GAS Using Varied & Fived Binary Chromosome Lengths and Real
Chromosomes for Low Sidelobe Spherical-Circular Array Pattern Synthesis
You Chung Chung*
Electrical and Computer Engineering
Utah State University
Logan, UT 84322-4120
youchung@helios.ece.usu.edu
Randy L. Haupt
Electrical and Computer Engineering
Utah State University
Logan, UT 84322-4120
haupt @ieee.org
1. Introduction
Low sidelobe array pattern synthesis is achieved by finding appropriate phase
shifter settings and amplitude weights for each element. The array pattem
synthesis using a genetic algorithm has been shown in many references, and the
various array types are used for the pattem synthesis [l-41. The performance of
GAS using binary and real continuous chromosomes is compared in reference [5,
61.
The performances of genetic algorithms (GAS) using various types of
chromosomes for low sidelobe spherical-circular array pattem synthesis are
compared. The spherical-circular array layout consists of eight circular arrays on
the surface of a sphere. The GAS control the phase shifters and the amplitude
weights of array elements, and minimize the peak sidelobe level while they
maintain the mainbeam direction and gain. Three methods are simulated. The
first method is a GA with varied binary chromosome lengths (GA-VBCL), and
the second method is a GA with a fixed binary chromosome length (GA-FBCL).
In addition, the last method is a GA with real continuous chromosomes (GARCC). The convergence speeds and the adapted peak sidelobe levels of these
three methods are compared when a good seed (-30dB amplitude taper) is
included/not included in the population.
The array shape of a spherical-circular array, and a GA with the spherical array
are shown in [4]. The spherical-circular array has 256 elements in 8 circular
arrays on a spherical surface with 10h radius. The array element spacing is
between 0.5h and 0.6h for the array. The inner circular elements of the sphericalcircular array are normalized to an amplitude of one and a phase of zero. The
amplitude and phase of spherical-circular elements on a circular array are
identical, so that the GAS only control 7 amplitude weights and 7 phase shifters of
the spherical-circular array to generate low sidelobe levels. The equation of a
spherical array pattern is also given by [4,7]:
E @ ,4) =
94.
fa
(0,4),exp[- jkR(CosS, - Cos&,
"4
)I
(1)
2. Genetic Algorithm with Varied Binary Chromosome Lengths
GA-VBCL, GA-FBCL and GA-RCC control the amplitude weights and phase
shifters to minimize the peak sidelobe level of the array pattern while it maintains
0-7803-6369-8100610.00 02000 IEEE
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the mainbeam direction and mainbeam gain. The CA-VBCL gradually reduces
the number of the phase shifter and the amplitude weight bits for control, as the
number of iteration increase. The flow chart of the amplitude & phase adaptive
genetic algorithm using varied binary chromosome lengths is shown in Figure 1.
The general processes of a CA are repeated until the CA reaches to the first
pseudo maximum generation. The pseudo maximum generation is the generation
when the numbers of amplitude and phase bits are reduced. The number of
amplitude and phase bits for control are reduced at the first pseudo maximum
generation in order to achieve fast convergence speed by reducing the current
searching space. A pseudo maximum generation can be set by a programmer, or
can be selected by the CA. A CA can set a pseudo maximum generation during
the optimization process when the CA with varied binary chromosomes has an
identical good population or the same best chromosome for a certain number of
generations. The number of bits for control are reduced at the second pseudo
maximum generation, and this process is repeated until the CA finds a
satisfactory solution. Therefore, the CA-VBCL reduces the size of the searching
area by reducing the number of amplitude weight and phase shifter bits for
control. Thus, the fast convergence speed is achieved with the CA with varied
binary chromosome lengths.
When an initial number of bits for control is 8 for the amplitude weights and 7 for
the phase shifters, the number of bits for control becomes 7 for the amplitude
weights and 6 for the phase shifters at the first pseudo maximum generation. The
pseudo maximum generation vs. the number of bits for control is shown in Figure
2. The pseudo maximum generation is set on every 100 generation in Figure 2.
The cost function of the genetic algorithm evaluates the peak sidelobe level
corresponding to the amplitude weights and phase shifter settings, and the cost
function is [sidelobe level (dB) - mainbeam power (dB)]. This cost function
allows the GAS to reduce peak sidelobe level while it keeps the mainbeam power
as high as possible.
3. Results and Conclusions
The CA controls amplitude weights and phase shifters of the spherical-circular
array element. The amplitude and phase of each element can vary from 0 to 1 and
from 0 to 2n, respectively for the continuous real chromosome. The amplitude
and phase ranges are encoded to the digital 8 bit weights and phase shifters for
binary chromosomes. The CA with varied binary chromosome lengths (GAVBCL) reduces one bit of the number of amplitude and phase bits for control
every 100 generations.
The pattern is normalized with mainbeam power using the initial -30dB
Chebyshev amplitude taper with uniform phase for all GAS. The population size
and mutation rate are 60 and 5%, respectively, and they are the same for all the
GAS. The two types of simulations are done. The first simulation includes a
good seed (Chebyshev amplitude taper with uniform phase) in the population, and
the second simulation does not includes the seed in the population. The two
different simulations show totally different results. The results are averaged from
6 independent runs for all GAS.
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I
Evaluation of Sidelobe level.
Measure peak sidelobe according to new amplitude and phase settings
I
~
itial Po dation.
Generation of
Generates amputild-random
with the specific number of bits for control.
1's and 0 ' s
4
Selection.
Rank chromosomes from best to worst & discard bottom 50%.
Matine and Creatine Offsuring.
Create new offspring settings from Selected top 50%
1
-
Mutation.
,
6
,
101 -200
,I"",
,-""I
++
Fig 2. Varied number of bits for control with pseudo maximum generation of a
GA using varied binary chromosome lengths.
Figures 3 and 4 show the cost vs. generation of the GAS for two different cases.
The GA with real continuous chromosome (GA-RCC) with the seed generates a
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I
lower cost than than the GA with fixed binary chromosome lengths (GA-FBCL).
In Figure 4, the GA-RCC without the seed does only reduces the sidelobe level by
2 dB, and the GA-VBCL outperforms other GAS (GA-RCC and GA-FBCL
Based on the results shown in Figures 3 and 4, the seed helps the GA-RCC to
converge fast since the searching areas of continuous real amplitude weights and
phase shifters are enormously large. Even though the seed helps GA-RCC to
converge fast at the beginning, the GA-VBCL generates a lower cost than the
GA-RCC does after the first pseudo maximum generation in Figure 3
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Fig 3. Cost vs. generation of GAS with
the seed.
Fig 4. Cost vs. generation of GAS w/o
the seed.
4. References
1. K. Yan and Y. Lu, “Sidelobe reduction in array-pattern synthesis using
genetic algorithm,” IEEE Trans. Antenna Propagat., vol. AP-45, pp. 11 171121, July 1997.
2. F. Ares, E. Villanueva and etc., “Application of genetic algorithm in the
design and optimization of array patterns,” IEEE Antennas and Propagation
Society Intemational Symposium Digest, vol. 3, pp. 1864-1867, Jul 13-18
1997.
3. F. J. Ares-Pena, J. A. Rodriguez-Gonzalez and etc., “Genetic algorithms in
the design and optimization of antenna array patterns,” IEEE Trans. Antenna
Popagat., vol. AP-47, pp. 506-510, March 1999.
4. Y. C. Chung and Randy L. Haupt, “ Adaptive nulling with spherical arrays
using a genetic algorithm,” IEEE Antennas and Propagation Society
International Symposium Digest, vol. 3, pp. 2000-2003, Orlando, FL, July
1999.
5. R. L. Haupt and S. E. Haupt, Practical Genetic Algorithms, John Wiley &
Sons Inc., New York, NY, 1998.
6. C. Z. Janikow and Z. Michalewicz, “An experimental comparison of binary
and floating point representations in genetic algorithms, ” Proceedings ofthe
Forth International Conference on Genetic Algorithms, pp. 3 1-36, San
Diego, CA, July 1991.
7. M. Hoffman, “Conventions for the analysis of spherical arrays ZEEE
Antennas and Propagation,” vol. AP-11, pp. 390-393, July 1963.
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