Synthesis of Cylindrical Antenna Arrays Using Simulated Annealing

SETIT 2007
4 th International Conference: Sciences of Electronic,
Technologies of Information and Telecommunications
March 25-29, 2007 – TUNISIA
Synthesis of Cylindrical Antenna Arrays Using
Simulated Annealing
Karim Y. Kabalan * , Elias Yaacoub * and Ali El-Hajj *
* American University of Beirut, Lebanon
kabalan@aub.edu.lb
Elias.Yaacoub@dargroup.com
elhajj@aub.edu.lb
Abstract: Pattern optimization of cylindrical antenna arrays is proposed using the Simulated Annealing method.
Although this method has been used in the literature for optimization of linear and circular arrays, the novelty in this
paper consists in using it to optimize cylindrical arrays by minimizing a given cost function. The emphasis is on antenna
arrays suitable for WCDMA base stations where the advantages of the optimized arrays were demonstrated by system
level simulations.
Key words: Antennas, cylindrical arrays, simulated annealing.
INTRODUCTION
Simulated Annealing (SA) is an optimization
technique originating from [1-3]. It has successfully
found applications in many fields, and particularly in
antenna pattern synthesis problems [4, 5]. Using SA,
excitation coefficients (magnitude and phase), inter
element spacing, and angular separations are varied in
a way to obtain a desired pattern or minimize a given
cost function (directivity increase, side lobe reduction
…). In the literature, SA has usually been applied to
linear and circular antenna arrays (e.g. in [4, 5]).
Traditional linear and circular arrays are discussed in
[6]. Stacking circular arrays one above the other will
lead to a cylindrical array where the elements along a
vertical line on the cylindrical surface form a linear
array and those lying in a transversal plane cutting the
surface constitute a circular array. In this case, the
total array factor would be the product of the array
factor of a linear array by that of a circular array [7].
Hence, the pattern synthesis reduces to two separate
pattern synthesis tasks: the synthesis of a circular
array and the synthesis of a vertical linear array. The
novelty in this paper is in applying SA to cylindrical
arrays. SA is applied to each array separately (linear
and circular), and then the multiplication property of
cylindrical arrays is used. Furthermore, the novelty
consists in designing, with SA, cylindrical arrays
suitable for WCDMA cellular systems. We will
consider both sector antennas, where SA is used to
generate a 120 degrees sector antenna for a 3-sectors
WCDMA network, and adaptive antennas for a
WCDMA system where smart antennas are used in
user specific beamforming.
This paper is divided into five sections. Section 1
presents an overview of the simulated annealing
technique. Cylindrical arrays are briefly introduced in
Section 2. In Section 3, antenna arrays for WCDMA
are designed using SA and their superiority over other
cylindrical antenna arrays is demonstrated via
simulations. Finally, conclusions are drawn in Section
4.
1. Overview of the Simulated Annealing
Algorithm
As its name implies, the Simulated Annealing (SA)
exploits an analogy between the way in which a metal
cools and freezes into a minimum energy crystalline
structure (the annealing process) and the search for a
minimum in a more general system. The algorithm is
based upon that of Metropolis et al. [1], which was
originally proposed as a means of finding the
equilibrium configuration of a collection of atoms at a
given temperature. The connection between this
algorithm and mathematical minimization was first
noted by Pincus [2], but it was Kirkpatrick et al. [3]
who proposed that it form the basis of an optimization
technique for combinatorial (and other) problems. The
algorithm employs a random search by applying
arbitrarily perturbation on the parameters of the
objective function (p(e)). And as the algorithm is
moving from one state to another, not only does it
keep all the perturbations that decrease p(e), but also it
accepts the perturbations that increase it with a
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probability p = exp( −∆f
/ T) , where ∆f is the
increase in f =p(e) due to the perturbation, and T is a
control parameter which by analogy with the original
application is
known
as the temperature of the system. By accepting some
changes that increase f, the SA algorithm avoids being
trapped in local minima. This method was applied
extensively in the literature to optimize the patterns of
linear arrays (e.g. in [4]), and circular arrays (e.g. in
[5]).
Below is a listing of a version of the SA algorithm
aimed to minimize an M variable function
f (x1,
x 2,
⋅⋅⋅,
x M ) over R
M .
Step 1: T = T 0
x i = x i0 i = 1,…, M
A = f(x 10 ,…, x M0 )
k = 0
Step 2: AcceptedCounter = 0
AttemptedCounter = 0
Step 3: Choose a random integer number r between 1
and M
Choose a random perturbation ∆x
B = f(x 1 ,…, x r + ∆x,…, x M )
AttemptedCounter AttemptedCounter + 1
If B ≤ A then
x r x r + ∆x
B A
k = 0
AcceptedCounter AcceptedCounter + 1
Else choose u a random real number in [0, 1]
If u ≤
e
⎛ ( B−
A)
⎞
⎜ − ⎟
⎝ T ⎠
then
x r x r + ∆x
B A
k = 0
AcceptedCounter AcceptedCounter + 1
Step 7: If k < Criterion3 then go to Step 2
Step 8: The End
Note that:
The initial temperature T 0 must be set at a high
value in order to guarantee acceptance of any system
perturbation at the beginning.
The temperature is decreased during the algorithm
(Step5), and no optimal decreasing strategy for T is
yet found. Anyway, the most common and simple rule
is T k+1 =α T k ( 0

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AF
circular
N
n=
1
( [sinθ cos( ϕ− ϕ )] + α )
j kr
n n
( θϕ , ) = ∑ I e
(2)
In (2), “r” is the radius, I n are the excitation
coefficients, α n are the excitation phases, and φ n is the
angle in the x-y plane between the x-axis and the n th
element.
Stacking circular arrays one above the other, with
an equal vertical separation between them, will lead to
a cylindrical array where the elements along a vertical
line on the cylindrical surface form a linear array, and
those lying in a transversal plane cutting the surface
constitute a circular array as illustrated in Figure 1.
Figure 1. Cylindrical Array.
In this case, the total array factor would be the
product of the array factor of a linear array by that of a
circular array [7]. The array factor of one circular
array in the x-y plane is given by (2), and the array
factor of a linear array on the z axis is given by (1).
n
c
jα
n ( kd cos m )
( )
j m θ + β
nm
θ I
ne
⋅ bme
In (4), the term
= (4)
b
m
exp( jκd
cosθ + β
comes
from the m
th vertical element and Inex(
jα n ) is the
excitation coefficient of the n th element of a circular
array. Substituting (4) into (3), it becomes:
m
M
N
j( kdmcos θ+ βm) jαn jkrsinθcos( ϕ−ϕn)
∑ m ∑ n
m= 1 n=
1
AF( θϕ , ) = b e I e e
=
M
N
jkd ( mcos θ+
βm)
∑be
m ∑ n
m= 1 n=
1
= AF ( θϕ , ) ⋅AF
( θϕ , )
linear
circular
Ie
m
)
jkr { sinθcos( ϕ− ϕ ) + α }
n
(5)
It is evident from (5) that the array factor of a
cylindrical array is equivalent to the multiplication of
the array factors of a linear array by that of a circular
array. Hence, the pattern synthesis reduces to two
separate pattern synthesis tasks: the synthesis of a
circular array and the synthesis of a vertical linear
array.
3. Simulated Annealing Applied to
Cylindrical Antenna Arrays
Using SA, excitation coefficients (magnitude and
phase), inter element spacing, and angular separations
are varied in a way to obtain a desired pattern or
minimize a given cost function (directivity increase,
side lobe reduction, pattern nulling, pattern shaping,
…). Figure 2 shows, as an example, the result of
optimizing a 10 element linear array with Chebychev
excitations and 20 dB side lobe level (sll) to obtain a
null at 50 degrees.
n
Using a cylinder of M identical circular arrays, and
with the help of (1) and (2), the total array factor is
given by the sum of the M individual array factors:
0
-5
-10
initial
SA
AF(
θ,
ϕ)
=
M
∑∑
m=
1
⎡
⎢
⎣
N
n=
1
c
nm
( θ ) e
jkr sinθ
cos( ϕ −ϕn
)
⎤
⎥
⎦
(3)
In (3), we consider that in the far field region, the
array factor of each circular array is the same as the
array factor of the circular array in the x-y plane.
However, the vertical distribution of the elements
introduces a phase difference between the circular
arrays identical to the phase difference between the
elements of a linear array, since the elements along the
vertical direction constitute a linear array. Hence, in
(3):
Array Factor (dB)
-15
-20
-25
-30
-35
-40
-45
-50
0 20 40 60 80 100 120 140 160 180
theta
Figure 2. Approximating a Chebyshev Antenna
Pattern with 20 dB sll and a Null at 50 degrees.
In fact, the solid curve shows that the array
obtained by SA approximates well the Chebyshev
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array, in addition to having a null at 50 degrees. The
optimized parameters and the cost function are shown
in Table 1.
Table 1. Parameters Obtained after SA optimization
of a 10 element Chebyshev Linear Array with 20 dB
sll to Obtain a Null at 50 Degrees.
Element No. 1 2 3 4 5
Excitation
Coefficients 1.15 1.36 1.34 1.22 1.57
Excitation
Phases
(degrees)
9 0 3 3 358
Figure 3. Array Factor Magnitude of a Sector
Antenna Generated by SA.
Distances
from the
Origin (in
wavelengths)
Element
No.
-
2.36
-1.75 -1.21 -0.75 -0.27
6 7 8 9 10
Excitation
Coefficients 1.76 1.39 0.55 1.11 1.03
Excitation
Phases
(degrees)
Distances
from the
Origin (in
wavelengths)
356 359 61 338 5
0.23 0.78 1.29 1.36 1.97
Table 2. Parameters Obtained after SA optimization
of a 10 element Circular Array to Obtain a Sector
Antenna for WCDMA.
Element No. 1 2 3 4 5
Excitation
Coefficients 0.21 0.3 0.06 0.85 0.75
Excitation Phases
(degrees)
Angle φ n (degrees)
5 188 337 358 157
0 5 71 214 219
Element No. 6 7 8 9 10
Cost Function
π
∫ ( | ( θ)| |
desired
( θ)| ) θ 100 ( | (50)| |
desired
(50)|)
0
2 2
Cost = AF − AF d + AF − AF
Excitation
Coefficients 0.6 1.34
Excitation Phases
(degrees)
-
0.71
1.45 0.38
291 349 232 144 336
Initial (Starting) Array
10 element linear Chebychev with 20 db sll and inter element
spacing d = 0.5 λ
Angle φ n (degrees)
Cost Function
232 263 269 282 288
The importance of the null at 50 degrees is
emphasized by the factor 100 in the cost function. In
Figure 3, a 10 element circular array is optimized to
cover a 120 degree sector in a wireless system. The
obtained parameters are shown in Table 2.
2π
∫ ( | ( θ 90, ϕ)| |
desired
( θ 90, ϕ)|
)
Cost = AF = − AF = dϕ
0
Initial (Starting) Array
10 element uniform circular array with radius r = 1 λ and
the main beam steered at max at θ = 90, φ = 0
2
Stacking circular arrays one above the other will
lead to a cylindrical array where the elements along a
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SETIT2007
vertical line on the cylindrical surface form a linear
array and those lying in a transversal plane cutting the
surface constitute a circular array. In this case, the
total array factor would be the product of the array
factor of a linear array by that of a circular array [7].
Applying this approach to the arrays of Figures 2 and
3 will lead to a cylindrical array that has a sector
pattern in the azimuth plane (same normalized pattern
as that of the circular antenna of Figure 3) and a Null
at 50 degrees in the elevation plane, as shown in
Figure 4.
demonstrated below.
In Figure 5, a 10 element circular array is
optimized by using SA to generate an array suitable
for being used in WCDMA base stations as a smart
antenna. The parameters of the obtained array are
shown in Table 3. In the cost function, the first term
corresponds to approximating the solid curve of
Figure 5, the second term corresponds to keeping the
sll below the constant part of the solid curve, and the
third term corresponds to increasing the peak value of
the array factor as much as possible.
Figure 4. Array Factor Magnitude of a Cylindrical
Antenna Array Obtained by Combining the Arrays of
Figs. 2 and 3 in the Plane φ = 180 degrees.
Hence, the pattern synthesis reduces to two
separate pattern synthesis tasks: the synthesis of a
circular array and the synthesis of a vertical linear
array. The novelty in this paper is in applying SA to
each array separately, then using the multiplication
property of cylindrical arrays. Although one might
argue that applying SA directly to the cylindrical array
would lead to more degrees of freedom, and hence
better results, using this approach would not allow
making use of the multiplication property of
cylindrical arrays. In fact, if we consider the
cylindrical array as an array of P elements instead of a
linear array with M elements, each element being a
circular array of N elements, and SA is applied by
allowing the location of each of the P elements to be
varied, the final disposition of the elements would still
be on a cylindrical surface, but their disposition in a
plane will not be a circular array, and their disposition
along a vertical line will not be a linear array. Hence,
the multiplication property would be lost. Although
this approach deserves further investigation and is
under current research, a much simpler one is
presented here: instead of having a problem of
complexity P = M x N, it is reduced to two problems
of complexities M and N, respectively. Hence, the
total complexity would be M+N. This reduction is
very useful when combined with careful design of the
patterns of the linear and circular arrays constituting
the cylindrical array, in order to take advantage of the
pattern multiplication property, as will be
Figure 5. Array Factor Magnitude of a Circular
Antenna Array Designed Using SA to be Used as a
Smart Antenna.
Table 3. Parameters Obtained after SA optimization
of a 10 element Circular Array to Obtain a Smart
Antenna for WCDMA.
Element No. 1 2 3 4 5
Excitation
Coefficients 1.26 2.83 2.61 2.13 2.48
Excitation Phases
(degrees)
Angle φ n
(degrees)
108 74 205 274 343
0 100 117 156 182
Element No. 6 7 8 9 10
Excitation
Coefficients 1.49 1.83 1.92 1.87 2.12
Excitation Phases
(degrees)
Angle φ n
(degrees)
32 175 277 64 73
217 240 288 314 344
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Cost Function
2π
∫ (
Cost = | AF( θ = 90, ϕ)| − | AF ( θ = 90, ϕ)|
dϕ
0
ϕ2
∫ ( | ( 90, ) |
desired
( 90, ))
ϕ1
− | AF( θ = 90, ϕ = 0) |
desired
+ AF θ = ϕ − AF θ = ϕ dϕ
[φ 1 , φ 2 ]: range of values where the solid curve is
constant
Initial (Starting) Array
10 element uniform circular array with radius r = 1 λ
and the main beam steered at max at θ = 90, φ = 0
)
2
Table 4. Parameters Obtained after SA optimization
of a 3 element Chebyshev Linear Array with 20 dB sll
to Approximate the peak at 90 degrees of a 10 Element
Chebyshev Linear Array with 20 dB sll.
Element
No.
Excitation
Coefficients
Excitation
Phases
(degrees)
Distances
from the
Origin (in
wavelengths)
1 2 3
2.42 -2.88 2.61
292 112 292
-2.90 -2.21 -1.51
To increase the directivity, three of the circular
array of Figure 5 are stacked to obtain a cylindrical
array. The linear array in the vertical direction is
optimized with SA, to approximate the pattern of a
Chebyshev linear array with 10 elements and 20 dB
sll. The result shown in Figure 6 can be considered as
very good for a 3 element array approximating the
pattern of a 10 element array! The optimized
parameters and the cost function are shown in Table 4.
Combining the arrays of Figures 5 and 6, a
cylindrical array with 30 elements is obtained. It has
the same normalized pattern in the azimuth plane as
that of the array of Figure 5.
Cost Function
π
∫ ( | ( θ) | |
desired
( θ) | ) θ 10 ( |
desired
(90) | | (90) | )
0
2 2
Cost = AF − AF d + AF − AF
Initial (Starting) Array
3 element linear Chebychev with 20 db sll and inter
element spacing d = 0.64 λ
However, in the elevation plane, the multiplication
property led to concentrating the radiated power in the
azimuth plane, as shown in Figure 7.
Figure 6. Array Factor Magnitude of a Linear
Antenna Array Designed Using SA to approximate
the Pattern of a 10 Element Chebyshev Linear Array
with 20 dB sll.
Figure 7. Array Factor Magnitude of the Cylindrical
Antenna Array Obtained by combining the Arrays of
Figs. 5 and 6 Compared to that of the Circular Array
of Fig. 5 in the Plane φ = 0.
This cylindrical array is used to simulate the
downlink user capacity in a WCDMA cellular system
using the same approach presented in [7]. The results
are compared to those obtained by using the
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cylindrical arrays of [7] in Figure 8.
in the azimuth plane. Hence, using SA, we designed a
cylindrical array that led to considerable improvement
in the downlink user capacity of a WCDMA cellular
system, although compared to arrays having three
times more elements!
However, it should be noted that beam broadening
and side lobe level increase due to steering were not
taken into account in the capacity simulations, i.e. the
pattern of the array designed with the SA method was
assumed unaltered when the beam is steered toward
the desired user. The other arrays in the simulation
have 360 degrees symmetry in the azimuth plane [7],
and consequently don’t have this problem.
Figure 8. Capacity vs. Maximum Base Station Power
for Various Antenna Configurations.
The simulated system is a symmetric network of
equivalently equipped base stations, i.e., all the sectors
are deployed with identical advanced antennas. It
consists of 7 hexagonal cells, and takes into account
path loss and shadow fading, i.e., fast fading is
assumed to be averaged out by perfect power control.
System functions such as soft and softer handover are
implemented in the model; a user located at a distance
greater than 80% of the radius of the cell from the
base station is considered in soft handover. It is in
softer handover when the angle at the base station
between this user and the boundary of the neighboring
sector is less than 10˚. The simulation parameters are
given in Table 5.
Table 5. Simulation Parameters.
Parameter
Value
Propagation model K = -50 dB, n = 4
Chip rate
Portion of downlink power
allocated for traffic channels
3.84Mcps
0.8
Orthogonality factor 0.4
Noise spectral density (N o )
Active set size 2
E b /I 0 target
Standard deviation for shadowing
σ
Speech Rate
-169dBm/Hz
7.6dB
8dB
Number of Iterations 100
8 kbps/sec
In Figure 8, Array1, Array2, and Array3
correspond to 99 element cylindrical arrays having,
respectively, uniform, Chebyshev, and Bessel patterns
4. Conclusion
The Simulated Annealing optimization technique
was applied to the pattern synthesis problem of
cylindrical antenna arrays. The process is divided into
two optimization problems: one for linear arrays and
one for circular arrays, then the multiplication
property of cylindrical arrays is used. Obtaining good
results with SA separately for linear and circular
arrays led to obtaining good results with cylindrical
antenna arrays. The emphasis was on arrays designed
for WCDMA cellular systems. Simulations
demonstrated that the cylindrical array obtained with
SA led to an increase in user capacity, when compared
to cylindrical arrays having three times more antenna
elements, in the case of smart antennas deployed in a
WCDMA cellular system.
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