Satellite Communications Part 11 pot

Introduction In cellular satellite communications systems, a given coverage area is typically filled with a network of contiguous spot beams, which carry concentrated radiation along pr

Guidelines for Satellite Tracking 291

In cases where perigee height is less than 220 kilometers, the equations for a and IL should

be truncated after the C1 term, and all terms involving C5,  , and  Mshould be

cos

18

sin4

u u

Guidelines for Satellite Tracking 293

In cases where perigee height is less than 220 kilometers, the equations for a and IL should

be truncated after the C1 term, and all terms involving C5, , and  Mshould be

2

cos

18

2

sin4

u u

N N N

2.2 Propagation models modifications

SGP propagation model was modified in time Several minor points in the original SGP4

paper emerged where performance of SGP4 could be improved To maximise the usefulness

of all of these features, one should ideally use Two Line Elements formed with differential

correction, using an identical model as well (Vallado, D et al 2006) Next chapter will shed

some light on what Two Line Elements are

3 Two Line Elements

Orbit tracking programs require information about the shape and orientation of satellite

orbits That information was available from different websites One of most common quoted

sources is CelesTrak.com website, maintained by the group of satellite tracking enthusiasts

As a result of legislation passed by the US Congress and signed into law on 2003 November

24 (Public Law 108-136, Section 913), which was updated in 2006 (US National Archives,

2006.), Air Force Space Command (AFSPC) has embarked on a three-year pilot program to

provide space surveillance data—including NORAD two-line element sets (TLEs)—to

non-US government entities (NUGE) Since non-US Public Law prohibits the redistribution of the

data obtained from this new NUGE service "without the express approval of the Secretary of

Defence“a lot of other sources were immediately shut down

CelesTrak has received continuing authority to redistribute Space Track data from US government and that way become one of the most useful information sources for the community

TLE’s are redistributed in a form shown in Fig 5 All relevant parameters are color-coded and explained in Table 1

Table 1 Two Line Elements Explained There are several things to consider The accuracy of the original TLEs is not known Some TLE data propagates into future quite well, while, the next set of elements can depart dramatically after only a day or less Methods to overcome this problem are explained in (Vallado, D et al 2006)

4 Integrating Mathematical Models

Our intention was to integrate all orbital propagation models into one C# program Integral version of the program can be downloaded from

http://medlab.elfak.ni.ac.rs/spacetrack/sgpsdp.rar It is important to highlight main program methods used for satellite position calculations:

publicvoid SGP(int IFLAG, double TSINCE) publicvoid SGP4(int IFLAG, double TSINCE) publicvoid SDP4(int IFLAG, double TSINCE) publicvoid SGP8(int IFLAG, double TSINCE) publicvoid SDP8(int IFLAG, double TSINCE) 

Guidelines for Satellite Tracking 295

z

N N

2.2 Propagation models modifications

SGP propagation model was modified in time Several minor points in the original SGP4

paper emerged where performance of SGP4 could be improved To maximise the usefulness

of all of these features, one should ideally use Two Line Elements formed with differential

correction, using an identical model as well (Vallado, D et al 2006) Next chapter will shed

some light on what Two Line Elements are

3 Two Line Elements

Orbit tracking programs require information about the shape and orientation of satellite

orbits That information was available from different websites One of most common quoted

sources is CelesTrak.com website, maintained by the group of satellite tracking enthusiasts

As a result of legislation passed by the US Congress and signed into law on 2003 November

24 (Public Law 108-136, Section 913), which was updated in 2006 (US National Archives,

2006.), Air Force Space Command (AFSPC) has embarked on a three-year pilot program to

provide space surveillance data—including NORAD two-line element sets (TLEs)—to

non-US government entities (NUGE) Since non-US Public Law prohibits the redistribution of the

data obtained from this new NUGE service "without the express approval of the Secretary of

Defence“a lot of other sources were immediately shut down

CelesTrak has received continuing authority to redistribute Space Track data from US government and that way become one of the most useful information sources for the community

TLE’s are redistributed in a form shown in Fig 5 All relevant parameters are color-coded and explained in Table 1

Table 1 Two Line Elements Explained There are several things to consider The accuracy of the original TLEs is not known Some TLE data propagates into future quite well, while, the next set of elements can depart dramatically after only a day or less Methods to overcome this problem are explained in (Vallado, D et al 2006)

4 Integrating Mathematical Models

Our intention was to integrate all orbital propagation models into one C# program Integral version of the program can be downloaded from

http://medlab.elfak.ni.ac.rs/spacetrack/sgpsdp.rar It is important to highlight main program methods used for satellite position calculations:

publicvoid SGP(int IFLAG, double TSINCE) publicvoid SGP4(int IFLAG, double TSINCE) publicvoid SDP4(int IFLAG, double TSINCE) publicvoid SGP8(int IFLAG, double TSINCE) publicvoid SDP8(int IFLAG, double TSINCE) 

Previous FORTRAN IV code produced by T.S Kelso in 1988 according to (Hoots, F R et al

1980) was not optimized and hard to execute on modern parallel (multi-core) architectures

The FORTRAN implementation of the SGP4 and SDP4 model in respective methods is

rudimentary for the propagation process It was necessary to produce functions which

would help us calculate position ݎԦ and velocity ݎԦ of a satellite at any given time by using

the TLE data Models specified in (Hoots, F R and al., 1980) from the original FORTRAN IV

code are ported to C# in respect to the corrections made during the years, especially in the

SDP4 subroutine DEEP C# code contains the same variable names and structures as in the

original FORTRAN routines to ensure compatibility and expandability  

Additional encapsulation was done with the creation of ActiveX component ready to be

integrated in any NET project

5 NAVSTAR Satellite Tracking Software

NAVSTAR satellite tracking software presented in this paper is also based on the

mathematical SGP4/SDP4 model Program uses two line elements set as an input to

calculate and visualize satellite’s position in Space It can be used to navigate telescopes to

space objects passing over certain point on Earth The complete mathematical model is

encapsulated in ActiveX control, so it acts like a black box The data is provided from TLEs

and on the other end viewport coordinates are calculated

NAVSTAR has three basic functions:

 Graphical display of satellite positions in real-time, simulation, and manual

modes;

 Tabular display of satellite information in the same modes;

 Generation of tables (ephemerides) of past or future satellite information for

planning or analysis of satellite orbits

Fig 6 Satellite selection dialog and Table Window

The principal feature of NAVSTAR is a series of Map Windows, which display the current

position of satellites and observers on a simple world map, together with information such

as bearing (azimuth), distance, and elevation above the observer's horizon The maps may

be updated in real time, simulated time, or manually set to show the situation at any given moment of time, past or future

An additional Table Window displays much more-detailed information about one or more satellites in a tabular form The tabulated items can be selected and rearranged to fit the screen This information can be updated in real-time, manual, or simulation modes as illustrated in Fig 6

Also, satellite 2D footprint (Fig.7) tracking is available, as well as a 3D view (Fig.8) Tracking algorithms SGP4 and SDP4 give considerable accuracy and opportunity of efficient computation of viewing opportunities It’s also possible in 3D view to make a prediction on satellites position in the future, or to see its position in the past All is based on the information gathered from TLE’s

The preciseness of visualization depends on accuracy and age of gathered TLE data

Fig 7 2D View Regarding the 3D View (Fig.8), options for variable view angle, zoom and time increment are implemented This gives a user the opportunity to view satellite from all angles and possibility to see its path (orbit), area on the Earth covered by its signal (in a form of beam) and real-time movement, as well as possible faster movement caused by a time speed up

Fig 8 3D Satellite tracking View

Guidelines for Satellite Tracking 297

Previous FORTRAN IV code produced by T.S Kelso in 1988 according to (Hoots, F R et al

1980) was not optimized and hard to execute on modern parallel (multi-core) architectures

The FORTRAN implementation of the SGP4 and SDP4 model in respective methods is

rudimentary for the propagation process It was necessary to produce functions which

would help us calculate position ݎԦ and velocity ݎԦ of a satellite at any given time by using

the TLE data Models specified in (Hoots, F R and al., 1980) from the original FORTRAN IV

code are ported to C# in respect to the corrections made during the years, especially in the

SDP4 subroutine DEEP C# code contains the same variable names and structures as in the

original FORTRAN routines to ensure compatibility and expandability  

Additional encapsulation was done with the creation of ActiveX component ready to be

integrated in any NET project

5 NAVSTAR Satellite Tracking Software

NAVSTAR satellite tracking software presented in this paper is also based on the

mathematical SGP4/SDP4 model Program uses two line elements set as an input to

calculate and visualize satellite’s position in Space It can be used to navigate telescopes to

space objects passing over certain point on Earth The complete mathematical model is

encapsulated in ActiveX control, so it acts like a black box The data is provided from TLEs

and on the other end viewport coordinates are calculated

NAVSTAR has three basic functions:

 Graphical display of satellite positions in real-time, simulation, and manual

modes;

 Tabular display of satellite information in the same modes;

 Generation of tables (ephemerides) of past or future satellite information for

planning or analysis of satellite orbits

Fig 6 Satellite selection dialog and Table Window

The principal feature of NAVSTAR is a series of Map Windows, which display the current

position of satellites and observers on a simple world map, together with information such

as bearing (azimuth), distance, and elevation above the observer's horizon The maps may

be updated in real time, simulated time, or manually set to show the situation at any given moment of time, past or future

An additional Table Window displays much more-detailed information about one or more satellites in a tabular form The tabulated items can be selected and rearranged to fit the screen This information can be updated in real-time, manual, or simulation modes as illustrated in Fig 6

Also, satellite 2D footprint (Fig.7) tracking is available, as well as a 3D view (Fig.8) Tracking algorithms SGP4 and SDP4 give considerable accuracy and opportunity of efficient computation of viewing opportunities It’s also possible in 3D view to make a prediction on satellites position in the future, or to see its position in the past All is based on the information gathered from TLE’s

The preciseness of visualization depends on accuracy and age of gathered TLE data

Fig 7 2D View Regarding the 3D View (Fig.8), options for variable view angle, zoom and time increment are implemented This gives a user the opportunity to view satellite from all angles and possibility to see its path (orbit), area on the Earth covered by its signal (in a form of beam) and real-time movement, as well as possible faster movement caused by a time speed up

Fig 8 3D Satellite tracking View

The part of Earth not covered with the Sun light is dimmed on the globe, so the user can predict when it will become possible to see the satellite by a naked eye

This software can be used to visualise the orbit trajectory of a satellite under different points of view It gives the user the freedom of being able to study the satellite’s ground conjunctions by tracking the satellite over the ground, or, with another approach, by calculating the elevation and azimuth angle of the satellite from a static ground station position

Another very useful functionality, previously mentioned is satellite’s footprint visualisation These functionalities can be used, from an engineering point of view, to adjust ground station dishes in order to establish reliable links to the satellite by calculating it’s precise position with the help of TLE data sets

3D view can be used to simulate satellite’s orbit around the Earth Time lapse function enables user to see the orbit in the future, and exact satellite’s position so we would be able

to see which orbit parameters the satellite has to have in order to fulfil its task

6 Conclusion

It is assumed that “space age” started with the first artificial satellite in the orbit around the planet Nowadays, satellites are used for various different purposes: telecommunications, broadcasting, observation, imaging and even espionage What they all have in common is the fact that they all must obey the rules of celestial mechanics To be able to visualise the motion, software presented in this chapter was created

For the satellite dynamics, the SGP4 and SDP4 models by NORAD were implemented Both SGP4 and SDP4 are based on fundamental laws stated by Newton and Kepler One of the biggest advantages of SGP4/SDP4 models is that they’ve been recognized and verified by NORAD thus providing a precise and manageable mathematical framework for the orbital calculations But bear in mind that those are not perfect models They work with mean values NORAD has removed periodic variations in a particular way, and the models, in their present form do not contain numerical integration methods

Future investigations and updates will improve propagation making it more precise This process will certainly increase the complexity, so the balance between complexity and preciseness must be kept

7 References

Binderink, A L.; Radomski, M.S.; Samii M V (1989) Atmospheric drag model calibrations

for spacecraft lifetime prediction; In NASA, Goddard Space Flight Center, Flight Mechanics/ Estimation Theory Symposium; 445-458

Bunnell, P (1981) Tracking Satellites in Elliptical Orbits; Ham Radio Magazine; 46-50

Hoots, F R.; Roehrich R L (1980) Models for propagation of NORAD element sets; Project

Spacecraft Report No 3; Aerospace Defence Command, United States Air Force, 3-6

King-Hele, D.G (1983), Observing Earth Satellites, Macmillan

Montenbruck, O.; Gill, E (2000) Real-Time estimation of SGP4 Orbital Elements from GPS

Navigation Data; International Symposium SpaceFlight Dynamics, MS00/28; 2-3

US National Archives and Records Administration.(2006) US Public Law, Section 109–364;

Oct 17, 2006, Stat 2355

Vallado A D.; Crawford P (2006) SGP4 Orbit Determination; American Institute of

Aeronautics and Astronautics publication; 19-21

Interference in Cellular Satellite Systems 299

Interference in Cellular Satellite Systems

Ozlem Kilic and Amir I Zaghloul

X

Interference in Cellular Satellite Systems

Ozlem Kilic(1) and Amir I Zaghloul(2,3)

(1)The Catholic University of America, Washington, DC, U.S.A

(2)Virginia Polytechnic Institute and State University, VA, U.S.A

(3)US Army Research Laboratory, MD, U.S.A

1 Introduction

In cellular satellite communications systems, a given coverage area is typically filled with a

network of contiguous spot beams, which carry concentrated radiation along preferred

directions The coverage regions for such applications are typically large areas, such as

continents and many beams need to be generated

Due to bandwidth limitations in cellular communications, the same bandwidth is allocated

to beams which are isolated spatially This is known as frequency reuse, and the beams

operating at the same frequency are referred as co-channel beams While this approach

allows a large coverage area with limited bandwidth, the co-channel beams have the

potential to interfere with each other This is known as co-channel interference and its

nature and how it could be reduced is the focus of this chapter

The interference in multiple beam satellite communications systems will be investigated

under two different approaches First approach, which is the conventional way of defining

beam coverage on earth, is discussed in Section 2 This will be referred to as spot beam

coverage as explained in further detail later The interference will be investigated for two

cases; first is the uplink where interference at the satellite antenna is the main concern, and

the second is the downlink where interference at the user terminal is calculated Section 3

discusses a new method of defining beam coverage on earth, referred to as sub-beam

coverage The motivation is to keep the coverage on earth identical but reduce the satellite

antenna size as much as 50% (Kilic & Zaghloul, 2009) The advantages and overall

performance of the sub-beam approach in terms of interference is the subject of Section 3.1

2 Interference in Cellular Satellite Systems

In multibeam satellite systems, the coverage area is divided into a number of beams often

referred to as spot beams, which are much smaller in size and cover the area contiguously

1 Copyright 2009 American Geophysical Union, This chapter has material substantially reproduced, with permission,

from Radio Science, Volume 44, No 1, January 2009, „Antenna Aperture Size Reduction Using Subbeam Concept in

Multiple Spot Beam Cellular Satellite Systems,, O Kilic and A.I Zaghloul.

15

Since satellite systems are bandwidth limited, the sub-division of beams into smaller

portions allow for frequency reuse to increase capacity The available bandwidth is shared

among these beams as depicted in Figure 1 for reuse factor of n

Fig 1 Frequency reuse in multi-beam satellite communications

The size of the antenna that generates these beams on earth is related directly to the peak

gain at the center of the spot beams and the smallest spot beam size The spot beams are

typically defined by the contours at 3 or 4 dB down from the peak power at the center of the

beam

2.1 Coverage on Earth

Achieving a contiguous coverage is important so that there are no regions without service in

the coverage area Since the beams are defined by the projection of antenna patterns on earth

at a certain contour value, they tend to be close to circular shape These circles on earth need

to be structured so that they overlap with each other to avoid any gaps in the coverage area

In order to have a systematic approach, these can be represented by various geometric

lattices that tessellate A few such possibilities are shown in Figure 2 It is often the hexagon

that is used in the system design as it closely represents a circle, i.e for the same distance

from the center to the edge, the hexagon area is closest to that of the circle that circumscribes

it Therefore the hexagon represents the beam which circumscribes it as shown in Figure 3

This assures that there are no gaps between beams Then the system is designed based on

this artificial hexagonal geometry on earth as depicted in Figure 4

Fig 2 Contiguous coverage on earth using tessalating shapes – hexagonal, square and

triangular lattices

Fig 3 Hexagonal representation of a circular beam

Fig 4 Hexagonal Coverage on Earth

2.2 Frequency Reuse

Since the satellite systems typically serve large ares such as countries or continents, a large number of beams need to share the available bandwidth Therefore, the available bandwidth within a beam becomes a very limited resource, as is implied in Figure 1 earlier To circumvent this, a frequency reuse scheme is often utilized This is based on reusing the same frequencies in spatially isolated beams Therefore, the available bandwidth is divided into a smaller number of beams than the total number of beams in the coverage area The set

of contiguous beams that share the total available bandwidth is known as a cluster The clusters are then repeated in the coverage area relying on the fact that the beams operating

at the same bandwidth will be separated from each other sufficiently so that they do not interfere with each other

Interference in Cellular Satellite Systems 301

Since satellite systems are bandwidth limited, the sub-division of beams into smaller

portions allow for frequency reuse to increase capacity The available bandwidth is shared

among these beams as depicted in Figure 1 for reuse factor of n

Fig 1 Frequency reuse in multi-beam satellite communications

The size of the antenna that generates these beams on earth is related directly to the peak

gain at the center of the spot beams and the smallest spot beam size The spot beams are

typically defined by the contours at 3 or 4 dB down from the peak power at the center of the

beam

2.1 Coverage on Earth

Achieving a contiguous coverage is important so that there are no regions without service in

the coverage area Since the beams are defined by the projection of antenna patterns on earth

at a certain contour value, they tend to be close to circular shape These circles on earth need

to be structured so that they overlap with each other to avoid any gaps in the coverage area

In order to have a systematic approach, these can be represented by various geometric

lattices that tessellate A few such possibilities are shown in Figure 2 It is often the hexagon

that is used in the system design as it closely represents a circle, i.e for the same distance

from the center to the edge, the hexagon area is closest to that of the circle that circumscribes

it Therefore the hexagon represents the beam which circumscribes it as shown in Figure 3

This assures that there are no gaps between beams Then the system is designed based on

this artificial hexagonal geometry on earth as depicted in Figure 4

Fig 2 Contiguous coverage on earth using tessalating shapes – hexagonal, square and

triangular lattices

Fig 3 Hexagonal representation of a circular beam

Fig 4 Hexagonal Coverage on Earth

2.2 Frequency Reuse

Since the satellite systems typically serve large ares such as countries or continents, a large number of beams need to share the available bandwidth Therefore, the available bandwidth within a beam becomes a very limited resource, as is implied in Figure 1 earlier To circumvent this, a frequency reuse scheme is often utilized This is based on reusing the same frequencies in spatially isolated beams Therefore, the available bandwidth is divided into a smaller number of beams than the total number of beams in the coverage area The set

of contiguous beams that share the total available bandwidth is known as a cluster The clusters are then repeated in the coverage area relying on the fact that the beams operating

at the same bandwidth will be separated from each other sufficiently so that they do not interfere with each other

There are only a discrete set of possible cluster sizes, N, to accommodate a contiguous

coverage for a hexagonal geometry [Mehrotra, 1994] The possible number of beams in a

cluster which would form a tessalating shape is given by:

j i j i

where N, is the number of beams in the cluster; i.e the number of beams that share the total

bandwidth, and i, j are non-negative integer numbers This results in possible cluster sizes of

3, 4, 7, 9, etc An example of how these clusters are formed is shown in Figure 5

Fig 5 Different cluster options for hexagonal lattice

2.3 Co-channel Beams and Tiers

The clusters as depicted in Figure 5 for reuse factors of 3, 4 and 7, are repeated to fill the

required coverage area on earth An example of how a cluster size of three (i.e i=1, j=1)

would be used to fill a given area is shown in Figure 6 A cluster size of three implies that

the total available bandwidth is shared between three beams The numbers 1, 2, 3 are used

to identify the beams using the corresponding bandwidth segments Therefore, beams with

the same number imply that they use the same frequency

Beams operating at the same frequency are known as channel beams In Figure 6, the

co-channel beams are shown in the same color and labeled with the same bandwidth segment

number It can be observed that the location of co-channel beams follow a pattern They can

be grouped by their distance with respect to a reference beam The set of co-channel beams

which have same distance from a reference beam are said to fall on a tier Therefore tiers

define a set of beams equidistant from a reference point In a hexagonal geometry, each tier

consists of six or twelve beams Figure 7 shows the first four tiers for the frequency reuse of

three

Fig 6 Cluster coverage, N=3

Fig 7 Tiers in frequency reuse The beams that lie on a tier are equidistant from the beam at the center of the tier, and for an azimuthally symmetric power distribution, beams on a tier would contribute the same amount

Interference in Cellular Satellite Systems 303

There are only a discrete set of possible cluster sizes, N, to accommodate a contiguous

coverage for a hexagonal geometry [Mehrotra, 1994] The possible number of beams in a

cluster which would form a tessalating shape is given by:

j i

j i

where N, is the number of beams in the cluster; i.e the number of beams that share the total

bandwidth, and i, j are non-negative integer numbers This results in possible cluster sizes of

3, 4, 7, 9, etc An example of how these clusters are formed is shown in Figure 5

Fig 5 Different cluster options for hexagonal lattice

2.3 Co-channel Beams and Tiers

The clusters as depicted in Figure 5 for reuse factors of 3, 4 and 7, are repeated to fill the

required coverage area on earth An example of how a cluster size of three (i.e i=1, j=1)

would be used to fill a given area is shown in Figure 6 A cluster size of three implies that

the total available bandwidth is shared between three beams The numbers 1, 2, 3 are used

to identify the beams using the corresponding bandwidth segments Therefore, beams with

the same number imply that they use the same frequency

Beams operating at the same frequency are known as channel beams In Figure 6, the

co-channel beams are shown in the same color and labeled with the same bandwidth segment

number It can be observed that the location of co-channel beams follow a pattern They can

be grouped by their distance with respect to a reference beam The set of co-channel beams

which have same distance from a reference beam are said to fall on a tier Therefore tiers

define a set of beams equidistant from a reference point In a hexagonal geometry, each tier

consists of six or twelve beams Figure 7 shows the first four tiers for the frequency reuse of

three

Fig 6 Cluster coverage, N=3

Fig 7 Tiers in frequency reuse The beams that lie on a tier are equidistant from the beam at the center of the tier, and for an azimuthally symmetric power distribution, beams on a tier would contribute the same amount

of interference As the tier’s number increases, the diameter hence the distance from the center

beam increases, reducing the contribution from the beams in that tier compared to the beams

on closer tiers assuming the radiation decreases away steadily from the antenna peak

As can be observed from Figures 6 and 7, higher frequency reuse numbers result in tiers

with larger diameters, thereby increasing the distance between co-channel beams and

reducing the total number of beams operating at the same frequency However, this is done

at the expense of reduced bandwidth within a beam, a trade off which needs to be decided

by system engineers based on the requirements of a particular system

2.4 Antenna Pattern and Spot Beam Generation

Due to their ability to generate multiple beams simultaneously, phased arrays are a natural

choice for multi-beam satellite antennas Each beam in the coverage is generated by

electronically scanning the beam A key parameter in satellite antenna design is the

directivity of antenna, which defines how well the antenna focuses the power in the desired

direction

The radiation pattern of an array antenna depends on the array factor For a MxN planar

array, the array factor is given by (Balanis, 2005)

( 1)( sin sin ) ( 1)( sin cos )

m n

 

where Imn are the voltages that feed the mnth element in the array The array factor is related

to the antenna directivity which measures how well the input power is focused along a

given direction and can be computed as follows (Stutzman, 1998)

0

max 0 0 0

0 0

d d sin )]

, ( AF )][

, ( AF [

| ) , ( AF )][

, ( AF [ 4

In the case of a large array with uniform excitation, this equation can be approximated by

20

Fig 8 A uniformly spaced 40x40 element planar array The plot of the directivity in two different planes for this particular array is shown in Figures 9-10 The 40x40 element antenna is sized to have an edge gain of 36 dB defined at its -4dB contour relative to the peak Thus, the peak gain of the antenna is 40 dB, with a beamwidth of 1.44 degrees at the - 4 dB down contour from the peak gain The spot beam in the coverage area is defined by this contour The contour plot for the same antenna is shown

in Figure 11 The beam is generated at the center as denoted by the white circle

As seen in the contour plot, the energy is radiated to the whole coverage area even though most of the power is focused in the spot beam This “leak” into other regions is not a problem for beams operating at different frequencies as filtering will eliminate this energy But for co-channel beams; i.e beams operating at the same frequency, this leaking energy creates the interference which is the focus of this chapter

Interference in Cellular Satellite Systems 305

of interference As the tier’s number increases, the diameter hence the distance from the center

beam increases, reducing the contribution from the beams in that tier compared to the beams

on closer tiers assuming the radiation decreases away steadily from the antenna peak

As can be observed from Figures 6 and 7, higher frequency reuse numbers result in tiers

with larger diameters, thereby increasing the distance between co-channel beams and

reducing the total number of beams operating at the same frequency However, this is done

at the expense of reduced bandwidth within a beam, a trade off which needs to be decided

by system engineers based on the requirements of a particular system

2.4 Antenna Pattern and Spot Beam Generation

Due to their ability to generate multiple beams simultaneously, phased arrays are a natural

choice for multi-beam satellite antennas Each beam in the coverage is generated by

electronically scanning the beam A key parameter in satellite antenna design is the

directivity of antenna, which defines how well the antenna focuses the power in the desired

direction

The radiation pattern of an array antenna depends on the array factor For a MxN planar

array, the array factor is given by (Balanis, 2005)

( 1)( sin sin ) ( 1)( sin cos )

m n

 

where Imn are the voltages that feed the mnth element in the array The array factor is related

to the antenna directivity which measures how well the input power is focused along a

given direction and can be computed as follows (Stutzman, 1998)

0 0

max 0

0 0

0 0

d d

sin )]

, (

AF )][

, (

AF [

| )

, (

AF )][

, (

AF [

4

In the case of a large array with uniform excitation, this equation can be approximated by

20

Fig 8 A uniformly spaced 40x40 element planar array The plot of the directivity in two different planes for this particular array is shown in Figures 9-10 The 40x40 element antenna is sized to have an edge gain of 36 dB defined at its -4dB contour relative to the peak Thus, the peak gain of the antenna is 40 dB, with a beamwidth of 1.44 degrees at the - 4 dB down contour from the peak gain The spot beam in the coverage area is defined by this contour The contour plot for the same antenna is shown

in Figure 11 The beam is generated at the center as denoted by the white circle

As seen in the contour plot, the energy is radiated to the whole coverage area even though most of the power is focused in the spot beam This “leak” into other regions is not a problem for beams operating at different frequencies as filtering will eliminate this energy But for co-channel beams; i.e beams operating at the same frequency, this leaking energy creates the interference which is the focus of this chapter