 7211 and 7212, respectively In these sections we brie y explain several techniques that belong to these families and also point out the vulnerabilities in each of these techniques 7211 Range-Dependent Localization We consider the range-dependent schemes in this section While protocols based on these techniques produce more accurate location estimations, they also generally result in more expensive systems due to the cost of the hardware needed for the measurements Range-dependent localization techniques involve measuring the physical properties such as time of arrival (TOA), time difference of arrival (TDOA), received signal strength (RSS), or angle of arrival (AOA) These measurements can then be used to compute the distance to the locators, taking advantage of physical properties We next consider these schemes in more detail, starting with the widely known global positioning system (GPS) 72111 GPS The rst thing that comes to mind when we talk of location determination is the concept of global positioning systems [143] This is an attestation to the widespread use of GPS these days Thus a natural question to ask is the reason for studying the problem of location determination given the concept of GPS Before addressing this question we look at the concept of GPS brie y and then point out the shortcomings of GPS GPS is a constellation of 27 Earth orbiting satellites of which 24 are in operation and three are redundant The satellite network was developed and deployed by the US military as a navigation system in 1993, but is now freely available Each of these 3000 4000 lb solar-powered satellites makes two rotations of the Earth every day at about 12,000 miles The orbits of the satellites are arranged such that at any time four satellites are visible anywhere on Earth Signals from at least four satellites are collected by a GPS receiver, which determines the distance to each satellite as we explain next and then uses this information to deduce its own location using trilateration (http://electronics howstuffworkscom) A typical GPS radio signal has a strength of about (1 10216) W at the Earth s surface This is roughly equivalent to seeing a 25 W light bulb present in Tokyo from Los Angeles [144] The GPS system provides a location resolution of approximately 5 6 m on average [144] GPS also provides devices with accurate time references LORAN [145] is a similar system but is based on the use of ground-based beacons instead of satellites We do not consider LORAN here We show the operation of GPS in Figure 73 Here we show the GPS receiver marked as R, receiving signals from four satellites From these signals the receiver can determine the range from each of the satellites given as d1, d2, d3, and d4 Additionally, the receiver also needs information about the positions of the four satellites, shown as X1, X2, X3, and X4 in the gure Thus, a GPS receiver needs to know two things in order to determine the location using trilateration:.

where the k are independent identically distributed, stable, symmetric, random variables (see 14 in [BRE 68]) L vy s symmetric -stable motion can be de ned as a limit of X(nx) when n + f (n) Thus, stochastic self-similar processes appear as natural limits of renormalization procedures Theorem 58 provides another example, which is neither Gaussian nor stable 516 A common structure for a stochastic (semi-)self-similar process We now wish to propose a uni ed structure for the known (semi-)self-similar processes The basic ingredients are as follows: a self-similar parameter H > 0; a vaguelette type basis (see [MEY 90]); a sequence of independent and identically distributed random variables Let Ed = {0, 1}d be the set of binary sequences of length d and Ed = Ed {1, 1, , 1} Let us note by d the set of{(n, k), n Z, k Zd } Let us nd observe that (x) = 2 2 u (2n x k), with x Rd and = (n, k, u), where n is the scale parameter, k the localization parameter and u the orientation parameter: if we have u = 0, u is the mother wavelet, if we have u = 1, u is the father wavelet This dilation and translation structure, in fact, uses a structure of subjacent binary trees Function is assumed to rapidly decrease at in nity, and be null and Lipschitzian in zero Let us observe that , with = (n, k, u) a random variable sequence such that, for any n, n , k, k : n ,k ,u = n,k,u

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One important message that this new model of project success should transmit is the complexity of what success really encompasses Rather than using the old, outmoded triple constraint as a metric, it is vital to raise our standards to consider a far wider, and more important focus Make no mistake, if our company s highest goal is simply to get the project done fast and cheaply, perhaps a traditional triple constraint approach is sufficient to model our behavior and rate performance On the other hand, if we have set larger goals, ones that challenge us to look toward long-term viability, customer satisfaction, and corporate renewal, we must recognize the greater challenges of customer-based project success

Let us then de ne the process X by: X(x) =

2 nH (x)

(57)

The process is semi-self-similar for the multiplicative group of power 2 Again, the number 2 does not play a crucial role Later on, we will see that Brownian fractional motions have this structure We observe that Weierstrass functions return to this framework by supposing that (x) = sin(x)1[0,2 ] (x) In fact: Wb,H (x) =

it needs the location of at least three satellites; and it needs to know the distance between itself and each of those satellites

b nH (bn x 2k )

The question of parameter identi ability for a semi-self-similar process is natural We will be dealing with it in the next section 517 Identifying Weierstrass functions 5171 Pseudo-correlation Subject to existence, let us de ne the pseudo-correlation function of a deterministic function f by (see [BAS 62]): f ( ) ( ) =

A function is said to be pseudo-random if lim + f ( ) ( ) = 0, and pseudo-stationary when f ( r) = f ( ) for any r It can be shown that Weierstrass functions are pseudo-random and pseudo-stationary The example of Weierstrass functions shows that a semi-self-similar phenomenon is not solely determined by the self-similar parameter H Is it possible to identify the parameters that generate these phenomena We will see later that generally it is possible To conclude this introduction, we will be focusing on Weierstrass functions, whose identi cation requires general tools, although their demonstration is elementary To this end, let us introduce the quadratic variations of a function f : V N (f ) = Let us de ne: RN = VN 1 2 log2 2 VN 1 N

Page 142 The key lies in identifying the prevailing goal for our company If that goal is long-term survival, then the path we must travel to attain it is equally clear

N 1

Based on this information, a receiver can draw spheres centered on each of the satellites with the radius of each sphere being equal to the distance between the GPS receiver and the satellite on which the sphere is centered The spheres will hopefully intersect at only one point on the Earth s surface which corresponds to the location of the GPS receiver

By using the pseudo-random character of Weierstrass functions, we can show that RN measures H, when N +

h(t) =

(739)

where si are complex poles in C \

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