![Mathematical Investigation and Analysis of Regional Signal Coverage in Mobile
Communication
QU Zhiming JIAO Lijun LIU Libo
School of Civil Engineering, Hebei University of Engineering, Handan, Hebei Province, 056038,
China
chinaqzm@163.com jiaolijun@hebeu.edu.cn liulbyjp@126.com
Abstract
According to the characteristics and actual conditions
of signal coverage, coverage is analyzed in different
regions such as plains, mountains and urban areas.
Signal is weak in plain, which needs to extend coverage.
Stations can be alternately erected in roads and their
surrounding areas. The signal being covered in mountain
area with less base stations is considered. Interference of
urban coverage should be reduced and coverage between
the high-rise building and the basement properly handled.
Coverage of the signal is mathematically analyzed, and
the rectangular mathematical coverage system and
physical coverage grid are formed. The coverage
functions are selected after the shape of coverage is
determined. In the process of displacement coverage of
effective mobile signal, the displacement function of the
physical coverage is deduced. The entire first and second
order approximations are carried out on constant
function during physical coverage.
1. Introduction
In order to maintain a superior position in the
increasingly fierce competition in the MCs market,
telecommunications industry's international
competitiveness should be enhanced. It is necessary to
improve the perception of mobile users and constantly
improve the network coverage and a lot of depth coverage
in network optimization projects [1-3]. The most
reasonable programme needs to be developed in depth
coverage optimization projects, which is economical and
convenient. It creates greater economic and social benefits
for enterprises. In order to deeply study the MCs coverage
optimization project from theory calculation to the actual
engineering experience, and in view of the deep coverage
communications characteristics and the actual situation,
the different regions of the coverage must be analyzed in
which environmental factors, the historical data and the
existing building experience should be considered [4-5].
If the environment is greatly different, the coverage
analysis must be adjusted.
2. Signal coverage strategies in mobile
communication
2.1. Plain areas
Signal coverage in plain region should pursue Hassle
Free, non-blocking, non-interference, clear switching line.
A base station is evenly established well-distributed plain
town. If more buildings are in the town and signal
penetration capability is insufficient, it should solve the
problem like that in urban areas. Carrier frequency or
additional base stations are added in the town with traffic
of calls. Repeater and micro-cellular may be increased to
extend the coverage. People highly demands for
communications in highway and main road. Firstly, it
should to ensure that there are no blind spots on highway.
Secondly, switch line should be kept vertical to the
highway as far as possible. Therefore, blind spots should
be covered and, according to the length of the section of
blind spots, repeaters and additional base stations should
also be decided.
2.2. Mountain areas
In mountain areas, the coverage of town, villages and
highway is considered. Possibly less base stations are
used to cover more areas to ensure no breakpoints on
highway. Town in rocky mountain areas is, at least, to set
up a base station which mainly provides more intensive
adequate channels for dense areas. Base stations can be
built in the nearby mountains which does not only ensure
the town coverage but maintain the coverage of some
villages and roads. Generally, mountain villages are
distributed at the end of the ravine and the road is
sandwiched between the two main mountains. Base
stations should be built in the main ridge of the
mountains, so it can cover both sides of the mountain
slope, the relative slope in the two main adjacent
mountain ranges and the village at the end of mountains.
Due to winding road, only ridge on the base station can
not guarantee continuous coverage. It must be based on
the actual situation to fill with the non-coverage sections.
2.3. Urban areas
The characteristics of urban coverage is to absorb
reasonable calls traffic, control switch line, reduce
interference and properly handle the coverage of the high-
rise building and basement. Because of the high-rise
2008 ISECS International Colloquium on Computing, Communication, Control, and Management
978-0-7695-3290-5/08 $25.00 © 2008 IEEE
DOI 10.1109/CCCM.2008.112
385
2008 ISECS International Colloquium on Computing, Communication, Control, and Management
978-0-7695-3290-5/08 $25.00 © 2008 IEEE
DOI 10.1109/CCCM.2008.112
385
Authorized licensed use limited to: IEEE Xplore. Downloaded on October 18,2011 at 21:39:59 UTC from IEEE Xplore. Restrictions apply.](https://image.slidesharecdn.com/mathematicalinvestigationandanalysisofregionalsignalcoverageinmobile-160922180256/85/Mathematical-investigation-and-analysis-of-regional-signal-coverage-in-mobile-communication-1-320.jpg)
![( )
( )
[ ]1 2 3 4 5 6
, 1 0 0 0
0 1 0 0,
i
i
i i i i i i
u x y x y
x yv x y
d d d d d d
⎧ ⎫ ⎡ ⎤⎪ ⎪
= ⋅⎨ ⎬ ⎢ ⎥
⎣ ⎦⎪ ⎪⎩ ⎭
′
(5)
(6)
And the general series expression is
(7)
Equation (4) to (7) can be expressed in equation (8).
( )
( )
,
,
i
i
i
u x y
SD
v x y
⎧ ⎫⎪ ⎪
=⎨ ⎬
⎪ ⎪⎩ ⎭
(8)
Where, S is the basic series, and
[ ]1 2, , ,i i i inD d d d ′= . n is the degree-of-freedom in
physical coverage which is decided by the number of S .
Substituting equations (8) and (3) into (1) and using
manifold method, ( ),u x y and ( ),v x y are expressed
as equation (9).
( )
( )
( )
( )
4
1
, ,
( , )
, ,
i
i
i i
u x y u x y
w x y TD
v x y v x y=
⎧ ⎫ ⎧ ⎫⎪ ⎪ ⎪ ⎪
= =⎨ ⎬ ⎨ ⎬
⎪ ⎪ ⎪ ⎪⎩ ⎭ ⎩ ⎭
∑ (9)
Where,
4
1
( , )i i
i
TD w x y SD
=
= ∑ , displacement
matrix [ ]1 2 4( , ) , ( , ) , , ( , )T w x y S w x y S w x y S= ,
vector of degree-of-freedom [ ]1 2 3 4, , ,D D D D D ′= .
The displacement function in the wholly selected area is
confirmed while the displacement functions in each
regional unit are deduced.
4. Conclusions
Based on the characteristics and actual conditions,
signal coverage in different regions such as the plains,
mountains and the urban areas of coverage are analyzed.
In Plain areas, the signal is weak which needs to be
extent. Station of highway and its surrounding areas can
be alternately setup. In mountain areas, the signal
coverage in town, villages and highway are mainly
considered, which is covered by reasonable base stations.
The characteristics in urban areas coverage is properly to
absorb traffic of calls, control switch line rightly, reduce
interference and reasonably handle the coverage between
high-rise building and the basement.
Mathematical analysis is carried out on the signal
coverage to form the rectangular coverage mathematical
system and physical coverage grids. After determining the
coverage shape, the coverage weight function can be
selected by various physical coverage. As to the
displacement coverage during effective mobile signal, the
displacement functions in each physical coverage in
selected area are deduced. Generally, it can be constant,
linear, and high-order polynomial or local series.
Lagrange interpolation function is utilized on constant
functions in physical coverage to carry out the entire 1-
order and 2-order approximation. Thus, each of the
regional units of displacement functions identified, the
displacement function in entirely selected region is
identified.
5. References
[1] WANG Baodong, “Coverage strategy of wireless
signal in mobile communication”, Mobile
communication, Oct., 2002, pp. 62-63.
[2] KONG Qinghua, Study on mobile communication
signals in-depth coverage optimization engineering
projects, Jiling University, June, 2006.
[3] SHU Huahong, Radio engineering in cellular mobile
communications, Beijing, Telecom press, Jan., 2005.
[4] DAI Meitai, Network optimization in GSM mobile
communications, Beijing, Telecom press, April, 2003.
[5] YANG Fengrei, Planning of communication network,
Beijing, Telecom press, July, 2005.
[6] HUA wei, Theory of wireless communication, Jan.,
2001.
388388
Authorized licensed use limited to: IEEE Xplore. Downloaded on October 18,2011 at 21:39:59 UTC from IEEE Xplore. Restrictions apply.](https://image.slidesharecdn.com/mathematicalinvestigationandanalysisofregionalsignalcoverageinmobile-160922180256/85/Mathematical-investigation-and-analysis-of-regional-signal-coverage-in-mobile-communication-4-320.jpg)
Mathematical investigation and analysis of regional signal coverage in mobile communication
- 1. Mathematical Investigation and Analysis of Regional Signal Coverage in Mobile Communication QU Zhiming JIAO Lijun LIU Libo School of Civil Engineering, Hebei University of Engineering, Handan, Hebei Province, 056038, China chinaqzm@163.com jiaolijun@hebeu.edu.cn liulbyjp@126.com Abstract According to the characteristics and actual conditions of signal coverage, coverage is analyzed in different regions such as plains, mountains and urban areas. Signal is weak in plain, which needs to extend coverage. Stations can be alternately erected in roads and their surrounding areas. The signal being covered in mountain area with less base stations is considered. Interference of urban coverage should be reduced and coverage between the high-rise building and the basement properly handled. Coverage of the signal is mathematically analyzed, and the rectangular mathematical coverage system and physical coverage grid are formed. The coverage functions are selected after the shape of coverage is determined. In the process of displacement coverage of effective mobile signal, the displacement function of the physical coverage is deduced. The entire first and second order approximations are carried out on constant function during physical coverage. 1. Introduction In order to maintain a superior position in the increasingly fierce competition in the MCs market, telecommunications industry's international competitiveness should be enhanced. It is necessary to improve the perception of mobile users and constantly improve the network coverage and a lot of depth coverage in network optimization projects [1-3]. The most reasonable programme needs to be developed in depth coverage optimization projects, which is economical and convenient. It creates greater economic and social benefits for enterprises. In order to deeply study the MCs coverage optimization project from theory calculation to the actual engineering experience, and in view of the deep coverage communications characteristics and the actual situation, the different regions of the coverage must be analyzed in which environmental factors, the historical data and the existing building experience should be considered [4-5]. If the environment is greatly different, the coverage analysis must be adjusted. 2. Signal coverage strategies in mobile communication 2.1. Plain areas Signal coverage in plain region should pursue Hassle Free, non-blocking, non-interference, clear switching line. A base station is evenly established well-distributed plain town. If more buildings are in the town and signal penetration capability is insufficient, it should solve the problem like that in urban areas. Carrier frequency or additional base stations are added in the town with traffic of calls. Repeater and micro-cellular may be increased to extend the coverage. People highly demands for communications in highway and main road. Firstly, it should to ensure that there are no blind spots on highway. Secondly, switch line should be kept vertical to the highway as far as possible. Therefore, blind spots should be covered and, according to the length of the section of blind spots, repeaters and additional base stations should also be decided. 2.2. Mountain areas In mountain areas, the coverage of town, villages and highway is considered. Possibly less base stations are used to cover more areas to ensure no breakpoints on highway. Town in rocky mountain areas is, at least, to set up a base station which mainly provides more intensive adequate channels for dense areas. Base stations can be built in the nearby mountains which does not only ensure the town coverage but maintain the coverage of some villages and roads. Generally, mountain villages are distributed at the end of the ravine and the road is sandwiched between the two main mountains. Base stations should be built in the main ridge of the mountains, so it can cover both sides of the mountain slope, the relative slope in the two main adjacent mountain ranges and the village at the end of mountains. Due to winding road, only ridge on the base station can not guarantee continuous coverage. It must be based on the actual situation to fill with the non-coverage sections. 2.3. Urban areas The characteristics of urban coverage is to absorb reasonable calls traffic, control switch line, reduce interference and properly handle the coverage of the high- rise building and basement. Because of the high-rise 2008 ISECS International Colloquium on Computing, Communication, Control, and Management 978-0-7695-3290-5/08 $25.00 © 2008 IEEE DOI 10.1109/CCCM.2008.112 385 2008 ISECS International Colloquium on Computing, Communication, Control, and Management 978-0-7695-3290-5/08 $25.00 © 2008 IEEE DOI 10.1109/CCCM.2008.112 385 Authorized licensed use limited to: IEEE Xplore. Downloaded on October 18,2011 at 21:39:59 UTC from IEEE Xplore. Restrictions apply.
- 2. buildings, basement and less signal penetration capability of thick concrete structure, multiple base station signals in high-level building interfere with each other. According to signal penetration capability by the base station coverage, the parameters of splicing sector are adjusted to ensure that there are not non-coverage corners. The mobile users in cars moving faster on the main street must be avoided by multi-sector consecutive switch in order to ensure their voice quality. The best line in the complex switch of the main street will be connected with sector coverage. Separate coverage is done on the need for blind spots. Repeater can be used if there is no traffic of calls. Micro- cellular is necessary and the antenna system is increased by the size, so is the basement. Elevator antenna can be installed with the staircase or fixed antenna in the elevator shaft. If there is no distribution system, the repeater should be increased. As to high-rise buildings higher than the base-station antennas, a number of signals will be received from multi-base station signal, in which contains the same frequency signal and even the adjacent channel. If wireless repeaters are used, receiving antennas should be located in the lower floors which can receive signals dominant position. Busy areas network technology will use double network technology to connect static and slow mobile users with lower net, and the rapid mobile network users are connected to the upper deck to reduce rapid mobile users’ switching frequency. 3. Mathematical analysis of signal coverage in mobile communication Using mathematical overlapping coverage to cover all the regional mobile signal, coverage in the sub-region can be differentiated and continuous in the contact line. The physical grids formed in the borders, summit, highways and different regions in covered regions split the mathematical coverage as physical coverage further in which the limited coverage system is constructed. In fig.1, two circles and rectangle are used to form mathematic coverage, , ,I II IIIV V V , to cover regions with the quadrilateral borders and a road running through it. Mathematical coverage IV , IIV , IIIV and their overlapping parts of regions are divided into two physical coverage II and III , ,I IIII II and ,I IIIII III respectively. At the same time, physical coverage is overlapped each another to form a manifold unit. I III II means the overlapping parts of the physical coverage ,I III II , and I II II II III the overlapping parts of , ,I II II II III and so on. Figure 1. Regional coverage with a highway As to the limited coverage system, the coverage function ( ),iu x y and ( ),iv x y , ( ),x y E∈ are defined respectively based on each physical coverage ( )1,2, ,iC i m= , then weighted and summed the functions to solve the field functions ( ),u x y and ( ),v x y in the whole area. In fig.2, if there is a public area E among m inter-overlapping coverage iC , then, in ( ),u x y and ( ),v x y , ( ),iu x y and ( ),iv x y can be expressed as follow. Figure 2. Public area, E, under m coverage area ( ) ( ) ( ) ( ) ( ) ( )1 , , , , , , n i i i i u x y u x y w x y v x y v x y v x y= ⎧ ⎫ ⎧ ⎫⎪ ⎪ ⎪ ⎪ =⎨ ⎬ ⎨ ⎬ ⎪ ⎪ ⎪ ⎪⎩ ⎭ ⎩ ⎭ ∑ (1) Where, ( ),iw x y is the coverage weight function which should satisfy with equation (2). ( ) ( ) ( ) ( ) ( ) ( ) 1 , 0, , , , 0, , , 1, , i i i i m i i w x y x y C w x y x y C w x y x y E = ≥ ∈ = ∉ ≡ ∈∑ (2) In order to determine the overall displacement function ( ),u x y and ( ),v x y in the selected regions, how to choose the shape of coverage, coverage displacement and weight function must be considered. In the following sections, the standard rectangular grid is used as a mathematical coverage to complete the theoretical derivation and implementation of manifold method. 3.1. Rectangular coverage in regional communication Two kinds of separate and independent mathematical and physical mesh are used. Mathematical grid defines the similar precision and physical grid as a practical 386386 Authorized licensed use limited to: IEEE Xplore. Downloaded on October 18,2011 at 21:39:59 UTC from IEEE Xplore. Restrictions apply.
- 3. material border, and then the communication region is defined. In fig.3, as to the arbitrary two-dimensional Figure 3. Rectangular coverage in selected communication region structure body Ψ , the first step is to establish the rectangular grids without relations to Ψ but to cover all communication region.. Each intersection in grids is the corresponding central point of mathematical coverage, recorded separately as mathematical coverage (1), (2),…, (16). Areas of each mathematical coverage contain all the rectangular area of this intersection. For example, mathematical coverage (5) is the rectangular area surrounded by (1), (5), (9), (10), (6) and (2). With the definition of mathematical coverage grid, the physical grid can be further divided mathematical coverage into more discontinuous area which is defined as physical regional coverage. As to communication area such as continuum calculation, the physical coverage is, actually, the intersection between regional border and mathematical coverage. For instance, the definition of 5th physical coverage in fig.3 is shown in fig.4. Figure 4. Area of 5th physical coverage The intersection of communication and mathematical coverage defines the physical coverage system in analytical domain. The regional units are the public area or intersection of all the physical coverage. regional units in rectangular grids must be the public area of four physical coverage, such as 3 4 7 8 and 5 6 9 10 etc. in fig.3. As the regional unit is the polygon, it can be discrete to the arbitrary geometry of structure, which is not used quadtree or octree tree to divide mesh and then using complex algorithm to form the triangle or quadrangle. Thus, it can simplify the process of the numerical analysis and has great advantages and broad applicability. Form the definition of the coverage and regional units by the method above, the unit is arbitrary shape. A regional unit is composed of the public area of four physical coverage, and each unit must have four physical coverage. The adjacent regional units along the border have the same public physical coverage nodes. 3.2. Selection of coverage function in Regional units The various weight functions in each coverage is further chosen after determining the coverage shape. To the rectangular coverage system, the weight functions can be directly defined as the shape function corresponding to rectangular units, which can satisfy with equation (2). The regional unit 1 5 6 2 in fig.5 is used to illustrate the coverage weight function. The four mathematical coverage nodes in the unit are (1), (5), (6) and (2) respectively, and the central point coordinates is 0 0( , )x y . Fig.5 Regional unit 1 5 6 2 (E1) 3.3. Displacement coverage of effective mobile signal In the selected region, there are many structural ways of ( ),iu x y and ( ),iv x y that are constant, linear, high-order polynomial or series. Presently, Lagrange points-interpolation function is often used. To constants function of ( )1,2, ,iC i m= , equation (4) is deduced. 0 0 1 0 0 2 0 0 5 0 0 6 1 ( , ) 1 1 4 1 ( , ) 1 1 4 1 ( , ) 1 1 4 1 ( , ) 1 1 4 x x y y w x y a b x x y y w x y a b x x y y w x y a b x x y y w x y a b − − ⎫⎛ ⎞⎛ ⎞ = − −⎜ ⎟⎜ ⎟ ⎪ ⎝ ⎠⎝ ⎠ ⎪ ⎪− −⎛ ⎞⎛ ⎞ = − + ⎪⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠⎪ ⎬ − −⎛ ⎞⎛ ⎞⎪= + −⎜ ⎟⎜ ⎟⎪⎝ ⎠⎝ ⎠ ⎪ − −⎛ ⎞⎛ ⎞⎪= + +⎜ ⎟⎜ ⎟⎪⎝ ⎠⎝ ⎠⎭ (3) Where, 1( , )x y E∈ . ( ) ( ) 1 2 , 1 0 0 1, i i ii u x y d dv x y ⎧ ⎫ ⎡ ⎤⎡ ⎤⎪ ⎪ =⎨ ⎬ ⎢ ⎥⎢ ⎥ ⎣ ⎦ ⎣ ⎦⎪ ⎪⎩ ⎭ (4) The entire first and second order approximations in physical coverage iC are expressed as 387387 Authorized licensed use limited to: IEEE Xplore. Downloaded on October 18,2011 at 21:39:59 UTC from IEEE Xplore. Restrictions apply.
- 4. ( ) ( ) [ ]1 2 3 4 5 6 , 1 0 0 0 0 1 0 0, i i i i i i i i u x y x y x yv x y d d d d d d ⎧ ⎫ ⎡ ⎤⎪ ⎪ = ⋅⎨ ⎬ ⎢ ⎥ ⎣ ⎦⎪ ⎪⎩ ⎭ ′ (5) (6) And the general series expression is (7) Equation (4) to (7) can be expressed in equation (8). ( ) ( ) , , i i i u x y SD v x y ⎧ ⎫⎪ ⎪ =⎨ ⎬ ⎪ ⎪⎩ ⎭ (8) Where, S is the basic series, and [ ]1 2, , ,i i i inD d d d ′= . n is the degree-of-freedom in physical coverage which is decided by the number of S . Substituting equations (8) and (3) into (1) and using manifold method, ( ),u x y and ( ),v x y are expressed as equation (9). ( ) ( ) ( ) ( ) 4 1 , , ( , ) , , i i i i u x y u x y w x y TD v x y v x y= ⎧ ⎫ ⎧ ⎫⎪ ⎪ ⎪ ⎪ = =⎨ ⎬ ⎨ ⎬ ⎪ ⎪ ⎪ ⎪⎩ ⎭ ⎩ ⎭ ∑ (9) Where, 4 1 ( , )i i i TD w x y SD = = ∑ , displacement matrix [ ]1 2 4( , ) , ( , ) , , ( , )T w x y S w x y S w x y S= , vector of degree-of-freedom [ ]1 2 3 4, , ,D D D D D ′= . The displacement function in the wholly selected area is confirmed while the displacement functions in each regional unit are deduced. 4. Conclusions Based on the characteristics and actual conditions, signal coverage in different regions such as the plains, mountains and the urban areas of coverage are analyzed. In Plain areas, the signal is weak which needs to be extent. Station of highway and its surrounding areas can be alternately setup. In mountain areas, the signal coverage in town, villages and highway are mainly considered, which is covered by reasonable base stations. The characteristics in urban areas coverage is properly to absorb traffic of calls, control switch line rightly, reduce interference and reasonably handle the coverage between high-rise building and the basement. Mathematical analysis is carried out on the signal coverage to form the rectangular coverage mathematical system and physical coverage grids. After determining the coverage shape, the coverage weight function can be selected by various physical coverage. As to the displacement coverage during effective mobile signal, the displacement functions in each physical coverage in selected area are deduced. Generally, it can be constant, linear, and high-order polynomial or local series. Lagrange interpolation function is utilized on constant functions in physical coverage to carry out the entire 1- order and 2-order approximation. Thus, each of the regional units of displacement functions identified, the displacement function in entirely selected region is identified. 5. References [1] WANG Baodong, “Coverage strategy of wireless signal in mobile communication”, Mobile communication, Oct., 2002, pp. 62-63. [2] KONG Qinghua, Study on mobile communication signals in-depth coverage optimization engineering projects, Jiling University, June, 2006. [3] SHU Huahong, Radio engineering in cellular mobile communications, Beijing, Telecom press, Jan., 2005. [4] DAI Meitai, Network optimization in GSM mobile communications, Beijing, Telecom press, April, 2003. [5] YANG Fengrei, Planning of communication network, Beijing, Telecom press, July, 2005. [6] HUA wei, Theory of wireless communication, Jan., 2001. 388388 Authorized licensed use limited to: IEEE Xplore. Downloaded on October 18,2011 at 21:39:59 UTC from IEEE Xplore. Restrictions apply.