# A Computer Simulation of Dilution of Precision in the Global Positioning System Using Matlab

**ABSTRACT** The aim of this study is to design a computer simulation of the Dilution of Precision (DOP) which affects the GPS accuracy. DOP simulator is a usefull interface which provides a fast and practical analysis of both the DOP parameters and satellite geometry. The simulator was designed as a graphical user interface in MATLAB.

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A COMPUTER SIMULATION OF DILUTION OF PRECISION IN THE

GLOBAL POSITIONING SYSTEM USING MATLAB

Fevzi Aytaç Kaya

e-mail: aytacfk@gmail.com

Gazi University, Faculty of Engineering & Architecture

Department of Electrical&Electronics Engineering

06570 - Maltepe, Ankara, Turkey

Müzeyyen Sarıtaş

e-mail: muzeyyen@gazi.edu.tr

Key words: Dilution of Precision, satellite geometry, simulation, user interface

ABSTRACT

The aim of this study is to design a computer simulation of

the Dilution of Precision (DOP) which affects the GPS

accuracy. DOP simulator is a usefull interface which

provides a fast and practical analysis of both the DOP

parameters and satellite geometry. The simulator was

designed as a graphical user interface in MATLAB.

I. INTRODUCTION

GPS devices are designed to send any positional data

they calculate, regardless of its accuracy. Indeed, the

latitude, longitude and altitude reported by GPS devices

are frequently inaccurate by three hundred meters. This

inaccuracy can lead to poor business intelligence, or,

worse still, get an end-user into a precarious situation.

Developers who wish to write location-based services

must get smart about GPS precision. Fortunately, GPS

devices provide information which can be used to make

300m inaccuracies a thing of the past. This information

is known as "Dilution of Precision," and is the key to

writing commercial GPS applications [1]. The Dilution

of Precision (DOP) is a value of probability for the

geometrical effect on

GPS devices calculate the position using a technique

called “3-D multilateration,” which is the process of

figuring out where a number of spheres intersect. In the

case of GPS, each sphere has a satellite at its center. The

radius of the sphere is the calculated distance from the

satellite to the GPS device. Ideally, these spheres would

intersect at exactly one point, causing there to be only

one possible solution to the current location, but in

reality, the intersection forms more of an oddly shaped

area. The device could be located within any point in the

area, forcing devices to choose from many possibilities.

Figure 1 shows such an area created from three satellites.

The current location could be any point within the gray-

colored area. Precision is said to be “diluted” when the

GPS accuracy.

area grows larger, which leads to this article’s focus:

dilution of precision. The monitoring and control of

dilution of precision is the key to writing high-precision

applications [1].

Figure 1: GPS devices must choose one of several

possible solutions to the current location [1].

Several external sources introduce errors into a GPS

position estimated by a GPS receiver. One important

factor in determining positional accuracy is the

constellation, or geometry, of the group of satellites from

which signals are being received. DOP only depends on

the position of the satellites: how many satellites you can

see, how high they are in the sky, and the bearing towards

them. This is often refered to as the geometry [2]. The

computed position can vary depending on which satellites

are used for the measurement. Different satellite

geometries can magnify or lessen the position error. A

greater angle between the satellites lowers the DOP, and

provides a better measurement. A higher DOP indicates

poor satellite geometry, and an inferior measurement

cofiguration, or in other words: the lower the value the

greater the confidence in the solution.

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II. THEORETICAL BACKGROUND

Satellite navigation depends on accurate range

measurements in order to determine the position of the

user. Because the receiver clock is generally not

synchronized with GPS system time, the range

measurements are erroneous and are therefore called

pseudoranges [3]. A pseudorange equation is thus:

(1)

where: PRi is the pseudorange to the ith satellite (m)

(X, Y, Z) is the unknown three dimensional user

position,

(Xi , Yi , Zi ) is the known three dimensional satellite

position (m),

tB is the unknown receiver clock offset (s),

c is the speed of light (m/s).

Because there are four unknowns, at least four

measurements are required to solve for X, Y, Z and tB.

Because the pseudorange equations are nonlinear, the

system must be linearized in order to derive a linear

relationship between pseudorange errors and position

errors. If we linearize the GPS pseudo-ranges by using a

first order Taylor Series expantion, we obtain a linear

equation of the form:

(2 )

where the hmn elements represent direction cosines to

each of the satellites.

In general, this equation can be written as:

∂Y = H ∂β ( 3 )

Thus, a receiver modifies an initial estimate of β using

∂β and iterates until convergence is achieved.

The column of ones in H shows that the receiver clock

offset (ctB) biases each pseudorange measurement by

exactly the same amount [3]. In practice, there may be

interchannel biases that affect each measurement

differently. However, great care is taken by receiver

manufacturers to calibrate these effects.

SATELLITE GEOMETRY

The GPS satellite geometry relates position errors to

range measurement errors. Considering equation 1 with

m pseudorange measurements. Then ∂Y is an m x 1

vector, H is an m x 4 matrix, and as usual ∂β is a 4 x 1

vector. If we think of ∂β as a zero-mean vector containing

the errors in the estimated user state, then we are

interested in the statistics of ∂β because that will

characterize the expected position errors. Using the

generalized inverse of H, we find the covariance of ∂β

[4]:

Now we have the covariance of ∂Y, the pseudorange

errors. These are assumed to be uncorrelated, Gaussian

random variables. As such, they are statistically

independent which results in a diagonal covariance

matrix. Furthermore, the range measurement errors are

assumed to have the same variance ( σr ) for each satellite.

So, we have:

cov (∂Y) = σr

2 I ( 5 )

which results in :

(6)

It can easily be shown that HTH is symmetric, so the

transpose is unnecessary:

cov (∂β) = σr

Now the relationship between range measurement errors

and position errors becomes clear. Let G=(HTH )-1 so that

cov (∂β) = σr

we get:

( 8 )

The elements of G give a measure of the satellite

geometry called the dilution of precision (DOP) [5].

Various DOP values can be calculated from the diagonal

elements of G. For example:

(9)

2 (HTH )-1 ( 7 )

2 G. Using B = ctB and expanding Eq. 7,

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where GDOP is the Geometric Dilution of Precision

defined as:

(10)

Other DOPs are as follows:

( 11 )

where:

PDOP is the Position Dilution of Precision

HDOP is the Horizontal Dilution of Precision

VDOP is the Vertical Dilution of Precision

TDOP is the Time Dilution of Precision

While each of these GDOP terms can be individually

computed, they are formed from covariances and so are

not independent of each other. A high TDOP (time

dilution of precision), for example, will cause receiver

clock errors which will eventually result in increased

position errors. Good GDOP, a small value representing

a large unit-vector-volume, results when angles from

receiver to satellites are different. Wheras poor GDOP, a

large value representing a small unit vector-volume,

results when angles from receiver to the set of satellites

Figure 2 : Good satellite geometry [1].

Figure 3 : Poor satellite geometry [1].

used are similar. Clustered satellites give poor GDOP

values (Figures 2,3). The higher the DOP, the weaker the

geometry.

ECEF COORDINATE REFERENCE FRAME

The Cartesian coordinate frame of reference used in GPS

is called Earth-Centered, Earth-Fixed (ECEF). ECEF uses

three-dimensional XYZ coordinates (in meters) to

describe the location of a GPS user or satellite. The term

"Earth-Centered" comes from the fact that the origin of

the axis (0,0,0) is located at the mass center of gravity.

The term "Earth-Fixed" implies that the axes are fixed

with respect to the earth (that is, they rotate with the

earth). The Z-axis pierces the North Pole, and the XY-

axis defines the equatorial plane [6].

III. COMPUTER SIMULATION

The fundamental aim in designing the Dilution of

Precision computer simulator, namely the DOP Simulator,

is to obtain a useful interface which provides a fast and

practical analysis of both the DOP parameters and the

satellite geometry. The simulator was designed as a

graphical user interface in MATLAB [7].

The DOP Simulator, when started, asks the user to enter

the ECEF coordinates of the four satellites and the GPS

receiver with respect to the ECEF coordinate reference

frame in meters. The user enters the coordinates to be

simulated and presses the Analyze button. The program

gets the coordinates as inputs and calculates the DOP

parameters according to the equations which were

mentioned above (Equations 1 to 11). The results are

displayed in the corresponding windows. All the

parameters, namely GDOP, PDOP, TDOP, HDOP, and

VDOP are calculated and displayed. Later the results are

rated according to levels between 1 and 50 (Table 1).

Table 1: DOP Ratings

DOP RATING

1 IDEAL

2-3 EXCELLENT

4-6 GOOD

7-8 MODERATE

9-20 FAIR

21-50 POOR

Finally the description of the rating associated with each

parameter is stated in an EXPLANATIONS window. To

see the explanation associated with each parameter, the

user should click the EXPLAIN button which is located

under the rating of each parameter (Figure 4). Finally, the

program shows the satellite geometry (on the right top of

the interface) associated with the coordinates that the user

has entered. This provides the opportunity to see both the

DOP values and the satellite geometry at the same time so

that a better imagination is created in the users

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Figure 4: The Dilution of Precision computer simulation user interface.

mind. The satellites are simulated with four spheres that

have equal volumes. The GPS device is simulated with a

sphere having a bigger volume than the four satellites.

From another point of view; the user can clear the

coordinates using the Clear Coordinates button and

enter different new coordinates to see how the DOP

values change with the satellite geometry and also how

the precision of the positioning is diluted. The user can

also decide on which parameter mostly causes the

dilution of precision by looking both at the RESULTs

and the RATINGs. That is; the user can consider

whether the dilution of precision occurred vertically,

horizontally, positional or a timing error.

The DOP Simulator has some other properties. One of

these is; the program detects invalid inputs and warns the

user to correct the mistake. That is, when a user enters

for example a letter between the numbers in one of the

coordinates by mistake, the program detects the invalid

input and warns the user with a message saying which of

the coordinates he has entered wrong and asks to correct

it. Another property is the additional explanation

shortcuts to each of the ratings. If the user wants to

compare the ratings, he can simply click the ? button

near the rating names which are located at the right

bottom of the interface. These shortcuts directly give the

explanations of the DOP ratings.

IV. RESULTS AND CONCLUSION

The graphical user interface, which is named as the DOP

Simulator was operated and various different coordinates

were tried. The results proved that the DOP parameters

change according to the satellite geometry. The DOP

values were decreased, as the angles between the satellites

were increased, as a result the accuracy was increased.

However the simulator is considered as a sufficient

interface, the user can plot the coordinates by writing the

necessary code in the MATLAB command window and

do detailed analysis using the camera view option. This

option is not used in the simulation since it will be a time

consuming process which is a contradiction to the design

criteria of the simulation.

REFERENCES

1. www.codepedia.com

2. www.environmental-studies.de/Precision_Farming/

GPS_E/5E.html

3. Milliken, R.J. and Zoller, C. J., "Principle of

Operation of NAVSTAR and System Characteristics

Navigation, Vol. 25, 1978

4. Brown R.G. and P.Y.C. Hwang (1992), Introduction

to Random Signals and Applied Kalman Filtering,

John Wiley & Sons Inc.

5. Milliken, R.J. & Zoller, C.J. (1980): Principle of

Operation of NAVSTAR and System Characteristics.

Navigation, Vol. 1

6. www.commlinx.com.au

7. Kaya, F. A. Gazi Univ. Graduation Project, 2005.

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