Journal of Geographic Information System, 2011, 3, 318-322
doi:10.4236/jgis.2011.34029 Published Online October 2011 (http://www.SciRP.org/journal/jgis)
Performance Improvement of GPS GDOP Approximation
Using Recurrent Wave let Neu ral Netw ork
Sadaf Tafazoli, Mohammad Reza Mosavi*
Department of Electrical Engineering, Iran University of Science and Technology, Tehran, Ir an
Received May 5, 2011; revised June 26, 2011; accepted July 14, 2011
Abstract
One of the most important factors affecting the precision of the performance of a GPS receiver is the relative
positioning of satellites to each other. Therefore, it is essential to choose appropriate accessible satellites
utilized in the calculation of GPS positions. Optimal subsets of satellites are determined using the least value
of their Geometric Dilution of Precision (GDOP). The most correct method of calculating GPS GDOP uses
inverse matrix for all combinations and selecting the lowest ones. However, the inverse matrix method, es-
pecially when there are so many satellites, imposes a huge calculation load on the processor of the GPS
navigator. In this paper, the rapid and precise calculation of GPS GDOP based on Recurrent Wavelet Neural
Network (RWNN) has been introduced for selecting an optimal subset of satellites. The method of NNs pro-
vides a realistic calculation approach to determine GPS GDOP without any need to calculate inverse matrix.
Keywords: Rapid and Precise Calculation, GPS GDOP, RWNN
1. Introduction
Global Positioning System (GPS) is a satellite based po-
sitioning system which was rapidly grown in the past two
decades. It can cover almost all around the world, using
satellite signals in order to measure accurate time, alti-
tude, longitude and latitude in every desirable point on
earth, sea or air as well as space.
The methods of calculating the coordinates in a GPS
receiver are based on the utilizing four visible satellites
in a set of visible satellites and errors may usually hap-
pen when one or more satellites are invisible or the in-
formation sent by them are unclear. There are various
algorithms for the classification of visible satellites that
can be used to classify appropriate satellites in one group.
Considering the geostationary position of satellites in
earth's orbit, five to eight GPS satellites are visible at any
position on earth. To calculate the coordinates of that
point on the earth the four-satellite groups can be used
differently [1,2].
In the common algorithm used for the most GPS re-
ceivers, the first selected satellite is the one whose con-
necting line joining the satellite to its receiver is more
vertical. After selecting the base satellite, the other three
satellites are determined based on their most appropriate
geometric configuration. At the beginning of locating
and after appropriate selection of four satellites accord-
ing to the said algorithm, the error occurring within a
specified limit is acceptable; but the increase in the time
of the original selection increases proportionally the oc-
curred er ror. Therefore, it is required that th e selection is
repeated within specified intervals. To reduce the errors
of calculation, it is necessary to reduce the temporal in-
terval existing between the classification and frequent
selections of appropriate satellites. The increase rate of
calculation error after selection, and fixation of satellites
depends on the different parameters including manufac-
turing technology of receiver, quality of receiver, applied
algorithms, and number of tracking channels. Any in-
crease in tracking channels of a receiver increases the
ability of the receiver in selecting and classifying satel-
lites. Therefore, the receivers having more tracking
channels cause less error than other receivers. As the
vehicles using the GPS moves, the initial selected satel-
lites disappear in horizon and become invisible to the
human eye. At this stage, other appropriate satellites
shall be selected [3,4].
The best method for calculating the Geometric Dilu-
tion of Precision (GDOP) of GPS satellites is to use in-
verse matrix for all configurations and selecting the
smallest one; but, the inversion of matrix imposes a huge
calculation load on the processor of the navigator [5].

S. TAFAZOLI ET AL.319
The use of Neural Networks (NNs) is a solution for
complicated issues, for which no mathematical models
are available. NNs use several simple computing units
called neuron (nervous cells) patterned after the cells of
human brain. The neurons of each NN process and con-
vert the stimulations or input data to send them to the
outputs. These outputs may be connected to the inputs of
other neurons. These neurons connecting to each other
form a NN. A NN consists of one input and one output
layer. The data are received by a NN through its input
layer and processes by all layers before receiving the
output layer [6].
The objective of this paper is to suggest a rapid
method for calculating GPS GDOP using the NNs. This
method strongly decreases the load of GPS GDOP clas-
sical method calculations for selecting an optimal subset
of satellites by GPS navigator processors. This paper has
been organized as follows. In the Section 2, the concept
of GPS GDOP has been studied in brief. In the Sections
3, 4 and 5, the manner of rapid and precise calculation of
GPS GDOP based on RWNN to select an appropriate
subset of navigator satellites. To study the workab ility of
this method, it has been tested and compared in the Sec-
tion 6. Finally, the Section 7 provides us with the con-
clusion.
2. The Concept of GPS GDOP
GPS receivers report usually the geometric quality of
satellites based on Position Dilution of Precision (PDOP).
PDOP shows the vertical and horizontal DOP, i.e. geo-
graphical longitude, latitude, and altitude. Low DOP
increases and high DOP decreases the probability of pre-
cision. If the sides of the pyramid formed by four satel-
lites are almost equal (equiangular pyramid), this con-
figuration leads to an appropriate GPS GDOP and vice
versa. GPS GDOP is a very effective tool for GPS. All
receivers use the algorithms based on GPS GDOP to find
the best subset of visible satellites used for tracking. To
locate satellites, azimuth (
A
Z) and angle of elevation
() can be used. To define the concept of GPS GDOP,
it is helpful to use sample calculations expressing a
compromise between precision and the position of satel-
lite. For this purpose, the matrix of
E
H
is defined as
follows [7]:
Cos(1) *Sin(1)Cos(1) * Cos(1)Sin(1)1
Cos( 2)*Sin(2)Cos( 2)*Cos(2)Sin( 2)1
Cos( 3)*Sin(3)Cos( 3)*Cos(3)Sin( 3)1
Cos( 4)*Sin(4)Cos( 4)*Cos(4)Sin( 4)1






EAzE AzE
EAzEAzE
HEAzE AzE
EAzEAzE



(1)
GPS GDOP shall be defined as follows:
1trace[adj( )]
GDOP trace()det()


T
T
T
H
H
HH HH (2)
GPS GDOP provides us with a simple interpretation of
the fact that how much a unit of measurement error par-
ticipates in the occurrence of positioning error for a
specified position, and it determines the factor of meas-
urement noise amplitude.
3. Rapid and Precise Calculation of GPS
GDOP Using Neural Networks
To prepare experimenting data, all input and output
variables are normalized within an interval of [0,1] to
reduce the experimenting time. whereas, T
H
H is a 4 ×
4 matrix, it has four eigenvalues i

We know that four eigenvalues for the matrix of
().
1, 2,3, 4i

1
T
HH

is equal to 1
i


. Considering that the trace of
a matrix is equal to the sum of its eigenvalues, the fol-
lowing equation is formed as follows [8]:
111
1234
GDOP 1




 (3)
The mapping with the definition of four variables is as
follows:
11234
trace( )   T
x
HH (4)
2222 2
1234
2trace 

 
T
xH()


H


H
(5)
3333 3
312 3 4
trace () 

 
T
xH (6)
41234
det( )
T
x
HH   (7)
GPS GDOP can be shown as a functional mapping of
from
4
RR1
f

directly to GPS GDOP; i.e.


1fn
f





:
Input:


1234
,,, T
x
xxx
Output: GPS GDOP

y
The mapping from f

to GPS GDOP is strictly
non-linear and cannot be determined analytically, but it
can be approximated using a NN. The NN used in this
paper has been designed for the mapping of
from
41
RR
f

to GPS GDOP. The Figure 1 shows the overall
diagram block of GPS GDOP approximation using NNs
including Recurrent NN (RNN), Wavelet NN (WNN),
and Recurrent Wavelet NN (RWNN).
4. RWNN Architecture
The RWNN topology is made of tree layers, one input
layer, one hidden layer, and one output layer. Figure 2
presents a kind of tree layer RWNN structure. The input
layer has
M
nodes. The output layer also has only one
neuron whose output is the signal represented by the

S. TAFAZOLI ET AL. 320
Figure 1. The overall diagram block of GPS GDOP
approximation using NNs.
Figure 2. RWNN architecture with (M + 1, N + 1, 1) struc-
ture.
weighted sum of several wavelets. The hidden layer is
composed of finite number of wavelets representing the
signal.
The components of the proposed RWNN essentially
include: k
x
-value of the -th input neuron; k
j

-output
of.the -th hidden.neuron;
j
I
j
k
w-interconnection weight
between the -th input neuron and -th hidden neuron;
kj
O
j
w-interconnection weight between the
j
-th hidden
neuron and the output neuron;
D
j
w-recurrent weight
for
j
-th hidden neuro n.
The net internal activity of neuron
j
at time n, is
given by:
)]1([
,
)()(
0
)( 


n
j
net
ba
D
j
wn
k
xn
I
jk
w
M
k
n
j
net

(8)
where, is the sum of input to the ()
j
net n
j
-th recurrent
neuron, ,j is the output of the -th recur-
rent neuron and is computed by passing
through the wavelet function
()n


ab net

j()
j
net n
,(.) abj

, obtaining:
,
() ()
() ()
jj
ab jj
netnbn
net nan 
 
 



(9)
where, and are the dilation and transla-
tion coefficients of the -th wavlon in hidden layer,
respectively. The RWNN output of the Figure 2 network
is computed:
()
j
an ()
j
bnj
 
,
0
() NO
jabj
J
ynwnnet n 

 


(10)
The wavelet function which we have considered here
is the called “Gaussian-derivative” function as:
2
1
2
()
x
xxe 

 (11)
5. Learning Algorithm for RWNN
The basic principle of the RWNN is to use the gradient
steepest descent method to minimize the cost function.
The training of RWNN is traditionally based on minimi-
zation of the cost function. Suppose a set of training
samples is available, the problem can be characterized as
choosing the weights (or coupling strengths) of a given
network such that the following total squared error is
minimized [9]:
∑

2
2
11
()()() ()
22
Ene ndnyn

(12)
where and represent the desired and actual
output of the output neuron , respectively. is a time
varying error. The output weights can be adjusted ac-
cording to:
()dn ()yn ()en
,
()
(1) ()()
()
() ()()()
()
OO
jj
O
j
OO
jja
O
j
En
wnwn wn
yn
wn enwennetn
wn


bj

 

 

(13)
where,

is a learning rate. The recurrent weights are
updated as follow:
() ()
(1)()() ()
() ()
DD D
jj j
DD
jj
En yn
wnwnwn en
wn wn


  

(14)
To determine the partial ()
()
D
j
yn
wn

 derivative, is dif-
ferentiated the network dynamics with respect to
as follow:
()
D
j
wn
,[()]
() () ()
ab j
O
j
DD
jj
net n
yn wn
ww
 

n
(15)
where ,()
()
abj
D
j
net n
wn






 is done by using the chain rule

S. TAFAZOLI ET AL.321
for differe ntiati on , obta ining:
,,
,
[()][()] (
() () ()
()
() ()
ab jab jj
DD
j jj
j
ab jD
j
net nnet nnet n
net n wn wn
netn
net nwn









)
(16)
Or:
,,
,
,
() ()
() ()
(1)
(1) ()()
ab jab j
Dj
j
ab j
D
ab jjD
j
net nnet n
an wn
net n
net nwnwn



 







 

 


(17)
Equation (17) is non-linear dynamic recursive equa-
tion and can be solved recursively with given initial con-
dition as:
,(0) 0
(0)
ab
D
j
w
 

 (18)
The inputs weight can be adjusted as follow:
,
()
(1) ()()
()
() ()()
()
()()()()
II
jkjk I
jk
I
jk I
jk
ab j
IO
jkj I
jk
En
wnwnwn
yn
wn enwn
netn
wn enwnwn





 





 
(19)
where ,()
()
abj
I
jk
net n
wn





is obtained as:
,,
,
,,
()() ()
() () ()
()
() ()
()( 1)
() ()
() ()
ab jab jj
II
j
jk jk
j
ab jI
jk
ab jab j
D
kj I
jjk
net nnet nnet n
netn wn wn
netn
net nwn
net nnet n
xnwn
an wn



 


 



 


 


 






(20)
With initial conditions:
,(0) 0
(0)
ab
I
jk
w
 
 (21)
The translation coefficient of the -th wavlon in
hidden layer can be adjusted according to:
j
',
()
(1) ()()
()
()()()
1
()()()[()] ()
jj
j
jj
O
jjabj
j
b
En
bnbn bn
yn
bn enn
bnenw nnetnan




 



(22)
The dilation coefficient of the -th wavlon in hidden
layer is updated as follow: j
',2
()
(1) ()()
()
()()()
() ()
()() ()[()]()
jj
j
jj
jj
O
jjabj
j
En
anan an
yn
anenan
netnbn
an enwnnetnan




 




(23)
6. Testing the Proposed Method
The parameters of the proposed NNs have been opti-
mized on the test data based on try and error method.
Figure 3 shows the approximation error values using
RWNN for 9 0 0 d ata.
The Table 1 shows four significant statistical charac-
teristics of approximation error including maximum,
minimum, average, and RMS for the approximation of
GPS GDOP for 900 test data using RWN N.
The approximation with RNN and WNN has been
done and the comparative results have been shown in
Table 2. Root Mean Square (RMS) was used to evaluate
approximations results [10]. RMS value is computed
using:

2
Real NN
1
1
RMSGDOP GDOP




iT
i
T (24)
where is number of tests. T
As it is visible over this Table, the RWNN is more
Table 1. Maximum, minimum, average and RMS error for
the approximation of GPS GDOP for 900 test data deter-
mined using RWNN.
Parameters Value
Maximum 3.5928
Minimum -3.4024
Average 0.0544
RMS 0.4590
Table 2. Comparison of RNN, WNN and RWNN perform-
ance for GPS GDOP approximation.
NN Type RNN WNN RWNN
RMS Value0.53380.5172 0.4590

S. TAFAZOLI ET AL. 322
Figure 3. The approximation error values of GPS GDOP
determined based on RWNN for 900 data with (4, 3, 1)
structure.
Table 3. Comparison CPU time of classical method and NN
approach for GPS GDOP approximation.
Model Name CPU Time [msec.]
Matrix inversion method 1.080
NN approach 0.029
efficiency in comparing with RNN and WNN; this is
because of the RMS approximation error shortage over
them.
Table 3 presents the comparison CPU time of classi-
cal method and NN approach for GPS GDOP approxi-
mation. The simulation results demonstrate that NN ap-
proach is accurate and faster than classical method.
7. Conclusions
In this paper, the rapid and precise calculation of GPS
GDOP using RWNN has been studied for the selection
of an appropriate subset of navigator satellites. The
method of NNs is a realistic computing approach used
for the calculation of GPS GDOP without any need to
inverse matrix, which imposes a huge computing load on
the processor of the navigator. The performance of the
proposed NN has been studied on the test data of the
paper. The results show that the proposed method is fully
capable to select an optimal subset of GPS satellites with
the best geometric configuration. The results of simula-
tion show that the efficiency of RWNN is better than
RNN and WNN.
8. References
[1] R. Yarlagadda, I. Ali, N. Al-Dhahir and J. Hershey, “GPS
GDOP Metric,” IEE Proceedings—Radar, Sonar and
Navigation, Vol. 147, No. 5, 2000, pp. 259-264.
doi:10.1049/ip-rsn:20000554
[2] M. Zhang and J. Zhang, “A Fast Satellite Selection Algo-
rithm: Beyond Four Satellites,” IEEE Journal of Selected
Topics in Signal Processing, Vol. 3, No. 5, 2009, pp. 740-
747. doi:10.1109/JSTSP.2009.2028381
[3] D. Simon and H. El-Sherief, “Navigation Satellite Selec-
tion Using Neural Networks,” Journal of Neurocomput-
ing, Vol. 7, 1995, pp. 247-258.
doi:10.1016/0925-2312(94)00024-M
[4] M. R. Mosavi, “High Performance Methods for GPS
GDOP Approximation Using Neural Network,” Journal of
Geoinformatics, Vol. 4, No. 3, 2008, pp. 9-16.
[5] D. J. Jwo and C. C. Lai, “Neural Network-Based Geome-
try Classification for Navigation Satellite Selection,”
Journal of GPS Navigation, Vol. 56, No. 2, 2003, pp.
291-304. doi:10.1017/S0373463303002200
[6] M. R. Mosavi, “GPS Receivers Timing Data Processing
using Neural Networks: Optimal Estimation and Errors
Modeling,” Journal of Neural Systems, Vol. 17, No. 5,
2007, pp. 383-393. doi:10.1142/S0129065707001226
[7] B. W. Parkinson, “G lobal Positioning S ystem: Th eory and
Applications,” The American Institute of Aeronautics and
Astronautics, Vol. 1, 1996.
[8] D. J. Jwo and C. C. Lai, “Neural Network-Based GPS
GDOP Approximation and Classification,” Journal of
GPS Solutions, Vol. 11, No. 1, 2007, pp. 51-60.
doi:10.1007/s10291-006-0030-z
[9] M. R. Mosavi, “An Adaptive Correction Technique for
DGPS using Recurrent Wavelet Neural Network,” IEEE
Conference on Systems, Man, and Cybernetics, 2007, pp.
3029-3033. doi:10.1109/ICSMC.2007.4413579
[10] O. Øvstedal, “Absolute Positioning with Single- Fre-
quency GPS Receivers,” Journal of GPS Solutions, Vol.
5, No. 4, 2002, pp. 33-44. doi:10.1007/PL00012910