CONCLUSIONES PARCIALES

TOA (time of arrival) and TDOA (time difference of arrival) methods use geometric relationships based on distances or distance differences between a mobile station and a number of fixed terminals to determine the position coordinates of the mobile target. Data for distance estimations are derived from the arrival times of radio signal epochs at one or more receivers. The TOA method uses the transit time between transmitter and receiver directly to find distance, whereas the TDOA method calculates location from the differences of the arrival times measured on pairs of transmission paths between the target and fixed terminals. Both TOA and TDOA are based on the time-of-flight (TOF) principle of distance measurement, where the sensed parameter, time interval, is converted to distance by multiplication by the speed of propagation. In TOA, location estimates are found by determining the points of intersection of circles or spheres whose centers are located at the fixed stations and the radii are estimated distances to the target. TDOA locates the target at intersections of hyperbolas or hyperboloids that are generated with foci at each fixed station of a pair.

Several methods of finding the time of flight have been discussed previously in Chapters 3, 4, and 5. TOA- and TDOA-based location systems may be unilateral or multilateral. In a unilateral system the target communicates with or merely receives fixed terminal transmissions to measure time durations. A multiplexing protocol must be employed since the target must acquire its time data separately from each base station, without those stations interfering with each other. Time, frequency, or code division multiplex techniques may be used. By contrast, a multilateral system performs the location calculations independently of the target, either at one of the base stations or at a separate network infrastructure computing function. The geometric principles of TOA and TDOA location methods are the same for unilateral and multilateral systems.

Ways of applying the two methods to locate a target are first explained while ignoring noise and other impairments. Then the causes of location accuracy deterio-ration are presented, followed by some algorithms that have been suggested for improving accuracy. Most examples are in two dimensions to simplify presentation and illustration, but extension to three dimensions can be done using the same concepts and methods.

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7.1 TOA Location Method

Designers of systems that estimate the distance between two terminals can choose among a number of different system technologies to get position coordinate esti-mates using TOA. Chapters 3, 4, and 5 detail how time of flight is measured using different communication concepts, and systems employing any of those could provide data for TOA location estimation. Figure 7.1 shows a simple geometric arrangement for determining the location of a target mobile station, MS, that is located on the same plane as base stations BS1 and BS2. The example uses the minimum of two base stations, which for simplification of the calculations are located on the x-axis, with BS1 at the origin. This choice of axes is completely general since coordinates in any rectangular reference frame can be mapped to the arrangement of the figure through coordinate transformation by translation and rotation [1]. The coordinates of BS1 and BS2 are known in advance, and distances d₁and d₂are found by multiplying the measured signal propagation time between each base station and the target by the speed of light. It should be noted that data for finding the solution by the TOA method does not have to be obtained from time-of arrival-measurements. The distance readings could be provided by calculating propagation distance directly from transmitter radiated power and received signal strength, when the propagation law is accurately known (Chapter 6). However, due to poor accuracy, distance is rarely measured directly using RSS methods.

The equations for the two intersecting circles with centers at the base stations and radii equal to distances from the target are

d₁²= x²+ y² (7.1)

d₂²= (x − x₂)²+ y² (7.2) These equations can be solved explicitly for x, y, the coordinates of the mobile station target:

x=d₁²− d₂²+ x²₂

2⭈ x₂ (7.3)

y

( , )x y

BS1 BS2

MS

(0,0) (x₂,0) x

d₁ d₂

Figure 7.1 Two-dimensional terminal deployment for target location by TOA.

y= ±

√

From (7.4) and Figure 7.1 it is evident that y has two possible solutions, one below and one above the x-axis in this example. The true location of the target can be resolved only if there is additional information, aside from the time of arrival data, about where it may be located. For example, it may be known that the target must be in the upper half plane. In this case the negative value for y in (7.4) can be excluded and the target’s coordinates are then known, with the value of x given by (7.3).

The TOA method gives the correct location of the target in two dimensions without ambiguity if at least three fixed base stations are used in the measurement.

Such an arrangement is shown in Figure 7.2 where an additional base station, BS3, has been added to the base stations BS1 and BS2 of Figure 7.1. The equation of the third circle centered on BS3 and passing through the target location MS is:

d₃²= (x − x₃)²+ (y − y₃)² (7.5) Solving (7.1), (7.2), and (7.5) gives the coordinates:

x= x²₂+ d₁²− d₂²

2⭈ x2 (7.6)

y=x²₃+ y²₃+ d₁²− d₃²− 2 ⭈ x ⭈ x₃

2⭈ y3 (7.7)

We see that the coordinates of the target can be estimated with no ambiguity since, as seen in Figure 7.2, the position they define is the only one where all three circles intersect.

7.1.1 Overdetermined TOA Equation Solution

Distance measurements that are the basis for TOA location are subject to various causes of imprecision, among them noise, channel interference, multipath, and

y

( , )x y

BS1 BS2

MS

(0,0) (x₂,0) x

d d

d3 BS3

(x₃,y₃)

1 2

Figure 7.2 Two-dimensional three base terminal deployment for unambiguous target location.

imprecise clocks. Positioning accuracy can be improved by incorporating in the location process a larger number of fixed stations than the minimum required for unambiguous location estimation. Figure 7.3 depicts a two-dimensional layout of four fixed terminals labeled P1, P2, P3, and P4, with known coordinates, and a target terminal P0 whose location is to be determined. If the true distances d₁ through d₄could be measured exactly, the coordinates of P0 would be at the point of intersection of the circles formed with the fixed stations at the centers and the radii equal to the distances to the target. However, the actual distance measurements, designated D₁, D₂, D₃, and D₄, are not exact, the circles do not cross at one point, and it is necessary to define a criterion for deciding on the estimated location coordinates.

The equations of the four circles defined by base station positions P1 (x₁, y₁) through P4 (x₄, y₄) and measured distances to target P0, D₁through D₄, are

(1) (x₁− x)²+ (y₁− y)²= D²₁

(2) (x₂− x)²+ (y₂− y)²= D²₂ (7.8) (3) (x₃− x)²+ (y₃− y)²= D²₃

(4) (x₄− x)²+ (y₄− y)²= D²₄

We describe here a method of estimating the position of P0 using a least squares (LS) error criterion. Using the least squares method, the position estimate has coordinates x_e, y_ethat minimize the function F :

F= ∑^M

i= 1冠

√

where in our example M= 4.

Coordinates x_e, y_ethat minimize the nonlinear expression (7.9) can be found by an iterative algorithm based on a Taylor series expansion or gradient descent [2, 3]. Such a method may be time consuming and inconvenient to implement for many applications. An alternative approach, described below, gives a closed form solution to the estimation problem. It works by first creating a set of linear equations from the equation set (7.8)

P1 P2

P3 P4

P0

x y

d₁ d₂

d₃ d₄

Figure 7.3 Deployment of terminals for overdetermined TOA location.

Expand the factors on the left side of the equations of (7.8) and subtract equations (2), (3), and (4) from (1), to give the following new set of M− 1, or in this case three, equations:

(1) (x₁− x₂)x+ (y₁− y₂)y_e=1

2冠^x21− x₂²+ y²₁− y²₂+ D²₂− D²₁冡

(2) (x₁− x₃)x+ (y₁− y₃)y_e=1

2冠^x21− x₃²+ y²₁− y²₃+ D²₃− D²₁冡 ^(7.10)

(3) (x₁− x₄)x+ (y₁− y₄)y_e=1

2冠^x21− x₄²+ y²₁− y²₄+ D²₄− D²₁冡

Equation (7.10) is an overdetermined set of linear equations in x, y. It can be expressed in matrix form as:

A⭈ P_e = b (7.11)

where

A =

冤

冥

b=1

2 ⭈

冤

冥

P_e =

冋

册

The closed form LS solution to (7.10) is [4–6]:

P_e =

冋

册

While this development uses four base stations, it can be extended logically to a larger number and also to three dimensions, when the number of base stations, M, must be equal to four or more.

The following example demonstrates how (7.15) is used.

Example 7.1

Four fixed base stations at P1 through P4 and a target at P0 are deployed as shown in Figure 7.3. Measured distances between base stations and target are:

Measured distances: D₁= 2.5 D₂= 3.2 D₃= 4.8 D₄= 2.5 The coordinates of the base stations are:

Base station coordinates: P1 = (3, 2) P2 = (−2, 3) P3 = (−3, −1) P4 = (2, −1) Find a least squares estimate of target position P0.

1. A and b according to (7.12) and (7.13):