Probably the most widespread system based on TOA is GPS. GPS is a three-dimensional location problem. In order to find three position coordinates, a GPS
D1 D2
D3
D4 P2 P1
P3 P4
x y
P0
−8−8
−6
−6
−4
−4
−2
−2 0
0 2
2 4
4 6
6 8
8
Figure 7.4 TOA circles in overdetermined solution of location equations. X marks the estimated location and the small square marks the true location.
receiver needs to measure the distance to at least three satellites that serve as reference stations, each of whose position in space at the time of epoch transmission is known or can be calculated. The GPS terminal receives satellite data messages that specify the time of transmission of a known epoch of the signal as well as information that the receiver uses to track satellite position. If the receiver had an accurate real-time clock, it could record the time of arrival of the reference signal epoch, then subtract the signal transmission time to get time of flight. The time of flight multiplied by the signal propagation speed is the distance to that particular satellite at the time the ranging message was transmitted. In systems discussed in Chapters 3, 4, and 5 for finding two-way distance between two terminals, it was necessary for each terminal to transmit and receive, one terminal acting as an initiator and the other as a responder, in order to calculate the two-way propagation time between them. In GPS, only one-way transmissions are possible, from the satellite to the receiver. The receiver clock is not accurate enough to use its time-of-arrival measurement to find propagation time by subtracting the satellite’s epoch transmission time. Instead, an initial distance calculation, based on the receiver’s time-of-arrival clock reading, is made. This distance is called pseudorange. The deviation of the pseudorange from the actual range is the same for all satellites because the same receiver clock is used to make all time measurements, and this deviation can be determined if the pseudorange to at least four satellites is measured.
The following description of how position coordinates may be obtained from pseudorange measurements is based on GPS. The same principles can be applied to a wholly earthbound system, either unilateral, as is GPS, or multilateral (multiple fixed station receivers and target transmitter), as long as it has the following characteristics: reference station coordinates are known at the time of reference station transmission or reception, reference station clocks are synchronized, and time of transmission is conveyed from transmitter to receiver.
The TOA equations in three dimensions are those of spheres whose centers are the known locations of the reference stations and the radii are the distances from each reference station to the target. The target is located on the locus of intersection of the spheres. For four reference stations with coordinates xi, yi, zi, i= 1 to 4, pseudoranges Ri, clock offset times propagation speed⌬, and unknown target coordinates x, y, z:
(R1− ⌬)2= (x − x1)2+ (y − y1)2+ (z − z1)2
(R2− ⌬)2= (x − x2)2+ (y − y2)2+ (z − z2)2 (7.16) (R3− ⌬)2= (x − x3)2+ (y − y3)2+ (z − z3)2
(R4− ⌬)2= (x − x4)2+ (y − y4)2+ (z − z4)2
We can get an insight into what to expect from solving these equations by manipulating them so that the unknown parameters, x, y, z, and⌬ are expressed explicitly [7]. This is accomplished as follows. Expand all the square terms in each of the equations (7.16). Then write three new equations that are the first expanded equation minus the second, the first minus the third, and the first minus the fourth.
The x2, y2, and z2 terms are eliminated giving three linear equations with four unknowns that can be rearranged as follows:
␦x2⭈ x +␦y2⭈ y + ␦z2⭈ z =␦R2⭈ ⌬ +2
2
␦x3⭈ x +␦y3⭈ y + ␦z3⭈ z =␦R3⭈ ⌬ +3
2 (7.17)
␦x4⭈ x +␦y4⭈ y + ␦z4⭈ z =␦R4⭈ ⌬ +4
2 which contains compacted constants whose values are expressed as:
␦xi= (xi− x1)
␦yi= (yi− y1)
␦zi= (zi − z1) (7.18)
␦Ri= (Ri− R1)
i= x2i + y2i + z2i − x21− y21− z21for i= 2 . . . 4
Equation (7.17) can be solved for x, y, and z in terms of⌬ and the symbolic values of (7.18). The following example problem demonstrates the use of (7.17) and (7.18) to get a numerical solution [7].
Example 7.2
Figure 7.5 is a simplified geometric representation of the earth and a GPS satellite orbit. The origin is at the center of the earth. The positive z-axis extends through the North Pole, positive x intercepts the equator at the prime meridian, that is, longitude 0, and the positive y-axis crosses the equator at latitude 90E. The radius of the earth is re and the radius of the satellite orbit is rs. Any point P can be specified in polar coordinates (r,,) or rectangular coordinates (x, y, z). In order to simplify the numbers in this example, all distances are normalized by dividing by the radius of the earth, whose mean value is 6,360 km. Thus the normalized earth radius Re = 1. The distance to a satellite directly overhead from the target terminal is 20,200 km, and its normalized height= 20,200 km/6,360 km = 3.176.
The normalized radius of the satellite orbit Rs= 3.176 + 1 = 4.176. Coordinates of four satellites that are derived from data contained in received messages and measured times of flight are shown in Table 7.1. The times of flight, offset by an unknown clock bias, are used to calculate the pseudoranges R1, R2, R3and R4. The values in the second and fourth columns of Table 7.1 can be used to find the constants defined in (7.18) and then substituted in (7.17). The result is
−0.288x + 1.073y + 0.932z = 0.078⌬ − 0.325
−1.17x − 0.819y − 1.256z = 0.134⌬ − 0.559 (7.19) 1.423x− 0.124y − 1.256z = 0.134⌬ − 0.559
z
x
Re,re y
P x,y,z( ) r θ
Earth φ
Orbit Rs,rs
Figure 7.5 Geometric representation of the Earth and a GPS satellite orbit.
Table 7.1 Satellite Data
Normalized Position Measured Time Normalized Satellite (i) (xi, yi, zi) of Flight (ms) Pseudorange Ri
1 (0.828,−3.09, 2.684) 87.378 4.119
2 (0.54,−2.017, 3.617) 89.04 4.197
3 (−0.342, 3.909, 1.428) 90.212 4.252
4 (2.251,−3.215, 1.428) 90.212 4.252
These linear equations can be solved for x, y, and z in terms of the distance bias⌬:
x= .060⌬ + .256
y= .229⌬ − .955 (7.20)
z= −.199⌬ + .829
In order to find⌬, (7.20) is substituted in any one of the expressions in (7.16), the first one for example, resulting in a quadratic equation in ⌬:
⌬2− 11.083⌬ + 9.553 = 0 (7.21)
This equation has two possible solutions for the range bias:
(1) ⌬ = 0.942 (7.22) (2) ⌬ = 10.141
These values, substituted for⌬ in (7.20), give alternative mathematical solutions to the target position shown in Table 7.2, Also shown are corresponding clock biases in milliseconds, which are the calculated values of (⌬ ⭈ 6,360 km/300 km/
ms). The normalized distance of the target from the center of the Earth, column 4, is found from the target coordinates by
√
x2+ y2+ z2.Solving the ambiguity about the actual target position is easy. Column 4 of Table 7.2 indicates that the target is either on the earth’s surface, solution 1, or well beyond it. Thus the coordinates calculated from (1) in (7.22) give the true position, which are shown in the third column of Solution 2 in Table 7.2.
The above example has demonstrated that the TOA method with four fixed terminals (the satellites) and a time bias can give a wrong solution for the target coordinates. The mathematical development demonstrated is not used in GPS receivers. They will most likely solve the expressions of (7.16) by an iterative process—substituting an estimate for x, y, and z, and then altering the estimate in steps until a solution is arrived at. Several algorithms are available for converging to the best estimate of target coordinates and receiver clock bias, among them the least-mean-square (LMS) and Newton’s method [2]. Most often, more than four satellites are in view at one time, or information is available to reduce the dimensions to be solved for, for example if altitude is known or the clock has been previously adjusted, so that there is no possibility of an ambiguity error in the solution.
7.2 TDOA
Another form of location estimation based on time of flight measurements is TDOA.
TOA, as we have seen above, needs some degree of coordination between fixed stations and target in order to determine absolute distances. In two-way distance measurement, initiated by one of two terminals, the second terminal serves as a responder that replies to the initiator after an agreed-upon time interval, or in the case of DSSS, after code synchronization. The time transfer method (Chapter 4) also entails a two-way communication sequence to exchange information on the local clock readings of signal epochs. GPS does not require two-way communication between satellites and receivers. However, the satellite transmitters must send at least their clock reading at the instant of epoch transmission, as well as accurate position information. For many applications, however, target location may be necessary when the target transmitter does not adapt its messages to the expectations
Table 7.2 Possible Solutions to GPS Ranging Example
Clock Bias (ms) Normalized Coordinates Normalized Distance
Solution (x, y, z) from the Earth Center
1 20 (0.198,−0.739, 0.642) 1.00
2 215 (−0.366, 1.366, −1.186) 1.85
of the receiver [8]. Clandestine transmitter location, radar location, and electronic warfare in general are some examples. In all of these examples, fixed station reference receivers locate a target transmitter, a multilateral situation. We have already encountered a classical TDOA unilateral system, Loran-C, in Chapter 2.
A receiver, which desires to estimate its own location, does not need real-time information from the very low-frequency, low-data-rate fixed transmitters in order to calculate its position. The fact that the TDOA location method can operate with transmitters using their normal communication protocol and with no modification of hardware or software, gives it more applications than TOA, except for GPS.
While TDOA transmissions do not need to include a special message for the purpose of the location function, they must have a modulated identity that includes a specific epoch that can be recognized by the receivers. TDOA cannot be used where transmitters emit unmodulated carriers.
Instead of measuring the time of flight of a transmission between two terminals, TDOA measures the difference in the times of flight between a target terminal and a pair of fixed reference terminals. Clock synchronization is required only on one side of the communication link—the side of the fixed terminals [6]. At least one additional fixed terminal is required for TDOA per dimension compared to TOA.
A TDOA system needs at least three fixed terminals for a two-dimensional location problem and at least four fixed terminals to estimate three-dimensional coordinates [9].