SYNTHESIS OF UNEQUALLY SPACED ANTENNA ARRAYS FOR WIDEBAND
APPLICATIONS IN RADIO ASTRONOMY
Sreepriya Prasannan
B.Tech., Mahatma Gandhi University, India, 2006
PROJECT
Submitted in partial satisfaction of the requirements for the degree of
MASTER OF SCIENCE in
ELECTRICAL AND ELECTRONIC ENGINEERING
at
CALIFORNIA STATE UNIVERSITY, SACRAMENTO
FALL
2010

SYNTHESIS OF UNEQUALLY SPACED ANTENNA ARRAYS FOR WIDEBAND
APPLICATIONS IN RADIO ASTRONOMY
A Project
By
Sreepriya Prasannan
Date
Approved by:
__________________________________, Committee Chair
Preetham B. Kumar, Ph.D.
__________________________________, Second Reader
Fethi Belkhouche, Ph.D.
____________________________ ii

Student: Sreepriya Prasannan
I certify that the student has 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 ________________
Date Preetham B. Kumar, Ph.D.
Department of Electrical & Electronics Engineering iii

Abstract of
SYNTHESIS OF UNEQUALLY SPACED ANTENNA ARRAYS FOR WIDEBAND
APPLICATIONS IN RADIO ASTRONOMY by
Sreepriya Prasannan
This project will focus on the design of wideband antenna arrays, which can operate in the 1-22 GHz band. The design process utilizes the principle of providing unequal spacing between adjacent elements of the array, which has the desired effect of significantly increasing the bandwidth of a default uniformly spaced array. This project will have applications such as radio astronomy. Both one-dimensional and twodimensional array geometries will be considered for simulation.
_______________________, Committee Chair
Preetham B. Kumar, Ph.D.
_______________________
Date iv

ACKNOWLEDGMENT
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. v

TABLE OF CONTENTS
Page
Acknowledgement……………………………………………………………………..v
List of Tables…………………………………………………………………………viii
List of Figures………………………………………………………………………….ix
Chapters:
1. INTRODUCTION…………………………………………………………………...1
2. UNIFIED ANALYTICAL TECHNIQUE FOR WIDEBAND NONLINEAR
SYNTHESIS OF ANTENNA ARRAYS……………………………………………...4
2.1. Linear Synthesis………………………………………………………………......5
2.2 .Nonlinear Synthesis……………………………………………………………...5
3. SYNTHESIS OF NONUNIFORMLY SPACED WIDEBAND ARRAYS…………..8
3.1. Synthesis of Linear Arrays…………………………………………………........8
3.2. Synthesis of Wideband Concentric Ring Arrays…...............................................10
3.3. Synthesis of Wideband Hemispherical Arrays…………………………….........12
4. SIMULATION RESULTS FOR WIDEBAND PERFORMANCE………………….15
4.1. Linear Array……………………………………………………………………..16
4.2. Concentric Ring Array……………………………………………………………19
4.3. Hemispherical Array……………………………………………………………..22
5. CONCLUSION………………………………………………………………………27 vi

Appendix……………………………………………………………………………28
References…………………………………………………………………………....40 vii

LIST OF TABLES:
Page
1. Linear Array Element Positions …………………………………………………..16
2. Concentric Ring Array Element Positions………………………………………...20
3. Hemispherical Ring Array Element Positions……………………………….........23 viii

LIST OF FIGURES
Page
1.
Geometry of 2N+1 element non-periodic symmetric linear array…………....8
2. Geometry of circular ring array……………………………………………...10
3. Coordinate geometry of hemispherical array………………………………...12
4. Simulation plot for uniform linear array radiation pattern………………....18
5. Simulation plot for nonuniform linear array radiation pattern……………..19
6. Simulation plot for uniform concentric ring array radiation pattern………. 21
7. Simulation plot for nonuniform concentric ring array radiation pattern…….22
8. Simulation plot for uniform hemipherical array radiation pattern…………..25
9. Simulation plot for nonuniform hemispherical array radiation pattern..........26 ix

1
Chapter 1
INTRODUCTION
Radio Astronomy applications primarily require high gain, wideband antennas with a capability to detect even very weak signals from distant sources. Antenna arrays are an effective solution to provide a highly focused beam, with low sidelobes[1]. However, the performance and efficiency of a uniformly spaced array can be significantly improved by employing non-uniform spacing [4]. In comparison with the classical uniform structure, this approach can yield decreased sidelobe levels and narrow beamwidth values over a very wide frequency range of 100 MHz to 25 GHz.
Additionally, such a structure provides practical advantages such as reductions in size, weight and number of antenna elements. The objective of this project is to extend the earlier unified mathematical design of multi-dimensional non-uniformly spaced array geometries to wideband applications [3]. The approach utilizes a class of limiting properties of sinusoidal, Bessel or Legendre functions that are dictated by the array geometry addressed. The method provides element spacing values directly by a single application of the earlier developed algorithm, and permits the array designer to place practical constraints on the array geometry, (e.g. the minimum/maximum spacing between adjacent elements).
This project will focus on the design of wideband antenna arrays, which can operate in the 1-22 GHz band. The design process utilizes the principle of providing unequal spacing between adjacent elements of the array [4], which has the desired effect

2 of significantly increasing the bandwidth of a default uniformly spaced array.
Nonuniformly spaced wideband arrays having linear, planar, and cylindrical or spherical geometries are considered for simulation for this project [3]
The project compares the simulation results of the three different geometries mentioned above and demonstrates the wideband performances of uniform and nonuniform spaced antenna arrays. For the simulation [3] a desired array factor synthesis pattern is specified (whether linear, planar and cylindrical). Then the Legendre algorithm is applied to the prescribed pattern and the set of non-uniformly spaced array elements is obtained [5]. In this project about 18-element array of sources is obtained by the algorithm. Finally the improvement in the array pattern due to non-uniform spacing over uniform spacing is illustrated graphically over a definite frequency range. It will be clear from the results obtained with the three different set of geometries that the sidelobe levels are far reduced in the case of the non-uniformly spaced array as compared to the uniformly spaced array.
This project is organized as follows. Chapter 2 discusses the review of wideband array synthesis problem which includes both the Linear Synthesis and Nonlinear
Synthesis. In Chapter 3 a brief description of Unified Analytical technique for wideband
Nonlinear Optimization of antenna arrays is presented. Chapter 4 deals with Synthesis of
Nonuniformly spaced wideband arrays which includes study of Linear, Concentric ring array and Hemispherical arrays one by one. In Chapter 5, Simulation Results of wideband
Performance is presented. Simulation results of uniformly and nonuniformly spaced linear, concentric ring and hemispherical arrays are discussed in this chapter as

3 subsections. Chapter 6 gives the conclusions of this project and mentions the future studies that can be performed in the area of the project. At last, the appendix lists
MATLAB codes written in this project to implement the synthesis of nonuniformly spaced arrays in linear, concentric ring and hemispherical arrays which clearly gives decreased sidelobe levels and narrow beamwidth values over a frequency range from 100
MHz to 25 GHz.

4
Chapter 2
UNIFIED ANALYTICAL TECHNIQUE FOR WIDEBAND NONLINEAR
SYNTHESIS OF ANTENNA ARRAYS
The general wideband antenna array synthesis problem can be described by the matrix equation








E k u
E k u
 
............. =
E k u
1
2
M
)
.............
)
)

















A k r u
1 1
A k r u
..........................
..............................................
........................................................................
A k r u
1
2
M
A k r u
2 1
A k r u  
A k r u
2
2
M
A k r u
)................ ( ,
N
,
N
1
,
2
)
A k r u
)
M
)










 
 
..
 
 
 
(1).
Where r = [r1 r
2 r
3
… r
N
] is the array element position vector, k = 2π ∕λ the propagation constant, and λ is the wavelength. [E(k, u
1
) E(k, u
2
) E(k, u
3
) …
E(k, u
M
)]
T
is the prescribed [M × 1] far-field array factor at M field points (u j
) where j= 1, 2, … , M and [ I
1 ,
I
2
, I
3
, … , I
N
]
T
is the [ N × 1] array element current vector, and [ A (k , r i
, u j
) ] is the M × N rectangular matrix, which is a function of frequency, element position (r i
) and field point u j
.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:

5
2.1. Linear 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 [I
1
, I
2
, I
3
, … ,I
N
] are obtained from eqn. (1) by employing linear synthesis. These synthesized currents may be real or complex.
2.2. Nonlinear 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 eqn. (1) [4] written in matrix form:
[E] = [A][I] (2).

6
Performing a linear transformation T on the array factor [E], equation (2) transforms as:
T[E]=T[A][I] (3).
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 eqn. (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 [I
1,
I
2,
I
3
, …, I
N
]
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 = [ r
1 r
2 r
3
…
.
r
N
] .
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
)


F (
2
..........
)





..........
F (
N
)








=








B r
B r

2
)
............................................................
........................................
B r
1
1
1



1
1
M
)
) B(r ,
2
) B(r ,
2

M
....................
B r
N

N
)
 
I
1

 
  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.

7
The following sections in the project will outline the application of this method to the nonlinear spacing synthesis of wideband arrays having linear, planar, and cylindrical or spherical geometries.

Chapter 3
SYNTHESIS OF NONUNIFORMLY SPACED WIDEBAND ARRAYS
3.1. Sythesis of Linear Arrays:
Consider a symmetric linear array of 2N + 1 elements as shown in the figure 1 below:
8
Figure 1: Geometry of 2N+1 element non-periodic symmetric linear array[1]
The array factor of this linear array is given as follows:
E(u) = n
N 

0
 nI n cos( kd n u ); (6). where u = cos(θ) , 0
   
radians, I n
and d n 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 I n are obtained using standard linear synthesis methods [1].
The nonlinear synthesis [1], is performed next. The nonlinear synthesis algorithm

9 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 E d
(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 :
E(u m
) = n
N 

0
 n
I n cos( m
 n
); (7). where m = 0 or M – 1, where u m
= m
 u , where
 u = 1/ (M - 1),
 n
= kd n
 u , k =
2

/

and element positions d n are shown in figure 1. The synthesis algorithm starts with the Legendre transformation of the desired array pattern E d
(u) as follows:
F(
 p
) =
M m


1  
0 m
E d
( u m
) P m

1 / 2
(cos
 p
) ; p = 0,1,2,3, … ,N (8).
Where
 m

1 , m

0 ;
 m

2 , m>0; and P m-1/2
is the Legendre function of half integer order. Finally, the synthesized position of the pth element is given by: 2I p
2
 p
= cos
-1







 F (
 p
)
2 I p
2
 p n

1 

0
I n f (
 p ,
 n
)


2
 cos(
 p
)






(9).
= [2/ (cos
 n
 cos
 p
) )]
1/2
, 0
  n

 p
and
 p
<
 n
<

.
In the equation f (
 p
,
 n
)= 0, and element position d p
=
 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

10 a constant and c > 0 is limited by the condition that the
 and

values intersperse one another as shown below:

Space:

0

1

2

3
………………………… 
N

Space:

0

1

2

3
…………………………… 
N
3.2. Sythesis of Wideband Concentric Ring Arrays:
The geometry of the concentric ring of array of point sources is shown in figure. 2.
Figure 2 : Geometry of circular ring array [1] .

11
The array factor of the concentric ring array is given as follows:
E(

,

) = i
M N 

1 j

1
I ij e jk
 i sin(

) cos(
   j ) (10). where N is the number of concentric rings, M i
is the number of elements in the ith ring, I ij 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. I ij
= 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
 
/ , the array factor can be simplified, after considerable manipulation, to yield
E (u) =
N  i

1
M i
J o
(k
 i
u) (11). where u = sin

and ‘ J o
’ 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:
a p
2
=





 b p
2 
M p
2

 F ( b p
)
 p n

1 

1
M n f ( b p
, a n
)


2






(12). and
 p
 a p
/( k ) , f(b p
, a n
)
 
[1/ ( a n
2  b p
2 1/2
)] for a n


0 for b p

 b p a n



where in Eqn(14) , F b p
)
 
0

E u d
( )sin( b u du p
) ; p=1,2,3, ... ,N the values of b p
, p = 1,2,3, … ,N are selected as follows: b p
 k (
 p
 
/ 2) .
3.3. Sythesis of Wideband Hemispherical Arrays :
The geometry of the hemispherical array of point sources is shown in Figure.3.
12
Figure 3: Coordinate geometry of hemispherical array[1]

13
The array factor of hemispherical array is given as follows:
E

N M  n n

1 m

1
I
 
' n m
) e jkr
0
  ' n
   ' n cos(
  ' m
)]
(13).
The array factor is defined in the interval 0

, 0

, where the primed coordinates refer to source points, whereas the unprimed coordinates refer to field observation points, and I
 
' refer to the two-dimensional current distribution of the array. Referring to Figure [3] and equation (15), the index n=1,N; defines the number of rings in the array, and the index m=1, M n
; defines the number of point sources in the nth ring. The two-dimensional current excitation is obtained from linear synthesis methods
[5]. The nonlinear synthesis [5] of unequally spaced hemispherical array comprises of six steps. Firstly, the Fourier transform of E
  is given by:
B
 
2

0

E ji
 d

(14).
This is followed by the Legendre transform of the above Fourier transform, to yield
C(l,i) =


0
B
 i P l i
  
(15). finally the design equation of pth elevation is given by:

' p
 cos

1





 cos( x p
)

I p
2
2


D x i p
, )

N  n p 1
I f n
 n
' x p
)


2






(16).

Where in Equation (16)
D( x i p
, )= l
L 

0
C l i l

0.5) x p
] f
 n
' x p
)

[2 / (cos
 n
'  cos x p
)]
1/2 for
0 for
0
 x
 p
 n
'  x p
<
 n
'



The ‘x’ and ‘  ’ values intersperse each another as shown below:
‘x Space’ : x x
1 2 x
3 x
4 x
N

Space:

1

2

3

4
................

N
14 and

15
Chapter 4
SIMULATION RESULTS FOR WIDEBAND PERFORMANCE
The following section studies and demonstrates the wideband performance of different antenna arrays corresponding to the three geometries (linear, ring and spherical) explained in the previous sections of this project. The simulation of each array is organized in the following steps:
1) A desired array factor synthesis pattern [7] is prescribed (either one-dimensional or two-dimensional, depending on the geometry of the array).
2) A default equally spaced array of point sources is defined.
3) Then the specific synthesis algorithm [7] described in the previous sections is applied to the prescribed pattern and the set of non-uniformly spaced array elements is obtained.
4) Finally the improvement in the array pattern due to non-uniform spacing is illustrated graphically over a wideband 25 decade frequency range from 100 MHz to 25 GHz. In all the simulation examples, the current distribution in all array geometries is assumed uniform; hence pencil beam broadside patterns are being synthesized.

16
4.1. Linear Array :
The prescribed pattern is a pencil beam pattern described by the following equation:
E(u) = 

E max
E min
; 0 u u m
; u <u 1.0
m


where u=cos ( ) and values of E and max
E min are:
E max
= 1.0, E min
= 0.001
, the dividing boundary u m
is assumed as the position of the first null (at u~ 2/2N+1) of the initial equally spaced array.
The synthesis algorithm [7] is applied to obtain the appropriate adjacent element spacings of a 19 element array of point sources, and the synthesized array spacings are presented in table [1] along with those of the uniform array, with the wavelength value corresponding to the lowest frequency of 100 MHz.
The table below shows the array element positions for uniform and non-uniform array generated by the synthesis algorithm which are given as input to the MATLAB© programs to generate the simulation plots for uniform and non-uniform array.
Table [1]: Linear Array Element Positions:
Nonuniform Array(

) Uniform Array(

)
0.50 0.50
1.00 1.00
1.50
2.00
2.50
3.00
3.50
4.00
4.58
5.23
5.91
1.50
2.00
2.50
3.00
3.50
4.00
4.50
5.00
5.50

17
6.61
7.33
8.07
8.82
9.58
10.34
11.11
6.00
6.50
7.00
7.50
8.00
8.50
9.00
Simulation Plots for Linear Array:
Color Schemes Used In simulation:
Since 3 different frequencies (100 MHz, 12.5 GHz and 25 GHz) are used in simulation, there different colors are used in representation .
Red- 100 MHz
Blue – 12.5 GHz
Yellow – 25 GHz.
Figure 4 compares the uniform array radiation patterns at three frequencies over the
100Mhz to 25 GHz frequency range: 100 MHz, 12.5 GHz and 25 GHz. Figure 5 compares the nonuniform array radiation patterns at three frequencies over the 100Mhz to
25 GHz frequency range: 100 MHz, 12.5 GHz and 25 GHz.

18
-2
-3
-4
-5
0
-1
-6
-7
-8
-9
-10
0
100 Mhz
12.5 Ghz
25 Ghz
20 40 60 80 theta, degrees
100 120 140
Figure 4: Simulation plot for uniform linear array radiation pattern
160

19
-2
-3
-4
-5
0
-1
-6
-7
-8
-9
100 Mhz
12.5 Ghz
25 Ghz
-10
0 20 40 60 80 theta, degrees
100 120 140 160
Figure 5: Simulation plot for nonuniform linear array radiation pattern
It is clearly seen that at almost all the mentioned frequencies the sidelobe levels are far reduced in the case of the nonuniformly spaced array, as compared to the uniformly spaced array [2].
4.2. Concentric Ring Array :
The application of the technique is directed to the synthesis of a pencil beam pattern given by the following expression: E(u) =


1 for 0 u u
0 for u m m
1.0



20
Where u = sin(

), and the dividing boundary u m
is assumed as the position of the first null of the initial equally spaced array. Following is the antenna array which is a planar circular array with 10 concentric rings obtained by synthesis algorithm [7]. The number of point source elements in each ring from center of array to the outermost ring is 1,
6,12,18,24,30,36,42, 48 and 54.
The designed values of adjacent spacing are given in Table [2], with the wavelength corresponding to the lowest frequency of 100 Mhz.
Table [2]: Concentric Ring Array Element Positions:
Nonuniform array (

)
0.50
1.00
1.50
2.00
2.50
3.00
3.68
4.44
5.26
6.11
Uniform array (

)
0.50
1.00
1.50
2.00
2.50
3.00
3.50
4.00
4.50
5.00
Simulation plots for Concentric Ring arrays :
Color Schemes Used In simulation plot:
Since 3 different frequencies (100 MHz, 12.5 GHz and 25 GHz) are used in simulation, there different colors are used in representation.
Red- 100 MHz
Blue – 12.5 GHz

21
Yellow – 25 GHz.
Figure 6 and Figure 7 compares the uniform and non-uniform array radiation patterns at the three wideband frequencies: 100 MHz, 12.5 GHz and 25 GHz.
-2
-3
-4
0
-1
-5
-6
-7
-8
-9
100 Mhz
12.5 Ghz
25 Ghz
-10
0 20 40 60 80 theta, degrees
100 120 140 160
Figure 6: Simulation plot for uniform concentric ring array radiation pattern

22
-6
-7
-8
-9
-4
-5
0
-1
-2
-3
100 Mhz
12.5 Ghz
25 Ghz
-10
0 20 40 60 80 theta,degrees
100 120 140 160
Figure 7: Simulation plot for nonuniform concentric ring array radiation pattern
While there are no grating lobes in both the uniformly spaced array and nonuniformly spaced arrays, the sidelobe levels are significantly lowered in the case of the nonuniformly spaced array [2].
4.3. Hemispherical Array :
The application of the technique is directed to the synthesis of a pencil beam having the following pattern:
E

1 when 0

 
0.01 when
 m m
and 0
 

and 0<

2


2




23
Details of the hemispherical array elements are as follows:

Number of elevation rings in array = N

Initial spacing of elevation rings in array =
 n
'
0
  n
' 
90
0
.
 0
N

1) in the range

Number of equally spaced point source elements in nth elevation ring = M n
 n

Total number of elements in array = (

1) / 2 . Uniform current distribution in each element: I (
 
' n m
) = 1 .
The hemispherical array has a radius of 2.72

(corresponding to the lowest frequency of
100 MHz), and has 7 rings with 28 elements (top ring with 1 element; 2 nd
ring with 2 elements; 3 rd ring with 3 elements; 4 th ring with 4 elements….. so on with 7 th ring with 7 elements). In the simulation plots given below, those compare the array factor patterns of the uniformly spaced array and non-uniformly spaced arrays.
Table [3] compares the ring radii of the synthesized non-uniform array.
Table 3: Hemispherical Ring Array Element Positions :
Uniform Array (

) Nonuniform Array (

)
0.1963 0.0253
0.3927 0.2830

0.5891
0.7854
0.9818
1.1781
1.3745
24
0.5871
0.8724
1.1357
1.1781
1.3745
Simulation plots :
Color Schemes Used In simulation:
Since 3 different frequencies (100 MHz, 12.5 GHz and 25 GHz) are used in simulation, there different colors are used in representation .
Red- 100 MHz
Blue – 12.5 GHz
Yellow – 25 GHz.

25
0
-1
-2
-3
-4
-5
-6
-7
-8
-9
-10
0
Figure 8 and Figure 9 compares the uniform and non-uniform array radiation patterns at the three wideband frequencies: 100 MHz, 12.5 GHz and 25 GHz.
20 40 60 80 theta, degrees
100 120
100 Mhz
12.5 Ghz
25 Ghz
140 160
Figure 8: Simulation plot for uniform hemispherical array radiation pattern

26
-4
-5
-6
-7
-2
-3
0
-1
-8
-9
-10
0
100 Mhz
12.5 Ghz
25 Ghz
20 40 60 80 theta, degrees
100 120 140 160
Figure 9: Simulation plot for nonuniform hemispherical array radiation pattern
The simulation patterns show that the sidelobe levels in the nonuniformly spaced arrays are significantly lowered as compared to uniformly spaced arrays [2].

27
Chapter 6
CONCLUSION
This PROJECT outlines a generalized method of wideband antenna array synthesis that is applicable to a broad class of antennas with linear, planar, cylindrical and spherical geometry. The method works on an original uniformly spaced array in any geometry, and synthesizes a target pencil beam pattern by altering the uniform spacing of the array. The basis of the method is the discovery and application of a general inversion process, existing in different geometries, which results in recursive determination of the new element positions of the array. Considerable improvement in peak sidelobe level is obtained in the all simulated geometries, in the frequency range of 100 MHz – 25 GHz.
Even in some cases grating lobes that appear in the uniformly spaced array are removed by the application of unequal spacing in the array. Future work in this area of project will focus on the experimental characterization of such array configurations for practical radio astronomy applications.

28
APPENDIX
MATLAB code for implementing the synthesis of unequally spaced arrays by Legendre polynomial method using pencil beam.
% SYNTHESIS OF UNEQUALLY SPACED ARRAYS BY LEGENDRE POLYNOMIAL METHOD
% IDEAL STARTING FUNCTIONS-PENCIL BEAM
clear;
na=18;
nu=361;
umm=0.01;
%read(*,*)na,nu,umm;
na1=na;
na2=2*na-1;
na3=2*4+1;
um1=2./na2;
% um1=0.637*pi/180;
% nu=4.*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.*sin(0.3*i1*pi)/(0.3*i1*pi);
% if(abs(c(i)).lt.0.25)c(i)=0.0;
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;
b(i)=1.0; if u > um1 b(i)=umm; end ; end ;
ndxa=10;
dxaa=1./ndxa;
%for nd=3:ndxa;

29
nd=input( 'enter value of nd between 1 and 11----=' ); % to get element spacings
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 ;
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 ;
% pssl1(nd)=pssl(b1,nu);
% pssl2(nd)=pssl(b2,nu);
% bw1=bw(b1,nu,du);
% bw2=bw(b2,nu,du);
% pss1=20.*log10(pssl1(nd));
% pss2=20.*log10(pssl2(nd));
% fss2=20.*log10(fssl);
% xnd=nd;
% for m1=1:nu;
% u=(m1-1)*du;
% bx1=b(m1);
%
% bx2=b2(m1);
% bx3=b1(m1);
%
%
% if bx1==0.0 bx1=0.001;
% end;
% bx1=20.*log10(bx1);
% if bx2==0.0 bx2=0.001;
% end;
% bx2=20.*log10(bx2);
% if bx3==0.0 bx3=0.001;
% end;
30

% bx3=20.*log10(bx3);
% end;
dx=dx/2/pi
alpha=(nd-1)*dxaa*pi/2/pi
MATLAB code Implementing the Simulation results for wideband performance:
For Linear Arrays:
%% Wideband dipole pattern clear ; f0=0.1e9; % frequencies entered are 100 Mhz, 12.5Ghz, 25 Ghz.
f1=12.5e9; f2=25e9; c=3e8; fc=f0; lambdac=c/fc; lambda0=c/f0; k0=2*pi/lambda0; lambda1=c/f1; k1=2*pi/lambda1; lambda2=c/f2; k2=2*pi/lambda2; dspac=lambdac/2;
M=18; ldipole=142.65/(f0/1e6); for i = 1:M;
a(i) =1.0;
du(i)= (i)*dspac;
31

32 end ; dnu=[0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00 5.50 6.00 6.50
7.00 7.50 8.00 8.50 9.00]*lambdac; % values entered for uniform array dnu=[0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.58 5.23 5.91 6.61 7.33
8.07 8.82 9.58 10.34 11.11]*lambdac; % values entered for nonuniform array
N=721; 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; s=s+ a(m)*exp(j*k0*du(m)*cos(theta)); s1=s1+ a(m)*exp(j*k1*du(m)*cos(theta)); s2=s2+ a(m)*exp(j*k2*du(m)*cos(theta)); end ; eth(i1)=s; eth1(i1)=s1; eth2(i1)=s2; end ; ethm0=max(abs(eth)); ethm1=max(abs(eth1)); ethm2=max(abs(eth2)); ethm=max(ethm0,ethm1); ethm=max(ethm,ethm2); figure (1); subplot(2,1,1); plot(theta1*180/pi,log10(abs(eth)/ethm)); hold on ; plot(theta1*180/pi,log10(abs(eth1)/ethm), '--' );

plot(theta1*180/pi,log10(abs(eth2)/ethm), '-.' ); axis([0 180 -50 0]); hold off ; x=[max((abs(eth))) max((abs(eth1))) max((abs(eth2)))]; for i1=1:N; theta=(i1-1)*dth; theta1(i1)=theta; s=0.0; s1=0.0; s2=0.0; for m=1:M; s=s+ a(m)*exp(j*k0*dnu(m)*cos(theta)); s1=s1+ a(m)*exp(j*k1*dnu(m)*cos(theta)); s2=s2+ a(m)*exp(j*k2*dnu(m)*cos(theta)); end ; eth(i1)=s; eth1(i1)=s1; eth2(i1)=s2; end ; ethm0=max(abs(eth)); ethm1=max(abs(eth1)); ethm2=max(abs(eth2)); ethm=max(ethm0,ethm1); ethm=max(ethm,ethm2); subplot(2,1,2); plot(theta1*180/pi,log10(abs(eth)/ethm)); hold on ; plot(theta1*180/pi,log10(abs(eth1)/ethm), '--' ); plot(theta1*180/pi,log10(abs(eth2)/ethm), '-.' ); axis([0 180 -50 0]); hold off ; y=[max((abs(eth))) max((abs(eth1))) max((abs(eth2)))]; n=[f1 f0 f2]/1.e9; figure(2); plot(n,x, '--' ,n,y, '-.' );
For Concentric Ring Arrays:
33

34
%% Wideband dipole pattern clear ; f0=0.1e9; % frequencies entered are 100 Mhz, 12.5Ghz, 25 Ghz.
f1=12.5e9; f2=25e9; c=3e8; lambda0=c/f0; k0=2*pi/lambda0; lambda1=c/f1; k1=2*pi/lambda1; lambda2=c/f2; k2=2*pi/lambda2; dspac=lambda0/2;
M=10; ldipole=142.65/(f0/1e6); a=[1 6 12 18 24 30 36 42 48 54]; for i = 1:M;
%a(i)=1;
du(i)= (i)*dspac; end ; dnu=[0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00]*lambda0; % values entered for uniform array dnu=[0.50 1.00 1.50 2.00 2.50 3.00 3.68 4.44 5.26 6.11]*lambda0; % values entered for nonuniform array
N=721; 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; s=s+ a(m)*besselj(0,k0*du(m)*cos(theta)); s1=s1+ a(m)*besselj(0,k1*du(m)*cos(theta)); s2=s2+ a(m)*besselj(0,k2*du(m)*cos(theta)); end ; eth(i1)=s; eth1(i1)=s1; eth2(i1)=s2; end ; ethm0=max(abs(eth)); ethm1=max(abs(eth1)); ethm2=max(abs(eth2)); ethm=max(ethm0,ethm1); ethm=max(ethm,ethm2); figure (1); subplot(2,1,1); plot(theta1*180/pi,log10(abs(eth)/ethm)); hold on ; plot(theta1*180/pi,log10(abs(eth1)/ethm), '--' ); plot(theta1*180/pi,log10(abs(eth2)/ethm), '-.' ); axis([0 180 -50 0]); hold off ; x=[max((abs(eth))) max((abs(eth1))) max((abs(eth2)))]; for i1=1:N; theta=(i1-1)*dth; theta1(i1)=theta; s=0.0; s1=0.0; s2=0.0; for m=1:M; s=s+ a(m)*besselj(0,k0*dnu(m)*cos(theta));;
35

s1=s1+ a(m)*besselj(0,k1*dnu(m)*cos(theta));; s2=s2+ a(m)*besselj(0,k2*dnu(m)*cos(theta));; end ; eth(i1)=s; eth1(i1)=s1; eth2(i1)=s2; end ; subplot(2,1,2); ethm0=max(abs(eth)); ethm1=max(abs(eth1)); ethm2=max(abs(eth2)); ethm=max(ethm0,ethm1); ethm=max(ethm,ethm2); plot(theta1*180/pi,log10(abs(eth)/ethm)); hold on ; plot(theta1*180/pi,log10(abs(eth1)/ethm), '--' ); plot(theta1*180/pi,log10(abs(eth2)/ethm), '-.' ); axis([0 180 -40 0]); hold off ; y=[max((abs(eth))) max((abs(eth1))) max((abs(eth2)))]; n=[f1 f0 f2]/1.e9; figure(2); plot(n,x, '--' ,n,y, '-.' );
For hemispherical Array :
%% Wideband spherical array clear ; f0=0.1e9; % frequencies entered are 100 Mhz, 12.5Ghz, 25 Ghz f1=12.5e9; f2=25e9; c=3e8; fc=f0; lambdac=c/fc;
36

37 lambda0=c/f0; k0=2*pi/lambda0; lambda1=c/f1; k1=2*pi/lambda1; lambda2=c/f2; k2=2*pi/lambda2; phi=pi/2; ro=2.72*lambda0;
M=7; for i=1:M;
a(i)=1; end ; du=[0.1963 0.3927 0.5891 0.7854 0.9818 1.1781 1.3745]* lambda0; %values entered for uniform array dnu=[0.0253 0.2830 0.5871 0.8724 1.1357 1.1781 1.37445]*lambda0; % values entered for nonuniform array
N=721; 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;
thetap=du(m);

38 for n=1:m;
phip=(n-1)*2*pi/m; s=s+ a(m)*exp(j*k0*ro*(cos(theta)*cos(thetap)+sin(theta)*sin(thetap)*cos(phi
-phip))); s1=s1+ a(m)*exp(j*k1*ro*(cos(theta)*cos(thetap)+sin(theta)*sin(thetap)*cos(phi
-phip))); s2=s2+ a(m)*exp(j*k2*ro*(cos(theta)*cos(thetap)+sin(theta)*sin(thetap)*cos(phi
-phip))); end ; end ; eth(i1)=s; eth1(i1)=s1; eth2(i1)=s2; end ; ethm0=max(abs(eth)); ethm1=max(abs(eth1)); ethm2=max(abs(eth2)); ethm=max(ethm0,ethm1); ethm=max(ethm,ethm2); figure (1); subplot(2,1,1); plot(theta1*180/pi,log10(abs(eth)/ethm)); hold on ; plot(theta1*180/pi,log10(abs(eth1)/ethm), '--' ); plot(theta1*180/pi,log10(abs(eth2)/ethm), '-.' ); axis([0 180 -50 0]); hold off ; x=[max((abs(eth))) max((abs(eth1))) max((abs(eth2)))]; for i1=1:N; theta=(i1-1)*dth; theta1(i1)=theta; s=0.0; s1=0.0; s2=0.0; for m=1:M; thetap=dnu(m); for n=1:m;
phip=(n-1)*2*pi/m;

39 s=s+ a(m)*exp(j*k0*ro*(cos(theta)*cos(thetap)+sin(theta)*sin(thetap)*cos(phi
-phip))); s1=s1+ a(m)*exp(j*k1*ro*(cos(theta)*cos(thetap)+sin(theta)*sin(thetap)*cos(phi
-phip))); s2=s2+ a(m)*exp(j*k2*ro*(cos(theta)*cos(thetap)+sin(theta)*sin(thetap)*cos(phi
-phip))); end ; end ; eth(i1)=s; eth1(i1)=s1; eth2(i1)=s2; end ; ethm0=max(abs(eth)); ethm1=max(abs(eth1)); ethm2=max(abs(eth2)); ethm=max(ethm0,ethm1); ethm=max(ethm,ethm2); subplot(2,1,2); plot(theta1*180/pi,log10(abs(eth)/ethm)); hold on ; plot(theta1*180/pi,log10(abs(eth1)/ethm), '--' ); plot(theta1*180/pi,log10(abs(eth2)/ethm), '-.' ); axis([0 180 -50 0]); hold off ; y=[max((abs(eth))) max((abs(eth1))) max((abs(eth2)))]; n=[f1 f0 f2]/1.e9; figure(2); plot(n,x, '--' ,n,y, '-.' );

40
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 Arrays), Vol. 47, pp. 511-523, March 1999.
[2]. Yu C., 'Sidelobe reduction of asymmetric linear array by spacing perturbation',
Electronics Letters , Vol. 33 (9), pp. 730-732, April 1997.
[3]. Kumar B.P. and Branner G.R., ‘Generalized Analytical Technique for the Synthesis of Unequally Spaced Arrays With Linear, Planar, Cylindrical or Spherical Geometry’,
IEEE Transactions on Antennas and Propagation, Vol. 53, pp. 621-634, Feb. 2005.
[4]. Bevelacqua, P.J.; Balanis, C.A., Geometry and Weight Optimization for Minimizing
Sidelobes in Wideband Planar Arrays, IEEE Transactions on Antennas and
Propagation,Volume 57, Issue 4, Part 2, April 2009 Page(s):1285 – 1289.
[5]. Ferreira J.A. and Ares F., ‘Radiation pattern synthesis for conformal antenna arrays”,
Journal of Electromagnetic Waves and Applications, Vol. 14, pp. 473-492, 2000.
[6]. Ouacha, A. Erickson, R. Gunnarsson, R. Carlegrim, B. Samuelsson, C.Leijon, S.,
Wideband Antenna Arrays with Reconfigurable Beamforming and Beamshaping,
IEEE/MTT-S International Microwave Symposium , 3-8 June 2007, page(s): 1867 - 1870
Honolulu, HI.
[7]. Skolnik M.I., Nemhauser G. and Sherman III J.W., ‘Dynamic Programming Applied to Unequally Spaced Arrays’,
IEEE Trans. Antennas Propagat. Vol.12, pp. 35-43, Jan.
1964.

41
[8]. Bevelacqua, P.J.; Balanis, C.A., Geometry and Weight Optimization for
Minimizing Sidelobes in Wideband Planar Arrays, IEEE Transactions on Antennas and
Propagation,Volume 57, Issue 4, Part 2, April 2009 Page(s):1285 – 1289.
[9]. Ishimaru, A. ‘Theory of unequallyspaced arrays’, IEEE Trans. Antennas Propagat.
Vol.11, pp. 691-702, Nov.1962.
[10]. Haupt R.L., ‘Thinned Arrays using Genetic Algorithms’,
IEEE Transaction on
Antennas and Propagation , Vol. 42,pp. 993-999, July 1994.