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Which examples of mathematics have you enjoyed most The chances are that, whatever your preferences, many other readers will beg to differ. Mathematicians are by no means agreed on what mathematics is attractive, or worthwhile. On 19 November 1989, at the University of Tokyo, Yasumasa Kanada and Yoshiaki Tamura used a Hitac S-820 80E computer to calculate 11: to 1,073,740,000 decimal places. They used some advanced mathematics to write the computer program, and they were in a long line of mathematicians who have researched more and more powerful methods of calculating 11: accurately. Yet many mathematicians would say that this is 'not mathematics', but a not-very-interesting exercise in using a supercomputer. What do you think While the Indian prodigy Ramanujan was still at high school, he discovered that em = - 1 and was later disappointed to learn that Euler had found the same result in 1743. At that time, this formula was considered an extraordinary and beautiful connection between 11:, the ratio of the circum-

1341 Traditional Techniques As we discussed earlier, TOA and RSS metrics are the most popular in wireless positioning systems for urban and indoor areas TOA metrics provides for a more accurate measure of distance but may need additional infrastructure The RSS is an easier metrics to measure and integrates well with the existing infrastructure, but it is less reliable and often needs more complex algorithms and additional calibration procedure Positioning algorithms using AOA metrics are often simple and use the principles shown in Fig 138 Therefore, our discussion of the algorithms will emphasize TOA and RSS systems With reliable TOA-based distance measurements using UWB technology, simple geometrical triangulation methods can be used to nd the location of a tag or a mobile terminal [Fon01].

Due to estimation errors of distances at RP receivers caused by inaccurate TOA measurement, the geometrical triangulation technique can only provide a region of uncertainly, instead of a single-position x, for the estimated location of the mobile terminal To obtain an estimate of location coordinates in the presence of measurement errors of location metrics, a variety of direct and iterative statistical positioning algorithms have been developed to solve the problem by formulating it into a set of nonlinear iterative equations In some wireless geolocation applications, the purpose of the positioning systems is to provide visualization of the possible mobile locations instead of an estimate of the location coordinates On the other hand, positional accuracy is not constant across the area of coverage, and poor geometry of the relative positions of the mobile terminal and RP can lead to high geometric dilution of precision.

The output of statistical methods is an estimate of mobile location coordinates, and changes in the shape of the region of uncertainty are not revealed by this method When both region-of-uncertainty information and an estimate of the location are needed, both geometric and statistical triangulation algorithms are used [Pah02] Example 134: LS and RGWH Algorithms for Indoor Geolocation In [Kan04a] the performance of least squares (LS) and residual weighted (RGWH) LS algorithms in a square area with four RPs in the corners of the room is evaluated The LS algorithm is a simple traditional gradient algorithm used in basic GPS systems which iteratively minimizes the square of the error in estimation of the position For two-dimensional applications, considered in this example, it needs a minimum of three RPs With more RPs the algorithm is expected to provide more accurate positioning.

The RGWH, originally designed for cellular positioning [Che99a, Che99b], is another version of the LS algorithm, which calculates all possible solutions for positioning, in our case all possible 3RF combinations and the 4RF solution The nal estimate of the location is made by weighted averaging of all the estimates The weighting factor is the inverse of the residual error Thus less reliable positioning with higher residual error is counted with less emphasis Using a random-number generator, a uniformly distributed location in the room is selected Using the distance measurement errors model provided in Example 132, the estimated distances from the four RPs when a TOA system is used are determined These distances are then used by LS or RGWH algorithms to determine the estimated location of the terminal in the area The difference between the estimated location and the actual randomly selected location is used as the positioning error.

By repeating this experiment, the statistics of the positioning error associated.

Fig. 3.20 opened to a fixed distance, and counting the number of steps. This is equivalent to approximating to the coastline by a polygon with many small equal sides.