ESTIMATION OF GAIN, BEAMWIDTH AND SIDELOBE LEVELS OF
NONUNIFORM LINEAR AND CONCENTRIC RING ANTENNA ARRAYS TO
OBTAIN BROADBAND PERFORMANCE
Prasanna Chinnappagari
B. Tech, Sri Krishnadevaraya University, India, 2007
Vineela Dasari
B. Tech, Jawaharlal Nehru Technological University, India, 2007
PROJECT
Submitted in partial satisfaction of
the requirements for the degrees of
MASTER OF SCIENCE
in
ELECTRICAL AND ELECTRONIC ENGINEERING
at
CALIFORNIA STATE UNIVERSITY, SACRAMENTO
FALL
2011
ESTIMATION OF GAIN, BEAMWIDTH AND SIDELOBE LEVELS OF
NONUNIFORM LINEAR AND CONCENTRIC RING ANTENNA ARRAYS TO
OBTAIN BROADBAND PERFORMANCE
A Project
by
Prasanna Chinnappagari
Vineela Dasari
Approved by:
__________________________________, Committee Chair
Preetham B. Kumar, Ph.D.
__________________________________, Second Reader
Fethi Belkhouche, Ph.D.
____________________________
Date
ii
Students: Prasanna Chinnappagari
Vineela Dasari
I certify that these students have met the requirements for format contained in the
University format manual, and that this project is suitable for shelving in the Library and
credit is to be awarded for the project.
______________________, Graduate Coordinator
Preetham B. Kumar, Ph.D.
________________
Date
Department of Electrical and Electronic Engineering
iii
Abstract
of
ESTIMATION OF GAIN, BEAMWIDTH AND SIDELOBE LEVELS OF
NONUNIFORM LINEAR AND CONCENTRIC RING ANTENNA ARRAYS TO
OBTAIN BROADBAND PERFORMANCE
by
Prasanna Chinnappagari
Vineela Dasari
This project will focus on the estimation of gain, beamwidth and sidelobe levels of linear
and concentric ring antenna arrays over the wide range of frequency band of 1-20GHz.
The potential applications of the current wideband array design will be in radio
astronomy and Ultra Wideband (UWB) wireless applications. In the project, bandwidth
performance is achieved by altering the inter-element spacing of a default equally spaced
array. This approach is based on a proven theory that a uniformly spaced antenna array is
resonant at a single center frequency, while the introduction of unequal spacing expands
the working bandwidth of the system. The system design in this work is a two-stage
process: the first part is to obtain the unequally spaced design from a default uniform
array, and the second part is to estimate the Gain, beamwidth and sidelobe levels for nonuniform array over the frequency range of 1-20GHz and compared the performance
improvement over uniform antenna arrays. In this work, both linear and concentric ring
iv
arrays have been designed, and significant increase in performance bandwidth has been
achieved, as compared to the default uniform arrays.
_______________________, Committee Chair
Preetham B. Kumar, Ph.D.
_______________________
Date
v
ACKNOWLEDGEMENTS
With sincere respect and gratitude, I would like to thank everyone who has helped
me in successful completion of this project. First and foremost I would like to thank my
project advisor, committee chair and graduate coordinator Dr. Preetham Kumar for his
guidance, monitoring and providing me the necessary resources for the project. His
experience in the field of this project has helped me a lot to understand and finish up the
project.
Here I take up the opportunity to thank Dr. Fethi Belkhouche for reviewing my
project report and giving valuable suggestions as second reader. I also extend my
gratitude to the faculty and staff of Electrical and Electronics department who were very
considerate and encouraged me to finish up the requirements for my graduation.
Lastly but not the least I express gratitude to my family and friends who directly
or indirectly helped in completion of this project with their healthy criticism and
encouragement.
vi
TABLE OF CONTENTS
Page
Acknowledgements…………………………………...…………………………………..vi
List of Tables……………………………………………………………………………..ix
List of Figures……………………………………………………………………………..x
Chapter
1. INTRODUCTION……………………………………………………………….…….1
2. EVOLUTION OF WIDEBAND ANTENNAS FROM UNEQUALLY SPACED
ARRAYS………………………………………………………...........………….……4
2.1 Wideband Systems………………………………………..…………….…….4
2.2 Wideband Antenna Design…………………………………..………….…….4
2.3 Wideband Antenna Array Design………………………………………...…...5
2.3.1 Linear Array Synthesis………………………………...……………6
2.3.2 Nonlinear Array Synthesis……………………………...…….……..6
2.4 Synthesis of Wideband Antenna Arrays with Linear and Cylindrical
Geometry………………………………….…………………..........………….8
2.4.1 Linear Arrays……………………………………………….……….8
2.4.2 Concentric Ring Arrays…………………………...……………….11
3. ESTIMATION OF GAIN, BEAMWIDTH AND SIDELOBE LEVELS OF
LINEAR ARRAY ANTENNA……………………………………………………….14
3.1 Generation of Unequal Spacing Starting from a Default Equally Spaced
Array………………………………………………………………............14
3.2 Generation of Unequally Spaced Array Design and Estimation of the Gain,
Beamwidth and Pssl....................................................................................16
vii
3.2.1 Generation of the Delta Differential between Maximum and
Minimum Parameter Values………………………...................…..17
3.2.2 Graphs for Comparing Equally and Unequally Spaced Arrays…...25
4. ESTIMATION OF GAIN, BEAMWIDTH AND SIDELOBE LEVELS
OF CONCENTRIC RING ARRAY ANTENNA…………………………………….42
4.1 Generation of Unequal Spacings Starting from a Default Equally Spaced
Array……………………………………………………..................….…42
4.2 Generation of Unequally Spaced Array Design and Estimation of the Gain,
Beamwdith and Pssl…………………………………………......………..44
4.2.1 Generation of the Delta Differential between Maximum and
Minimum Parameter Values………….......................................…..44
4.2.2 Graphs for Comparing Equally and Unequally Spaced Arrays……49
5. CONCLUSION………………………………………………………………………..62
Appendix I MATLAB Code for Linear Antenna Arrays……..................................…….63
Appendix II MATLAB Code for Concentric Ring Antenna Arrays.................…............72
References………………………………………………………………………………79
viii
LIST OF TABLES
Page
1. Table 1 Alpha Values for Nd= 1 to 11 of Linear Array………………................15
2.
Table 2 Size of Dipoles for Different Frequencies………………….…………..16
3. Table 3 Delta Values for Dipole Length 0.15m of Linear Array……….……..…19
4. Table 4 Delta Values for Dipole Length 0.12m of Linear Array………….…..…20
5. Table 5 Delta Values for Dipole Length 0.1m of Linear Array………….…..…..21
6. Table 6 Delta Values for Dipole Length 0.08m of Linear Array………..…….…22
7. Table 7 Delta Values for Dipole Length 0.075m of Linear Array…………….....23
8. Table 8 Delta Values for Dipole Length 0.06m of Linear Array……………...…24
9. Table 9 alm0 Values for Nd= 2 to 10 of Ring Array………………...……...…....43
10. Table 10 Delta Value for Dipole Length 0.15m of Concentric Ring Array……..44
11. Table 11 Delta Values for Dipole Length 0.12m of Concentric Ring Array…….45
12. Table 12 Delta Values for Dipole Length 0.1m of Concentric Ring Array….......46
13. Table 13 Delta Values for Dipole Length 0.08m of Concentric Ring Array...….46
14. Table 14 Delta Values for Dipole Length 0.075m of Concentric Ring Array…...47
15. Table 15 Delta Values for Dipole Length 0.06m of Concentric Ring Array…….48
ix
LIST OF FIGURES
Page
1. Figure 1 Geometry of 2n+1 Element Non-Periodic Symmetric Linear Array…....8
2. Figure 2 Geometry of Circular Ring Array……………………............................11
3. Figure 3 Plot Shown to Calculate Delta Values……………………………….…18
4. Figure 4 Simulation Plot for gain of Linear Array of Length 0.15m….….……...25
5. Figure 5 Simulation Plot for gain of Linear Array of Length 0.12m……..…..…26
6. Figure 6 Simulation Plot for gain of Linear Array of Length 0.1m ………...….27
7. Figure 7 Simulation Plot for gain of Linear Array of Length 0.08m .................28
8. Figure 8 Simulation Plot for gain of Linear Array of Length 0.075m……..…...28
9. Figure 9 Simulation Plot for gain of Linear Array of Length 0.06m ………......29
10. Figure 10 Simulation Plot for Beamwidth of Linear Array of
Length 0.15m........................................................................................................30
11. Figure 11 Simulation Plot for Beamwidth of Linear Array of
Length 0.12m.......................................................................................................31
12. Figure 12 Simulation Plot for Beamwidth of Linear Array of
Length 0.1m ........................................................................................................32
13. Figure 13 Simulation Plot for Beamwidth of Linear Array of
Length 0.08m.......................................................................................................33
14. Figure 14 Simulation Plot for Beamwidth of Linear Array of
Length 0.075m.....................................................................................................34
15. Figure 15 Simulation Plot for Beamwidth of Linear Array of
Length 0.06m.......................................................................................................35
x
16. Figure 16 Simulation Plot for Peak Sidelobe Levels of
Linear Array of Length 0.15m …………………………………………………36
17. Figure 17 Simulation Plot for Peak Sidelobe Levels of
Linear Array of Length 0.12m………………………………………………….37
18. Figure 18 Simulation Plot for Peak Sidelobe Levels of
Linear Array of Length 0.1m...…………………………………………………38
19. Figure 19 Simulation Plot for Peak Sidelobe Levels of
Linear Array of Length 0.08m.............................................................................39
20. Figure 20 Simulation Plot for Peak Sidelobe Levels of
Linear Array of Length 0.075m…………………………………………….......40
21. Figure 21 Simulation Plot for Peak Sidelobe Levels of
Linear Array of Length 0.06m ………………………………………...……….41
22. Figure 22 Simulation Plot for Gain of
Concentric Ring Array of Length 0.15m…………..............................................49
23. Figure 23 Simulation Plot for Gain of
Concentric Ring Array of Length 0.12m…………..............................................50
24. Figure 24 Simulation Plot for Gain of
Concentric Ring Array of Length 0.1m…………................................................50
25. Figure 25 Simulation Plot for Gain of
Concentric Ring Array of Length 0.08m…………..............................................51
26. Figure 26 Simulation Plot for Gain of
Concentric Ring Array of Length 0.075m………................................................51
27. Figure 27 Simulation Plot for Gain of
Concentric Ring Array of Length 0.06m…………..............................................52
28. Figure 28 Simulation Plot for Beamwidth of
Concentric Ring Array of Length 0.15m..............................................................53
29. Figure 29 Simulation Plot for Beamwidth of
Concentric Ring Array of Length 0.12m.............................................................53
xi
30. Figure 30 Simulation Plot for Beamwidth of
Concentric Ring Array of Length 0.1m............................................................54
31. Figure 31 Simulation Plot for Beamwidth of
Concentric Ring Array of Length 0.08m..........................................................55
32. Figure 32 Simulation Plot for Beamwidth of
Concentric Ring Array of Length 0.075m........................................................55
33. Figure 33 Simulation Plot for Beamwidth of
Concentric Ring Array of Length 0.06m..........................................................56
34. Figure 34 Simulation Plot for Peak Sidelobe Levels of
Concentric Ring Array of Length 0.15m..........................................................57
35. Figure 35 Simulation Plot for Peak Sidelobe Levels of
Concentric Ring Array of Length 0.12m..........................................................58
36. Figure 36 Simulation Plot for Peak Sidelobe Levels of
Concentric Ring Array of Length 0.1m...........................................................58
37. Figure 37 Simulation Plot for Peak Sidelobe Levels of
Concentric Ring Array of Length 0.08m.........................................................59
38. Figure 38 Simulation Plot for Peak Sidelobe Levels of
Concentric Ring Array of Length 0.075m.......................................................60
39. Figure 39 Simulation Plot for Peak Sidelobe Levels of
Concentric Ring Array of Length 0.6m...........................................................61
xii
1
Chapter 1
INTRODUCTION
An antenna is an electronic device which converts the electrical signals into radio
waves and radio waves into electrical signals; however the performance of single element
antenna is limited. They cannot meet the gain and radiation pattern requirements, and in
many applications, it is required to design antennas which are suitable for long distance
communication, highly directive and also wideband in nature.
A possible solution to obtaining a high-gain, high bandwidth antenna is by using
the array concept, which is essentially an assembly of several radiating antennas to form
a single antenna system in an electrical and geometrical configuration [5]. The
performance of the array increases with the number of elements in the array, but the cost
and complexity increases with higher number of elements. Now-a-days antenna arrays
are more prominent in wireless terminals, smart antenna and they are widely used in
applications like radar, sonar and communications.
Antenna arrays have the advantages of providing high overall gain by using large
number of array elements, capability of steerable beam like in smart antennas, provide
diversity gain in multipath reception, cancel the interference from particular direction and
also enable array signal processing. Additionally, introducing unequal spacing between
the antenna elements can further increase the performance, by widening the bandwidth of
operation. Uniform antenna arrays will enable the operation at one particular frequency,
but by employing unequal spacing between the elements of array, we can achieve fairly
2
constant gain and low sidelobe level over a wide frequency range. In this project, we
focused our efforts on the design of wideband non-uniformly spaced antenna arrays and
compared the gain, beamwidth and sidelobe levels with those of uniform array antenna
over the frequency range of 1-20GHz.
There are different possible geometries for antenna arrays such as linear, circular,
concentric ring, spherical and hemispherical arrays. In this project we concentrated the
goals on designing and estimating the gain, beamwidth and sidelobe levels of linear and
concentric ring antenna arrays of cylindrical dipoles. The total number of elements used
in the linear antenna array was 18 and in ring array was 10. The gain, beamwidth and
sidelobe responses were found to change with the spacing variation between the antenna
elements, and also with the choice of the dipole element length. By using the several trial
sets of element spacing and dipole length values, the gain, sidelobe and beamwidth
parameters mentioned above were estimated over the frequency range of 1-20 GHz and
compared with that of equally spaced antenna arrays. The simulation results for both
linear and concentric ring arrays show significant improvement in wideband gain
performance, as compared with uniform arrays.
The project report is organized as follows: Chapter 1 provides the introduction to
the report; Chapter 2 describes the background of wideband antennas, models and
techniques for optimization of linear and concentric ring antenna arrays. Chapter 3 gives
the simulation results and analysis of equally and unequally spaced linear antenna,
Chapter 4 discusses the simulation results of uniform and non uniform concentric ring
antenna, Chapter 5 shows the conclusion and future scope in this area of study. Finally,
3
Appendix I list the MATLAB programs for generating spacing for linear array and for
plotting parameters. APPENDIX II lists the MATLAB programs for generating spacing
for concentric ring array antennas and to plot the gain, beamwidth and sidelobe levels.
4
Chapter 2
EVOLUTION OF WIDEBAND ANTENNAS FROM UNEQUALLY SPACED
ARRAYS
2.1. WIDEBAND SYSTEMS
Wideband antennas operate over a wide range of frequencies in which
approximately or exactly the same operating characteristics over a very wide passband. A
system is typically described as wideband if the message bandwidth significantly exceeds
the channel's coherence bandwidth. Some communication links, like Ultra-Wideband
(UWB) systems, have such a high data rate that they are forced to use a wide bandwidth;
other links may have relatively low data rates, but deliberately use a wider bandwidth
than necessary for that data rate in order to gain other advantages [12] .
Wideband communication systems also have additional advantages due to their
high bandwidth, such as reduced fading from multipath and low power requirements.
UWB systems, for example, transmit pulses of very short duration, compared to
traditional communication schemes, which send sinusoidal waves [13].
2.2 WIDEBAND ANTENNA DESIGN
The general goal of a wideband antenna system is to maintain radiation characteristics
over the desired wideband frequency range. Primarily, the radiation properties to
maintain are antenna gain, sidelobe level and beamwidth of the primary antenna beam.
Traditionally, wideband antennas include log-periodic and helical antennas [14];
5
however, a newer concept of unequally spaced array design to increase bandwidth is
currently being studied and researched [4].
2.3 WIDEBAND ANTENNA ARRAY DESIGN
The general wideband antenna array synthesis problem can be described by the matrix
equation:
 A(k , r1 , u1 ) A(k , r2 , u1 )... A(k , rN , u1 )   I1 
 E (k , u1 ) 
 A(k , r , u ) A(k , r , u )... A(k , r , u )   
 E (k , u ) 
1
2
2
2
N
2
2 

 I2 

...
 ... 
...


 


 ... 
...
 = ...

 



 



 


 A(k , r , u ) A(k , r , u )... A(k , r , u )   I 
 E (k , u ) 
M 
1
M
2
M
N
M  N


(1)
Where r = [r1 r2 r3 …rN] is the array element position vector, k = 2π ∕λ the propagation
constant, and λ is the wavelength. [E (k, u1) E (k, u2) E (k, u3) …
E(k, uM)]T is the prescribed [M × 1] far-field array factor at M field points (uj) where
j= 1, 2, …,M and [ I1 , I2 , I3 ,…,IN ] T is the [ N × 1] array element current vector, and
[ A (k , ri , uj ) ] is the M × N rectangular matrix, which is a function of the frequency,
element position (ri) and field point uj .In this project a unified analytical synthesis
approach is presented for the synthesis of frequency-dependent, multi-dimensional array
geometrics. The technique utilizes both the linear [3] and nonlinear processes [3] to yield
closed form solutions to the optimal array element positions. The procedural steps for
both linear and nonlinear synthesis are discussed below:
6
2.3.1 LINEAR ARRAY SYNTHESIS
Step 1: The linear array synthesis procedure requires specification of a desired pattern
and a prescribed finite array geometry, which has all element positions, fixed.
Step 2: Given the field pattern and element spacing, the vector currents [I1, I2, I3, …,IN]
are obtained from equation 1 by employing linear synthesis. These synthesized currents
may be real or complex.
2.3.2 NONLINEAR ARRAY SYNTHESIS
Step 1: In the nonlinear synthesis procedure [3], the array geometry, the element currents
(from linear synthesis) and desired array pattern improvement are prescribed.
Step 2: The nonlinear synthesis process [3] is then performed to yield the optimum array
element positions. These positions are optimum in the sense that the array response
approaches specified performance. Element center positions (x, y, z coordinates) must be
real and positive while being limited by minimum and maximum adjacent element
spacing constraints (mutual coupling and grating lobes respectively).
The generalized array synthesis technique applicable to the wideband case can be derived
from equation (1) [4] written in matrix form:
[E] = [A] [I]
(2)
Performing a linear transformation T on the array factor [E], equation (2) transforms as:
T [E] = T [A] [I]
(3)
7
The linear transformation depends on the array geometry, which defines the governing
function in the array factor expression. The significant observation is that the specific
linear transformation described in equation (3), exists for arrays having rectangular,
cylindrical and spherical geometry and that each case provides for wideband operation.
Equation (3) can be simplified to obtain:
[F] = [B] [I]
(4)
Where [F] is the prescribed M  1 transformed array factor vector at N points:
[F(  1) ,F(  2 ) ,F(  3) , …, F(  N ) ] T, [I] is the N  1 current vector [I1,I2, I3,…,IN]T and
[B] is the M  N triangular matrix, which is a function of both the transformation vector,
 = [  1  2  3 … N ], and the array element position vector r = [ r1 r2 r3 … rN] .
The linear transformation T is derived from limiting functions of sinusoidal, Bessel and
Legendre functions and yields the following triangular form of matrix.
 F (1 )   B(r1 , 1 )

 F ( )   B(r ,  ) B(r , )

2 
2
2

 1 1

...
 ...


 = 

...
 ...


 


 

 F ( N )   B(r1 ,  M ) B(r2 , M )...B(rN ,  N ) 
 I1 
I 
 2
... 
 
... 
 
 
 I N 
(5)
Hence it is possible to solve the square equation (5) recursively to yield solutions for the
position vector ‘r’ of the different element positions. The following sections in the project
8
will outline the application of this method to the nonlinear spacing synthesis of wideband
arrays having linear, planar, and cylindrical or spherical geometries.
2.4 SYNTHESIS OF WIDEBAND ANTENNA ARRAYS WITH LINEAR AND
CYLINDRICAL GEOMETRY
In this project, design of wideband antenna arrays has focused on the following
geometries:

Linear arrays

Concentric ring arrays
2.4.1 LINEAR ARRAYS
Consider a symmetric linear array of 2N + 1 element [1] as shown in the figure 1 below:
Figure 1: Geometry of 2n+1 Element Non-Periodic Symmetric Linear Array [3]
Array factor quantifies the effect of combining radiating elements in an array without the
element specific radiation pattern taken into account [11]. The overall radiation pattern of
an array is determined by this array factor combined with the radiation pattern of the
antenna element [11]. The array factor of this linear array is given as follows:
9
N
E (u) =

n 0
where u = cos(θ) , 0  
I cos( kdnu );
n n
(6)
  radians, In and dn are the current excitation and element
position, respectively of the nth element of the array and the factor  n = 1, n = 0 and  n =
2, n  0. The currents In are obtained using standard linear synthesis methods [1]. The
nonlinear synthesis [1] is performed next. The nonlinear synthesis algorithm consists of a
four-step process, originating with the desired array field pattern, and culminating in the
generation of array element positions.
The desired pattern Ed (u) is first approximated with the
reconstructed pattern of a 2N + 1 element nonuniform symmetric array of point sources,
shown in Figure [1], and this response is in turn uniformly sampled at M points (M>>1)
in the interval 0  u  1 to yield :
N
E (um) =

n 0
Where m = 0 or
 /
I cos( mn);
n n
M – 1, where um = m u , where
(7)
u = 1/ (M - 1),  n = kdn u , k = 2
and element positions dn are shown in figure 1. The synthesis algorithm starts with
the Legendre transformation of the desired array pattern Ed (u) as follows:
M 1
F (  p ) =   m Ed (um)Pm  1 / 2(cos p ) ; p = 0, 1, 2, 3, …,N.
m 0
(8)
10
Where  m  1, m  0;  m  2 , m>0; and Pm-1/2 is the Legendre function of half integer
order. Finally, the synthesized position of the pth element is given by:




2 Ip 2


 cos(p )
 p = cos-1 
2
p 1
  F (p )  Inf (p , n)



 

n 0

(9)
= [2/ (cos n  cos p ) )] 1/2, 0   n  p and  p <  n <  .
In the equation f ( p,  n ) = 0, and element position dp =  p /( ku ) . The selection of the 
grid is the important constituent in the reconstruction of the array currents and positions
in the recursive form. The following relation defines the grid:  p =  p + c, where ‘c’ is
a constant and c > 0 is limited by the condition that  and  values intersperse one
another as shown below:
 Space: [  0
1
2
 3…  N]
[ 0
1
2
 3…  N]
 Space:
11
2.4.2 CONCENTRIC RING ARRAYS
The geometry of the concentric ring of an array of point sources [1] is shown in figure. 2.
Ring antennas have been mainly used in mobile communications, since they can radiate
power at low elevation angles.
Figure 2 : Geometry of Circular Ring Array [3]
The array factor of the concentric ring array is given as follows:
12
E ( , ) =
N
M
 I e
ij
jki sin( ) cos( j )
(10)
i 1 j 1
Where N is the number of concentric rings, Mi is the number of elements in the ith ring,
Iij is the excitation current of the element ‘j’ in the ring ‘i’ and i is the radial distance to
the ith ring from the center of the array. The current excitation, obtained from linear
synthesis method is uniform i.e. Iij = 1, for a broadside array pattern.
The elements in each ring are also assumed to be uniformly distributed around the
circular ring, i.e. j  2 / N , the array factor can be simplified, after considerable
manipulation, to yield
N
E (u) =
M
i
Jo (k i u)
(11)
i 1
Where u = sin

and ‘Jo’ is the Bessel function of zero order. Again, the nonlinear
synthesis algorithm [5] is a four step process, originating with the desired array field
pattern, and culminating with the generation of array element positions. Finally the
design equation of pth ring position of the array becomes:




Mp 2
 2

2
ap = bp 
2

p 1

 

 F (bp )   Mnf (bp, an) 

n 1

 
and p  ap /(k ) ,
f(bp, an)
2
2 1/2

 [1/ ( an  bp )] for a n  bp 



for b p  an 
 0


(12)
13

where in equation (14), F (bp )   Ed (u)sin(bpu)du; p=1,2,3,...,N
0
The values of b p , p = 1, 2, 3,…,N are selected as follows: bp  k (  p   / 2) .
14
Chapter 3
ESTIMATION OF GAIN, BEAMWIDTH AND SIDELOBE LEVELS OF
LINEAR ARRAY ANTENNA
This chapter presents the results of the two-step linear array design process that
was outlined in the previous chapter. The final data for the linear unequally spaced
antenna array includes the gain, beamwidth and sidelobe levels, as compared with the
uniform antenna array, in order to demonstrate the performance improvement. The
simulation is performed in a two-step process,
1) Generated the spacing between the antenna elements from the first MATLAB
program listed in the APPENDIX I.
2) Spacing values obtained from the first program are used in the second program in
APPENDIX I to calculate the difference of maximum and minimum values of
gain, beamwidth and sidelobe levels of equally and unequally spaced antennas.
The two steps listed above are elaborated in the sections below.
3.1 GENERATION OF UNEQUAL SPACINGS STARTING FROM A DEFAULT
EQUALLY SPACED ARRAY
The first MATLAB program estimates the optimized unequal spacing from the given
default array, in which the elements are equally spaced at half-wavelength. With the
change in the spacing between the elements, response of gain, beamwidth and peak
sidelobe level (PSSL) of the array changes, and the aim is to determine the optimum set
of spacing which will maximize the gain, for example.
15
The optimization process in the first step of the wideband array design starts with
the equally spaced array, with spacing between each adjacent element at half a
wavelength. Then the program specifies a design parameter, alpha, which specifies the
maximum shift allowed in the design of the new element spacing.
In the first program, the tolerance given for spacing (alpha) is varied in 10 steps
from one-tenth of a wavelength to half a wavelength and the array spacing are generated
in each case. The index of the shift is nd, and for each value of nd, the first program in the
Appendix I gives corresponding values of the parameter alpha, based on the formula
below:
alpha=(nd-1)*dxaa*pi/2/pi
where dxaa= 1/10 and pi=3.14
The following table gives the alpha values for each values of nd, for an 18-element array.
nd
1
2
3
4
5
6
7
8
9
10
11
alpha
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Table 1: Alpha Values for Nd= 1 To 11 of Linear Array
The antennas used in the array design are identical dipoles of half a wavelength in size.
Since the aim of the design is to create a wideband design from 1- 20 GHz, an interesting
16
question arises as to which wavelength to be utilized in the array spacing, and also
element size values. In order to test the variability of the design, different frequencies
were used in the 1- 20 GHz range:
1GHz, 1.25GHz, 1.5GHz, 1.75GHz, 2GHz, and 2.5GHz. The frequency values and
corresponding wavelength values are show in Table 2 below.
frequency(f GHz)
1
1.25
1.5
1.75
2
2.5
Wavelength (, meter)
0.3
0.24
0.2
0.16
0.15
0.12
Size of Dipole(/2) m
0.15
0.12
0.1
0.08
0.075
0.6
Table 2: Size of Dipoles for Different Frequencies
3.2 GENERATION OF THE UNEQUALLY SPACED ARRAY DESIGN AND
ESTIMATION OF THE GAIN, BEAMWDITH AND PSSL
The second MATLAB program, listed in Appendix I, applies the unequal element
spacing values, obtained from the first MATLAB program, and generates the array
parameters, such as gain beamwdith and PSSL, for each value of alpha. The simulation
results are divided into two main categories:
17
3.2.1 GENERATION OF THE DELTA DIFFERENTIAL BETWEEN MAXIMUM
AND MINIMUM PARAMETER VALUES
The following section gives the delta values for each case of unequally spaced array
design, corresponding to a specific value of the parameter alpha. The different delta
values are explained below:
delta1 is difference between maximum and minimum gain of the default uniform array.
delta2 is the difference between maximum and minimum gain of non-uniform array.
delta3 is the difference between maximum and minimum of beamwdith of uniform array.
delta4 is the difference between maximum and minimum of beamwidthwidth of nonuniform array.
delta5 is the difference between maximum and minimum of peak sidelobe level (PSSL)
of uniform array.
delta6 is the difference between maximum and minimum of peak sidelobe level (PSSL)
of the non-uniform array.
An example of gain, beamwidth and sidelobe levels of the uniform and non-uniform
array design, at nd = 11 => alpha = is shown below:
18
Gain
Beam
width
PSSL
frequency (Hz)
Figure 3: Plot Shown to Calculate Delta Values
…….. Uniform
---------- Nonuniform
The first graph shows the gain over frequency range of 1 to 20GHz, the second graph
shows the beamwidth over frequency range 1 to 20GHz, and finally, the third figure
shows the sidelobe levels over frequency range 1 to 20GHz.
Table 3 below, lists the delta values for different values of the shift parameter, alpha, at a
given array element length of 0.15 m, which corresponds, from Table 2, to a frequency of
1 GHz.
19
nd
delta4
delta1
delta2
delta3
BW
Gain(Equal) Gain(Unequal) BW(Equal)
(Unequal)
delta6
delta5
PSSL
PSSL(Equal)
(Unequal)
1
32.2567
32.2567
2.5
2.5
45.2987
45.2987
2
32.2567
39.1828
2.5
2.5
45.2987
47.7122
3
32.2567
22.0505
2.5
2.5
45.2987
28.9935
4
32.2567
26.9066
2.5
2.5
45.2987
39.9686
5
32.2567
23.4218
2.5
2.5
45.2987
36.1069
6
32.2567
21.7789
2.5
2
45.2987
27.9137
7
32.2567
22.9925
2.5
2
45.2987
28.733
8
32.2567
21.9605
2.5
2
45.2987
28.2963
9
32.2567
21.5641
2.5
2
45.2987
28.9425
10 32.2567
21.1303
2.5
1.5
45.2987
26.4501
11 32.2567
21.4665
2.5
1.5
45.2987
26.1597
Table 3: Delta Values for Dipole Length 0.15m of Linear Array
As, we can see from the above table 3, minimum value of delta1 is 32.25 and delta2 is
21.1. There is an improvement from 32.25 to 21.1 in gain for uniform and non-uniform
arrays respectively. There is an improvement from 2.5 to 1.5 in beamwidth, and 45.29 to
26.15 in PSSL.
20
Similarly, Table 4 below lists the delta values for different values of the shift parameter,
alpha, at a given array element length of 0.12 m, which corresponds, from Table 2, to a
frequency of 1.25 GHz.
nd
delta4
delta1
delta2
delta3
BW
Gain(Equal) Gain(Unequal) BW(Equal)
(Unequal)
delta6
delta5
PSSL
PSSL(Equal)
(Unequal)
1
38.867
38.867
3.5
3.5
47.8234
47.409
2
38.867
45.2372
3.5
3
47.8234
48.7185
3
38.867
24.6019
3.5
3
47.8234
25.8454
4
38.867
31.0981
3.5
3
47.8234
40.3027
5
38.867
25.5529
3.5
3
47.8234
33.5679
6
38.867
24.6019
3.5
2.5
47.8234
27.9693
7
38.867
24.6019
3.5
2.5
47.8234
27.3665
8
38.867
24.6019
3.5
2.5
47.8234
30.4195
9
38.867
24.6019
3.5
2.5
47.8234
30.6346
10
38.867
24.6019
3.5
2
47.8234
25.7316
11
38.867
24.6019
3.5
2
47.8234
27.9371
Table 4: Delta Values for Dipole Length 0.12m of Linear Array
From the above table 4, minimum value of delta1 is 38.86 and delta2 is 24.6. There is an
improvement from 38.86 to 24.6 in gain for uniform and non-uniform arrays respectively.
There is an improvement from 3.5 to 2 in Beamwidth, and 47.82 to 25.7 in PSSL.
Table 5 below lists the delta values for different values of the shift parameter, alpha, at a
given array element length of 0.1 m, which corresponds, from Table 2, to a frequency of
1.5 GHz.
21
1
delta4
delta1
delta2
delta3
BW
Gain(Equal) Gain(Unequal) BW(Equal) (Unequal
)
42.1654
42.1654
4
4
2
42.1654
43.6836
4
3.5
48.6408
50.0982
3
42.1654
27.7679
4
4
48.6408
26.9926
4
42.1654
34.7777
4
4
48.6408
41.4246
5
42.1654
29.2481
4
3.5
48.6408
35.8773
6
42.1654
27.7679
4
3.5
48.6408
32.309
7
42.1654
27.7679
4
3
48.6408
29.9429
8
42.1654
27.7679
4
3
48.6408
24.7264
9
42.1654
27.7679
4
2.5
48.6408
30.5713
10 42.1654
27.7679
4
2.5
48.6408
26.5054
11 42.1654
27.7679
4
2
48.6408
28.3114
nd
delta5
PSSL(Equal
)
delta6
PSSL
(Unequal)
48.6408
48.6408
Table 5: Delta Values for Dipole Length 0.1m of Linear Array
As, we can see from the above table 5, minimum value of delta1 is 42.16 and delta2 is
27.7. There is an improvement from 42.16 to 27.7 in gain for uniform and non-uniform
respectively. Similarly there is an improvement from 4 to 2 in Beamwidth, and 48.6 to
24.7 in PSSL.
22
Table 6 below, lists the delta values for different values of the shift parameter, alpha, at a
given array element length of 0.08 m, which corresponds, from Table 2, to a frequency of
1.75 GHz.
nd
delta4
delta1
delta2
delta3
BW
Gain(Equal) Gain(Unequal) BW(Equal)
(Unequal)
delta6
delta5
PSSL
PSSL(Equal)
(Unequal)
1
42.1654
42.1654
4
4
48.6408
48.6408
2
42.1654
43.6836
4
3.5
48.6408
50.0982
3
42.1654
27.7679
4
4
48.6408
26.9926
4
42.1654
34.7777
4
4
48.6408
41.4246
5
42.1654
29.2481
4
3.5
48.6408
35.8773
6
42.1654
27.7679
4
3.5
48.6408
32.309
7
42.1654
27.7679
4
3
48.6408
29.9429
8
42.1654
27.7679
4
3
48.6408
24.7264
9
42.1654
27.7679
4
2.5
48.6408
30.5713
10 42.1654
27.7679
4
2.5
48.6408
26.5054
11 42.1654
27.7679
4
2
48.6408
28.3114
Table 6: Delta Values for Dipole Length 0.08m of Linear Array
As, we can see from the above table 6, minimum value of delta1 is 42.16 and delta2 is
27.7. There is an improvement from 42.16 to 27.7 in gain for uniform and non-uniform
respectively. Similarly there is an improvement from 4 to 2 in Beamwidth, and 48.64 to
26.5 in PSSL.
Table 7 below, lists the delta values for different values of the shift parameter, alpha, at a
given array element length of 0.075 m, which corresponds, from Table 2, to a frequency
of 2 GHz.
23
nd
delta4
delta1
delta2
delta3
BW
Gain(Equal) Gain(Unequal) BW(Equal)
(Unequal)
delta6
delta5
PSSL
PSSL(Equal)
(Unequal)
1
43.4002
43.4002
5.5
5.5
46.9694
46.9694
2
43.4002
32.5621
5.5
5
46.9694
35.2127
3
43.4002
31.0771
5.5
5
46.9694
27.097
4
43.4002
33.9622
5.5
5.5
46.9694
37.8601
5
43.4002
32.1185
5.5
5
46.9694
36.002
6
43.4002
31.0771
5.5
4.5
46.9694
26.4177
7
43.4002
31.0771
5.5
4
46.9694
30.5658
8
43.4002
31.0771
5.5
4
46.9694
29.7221
9
43.4002
31.0771
5.5
3.5
46.9694
22.9357
10 43.4002
31.0771
5.5
3
46.9694
22.2618
11 43.4002
31.0771
5.5
3
46.9694
31.4418
Table 7: Delta Values for Dipole Length 0.075m of Linear Array
As, we can see from the above table 7, minimum value of delta1 is 43.4 and delta2 is
31.07. There is an improvement from 43.4 to 31.07 in gain for uniform and non-uniform
respectively. Similarly there is an improvement from 5.5 to 3 in Beamwidth, and 46.9 to
22.2 in PSSL.
Table 8 below, lists the delta values for different values of the shift parameter, alpha, at a
given array element length of 0.06 m, which corresponds, from Table 2, to a frequency of
2.5 GHz.
24
nd
delta4
delta1
delta2
delta3
BW
Gain(Equal) Gain(Unequal) BW(Equal)
(Unequal)
delta6
delta5
PSSL
PSSL(Equal)
(Unequal)
1
42.0265
42.0265
7
7
42.5643
42.5643
2
42.0265
34.1867
7
6
42.5643
36.0467
3
42.0265
32.874
7
6
42.5643
28.0064
4
42.0265
32.874
7
6
42.5643
34.7223
5
42.0265
32.874
7
6
42.5643
29.1166
6
42.0265
32.874
7
5
42.5643
26.5809
7
42.0265
32.874
7
4.5
42.5643
31.3999
8
42.0265
32.874
7
4.5
42.5643
24.6044
9
42.0265
32.874
7
4
42.5643
19.5186
10 42.0265
32.874
7
4
42.5643
25.2043
11 42.0265
32.874
7
4
42.5643
27.9664
Table 8: Delta Values for Dipole Length 0.06m of Linear Array
As, we can see from the above table 8, minimum value of delta1 is 42.02 and delta2 is
32.874. There is an improvement from 42.02 to 32.87 in gain for uniform and nonuniform respectively. Similarly there is an improvement from 7 to 4 in Beamwidth, and
42.564 to 19.51 in PSSL.
From the results, least value of delta2 (gain) is 21.1, when nd=10 and dipole length of
0.15m. Least value of delta4 (beamwidth) is 1.5 for nd=10 and dipole length of 0.15m.
Least value of delta6 (sidelobe level) is 19.5 when nd= 9 ad dipole length of 0.06m.
25
3.2.2 GRAPHS FOR COMPARING EQUALLY AND UNEQUALLY SPACED
ARRAYS
The following graphs in this section show the gain for uniform and non uniform array
antennas over the frequency range of 1-20GHz for different lengths of dipole. Each figure
below has response for equally spaced antenna designed as Uniformly_spaced_nd=1 and
nonuniformly
spaced
response
for
nd
=
2
to
11
designated
as
Non_uniformly_spaced_nd=2 to 11.
Figure 4 below is the response of gain over 1-20GHz of dipole length 0.15m and nd as 1
to 11. X-axis is frequency in GHz, Y-axis is gain in db.
60
Uniformly_spaced_Nd=1
Non_Uniformly_spaced_Nd=1
50
Non_Uniformly_spaced_Nd=2
Gain
Non_Uniformly_spaced_Nd=3
40
Non_Uniformly_spaced_Nd=4
Non_Uniformly_spaced_Nd=5
30
Non_Uniformly_spaced_Nd=6
Non_Uniformly_spaced_Nd=7
20
Non_Uniformly_spaced_Nd=8
Non_Uniformly_spaced_Nd=9
10
Non_Uniformly_spaced_Nd=10
0
0
5
10
15
20
Non_Uniformly_spaced_Nd=11
25
frequency (GHz)
Figure 4: Simulation Plot for Gain of Linear Array of Length 0.15m
26
Optimum wideband performance is defined by the flattest gain curve, which shows
relatively constant gain over the frequency range of interest. When comparing gains from
nd = 2 to 11, optimum performance is observed at nd = 7, which corresponds to an alpha
of 0.3.
Figure 5 below is the response of gain over 1-20GHz of dipole length 0.12m and nd as 1
to 11.
60
Uniformly_spaced_Nd=1
Non_Uniformly_spaced_Nd=1
50
Non_Uniformly_spaced_Nd=2
40
Non_Uniformly_spaced_Nd=3
Gain
Non_Uniformly_spaced_Nd=4
30
Non_Uniformly_spaced_Nd=5
Non_Uniformly_spaced_Nd=6
20
Non_Uniformly_spaced_Nd=7
Non_Uniformly_spaced_Nd=8
10
Non_Uniformly_spaced_Nd=9
Non_Uniformly_spaced_Nd=10
0
0
5
10
15
20
Non_Uniformly_spaced_Nd=11
25
frequency (GHz)
Figure 5: Simulation Plot for Gain of Linear Array of Length 0.12m
From the above figure 5, when comparing gains from nd = 2 to 11, optimum performance
is observed at nd = 10, which corresponds to an alpha of 0.45.
27
Figure 6 below is the response of gain over 1-20GHz of dipole length 0.1m and nd as 1 to
11
60
Uniformly_spaced_Nd=1
Non_Uniformly_spaced_Nd=1
50
Non_Uniformly_spaced_Nd=2
40
Non_Uniformly_spaced_Nd=3
Non_Uniformly_spaced_Nd=4
Gain
30
Non_Uniformly_spaced_Nd=5
Non_Uniformly_spaced_Nd=6
20
Non_Uniformly_spaced_Nd=7
Non_Uniformly_spaced_Nd=8
10
Non_Uniformly_spaced_Nd=9
Non_Uniformly_spaced_Nd=10
0
0
5
10
15
20
Non_Uniformly_spaced_Nd=11
25
frequency (GHz)
Figure 6: Simulation Plot for Gain of Linear Array of Length 0.1m
From the above figure 6, when comparing gains from nd = 2 to 11, optimum performance
is observed at nd = 11, which corresponds to an alpha of 0.5.
The figure 7 below is the response of gain over 1-20GHz of dipole length 0.08m and nd
as 1 to 11
28
Uniformly_spaced_Nd=1
50
Gain
45
Non_Uniformly_spaced_Nd=1
40
Non_Uniformly_spaced_Nd=2
35
Non_Uniformly_spaced_Nd=3
Non_Uniformly_spaced_Nd=4
30
Non_Uniformly_spaced_Nd=5
25
Non_Uniformly_spaced_Nd=6
20
Non_Uniformly_spaced_Nd=7
15
Non_Uniformly_spaced_Nd=8
10
Non_Uniformly_spaced_Nd=9
5
Non_Uniformly_spaced_Nd=10
0
Non_Uniformly_spaced_Nd=11
0
5
10
15
20
25
frequency (GHz)
Figure 7: Simulation Plot for Gain of Linear Array of Length 0.08m
From the above figure 7, when comparing gains from nd = 2 to 11, optimum performance
is observed at nd = 9, which corresponds to an alpha of 0.4.
The figure 8 below is the response of gain over 1-20GHz of dipole length 0.075m and nd
as 1 to 11
60
Uniformly_spaced_Nd=1
Gain
Non_Uniformly_spaced_Nd=1
50
Non_Uniformly_spaced_Nd=2
Non_Uniformly_spaced_Nd=3
40
Non_Uniformly_spaced_Nd=4
30
Non_Uniformly_spaced_Nd=5
Non_Uniformly_spaced_Nd=6
20
Non_Uniformly_spaced_Nd=7
Non_Uniformly_spaced_Nd=8
10
Non_Uniformly_spaced_Nd=9
Non_Uniformly_spaced_Nd=10
0
0
5
10
15
20
25
Non_Uniformly_spaced_Nd=11
frequency (GHz)
Figure 8: Simulation Plot for Gain of Linear Array of Length 0.075m
29
From the above figure 8, when comparing gains from nd = 2 to 11, optimum performance
is observed at nd = 10, which corresponds to an alpha of 0.45.
The figure 9 below is the response of gain over 1-20GHz of dipole length 0.06m and nd
as 1 to 11
50
Gain
45
Uniformly_spaced_Nd=1
Non_Uniformly_spaced_Nd=1
40
Non_Uniformly_spaced_Nd=2
35
Non_Uniformly_spaced_Nd=3
30
Non_Uniformly_spaced_Nd=4
25
Non_Uniformly_spaced_Nd=5
20
Non_Uniformly_spaced_Nd=6
15
Non_Uniformly_spaced_Nd=7
10
Non_Uniformly_spaced_Nd=8
Non_Uniformly_spaced_Nd=9
5
Non_Uniformly_spaced_Nd=10
0
0
5
10
15
20
Non_Uniformly_spaced_Nd=11
25
frequency (GHz)
Figure 9: Simulation Plot for Gain of Linear Array of Length 0.06m
From the above figure 6, when comparing gains from nd = 2 to 11, optimum performance
is observed at nd = 10, which corresponds to an alpha of 0.45. From the above graphs for
gain, it is clear that gain of unequally spaced antennas is more flat compared to gain of
equally spaced array.
30
The following graphs in this section show the beamwidth for uniform and non uniform
array antennas over the frequency range of 1-20GHz for different lengths of dipole.
Figure 10 below is the response of beamwidth over 1-20GHz of dipole length 0.15m and
nd as 1 to 11. X-axis is frequency in GHz, Y-axis is beamwidth in db.
3
Beam Width
2.5
2
1.5
1
0.5
0
0
5
10
15
20
-0.5
Uniformly_spaced_Nd=1
Non_Uniformly_spaced_Nd=1
Non_Uniformly_spaced_Nd=2
Non_Uniformly_spaced_Nd=3
Non_Uniformly_spaced_Nd=4
Non_Uniformly_spaced_Nd=5
Non_Uniformly_spaced_Nd=6
Non_Uniformly_spaced_Nd=7
Non_Uniformly_spaced_Nd=8
Non_Uniformly_spaced_Nd=9
Non_Uniformly_spaced_Nd=10
25 Non_Uniformly_spaced_Nd=11
frequency (GHz)
Figure 10: Simulation Plots for Beamwidth of Linear Array of Length 0.15m
From the above figure 10, when comparing beamwidth from nd = 2 to 11, optimum
performance is observed at nd = 10, which correspond to an alpha of 0.45.
31
The figure 11 below is the response of beamwidth over 1-20GHz of dipole length 0.12m
and nd as 1 to 11
4
Beam Width
Uniformly_spaced_Nd=1
3.5
Non_Uniformly_spaced_Nd=1
3
Non_Uniformly_spaced_Nd=2
2.5
Non_Uniformly_spaced_Nd=3
Non_Uniformly_spaced_Nd=4
2
Non_Uniformly_spaced_Nd=5
1.5
Non_Uniformly_spaced_Nd=6
1
Non_Uniformly_spaced_Nd=7
Non_Uniformly_spaced_Nd=8
0.5
Non_Uniformly_spaced_Nd=9
0
-0.5
Non_Uniformly_spaced_Nd=10
0
5
10
15
20
25
Non_Uniformly_spaced_Nd=11
frequency (GHz)
Figure 11: Simulation Plot for Beamwidth of Linear Array of Length 0.12m
From the above figure 11, when comparing beamwidths from nd = 2 to 11, optimum
performance is observed at nd = 10, which corresponds to an alpha of 0.45.
32
The figure 12 below is the response of beamwidth over 1-20GHz of dipole length 0.1m
and nd as 1 to 11
4.5
Beam Width
4
Uniformly_spaced_Nd=1
Non_Uniformly_spaced_Nd=1
3.5
Non_Uniformly_spaced_Nd=2
3
Non_Uniformly_spaced_Nd=3
2.5
Non_Uniformly_spaced_Nd=4
2
Non_Uniformly_spaced_Nd=5
Non_Uniformly_spaced_Nd=6
1.5
Non_Uniformly_spaced_Nd=7
1
Non_Uniformly_spaced_Nd=8
0.5
Non_Uniformly_spaced_Nd=9
Non_Uniformly_spaced_Nd=10
0
-0.5
0
5
10
15
20
25
Non_Uniformly_spaced_Nd=11
frequency (GHz)
Figure 12: Simulation Plot for Beamwidth of Linear Array of Length 0.1m
From the above figure 12, when comparing beamwidths from nd = 2 to 11, optimum
performance is observed at nd = 9, which corresponds to an alpha of 0.4.
33
The figure 13 below is the response of beamwidth over 1-20GHz of dipole length 0.08m
and nd as 1 to 11
5
Uniformly_spaced_Nd=1
Beam Width
Non_Uniformly_spaced_Nd=1
4
Non_Uniformly_spaced_Nd=2
Non_Uniformly_spaced_Nd=3
3
Non_Uniformly_spaced_Nd=4
Non_Uniformly_spaced_Nd=5
2
Non_Uniformly_spaced_Nd=6
Non_Uniformly_spaced_Nd=7
1
Non_Uniformly_spaced_Nd=8
Non_Uniformly_spaced_Nd=9
0
0
5
10
15
20
25 Non_Uniformly_spaced_Nd=10
Non_Uniformly_spaced_Nd=11
-1
frequency (GHz)
Figure 13: Simulation Plot for Beamwidth of Linear Array of Length 0.08m
From the above figure 13, when comparing beamwidths from nd = 2 to 11, optimum
performance is observed at nd = 6, which corresponds to an alpha of 0.25.
The figure 14 below is the response of beamwidth over 1-20GHz of dipole length 0.075m
and nd as 1 to 11
34
6
Beam Width
Uniformly_spaced_Nd=1
5
Non_Uniformly_spaced_Nd=1
Non_Uniformly_spaced_Nd=2
4
Non_Uniformly_spaced_Nd=3
Non_Uniformly_spaced_Nd=4
3
Non_Uniformly_spaced_Nd=5
Non_Uniformly_spaced_Nd=6
2
Non_Uniformly_spaced_Nd=7
1
Non_Uniformly_spaced_Nd=8
Non_Uniformly_spaced_Nd=9
0
0
5
10
15
20
25
Non_Uniformly_spaced_Nd=10
Non_Uniformly_spaced_Nd=11
-1
frequency (GHz)
Figure 14: Simulation Plot for Beamwidth of Linear Array of Length 0.075m
From the above figure 14, when comparing beamwidths from nd = 2 to 11, optimum
performance is observed at nd = 7, which corresponds to an alpha of 0.3.
35
The figure 15 below is the response of beamwidth over 1-20GHz of dipole length 0.06m
and nd as 1 to 11
8
Beam Width
7
Uniformly_spaced_Nd=1
Non_Uniformly_spaced_Nd=1
6
Non_Uniformly_spaced_Nd=2
Non_Uniformly_spaced_Nd=3
5
Non_Uniformly_spaced_Nd=4
4
Non_Uniformly_spaced_Nd=5
3
Non_Uniformly_spaced_Nd=6
Non_Uniformly_spaced_Nd=7
2
Non_Uniformly_spaced_Nd=8
Non_Uniformly_spaced_Nd=9
1
Non_Uniformly_spaced_Nd=10
0
-1
0
5
10
15
20
25
Non_Uniformly_spaced_Nd=11
frequency (GHz)
Figure 15: Simulation Plot for Beamwidth of Linear Array of Length 0.06m
From the above figure 15, when comparing beamwidths from nd = 2 to 11, optimum
performance is observed at nd = 7, which corresponds to an alpha of 0.3.
From the above set of graphs it is clear that beamwidth of unequally spaced
antennas(Non_Uniformly_spaced_ nd =2,3,4,5,6,7,8,9,10,11) is more flat compared to the
beamwidth of equally spaced array (Uniformly_spaced_ nd =1).
36
The following graphs in this section show the peak sidelobe levels (PSSL) for uniform
and non-uniform array antennas over the frequency range of 1-20GHz for different
lengths of dipole.
Figure 16 below is the response of PSSL over 1-20GHz of dipole length 0.15m and nd as
1 to 11. X-axis is frequency in GHz; Y-axis is PSSL in db.
60
Uniformly_spaced_Nd=1
PSSL
Non_Uniformly_spaced_Nd=1
50
Non_Uniformly_spaced_Nd=2
40
Non_Uniformly_spaced_Nd=3
Non_Uniformly_spaced_Nd=4
30
Non_Uniformly_spaced_Nd=5
Non_Uniformly_spaced_Nd=6
20
Non_Uniformly_spaced_Nd=7
Non_Uniformly_spaced_Nd=8
10
Non_Uniformly_spaced_Nd=9
Non_Uniformly_spaced_Nd=10
0
0
5
10
15
20
25
Non_Uniformly_spaced_Nd=11
frequency (GHz)
Figure 16: Simulation Plot for Peak Sidelobe Levels of Linear Array of Length
0.15m
From the above figure 16, when comparing PSSL from nd = 2 to 11, optimum
performance is observed at nd = 8, which corresponds to an alpha of 0.35.
37
Figure 17 below is the response of PSSL over 1-20GHz of dipole length 0.12m and nd as
1 to 11.
60
Uniformly_spaced_Nd=1
PSSL
Non_Uniformly_spaced_Nd=1
50
Non_Uniformly_spaced_Nd=2
Non_Uniformly_spaced_Nd=3
40
Non_Uniformly_spaced_Nd=4
Non_Uniformly_spaced_Nd=5
30
Non_Uniformly_spaced_Nd=6
Non_Uniformly_spaced_Nd=7
20
Non_Uniformly_spaced_Nd=8
10
Non_Uniformly_spaced_Nd=9
Non_Uniformly_spaced_Nd=10
0
0
5
10
15
20
Non_Uniformly_spaced_Nd=11
25
frequency (GHz)
Figure 17: Simulation Plot for Peak Sidelobe Levels of Linear Array of
Length 0.12m
From the above figure 17, when comparing PSSL from nd = 2 to 11, optimum
performance is observed at nd = 9, which corresponds to an alpha of 0.4.
38
The figure 18 below is the response of PSSL over 1-20GHz of dipole length 0.1m and nd
as 1 to 11.
60
PSSL
Uniformly_spaced_Nd=1
Non_Uniformly_spaced_Nd=1
50
Non_Uniformly_spaced_Nd=2
40
Non_Uniformly_spaced_Nd=3
Non_Uniformly_spaced_Nd=4
30
Non_Uniformly_spaced_Nd=5
Non_Uniformly_spaced_Nd=6
20
Non_Uniformly_spaced_Nd=7
Non_Uniformly_spaced_Nd=8
10
Non_Uniformly_spaced_Nd=9
Non_Uniformly_spaced_Nd=10
0
0
5
10
15
20
Non_Uniformly_spaced_Nd=11
25
frequency (GHz)
Figure 18: Simulation Plot for Peak Sidelobe Levels of Linear Array of
Length 0.1m
From the above figure 18, when comparing PSSL from nd = 2 to 11, optimum
performance is observed at nd = 4, which corresponds to an alpha of 0.15.
39
The figure 19 below is the response of PSSL over 1-20GHz of dipole length 0.08m and
nd as 1 to 11
50
Uniformly_spaced_Nd=1
PSSL
45
Non_Uniformly_spaced_Nd=1
40
Non_Uniformly_spaced_Nd=2
35
Non_Uniformly_spaced_Nd=3
30
Non_Uniformly_spaced_Nd=4
25
Non_Uniformly_spaced_Nd=5
Non_Uniformly_spaced_Nd=6
20
Non_Uniformly_spaced_Nd=7
15
Non_Uniformly_spaced_Nd=8
10
Non_Uniformly_spaced_Nd=9
5
Non_Uniformly_spaced_Nd=10
0
0
5
10
15
20
25
Non_Uniformly_spaced_Nd=11
frequency (GHz)
Figure 19: Simulation Plot for Peak Sidelobe Levels of Linear Array of
Length 0.08m
From the above figure 19, when comparing PSSL from nd = 2 to 11, optimum
performance is observed at nd = 9, which corresponds to an alpha of 0.4.
40
The figure 20 below is the response of PSSL over 1-20GHz of dipole length 0.075m and
nd as 1 to 11
60
PSSL
Uniformly_spaced_Nd=1
50
Non_Uniformly_spaced_Nd=1
Non_Uniformly_spaced_Nd=2
40
Non_Uniformly_spaced_Nd=3
Non_Uniformly_spaced_Nd=4
30
Non_Uniformly_spaced_Nd=5
Non_Uniformly_spaced_Nd=6
20
Non_Uniformly_spaced_Nd=7
Non_Uniformly_spaced_Nd=8
10
Non_Uniformly_spaced_Nd=9
Non_Uniformly_spaced_Nd=10
0
.
0
5
10
15
20
25 Non_Uniformly_spaced_Nd=11
frequency (GHz)
Figure 20: Simulation Plot for Peak Sidelobe Levels of Linear Array of
Length 0.075m
From the above figure 20, when comparing PSSL from nd = 2 to 11, optimum
performance is observed at nd = 10, which corresponds to an alpha of 0.45.
41
The figure 21 below is the response of PSSL over 1-20GHz of dipole length 0.06m and
nd as 1 to 11
50
45
Uniformly_spaced_Nd=1
40
Non_Uniformly_spaced_Nd=1
Non_Uniformly_spaced_Nd=2
35
Non_Uniformly_spaced_Nd=3
30
Non_Uniformly_spaced_Nd=4
25
Non_Uniformly_spaced_Nd=5
PSSL
20
Non_Uniformly_spaced_Nd=6
15
Non_Uniformly_spaced_Nd=7
10
Non_Uniformly_spaced_Nd=8
Non_Uniformly_spaced_Nd=9
5
Non_Uniformly_spaced_Nd=10
0
.
0
5
10
15
20
Non_Uniformly_spaced_Nd=11
25
frequency (GHz)
Figure 21: . Simulation Plot for Peak Sidelobe Levels of Linear Array of
Length 0.06m
From the above figure 21, when comparing PSSL from nd = 2 to 11, optimum
performance is observed at nd = 9, which corresponds to an alpha of 0.4.
From the above set of graphs it is clear that PSSL of unequally spaced
antennas(Non_Uniformly_spaced_nd=2,3,4,5,6,7,8,9,10,11) is flatter compared to the
PSSL of equally spaced array (Uniformly_spaced_nd =1).
42
Chapter 4
ESTIMATION OF GAIN, BEAMWIDTH AND SIDELOBE LEVELS
OF CONCENTRIC RING ARRAY ANTENNA
This chapter presents the results of the two-step concentric ring array design process
that was outlined in chapter 2. The final data for the concentric ring unequally spaced
antenna array includes the gain, beamwidth and sidelobe levels, as compared with the
uniform antenna array, in order to demonstrate the performance improvement. The
simulation is performed in a two-step process,
1) Generation of spacing between the antennas elements from the first MATLAB
program listed in the APPENDIX II.
2) Spacing values obtained from the first program are used in the second program in
APPENDIX II to calculate the difference of maximum and minimum values of gain,
beamwidth and sidelobe levels of equally and unequally spaced antennas.
4.1 GENERATION OF UNEQUAL SPACINGS STARTING FROM A DEFAULT
EQUALLY SPACED ARRAY
The first MATLAB program estimates the optimized unequal spacing from the given
default array, in which the elements are equally spaced at half-wavelength. With the
change in the spacing between the elements, response of gain, beamwidth and peak
43
sidelobe level (PSSL) of the array changes, and the aim is to determine the optimum set
of spacing which will maximize the gain, for example.
The optimization process in the first step of the wideband array design starts with the
equally spaced array, with spacing between each adjacent element at half a wavelength.
Then the program specifies a design parameter, alm0, which specifies the maximum shift
allowed in the design of the new element spacing.
In the first program, the tolerance given for spacing (alm0) is varied in 10 steps
from one-tenth of a wavelength to half a wavelength and the array spacing are generated
in each case. The index of the shift is nd, and for each value of nd, the first program in the
Appendix II gives corresponding values of the parameter alm0, based on the formula
below:
alm0=dx(1)+dxa*dxx
dxx= dxxx*tpi
dxa=(nd-1)*dxaa
ndxa=10, dxaa=2./ndxa
where dxxx= Default Spacing between Adjacent Rings (In Wavelengths)
dx (I)= Radius of I'th Ring In Default Uniform Array
The following table 9 gives alm0 values for different values of nd, for a 10 element array.
nd
2
3
4
5
6
7
8
9
10
alm0
3.76
4.39
5.02
5.65
6.28
6.91
7.53
8.1
8.79
44
Table 9: alm0 Values for nd= 2 to 10 of Ring Array
4.2 GENERATION OF THE UNEQUALLY SPACED ARRAY DESIGN AND
ESTIMATION OF THE GAIN, BEAMWDITH AND PSSL
The second MATLAB program, listed in Appendix II, applies the unequal element
spacing values, obtained from the first MATLAB program, and generates the array
parameters, such as gain beamwdith and PSSL, for each value of alpha. The simulation
results are divided into two main categories:
4.2.1 GENERATION OF THE DELTA DIFFERENTIAL BETWEEN MAXIMUM
AND MINIMUM PARAMETER VALUES
The following section gives the delta values for each case of unequally spaced array
design, corresponding to a specific value of the parameter alm0.
Table 10 below, lists the delta values for different values of the shift parameter, alm0, at a
given array element length of 0.15 m, which corresponds, from Table 2, to a frequency of
1 GHz.
nd
2
3
4
5
6
7
8
9
10
delta1
delta2
delta3
Gain(Equal) Gain(Unequal) BW(Equal)
496.2518
496.2518
496.2518
496.2518
496.2518
496.2518
496.2518
496.2518
496.2518
496.2518
508.3305
515.155
515.4266
506.3319
521.411
517.0018
521.0067
514.7109
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
delta4
BW
(Unequal)
2.5
2.5
2.5
2
2.5
2
1.5
1.5
1.5
delta5
PSSL(Equal)
75.7998
75.7998
75.7998
75.7998
75.7998
75.7998
75.7998
75.7998
75.7998
delta6
PSSL
(Unequal)
75.7998
78.8943
65.1775
83.8732
81.0041
81.1173
26.7436
28.2859
22.6819
Table 10: Delta Value for Dipole Length 0.15m of Concentric Ring Array
45
As we can see from the above table 10, there is an improvement in beamwidth from 2.5
to 1.5 for uniform and non-uniform respectively and 75.7998 to 22.6819 in PSSL, but no
significant improvement in gain.
Table 11 below lists the delta values for different values of the shift parameter, alm0, at a
given array element length of 0.12 m, which corresponds, from Table 2, to a frequency of
1.25 GHz.
delta1
delta2
500.8132
500.8132
500.8132
500.8132
500.8132
500.8132
500.8132
500.8132
500.8132
500.8132
498.7299
516.9039
510.6937
507.0474
520.7898
517.5651
520.3748
519.8909
delta3
nd Gain(Equal) Gain(Unequal) BW(Equal
2
3
4
5
6
7
8
9
10
)
3
3
3
3
3
3
3
3
3
delta4
BW
(Unequal)
3
3
3
2
3
2.5
2.5
2
2
delta5
PSSL(Equal)
73.0902
73.0902
73.0902
73.0902
73.0902
73.0902
73.0902
73.0902
73.0902
delta6
PSSL
(Unequal)
73.0902
75.3982
63.0188
85.2344
80.6616
78.4412
30.514
33.7083
26.4636
Table 11: Delta Value for Dipole Length 0.12m of Concentric Ring Array
As we can see from the above table 11, there is an improvement in beamwidth from 3 to
2 for uniform and non-uniform respectively and 73.0902 to 26.4636 inPSSL, but no
significant improvement in gain.
Table 12 below lists the delta values for different values of the shift parameter, alm0, at a
given array element length of 0.1 m, which corresponds, from Table 2, to a frequency of
1.5 GHz.
46
delta1
delta2
delta3
500.2443
500.2443
500.2443
500.2443
500.2443
500.2443
500.2443
500.2443
500.2443
500.2443
496.0851
511.9678
510.9196
508.8025
515.2386
519.9909
520.4758
516.1754
4
4
4
4
4
4
4
4
4
nd Gain(Equal) Gain(Unequal) BW(Equal)
2
3
4
5
6
7
8
9
10
delta4
BW
(Unequal)
4
4
3.5
3.5
3.5
3
3
2.5
2.5
delta5
PSSL(Equal)
75.4441
75.4441
75.4441
75.4441
75.4441
75.4441
75.4441
75.4441
75.4441
delta6
PSSL
(Unequal)
75.4441
72.7686
62.3989
84.2353
79.2695
74.526
33.8923
36.6763
30.8117
Table 12: Delta Value for Dipole Length 0.1m of Concentric Ring Array
As we can see from the above table 12, there is an improvement in beamwidth from 4 to
2.5 for uniform and non-uniform respectively and 75.4441 to 30.8117 in PSSL, but no
significant improvement in gain.
Table 13 below lists the delta values for different values of the shift parameter, alm0, at a
given array element length of 0.08 m, which corresponds, from Table 2, to a frequency of
1.75 GHz.
nd
2
3
4
5
6
7
8
9
10
delta1
delta2
delta3
Gain(Equal) Gain(Unequal) BW(Equal)
492.5462
492.5462
492.5462
492.5462
492.5462
492.5462
492.5462
492.5462
492.5462
492.5462
496.4932
507.4069
510.0159
496.5909
515.5424
513.3345
512.3381
513.4861
4.5
4.5
4.5
4.5
4.5
4.5
4.5
4.5
4.5
delta4
BW
(Unequal)
4.5
4.5
4.5
4
4.5
3.5
3.5
3
3
delta5
PSSL(Equal)
78.9391
78.9391
78.9391
78.9391
78.9391
78.9391
78.9391
78.9391
78.9391
delta6
PSSL
(Unequal)
78.9391
76.0635
67.5353
79.9559
82.2082
74.1911
36.8218
36.5545
34.239
Table 13: Delta Value for Dipole Length 0.08m of Concentric Ring Array
47
As we can see from the above table 13, there is an improvement in beamwidth from 4.5
to 3 for uniform and non-uniform respectively and 78.9391 to 34.239 in PSSL, but no
significant improvement in gain.
Table 14 below lists the delta values for different values of the shift parameter, alm0, at a
given array element length of 0.075 m, which corresponds, from Table 2, to a frequency
of 2 GHz.
nd
delta1
Gain(Equal)
delta2
Gain
(Unequal)
delta3
BW(Equal)
delta4
BW
(Unequal)
delta5
PSSL(Equal)
2
3
4
5
6
7
8
9
10
486.5532
486.5532
486.5532
486.5532
486.5532
486.5532
486.5532
486.5532
486.5532
486.5532
493.2451
503.0829
508.2444
490.7175
511.0848
517.6317
518.4684
513.9701
5
5
5
5
5
5
5
5
5
5
5
4.5
4.5
5
4
4
3.5
3.5
84.4795
84.4795
84.4795
84.4795
84.4795
84.4795
84.4795
84.4795
84.4795
delta6
PSSL
(Unequa
l)
84.4795
80.0598
69.3897
86.3471
84.0257
79.9671
40.53
39.4433
32.3845
Table 14: Delta Value for Dipole Length 0.075m of Concentric Ring Array
As we can see from the above table 14, there is an improvement in beamwidth from 5 to
3.5 for uniform and non-uniform respectively and 84.4795 to 32.3845 in PSSL, but no
significant improvement in gain.
48
Table 15 below lists the delta values for different values of the shift parameter, alm0, at a
given array element length of 0.06 m, which corresponds, from Table 2, to a frequency of
2.5 GHz.
delta1
nd Gain(Equal)
2
3
4
5
6
7
8
9
10
494.9356
494.9356
494.9356
494.9356
494.9356
494.9356
494.9356
494.9356
494.9356
delta2
Gain
(Unequal)
494.9356
494.9356
498.9885
503.8146
494.9356
499.1044
501.9388
503.0715
506.5183
delta3
BW(Equal)
6.5
6.5
6.5
6.5
6.5
6.5
6.5
6.5
6.5
delta4
BW
(Unequal)
6.5
6.5
6
5.5
6.5
5
5
4
4
delta5
PSSL(Equal)
88.7438
88.7438
88.7438
88.7438
88.7438
88.7438
88.7438
88.7438
88.7438
delta6
PSSL
(Unequal)
88.7438
83.9844
74.3613
90.9406
90.2297
84.1557
42.2426
40.009
35.6124
Table 15: Delta Value for Dipole Length 0.06m of Concentric Ring Array
As we can see from the above table, there is an improvement in beamwidth from 6.5 to 4
for uniform and non-uniform respectively and 88.7438 to 35.6124 in PSSL, but no
significant improvement in gain.
From the above set of tables for concentric arrays, there is performance improvement in
beamwidth and PSSL but there is no significant improvement in gain. Least value of
delta4 (beamwidth) is 1.5 for nd=10 and dipole length of 0.15m. Least value of delta6
(PSSL) is 22.6 when nd= 10 at dipole length of 0.15m.
49
4.2.2 GRAPHS FOR COMPARING EQUALLY AND UNEQUALLY SPACED
ARRAYS
The following graphs in this section show the gain for uniform and non-uniform array
antennas over the frequency range of 1-20GHz for different lengths of dipole.
The figure 22 below is the response of gain over 1-20GHz for nd= 2 to 10 of dipole length
0.15m. X-axis is frequency in GHz, Y-axis is gain in db.
600
Gain
400
200
0
0
5
10
15
20
25
Uniformly_spaced_Nd=2
Non_Uniformly_spaced_Nd=2
Non_Uniformly_spaced_Nd=3
Non_Uniformly_spaced_Nd=4
Non_Uniformly_spaced_Nd=5
Non_Uniformly_spaced_Nd=6
Non_Uniformly_spaced_Nd=7
Non_Uniformly_spaced_Nd=8
Non_Uniformly_spaced_Nd=9
Non_Uniformly_spaced_Nd=10
frequency (GHz)
Figure 22: Simulation Plot for Gain of Concentric Ring Array of Length 0.15m
Optimum wideband performance is defined by the flattest gain curve, which shows
relatively constant gain over the frequency range of interest. From figure 22, when
comparing gains from nd = 3 to 10, there is no significant improvement in performance.
Figure 23 below is the response of gain over 1-20GHz of dipole length 0.12m for nd= 2 to
10
50
600
400
Gain
200
0
0
5
10
15
20
Uniformly_spaced_Nd=2
Non_Uniformly_spaced_Nd=2
Non_Uniformly_spaced_Nd=3
Non_Uniformly_spaced_Nd=4
Non_Uniformly_spaced_Nd=5
Non_Uniformly_spaced_Nd=6
Non_Uniformly_spaced_Nd=7
Non_Uniformly_spaced_Nd=8
25 Non_Uniformly_spaced_Nd=9
Non_Uniformly_spaced_Nd=10
frequency (GHz)
Figure 23: Simulation Plot for Gain of Concentric Ring Array of Length 0.12m
From figure 23, when comparing gains from nd = 3 to 10, there is no significant
improvement in performance.
Figure 24 below is the response of gain over 1-20GHz of dipole length 0.1m for nd= 2 to
10
Gain
600
500
400
300
200
100
0
0
5
10
15
20
25
Uniformly_spaced_Nd=2
Non_Uniformly_spaced_Nd=2
Non_Uniformly_spaced_Nd=3
Non_Uniformly_spaced_Nd=4
Non_Uniformly_spaced_Nd=5
Non_Uniformly_spaced_Nd=6
Non_Uniformly_spaced_Nd=7
Non_Uniformly_spaced_Nd=8
Non_Uniformly_spaced_Nd=9
Non_Uniformly_spaced_Nd=10
frequency (GHz)
Figure 24: Simulation Plot for Gain of Concentric Ring Array of Length 0.1m
From figure 24, when comparing gains from nd = 3 to 10, there is no significant
improvement in performance. Figure 25 below is the response of gain over 1-20GHz of
dipole length 0.08m for nd= 2 to 10
51
Uniformly_spaced_Nd=2
600
Non_Uniformly_spaced_Nd=2
500
Non_Uniformly_spaced_Nd=3
400
Non_Uniformly_spaced_Nd=4
Gain
Non_Uniformly_spaced_Nd=5
300
Non_Uniformly_spaced_Nd=6
200
Non_Uniformly_spaced_Nd=7
100
Non_Uniformly_spaced_Nd=8
0
Non_Uniformly_spaced_Nd=9
0
5
10
15
20
25
Non_Uniformly_spaced_Nd=10
frequency (GHz)
Figure 25: Simulation Plot for Gain of Concentric Ring Array of Length 0.08m
From figure 25, when comparing gains from nd = 3 to 10, there is no significant
improvement in performance.
Figure 26 below is the response of gain over 1-20GHz of dipole length 0.075m for nd= 2
to 10
Gain
600
500
400
300
200
100
0
0
5
10
15
20
25
Uniformly_spaced_Nd=2
Non_Uniformly_spaced_Nd=2
Non_Uniformly_spaced_Nd=3
Non_Uniformly_spaced_Nd=4
Non_Uniformly_spaced_Nd=5
Non_Uniformly_spaced_Nd=6
Non_Uniformly_spaced_Nd=7
Non_Uniformly_spaced_Nd=8
Non_Uniformly_spaced_Nd=9
Non_Uniformly_spaced_Nd=10
frequency (GHz)
Figure 26: Simulation Plot for Gain of Concentric Ring Array of Length 0.075m
52
From figure 26, when comparing gains from nd = 3 to 10, there is no significant
improvement in performance.
Figure 27 below is the response of gain over 1-20GHz of dipole length 0.06m for nd= 2 to
10
Gain
600
500
400
300
200
100
0
0
5
10
15
20
Uniformly_spaced_Nd=2
Non_Uniformly_spaced_Nd=2
Non_Uniformly_spaced_Nd=3
Non_Uniformly_spaced_Nd=4
Non_Uniformly_spaced_Nd=5
Non_Uniformly_spaced_Nd=6
Non_Uniformly_spaced_Nd=7
Non_Uniformly_spaced_Nd=8
Non_Uniformly_spaced_Nd=9
Non_Uniformly_spaced_Nd=10
25
frequency (GHz)
Figure 27: Simulation Plot for Gain of Concentric Ring Array of Length 0.06m
From figure 27, when comparing gains from nd = 3 to 10, there is no significant
improvement in performance.As we can see from the above set of graphs, there is no
significant improvement in the gain for non uniform array when compared to uniform
ring array antenna.
The following graphs in this section show the beamwidth for uniform and non-uniform
array antennas over the frequency range of 1-20GHz for different lengths of dipole.
Figure 28 below is the response of beamwidth over 1-20GHz of dipole length 0.15m for
nd= 2 to 10. X-axis is frequency in GHz, Y-axis is beamwidth in db.
53
3
2.5
2
Beam
Width
1.5
1
0.5
0
-0.5
0
5
10
15
20
Uniformly_spaced_Nd=2
Non_Uniformly_spaced_Nd=2
Non_Uniformly_spaced_Nd=3
Non_Uniformly_spaced_Nd=4
Non_Uniformly_spaced_Nd=5
Non_Uniformly_spaced_Nd=6
Non_Uniformly_spaced_Nd=7
Non_Uniformly_spaced_Nd=8
Non_Uniformly_spaced_Nd=9
25 Non_Uniformly_spaced_Nd=10
frequency (GHz)
Figure 28: Simulation Plot for Beamwidth of Concentric Ring Array of Length
0.15m
From the above figure 28, when comparing beamwidth from nd = 3 to 10, optimum
performance is observed at nd = 10, which corresponds to an alm0 of 8.79.
The figure 29 below is the response of beamwidth over 1-20GHz of dipole length 0.12m
for nd= 2 to 10.
3.5
3
Beam 2.5
2
Width
1.5
1
0.5
0
-0.5 0
5
10
15
20
Uniformly_spaced_Nd=2
Non_Uniformly_spaced_Nd=2
Non_Uniformly_spaced_Nd=3
Non_Uniformly_spaced_Nd=4
Non_Uniformly_spaced_Nd=5
Non_Uniformly_spaced_Nd=6
Non_Uniformly_spaced_Nd=7
Non_Uniformly_spaced_Nd=8
Non_Uniformly_spaced_Nd=9
25 Non_Uniformly_spaced_Nd=10
frequency (GHz)
Figure 29: Simulation Plot for Beamwidth of Concentric Ring Array of Length
0.12m
54
From the above figure 29, when comparing beamwidth from nd = 3 to 10, optimum
performance is observed at nd = 10, which corresponds to an alm0 of 8.79.
The figure 30 below is the response of beamwidth over 1-20GHz of dipole length 0.1m
for nd= 2 to 10.
4.5
Beam
Width
Uniformly_spaced_Nd=2
4
Non_Uniformly_spaced_Nd=2
3.5
Non_Uniformly_spaced_Nd=3
3
Non_Uniformly_spaced_Nd=4
2.5
Non_Uniformly_spaced_Nd=5
2
Non_Uniformly_spaced_Nd=6
1.5
Non_Uniformly_spaced_Nd=7
1
Non_Uniformly_spaced_Nd=8
0.5
Non_Uniformly_spaced_Nd=9
0
-0.5 0
5
10
15
20
25
Non_Uniformly_spaced_Nd=10
frequency (GHz)
Figure 30: Simulation Plot for Beamwidth of Concentric Ring Array of Length 0.1m
for Nd= 2 To 10
From the above figure 30, when comparing beamwidth from nd = 3 to 10, optimum
performance is observed at nd = 2, which corresponds to an alm0 of 3.76.
The figure 31 below is the response of beamwidth over 1-20GHz of dipole length 0.08m
for nd= 2 to 10.
55
5
Uniformly_spaced_Nd=2
Non_Uniformly_spaced_Nd=2
Non_Uniformly_spaced_Nd=3
Non_Uniformly_spaced_Nd=4
Non_Uniformly_spaced_Nd=5
Non_Uniformly_spaced_Nd=6
Non_Uniformly_spaced_Nd=7
Non_Uniformly_spaced_Nd=8
Non_Uniformly_spaced_Nd=9
25
Non_Uniformly_spaced_Nd=10
4
3
Beam
2
Width
1
0
0
5
10
15
20
-1
frequency (GHz)
Figure 31: Simulation Plot for Beamwidth of Concentric Ring Array of Length
0.08m
From the above figure 31, when comparing beamwidth from nd = 3 to 10, optimum
performance is observed at nd = 9, which corresponds to an alm0 of 8.1.
The figure 32 below is the response of beamwidth over 1-20GHz of dipole length 0.075m
for nd= 2 to 10.
Beam
Width
6
Uniformly_spaced_Nd=2
5
Non_Uniformly_spaced_Nd=2
Non_Uniformly_spaced_Nd=3
4
Non_Uniformly_spaced_Nd=4
Non_Uniformly_spaced_Nd=5
3
Non_Uniformly_spaced_Nd=6
2
Non_Uniformly_spaced_Nd=7
1
Non_Uniformly_spaced_Nd=8
0
Non_Uniformly_spaced_Nd=9
-1
0
5
10
15
20
Non_Uniformly_spaced_Nd=10
25
frequency (GHz)
Figure 32: Simulation Plot for Beamwidth of Concentric Ring Array of Length
0.075m
56
From the above figure 32, when comparing beamwidth from nd = 3 to 10, optimum
performance is observed at nd = 9, which corresponds to an alm0 of 8.1.
The figure 33 below is the response of beamwidth over 1-20GHz of dipole length 0.06m
for nd= 2 to 10
7
6
5
Beam 4
Width 3
2
1
0
-1 0
5
10
15
20
Uniformly_spaced_Nd=2
Non_Uniformly_spaced_Nd=2
Non_Uniformly_spaced_Nd=3
Non_Uniformly_spaced_Nd=4
Non_Uniformly_spaced_Nd=5
Non_Uniformly_spaced_Nd=6
Non_Uniformly_spaced_Nd=7
Non_Uniformly_spaced_Nd=8
Non_Uniformly_spaced_Nd=9
Non_Uniformly_spaced_Nd=10
25
frequency (GHz)
Figure 33: Simulation Plot for Beamwidth of Concentric Ring Array of Length
0.06m
From the above figure 33, when comparing beamwidth from nd = 3 to 10, optimum
performance is observed at nd = 10, which corresponds to an alm0 of 8.79.
From the above set of graphs it is clear that beamwidth of unequally spaced antennas
(Non_Uniformly_spaced_nd=3, 4, 5, 6, 7, 8, 9, 10, 11) is flatter compared to the
beamwidth of equally spaced array (Uniformly_spaced_nd=2).
The following graphs in this section show the sidelobe levels for uniform and non
uniform array antennas over the frequency range of 1-20GHz for different lengths of
dipole.
57
The figure 34 below is the response of PSSL over 1-20GHz of dipole length 0.15m for
nd= 2 to 10. X-axis is frequency in GHz; Y-axis is PSSL in db.
120
100
80
PSSL 60
40
20
0
0
5
10
15
20
Uniformly_spaced_Nd=2
Non_Uniformly_spaced_Nd=2
Non_Uniformly_spaced_Nd=3
Non_Uniformly_spaced_Nd=4
Non_Uniformly_spaced_Nd=5
Non_Uniformly_spaced_Nd=6
Non_Uniformly_spaced_Nd=7
Non_Uniformly_spaced_Nd=8
Non_Uniformly_spaced_Nd=9
25 Non_Uniformly_spaced_Nd=10
frequency (GHz)
Figure 34: Simulation Plot for PSSL of Concentric Ring Array of Length 0.15m
From the above figure 34, when comparing PSSL from nd = 3 to 10, optimum
performance is observed at nd = 10, which corresponds to an alm0 of 8.79.
The figure 35 below is the response of PSSL over 1-20GHz of dipole length 0.12m for
nd= 2 to 10.
58
120
Uniformly_spaced_Nd=2
100
Non_Uniformly_spaced_Nd=2
Non_Uniformly_spaced_Nd=3
80
PSSL
Non_Uniformly_spaced_Nd=4
60
Non_Uniformly_spaced_Nd=5
Non_Uniformly_spaced_Nd=6
40
Non_Uniformly_spaced_Nd=7
20
Non_Uniformly_spaced_Nd=8
0
Non_Uniformly_spaced_Nd=9
0
5
10
15
20
25 Non_Uniformly_spaced_Nd=10
frequency (GHz)
Figure 35: Simulation Plot for PSSL of Concentric Ring Array of Length 0.12m
From the above figure 35, when comparing PSSL from nd = 3 to 10, optimum
performance is observed at nd = 9, which corresponds to an alm0 of 8.1.
The figure 36 below is the response of PSSL over 1-20GHz of dipole length 0.1m for nd=
2 to 10.
120
Uniformly_spaced_Nd=2
PSSL
100
Non_Uniformly_spaced_Nd=2
80
Non_Uniformly_spaced_Nd=3
Non_Uniformly_spaced_Nd=4
60
Non_Uniformly_spaced_Nd=5
40
Non_Uniformly_spaced_Nd=6
Non_Uniformly_spaced_Nd=7
20
Non_Uniformly_spaced_Nd=8
0
Non_Uniformly_spaced_Nd=9
0
5
10
15
20
25 Non_Uniformly_spaced_Nd=10
frequency (GHz)
Figure 36: Simulation Plot for PSSL of Concentric Ring Array of Length
0.1m
59
From the above figure 36, when comparing PSSL from nd = 3 to 10, optimum
performance is observed at nd = 7, which corresponds to an alm0 of 6.91.
The figure 37 below is the response of PSSL over 1-20GHz of dipole length 0.08m for
nd= 2 to 10.
PSSL
120
Uniformly_spaced_Nd=2
100
Non_Uniformly_spaced_Nd=2
Non_Uniformly_spaced_Nd=3
80
Non_Uniformly_spaced_Nd=4
Non_Uniformly_spaced_Nd=5
60
Non_Uniformly_spaced_Nd=6
40
Non_Uniformly_spaced_Nd=7
Non_Uniformly_spaced_Nd=8
20
Non_Uniformly_spaced_Nd=9
Non_Uniformly_spaced_Nd=10
0
0
5
10
15
20
25
frequency (GHz)
Figure 37: Simulation Plot for PSSL Of Concentric Ring Array Of Length
0.08m
From the above figure 37, when comparing PSSL from nd = 3 to 10, optimum
performance is observed at nd = 7, which corresponds to an alm0 of 6.91.
60
The figure 38 below is the response of PSSL over 1-20GHz of dipole length 0.075m for
nd= 2 to 10.
PSSL
120
Uniformly_spaced_Nd=2
Non_Uniformly_spaced_Nd=2
100
Non_Uniformly_spaced_Nd=3
80
Non_Uniformly_spaced_Nd=4
Non_Uniformly_spaced_Nd=5
60
Non_Uniformly_spaced_Nd=6
40
Non_Uniformly_spaced_Nd=7
Non_Uniformly_spaced_Nd=8
20
Non_Uniformly_spaced_Nd=9
0
0
5
10
15
20
25
Non_Uniformly_spaced_Nd=10
frequency (GHz)
Figure 38: Simulation plot for PSSL of concentric ring array of length
0.075m
From the above figure 38, when comparing PSSL from nd = 3 to 10, optimum
performance is observed at nd = 7, which corresponds to an alm0 of 6.91.
61
The figure 39 below is the response of PSSL over 1-20GHz of dipole length 0.06m for
nd= 2 to 10.
PSSL
120
Uniformly_spaced_Nd=2
100
Non_Uniformly_spaced_Nd=2
Non_Uniformly_spaced_Nd=3
80
Non_Uniformly_spaced_Nd=4
60
Non_Uniformly_spaced_Nd=5
Non_Uniformly_spaced_Nd=6
40
Non_Uniformly_spaced_Nd=7
Non_Uniformly_spaced_Nd=8
20
Non_Uniformly_spaced_Nd=9
0
0
5
10
15
20
25
Non_Uniformly_spaced_Nd=10
frequency (GHz)
Figure 39: Simulation Plot for PSSL of Concentric Ring Array of Length 0.06m
From the above figure 39, when comparing PSSL from nd = 3 to 10, optimum
performance is observed at nd = 7, which corresponds to an alm0 of 6.91.
From the above set of graphs it is clear that PSSL of unequally spaced antennas
(Non_Uniformly_spaced_nd=3, 4, 5, 6, 7, 8, 9, 10, 11) is flatter compared to the PSSL of
equally spaced array (Uniformly_spaced_nd =2)
62
Chapter 5
CONCLUSION
The main scope of this project was to observe the wideband behavior of non uniform
antenna arrays. This study is an extension to previous work on design of nonuniform
antenna array. Wideband behavior is observed by employing unequal spacing between
traditional uniform antennas. Antenna properties like gain, beamwidth and sidelobe levels
are observed over the wide range of frequencies of range of 1-20GHz. These properties
are plotted on the same graph to observe the improvement. From these graphs, we
observed that the gain, beamwidth and sidelobe levels are more uniform over the
frequency range compared to that of uniform antenna array. There is significant
improvement in all the antenna properties for linear antenna array. There is improvement
in sidelobe levels and beamwidth for concentric ring antenna array.
The future scope of the project could be in further improving the performance.
Gain, beamwidth and sidelobe levels are not completely flat over the wide frequency
range for linear and concentric ring antennas.
63
APPENDIX I
MATLAB Code for Linear Antenna Arrays
PROGRAM 1
This program generates the spacing between the antenna elements in Linear Array
%
SYNTHESIS OF UNEQUALLY SPACED ARRAYS BY LEGENDRE
POLYNOMIAL METHOD
% IDEAL STARTING FUNCTIONS-PENCIL BEAM
clear;
na=18;
nu=361;
umm=0.01;
na1=na;
na2=2*na-1;
na3=2*4+1;
um1=2./na2;
tpi=pi*2.;
ipath=1;
um=1.;
du=um/(nu-1);
nu1=nu;
du1=du;
xna=na;
dx(1)=0;
dx1(1)=0;
dxx=pi;
c(1)=1.0;
for i=2:na;
i1=i-1;
c(i)=2.0;
end;
for i=2:na1;
dx1(i)=dx1(i-1)+pi;
end;
for i=1:nu;
xi=i;
u=(xi-1)*du;
64
b(i)=1.0;
if u > um1 b(i)=umm;
end;
end;
ndxa=10;
dxaa=1./ndxa;
nd=input('enter value of nd between 1 and 11----=');
nu=nu1;
du=du1;
dxa=(nd-1)*dxaa;
alm=du*(dx(1)+dxa*pi);
thn(1)=du*dx(1);
c0=f2(alm,b,pi,nu)/f1(alm,thn(1));
for n1=1:na;
c1(n1)=c(n1)*c0/c(1);
end;
for n=2:na;
alm=du*(dx(n-1)+pi+dxa*pi);
s=0.0;
for m=1:n-1;
s=s+c1(m)*f1(alm,thn(m));
end;
s1=f2(alm,b,pi,nu)-s;
thn(n)=((2*c1(n)*c1(n)/s1/s1)+cos(alm));
if abs(thn(n))>1 thn(n)=cos(du*(dx(n-1)+pi));
end;
thn(n)=acos(thn(n));
if thn(n)>= alm thn(n)=(du*(dx(n-1)+pi));
end;
dx(n)=thn(n)/du;
dxx=dx(n)-dx(n-1);
if dxx<pi dx(n)=dx(n-1)+pi;
end;
if dxx<pi thn(n)=du*dx(n);
end;
end;
65
nu=361;
du=um/(nu-1);
for i=1:nu;
xi=i;
u=(xi-1)*du;
if u>1 u=2.-u;
end;
s1=0.0;
for l=1:na;
l1=l-1;
s1=s1+c(l)*cos(dx1(l)*u);
end;
b1(i)=s1;
end;
for i=1:nu;
u=(i-1)*du;
s=0.0;
for l=1:na;
l1=l-1;
s=s+c(l)*cos(dx(l)*u);
end ;
b2(i)=s;
end;
bm2=samax(b2,nu);
bm1=samax(b1,nu);
for m=1:nu;
u=(m-1)*du;
b2(m)=abs(b2(m)/bm2);
b1(m)=abs(b1(m)/bm1);
end;
dx=dx/2/pi
alpha=(nd-1)*dxaa*pi/2/pi
function bw=bw(a,n,du)
for i=2:n;
bwx=1./sqrt(2.);
if a(i+1)<bwx & a(i-1)>bwx k=i;
end;
bw=(k-1)*du;
66
bw=acos(bw)*180./pi;
bw=bw-90;
end;
function f1=f1(alm,thn)
pr=sqrt(2.);
c1=cos(thn)-cos(alm);
if alm<thn f1=0.0;
elseif alm>=thn;
c1=sqrt(c1);
f1=pr/c1;
end;
function f2=f2(alm,b,pi,nu)
x=cos(alm);
x2=(1.-x)/2.;
x1=sqrt(x2);
x1=sin(x1)^2;
[k,e] = ellipke(x1);
p1=2.*e/pi;
p(1)=2.*(2.*e-k)/pi;
p(2)=4.*x*p(1)/3.-p1/3.;
for l=3:nu;
l1=l-1;
xl=2.*l1;
xl1=(-0.5+l1);
xl2=0.5+l1;
p(l)=(xl*x*p(l-1)-p(l-2)*xl1)/xl2;
end;
s=b(1)*p1;
for l=1:nu-1;
s2=b(l+1)*p(l);
s=s+2.*s2;
end;
f2=s;
function pssl=pssl(a,n)
k=0;
for i=2,n;
67
if a(i-1)<a(i)& a(i+1)< a(i)k=k+1;
end;
if a(i-1)< a(i)& a(i+1)< a(i) a1(k)=a(i);
end;
fssl=a1(1);
pssl=samax(a1,k);
end;
function samax=samax(a,nn)
s=a(1);
for i=2:nn;
s=max(s,a(i));
end;
samax=s;
68
PROGRAM 2
This program plots the gain, beamwidth and sidelobe levels over the frequency range of 1
to 20 GHz
%% Wideband dipole pattern
clear ;
f0=1e9;
f1=1e9;
f2=20e9;
f3=5e9;
c=3e8;
fc=f0;
lambdac=c/f3;
lambda0=c/f3;
dspac=lambdac/2;
ldipole=142.65/(f3/1e6);
M=18;
for i = 1:M;
a(i) =1.0;
du(i)= (i)*dspac;
end;
dnu=[0 0.5 1.0 1.5 2.0 2.75 3.5706 4.4343 5.3304 6.2507 7.1888 8.1400 9.1007 10.0684
11.0411 12.0176 12.9968 13.9780]*lambdac;
N1 = 50;
df=(f2-f1)/N1;
for fn=1:N1;
69
f=f1+(fn-1)*df;
ff(fn)=f;
lambda=c/f;
k=2*pi/lambda;
N=361;
dth=pi/(N-1);
% Generation of two-dimensional electric field E(theta, phi)
for i1=1:N;
theta=(i1-1)*dth;
theta1(i1)=theta;
s=0.0;
s1=0.0;
s2=0.0;
for m=1:M;
s1=s1+ a(m)*exp(j*k*du(m)*cos(theta));
s2=s2+ a(m)*exp(j*k*dnu(m)*cos(theta));
end
eth1(i1)=s1*(cos(pi*ldipole*cos(theta)/lambda)-cos(pi*ldipole/lambda))/sin(theta);
eth2(i1)=s2*(cos(pi*ldipole*cos(theta)/lambda)-cos(pi*ldipole/lambda))/sin(theta);
end;
[y,im1]=max(abs(eth1));
ethm1(fn)=y;
[y,im2]=max(abs(eth2));
ethm2(fn)=y;
bwx=abs(ethm1(fn))/sqrt(2.);
for i=im1:N-1;
if abs(eth1(i+1))< bwx & abs(eth1(i-1))>bwx k=i
break
70
end;
end;
bw1(fn)=(k-im1)*dth*180/pi;
%bw1(fn)=acos(bw)*180./pi;
bwx=abs(ethm2(fn))/sqrt(2.);
for i=im2:N-1;
if abs(eth2(i+1))< bwx & abs(eth2(i-1))>bwx k=i
break;
end;
end;
bw2(fn)=(k-im2)*dth*180/pi;
% Peak Sidelobe level
k1=0;
for i=im1+1:N-1;
if abs(eth1(i+1))< abs(eth1(i)) & abs(eth1(i-1))< abs(eth1(i)) k1=k1+1;
s11(k1)=abs(eth1(i));
end;
end;
pssl1(fn)=max(s11);
k1=0;
for i=im2+1:N-1;
if abs(eth2(i+1))< abs(eth2(i)) & abs(eth2(i-1))< abs(eth2(i)) k1=k1+1;
s12(k1)=abs(eth2(i));
end;
end;
pssl2(fn)=max(s12);
end;
%ethm=max(ethm0,ethm1);
%ethm=max(ethm,ethm2);
subplot(3,1,1);
plot(ff,ethm1,'.',ff,ethm2,'-');
subplot(3,1,2);
plot(ff,bw1,'.',ff,bw2,'-');
subplot(3,1,3);
plot(ff,pssl1,'.',ff,pssl2,'-');
71
delta1=max(ethm1)-min(ethm1)
delta2=max(ethm2)-min(ethm2)
delta3=max(bw1)-min(bw1)
delta4=max(bw2)-min(bw2)
delta5=max(pssl1)-min(pssl1)
delta6=max(pssl2)-min(pssl2)
72
APPENDIX II
MATLAB Code for Concentric Ring Antenna Arrays
PROGRAM 1
This program generates the spacing between the antenna elements in Ring Array
% SYNTHESIS OF UNEQUALLY SPACED CONCENTRIC RING ARRAY
% NA=NUMBER OF RINGS IN THE ARRAY
% NU=NUMBER OF POINTS IN THE ARRAY FACTOR PATTERN
%
DXXX=DEFAULT SPACING BETWEEN ADJACENT RINGS (IN
WAVELENGTHS)
% C(I)=CURRENT IN I'TH RING OF THE ARRAY
% DX1(I)=RADIUS OF I'TH RING IN NONUNIFORM ARRAY
% DX(I)=RADIUS OF I'TH RING IN DEFAULT UNIFORM ARRAY
% UM1=WIDTH OF MAINLOBE IN THE DESIRED TARGET PATTERN
% UMM=MAGNITUDE OF DESIRED TARGET PATTERN OUTSIDE MAINLOBE
clear;
na=10;
nu=721;
umm=0.01;
dxxx=0.5;
na1=na;
na2=2*na-1;
na3=2*4+1;
tpi=pi*2.;
ipath=1;
um=1.;
du=um/(nu-1);
nu1=nu;
du1=du;
xna=na;
dx(1)=dxxx*tpi;
dx1(1)=dxxx*tpi;
dxx=dxxx*tpi;
c(1)=1.0;
for i=2:na;
i1=i-1;
c(i)=i;
end ;
73
for i=2:na1;
dx1(i)=dx1(i-1)+dxx;
end ;
for i=1:nu;
u=(i-1)*du;
s=0.0;
for n=1:na;
u1=dx1(n)*u;
s=s+c(n)*besselj(0,u1);
end;
b2(i)=s;
end ;
b2m=max(b2);
for i=1:nu;
b2(i)=b2(i)/b2m;
b2(i)=abs(b2(i));
end ;
k=1;
for i=2:n;
if(b2(i+1)>b2(i)& b2(i-1)>b2(i))k=i;
end;
zero1=(k-1)*du;
end;
um1=zero1;
for i=1:nu;
xi=i;
u=(xi-1)*du;
b(i)=1.0;
if(u > um1)b(i)=umm;
end;
b(i)=b(i);
end;
ndxa=10;
dxaa=2./ndxa;
nd=1; % number between 1 and 10
74
nu=nu1;
du=du1;
dxa=(nd-1)*dxaa;
alm0=dx(1)+dxa*dxx;
alm=alm0;
thn(1)=dx(1);
thnn=thn(1);
for l=1:nu;
u=(l-1)*du;
uu(l)=u;
p(l)=b(l)*sin(alm0*u);
end;
c1=alm*alm-thnn*thnn;
if(alm <= thnn)f1=0.0;
else f1=1./sqrt(c1);
end;
s=trapz(uu,p);
f2=s;
c0=f2/f1;
for n1=1:na;
c2(n1)=c(n1)*c0/c(1);
end ;
for n=2:na;
alm=dx(n-1)+dxx+dxa*dxx;
s=0.0;
for m=1:n-1;
c1=alm*alm-thnn*thnn;
if(alm <= thnn)f1=0.0;
else f1=1./sqrt(c1);
end;
s=s+c2(m)*f1;
end ;
for l=1:nu;
u=(l-1)*du;
uu(l)=u;
p(l)=b(l)*sin(alm*u);
end;
f2=trapz(uu,p);
75
s1=f2-s;
thn1=dx(n-1)+dxx;
thn(n)=alm*alm-c2(n)*c2(n)/s1/s1;
if(thn(n)< 0.0)thn(n)=thn1*thn1;
end;
thn(n)=sqrt(thn(n));
if(thn(n) >= alm)thn(n)=dx(n-1)+dxx;
end;
dx(n)=thn(n);
dxy=dx(n)-dx(n-1);
if(dxy < dxx)dx(n)=dx(n-1)+dxx;
end;
if(dxy < dxx)thn(n)=dx(n);
end;
end;
dx=thn/2/pi
76
PROGRAM 2
This code plots the gain, beamwidth and sidelobe levels for concentric ring array
%% Wideband dipole pattern
clear ;
f0=1e9;
f1=1e9;
f2=20e9;
f3=5e9;
c=3e8;
fc=f0;
lambdac=c/f3;
lambda0=c/f3;
dspac=lambdac/2;
M=10;
ldipole=142.65/(f3/1e6);
a=[1 6 12 18 24 30 36 42 48 54];
for i = 1:M;
%a(i)=1;
du(i)= (i)*dspac;
end;
dnu=[0.5 1.0 1.5 2.0 2.5 3.0 3.68 4.44 5.26 6.11]*lambda0;
N1 = 50;
df=(f2-f1)/N1;
for fn=1:N1;
f=f1+(fn-1)*df;
ff(fn)=f;
77
lambda=c/f;
k=2*pi/lambda;
N=361;
dth=pi/(N-1);
% Generation of two-dimensional electric field E(theta, phi)
for i1=1:N;
theta=(i1-1)*dth;
theta1(i1)=theta;
s=0.0;
s1=0.0;
s2=0.0;
for m=1:M;
s1=s1+ a(m)*besselj(0,k*du(m)*cos(theta));
s2=s2+ a(m)*besselj(0,k*dnu(m)*cos(theta));
end;
eth1(i1)=s1*(cos(pi*ldipole*cos(theta)/lambda)-cos(pi*ldipole/lambda))/sin(theta);
eth2(i1)=s2*(cos(pi*ldipole*cos(theta)/lambda)-cos(pi*ldipole/lambda))/sin(theta);
end;
[y,im1]=max(abs(eth1));
ethm1(fn)=y;
[y,im2]=max(abs(eth2));
ethm2(fn)=y;
bwx=abs(ethm1(fn))/sqrt(2.);
for i=im1:N-1;
if abs(eth1(i+1))< bwx & abs(eth1(i-1))>bwx k=i
break
end;
end;
bw1(fn)=(k-im1)*dth*180/pi;
78
bwx=abs(ethm2(fn))/sqrt(2.);
for i=im2:N-1;
if abs(eth2(i+1))< bwx & abs(eth2(i-1))>bwx k=i
break;
end;
end;
bw2(fn)=(k-im2)*dth*180/pi;
% Peak Sidelobe level
k1=0;
for i=im1+1:N-1;
if abs(eth1(i+1))< abs(eth1(i)) & abs(eth1(i-1))< abs(eth1(i)) k1=k1+1;
s11(k1)=abs(eth1(i));
end;
end;
pssl1(fn)=max(s11);
k1=0;
for i=im2+1:N-1;
if abs(eth2(i+1))< abs(eth2(i)) & abs(eth2(i-1))< abs(eth2(i)) k1=k1+1;
s12(k1)=abs(eth2(i));
end;
end;
pssl2(fn)=max(s12);
end;
%ethm=max(ethm0,ethm1);
%ethm=max(ethm,ethm2);
subplot(3,1,1);
plot(ff,ethm1,'x',ff,ethm2,'o');
subplot(3,1,2);
plot(ff,bw1,'x',ff,bw2,'o');
subplot(3,1,3);
plot(ff,pssl1,'x',ff,pssl2,'o');
delta1=max(ethm1)-min(ethm1)
delta2=max(ethm2)-min(ethm2)
delta3=max(bw1)-min(bw1)
delta4=max(bw2)-min(bw2)
delta5=max(pssl1)-min(pssl1)
delta6=max(pssl2)-min(pssl2)
79
REFERENCES
[1]. Kumar B.P. and G. R. Branner, ‘Design of unequally spaced arrays for performance
improvement’, IEEE Transactions on Antennas and Propagation (Special Issue on Phased
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[2]. Yu C., 'Sidelobe reduction of asymmetric linear array by spacing perturbation',
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[3]. Kumar B.P. and Branner G.R., ‘Generalized Analytical Technique for the Synthesis
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IEEE Transactions on Antennas and Propagation, Vol. 53, pp. 621-634, Feb. 2005.
[4]. Prasannan Sreepriya,’synthesis of unequally spaced antenna arrays for wideband
applications in radio astronomy’, Master of Science Project, Department of Electrical and
Electronic Engineering, CSUS, December 2010.
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genetic algorithm’, Advanced Communication Technology (ICACT), 2011 13th
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[6]. Ming-Iu Lai, Chin-Feng Liu, Shyh-Kang Jeng., ‘Design of a multifunctional and
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[7]. lee, A., Chen, L, Song, A., Wei, J., Hwang, H.K.,’Simulation Study of Wideband
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80
[8]. Yuan Shen, Win, M.Z., ‘Performance of Localization and Orientation Using
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[11]. “The Basics of Antenna Arrays”, retrieved from
http://www.orbanmicrowave.com/The_Basics_of_Antenna_Arrays.pdf
[12]. “Wideband corporation website”, retrieved from
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[13]. Peter Joseph Bevelacqua., ‘Antenna arrays: performance limits and geometry
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