THINNED ARRAYS USING GENETIC ALGORITHMS
Randy L. Haupt'
John J. Menozzi
Christopher J. McCormack
Department of Electrical Engineering
USAF Academy, CO 80840-6236
1.0 INTRODUCTION
A thinned array has uniformly spaced elements: most elements
fed with equal amplitude currents while selected others are terminated in a matched load (tumed off).
The thinned array has a spatial aperture taper that produces low sidelobes in the far field pattem. Traditional methods of array thinning include mapptng an amplitude taper to nonuniform spacing of the elements and randomly thinning the array [l]. This paper shows how to apply genetic alogrithms (GAS) to anive at an optimally thinned array [ 2 ] .
2.0 GENETIC ALGORITHMS
Some of the fundamental differences between GAS and more traditional optimization routines include [3]
1.
2.
3.
4.
GAS work with a coding of the parameter set, not the parameters themselves.
GAS search from a population of points, not a single point.
GAS use payoff(objective function) information, not derivatives or auxiliary knowledge.
GAS use probabilistic transition rules, not determirustic tules.
GAS are modeled after genetic recombination and evolution in nature. A parameter set of the function to be minimized is encoded in a binary string called a gene.
The genes reproduce and evolve until the superior genetic combmation (optimum solution) is found. The GA uses three basic genetic operators on the population of coded parameter strings: reproduction. crossover, and mutation. The value of the highest sidelobe level for an array of isotropic point sources serves as the measure of a particular string's fitness relative to others in the population. This relative fitness level is the baseline for deciding which members of a given population are chosen to take part in the three basic genetic operations listed above.
The most logical method for coding the objective furrction parameters is to use an unsigned binary code, with one bit representing each element in the array. A " I " for a given bit signifies that this particular element is "on", while ih
" 0 it is "off". All of the elements are equally spaced, equally weighted (if "on"), in-phase isotropic radiators- thus the fact that a given element is "on" or "off' is the only variable in the antenna pattem (objective function) calculation. Elements are inmed on or off symmetric about the physical center of the array. Initially, a normally distributed population, consisting of binary coded strings (genes) representing the element excitation scheme for a given array is generated. A sample initial population, consisting of eight initial members (arrays) and ten elements per a m y is shown in Figurr: 1 . Because of symmetry, as mentioned previously, each bit string represents half of the (elements in the array. The physical and phase center fall to the left of the first bit (element) and the following bits represent consecutively equally spaced elements out f "
U.S Government work not protected by US. Copyright.
,,

I
Sample Array Element Excitation Matrix
1 1 1 0 1 1 0 0 1 1
1 1 1 1 0 0 1 1 0 0
1 1 1 1 1 0 1 0 1 1
1 1 1 1 0 1 0 1 0 1
1 1 1 1 1 1 0 1 0 0
1 1 1 1 0 0 1 0 0 1
1 1 0 1 1 1 0 0 1 0 phase center
A
1 1 1 1 0 0 1 0 0 1
* elid element
Figure 1.
Distributed
Sample Population, Init-ally Normally
I
Once the initial population matrix is generated, the three genetic operations are performed on its members. The first procedure conducts the evaluation of the objective function for each of the possible sets of array excitation schemes (genes). Once each of the array fadield pattems is calculated, a procedure sorts each of the pattems for its respective highest sidelobe level. Next, the gene strings
ranked from the highest to the lowest sidelobe level. This ranking procedure can be compared to the natural selection that occurs in a population in nature. The "fittest" members (those with the best relative objective function performance) are selected both to live on to the next generation and also to crossover or "mate" with other fit members of the population and produce offspring. Natural selection of
the array thinning is shown in Figure 2.
Ranked Array Excitation Fitness V a l 2 Selected
1 1 1 0 1 1 1 0 1 0
1 1 1 1 1 0 1 1 0 1
1 1 1 1 0 1 0 1 0 1
1 : 1 0 1 1 0 0 1 0
1 1 1 0 1 1 0 0 1 0
1 1 1 1 0 1 0 1 0 0
1 1 1 1 0 0 1 1 0 1
1 1 0 1 1 1 0 0 1 1
-15.678
1
-15.274
-15.175
-14.125
-14.125
-13.987
-13.667
L YES
YES
YES
YES
NO
NO
NO
NO
I
F i g u r e 2 . Ranking/ Elitist Selection Irocedure
I
Following the ranking of the population, the bottom .50% of the members are discarded. Next, the furiction of crossover is performed on the remaining 50%. This process involves the pairing of altemating gene strings, such
;is the first and the third, second and fourth, etc., and then randomly selecting a crossover point for each pair of genes. It is at this crossover point that the two genes break into partial strings and exchange genetic material, forming two new and slightly different "children".
Graphically, the crossover operation is shown below in Figure 3.
The next generation, then, consists of the top 50% of the previous generation (the parents) and the other 50%
71 3

Crossover Operation on tihe
Top 50% of the Ranked Array Excitation Matrix
1 1 1 0 1 e 1 1 0 1 0
1 1 1 ~ 1 1 0 1 1 0 1
1 :I 1 1 e
1 0 l e 0 0 1 0 indicates the random crossover point
Figure 3 . Crossover Operation Performed on the Top 50! of the Gene Population is made up of the results of the crossover interaction hetweeii these elite parents (their children). This is shown in Figure 4.
New Array Excitation Matrix
Followinq the Crossover and Matina Procedures
I
I 1 1 1 0 1 1 1 0 1 0 \
1 1 1 1 1 0 1 1 0 1 I parents
1 1 1 1 0 1 0 1 0 1 I
I
1 1 1 0 1 1 0 0 1 0 /
1 1 1 O l 1 0 1 0 1 \
1 1 1 1 0 1 1 0 1 0 I offspring
1 1 1 0 1 1 0 0 1 0 I
1 1 1 1 1 0 1 1 0 1 /
Figure 4 . Next Generation Excitation Natrix,
Crossover and Reproduction
Following
I
I
1
Finally, one last genetic operation is performed on thi: gene strings: mutation.
Similar to mutation of a species in nature, GA mutation randomly selects and changes about out of every thousand bits of genetic material [3]. This random mutation keeps the system from settling into a local minimum.
3.0 RESULTS
The following table shows the consistent GA results from five array thinning optimizations for a 50 element linear m a y of isotropic point sources. Note that all of the runs are within 0.8dJ3 A sample farfield pattem for the 50 element array resulting from the GA optimization is shown in Figure 8. The array patterns are optimized for the lowest maximum sidelobe level.
A GA is ideal for optimizing the tllinning of an array. The bits in a gene correspond to turning the element on or o f f . Although a GA i G slow, it can handle very large problems involving many antenna elements.
714

TEST RUN NUMBER
I
2
7
4
5
-21 73 igure 5. G A Optimized 5 0 Element Array Thinning
Problem
References
I . Y.T. Lo and S.W. Lee, Editors. Antenna
and Design.
Van Nostrand Reinhold. New York: 1988.
2.
July, 1992, pp. 66-72.
3. Goldberg. David E. Genetic Algorithms. Addison-Wesley, New York: 1989. (Ch.
71 5