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Ph.D Thesis ; Rajashree J W ; 2011 Page 12

CHAPTER 2

LITERATURE REVIEW AND APPROACH TO THE PROBLEM

In this chapter a review of the technical literature on solving the antenna array thinning

problem is carried out. This is followed by a brief description of the problem definition

and the investigative approach taken up for the study.

Background information about EAA and related mathematical relationships are given in

Annexure A.

2.1 LITERATURE REVIEW

2.1.1 THINNING PROCESS An EAA consists of a large number of discrete antenna elements placed in a regular grid.

In many communication and Radar systems these antenna elements are in a regular linear

or planar grid. Characterization of these antenna arrays is generally done through their

Radiation Patterns. Typical metrics include Radiation Gain, Side lobe level (SLL), Half-

Power Beam width (HPBW), Directivity etc. These aspects are discussed in Appendix A.

The process of ‘Thinning’ involves reducing total number of active elements in the

antenna array without causing major degradation in system performance as measured

through these metrics. By thinning, periodicity in the array is broken and hence it can

also be considered as a subset of non-uniformly spaced array. Historically, a vast

majority of research efforts in this context were concentrated on non-uniformly spaced

arrays during 1960s. The major focus then was to find antenna element positions or inter

element spacings using various techniques.

Most of these methods were deterministic in nature, using measurable and quantifiable

knowledge of the array physics to determine an optimal result. The user develops cost

function criteria based upon array parameters of interest to exploit knowledge of the

deterministic qualities of the array. Typical deterministic optimization algorithms applied

to the synthesis problem include Newton’s Method, Simplex Method, LMS, Conjugate


Gradient, as well as many other constrained and unconstrained linear/non-linear

programs.

The trends found in the literature related to thinning in antenna arrays can be covered

under the following heads

a) Analytical methods

b) Numerical algorithms

c) Stochastic Search

d) Others

2.1.2 ANALYTICAL METHODS

Unz [2.1] studied a linear array with general arbitrarily distributed elements. The far zone

pattern was expressed using Jacobi expansion. The approach used a matrix form solution

which requires the manipulation of matrices of the order of number of elements. It

involves expressing the radiation pattern in a series expansion, truncating the expansion

and finding the solution of the matrix to find the desired spacing. He found that the

additional degree of freedom created by the random distribution of elements allowed him

to achieve the same performance as of an equally spaced array with fewer elements. The

complexity of this formulation, has limited its practical applications.

An analysis of various directional arrays with variable spacings was made by D D King,

R F Packard and R K Thomas [2.2]. It was an empirical approach to select a set of

spacings according to a specified law such as logarithmic spacing. Non-monotonically

increasing spacings were also investigated and found that they seem to offer promise of a

reasonable radiation pattern. It was found, fewer elements were required for a desired

beam width compared to an equally spaced array. Lacking better method during 1960’s,

this approach gave a better start to the design of unequally spaced arrays. However, it

provided little guidance for new designs other than those studied earlier.

Another approach using Poisson’s Sum formula and the introduction of a new role to the

source function was proposed for unequally spaced array antennas by Ishimaru [2.3].

This is done by converting the total radiation from the set of radiators from series

summation to an integral and applying the change of variables of integration using source


function and source number function. By this, the linear array is replaced by a series of

continuous source distribution, each of which has such a phase variation that the peak

occurs at different angles. Reduction of side lobes and suppression of grating lobes are

then attempted based on one of the conventional techniques such as Taylor.

Sandler [2.4] tried to find the equivalence between equally and unequally spaced arrays.

Lo [2.5] used a probabilistic approach to analyze large randomly spaced arrays. It was

found that sidelobe level is closely linked with number of elements in the array but

weakly connected to the aperture size whereas directivity was related to aperture size.

Design technique used by Mafett[2.6], Willey[2.7], Ogg [2.8] ,and Allen[2.9] for

unequally spaced arrays are based on density tapering, or space tapering, in which the

density of equal-amplitude elements is made proportional to the amplitude of the aperture

illumination of a conventionally designed array. The primary reason for employing

density tapering is to achieve low side lobes without the need for amplitude tapering. The

density tapered technique, when applied to large arrays, seems to give satisfactory results.

In a space tapered array that is derived from a reference array using an illumination taper

which has a maximum in the centre of the array, the minimum active element spacing

will occur at the centre. As the number of elements increases, the minimum spacing

decreases; conversely, as the number of elements decreases, the minimum spacing

increases.

The minimum allowable spacing will be determined principally by the physical size of

the element and, perhaps, by the effects of mutual coupling between elements. The

corresponding maximum number of elements will provide the best approximation to the

desired illumination function. This method is based on the fact that the illumination taper

can be expressed as a function of its moments.

An analytical iterative method was suggested by Harrington[2.10]. This approach starts

with an a priori reasonable set of element spacings by some means; these spacings are

later perturbed about their initial value and effects studied. The side lobes can be reduced

in height to approximately 2/N times the main lobe level, where N is the number of


elements, with main beam width remaining essentially the same as for the uniform array.

However, this method seems to be applicable only to control side lobes close to the main

beam.

A second iterative method reported by Andreasen [2.11] differs from the first in that it

does not start with a a priori set of spacings; it builds up the design one element at a time.

One of the arrays designed using this method is reported to have 21 elements and is 76

wavelengths long when used as a broadside array. The 3-db beam width is 0.74 degree,

the side lobe level -7.4 db. Such an approach is probably not suitable for applications

such as radar.

The method of synthesizing a space tapered linear array involves approximation of a

desired current distribution over the aperture. For an antenna possessing a continuous

aperture, the far-field pattern can be calculated from a knowledge of the currents in the

aperture. The side lobe structure and beam width of the radiation pattern are controlled by

the use of an appropriate illumination function which tapers the aperture current density

[2.12]. In amplitude tapered array, the current sheet of the continuous aperture is

approximated by discrete current sources, the elements. The illumination taper is

obtained by varying the relative amplitudes of the elements in the array. For arrays, in

which the elements are closely spaced (about 0.5 wavelength), there is little difference

between the patterns formed by the continuous aperture and by the discrete array

aperture. The statistically tapered arrays are useful when the number of elements is large

and when it is not practical to employ an amplitude taper to achieve low side lobes.

2.1.3 NUMERICAL METHODS

A major milestone paper in applying numerical technique in the form of dynamic

programming to unequally spaced arrays was done by M. Skolnik, G. Nemhauser, and J.

Sherman, III [2.13].

In this approach, the single N dimensional optimization problem is converted into a

sequence of N one-dimensional optimization problems. Thus instead of examining mN

cases for the Brute–Force approach, approximately (N-1) m2cases only need to be

examined, where N elements of the array can occupy any one of the m possible locations


in the aperture. The design of the complete array is built up from successive designs of

partial arrays. In general, this method will not give an optimal solution for the array, since

the inter-element dependence is not fully accounted for in the procedure. Nevertheless,

the technique, even with this shortcoming provided the much needed base for the design

of one class of antennas.

Klimczak, William N [2.14] proposed deterministically thinned aperture phased antenna

array. According to this study the grating lobes are suppressed and minimized when

unequal sized elements are employed in the array. By varying the size of the radiating

elements, the positions of the elements will not be periodic and the spacing between

adjacent elements in general will not be equal. Byon Kun Chang et al [2.15] proposed

minimax-maxmini algorithm to obtain uniform side lobe in the sense of Dolph-

Chebyshev array pattern. The minimax-maxmini algorithm is solved iteratively by the

revised simplex method in linear programming. The same authors, yet in another paper

[2.16], proposed weighted least square method for optimum thinned antenna arrays. In

this paper, a weighted least square approach is proposed to find an optimum beam pattern

in a Dolph-Chebyshev sense. The square of the error between a thinned array pattern and

a desired pattern is minimized in such a way that some sections of the array visual range

were emphasized than other ones to find an optimum beam pattern. It was shown that an

exponential weighting function is appropriate for achieving a uniform side lobe. Also, it

was found that the minimum mean square error does not necessarily yield an optimum

beam pattern based on convergence characteristics.

Minimax is an optimization method which does not require derivative information; it can

be applied to nonlinear problems for which analytical solutions are not easily possible

[2.17,2.18]. The method also lends to a situation which has sure-convergence properties.

The main objective in this method is to minimize the maximum error and to maximize

the minimum error with respect to the desired result. Success in applying the method to

antenna synthesis is based on how well the residues, which represent the error between

the actual and desired pattern at the sample points are chosen and calculated. Schjer-

Jacobsen and Madsen[2.19] was successful in applying this technique to different types

of antenna synthesis problems including optimization of Dolph-Cbebyshev arrays by


spacing variation and to find the element positions in non-uniformly spaced linear arrays

with uniform excitation that produce minimized (equal) sidelobe levels. The iterative

method for minimization is based on successive linear/ approximately linear

approximations to the nonlinear residuals.

Various programming approaches have been attempted for minimizing the maximum

error α and maximizing the minimum error β as a multi-objective linear programming

problem. The simplest approach is to minimize α and maximize β alternatively, subject to

the inequality constraints and updating the spacings at each iteration.

In Adaptive goal programming approach, the problem is converted into solving a multi-

objective linear programming problem with a fixed goal [2.20]. Since a fixed goal for the

sidelobe level is generally not easy to determine, an adaptive version of the goal

programming is taken. In this approach, the goals were set for α and β iteratively based

on the difference between α and β. If the goals for α and β are g1(k) and g2(k)

respectively, the methodology minimized the total error of α, β with respect to their goals,

where k is iteration index.

Chang et al [2.21] has compared the conventional minimax algorithm with other

approaches for a 51 element array with the array length that of 101-element equi-spaced

linear array of half-wavelength element spacing. It is shown that the minimax- algorithm

with adaptive goal programming performs best with respect to sidelobe. The maximum

sidelobe level is shown to be -23.5 dB. It is found that the sidelobe performance is much

closer to the Dolph-Chebyshev pattern than results obtained with the conventional

minimax algorithm. For array steered to 30 degrees from normal, sidelobes achieved are

about -19 dB.

2.1.4 STOCHASTIC SEARCH AND OTHER APPROACHES

Search based on stochastic approach is through probability rules, working in an “oriented

random” manner. These methods use only information from the objective function, not

requiring knowledge of its derivatives or possible discontinuities [2.22,2.23].


These techniques gained popularity with the recent progress in computer systems, since

they require simultaneous working through a large number of solutions of the proposed

problem. That is necessary in order to let the method explore properly all regions from

the search domain containing the optimal solution. Since these techniques work with

probability rules, it is less likely that they will converge to a local minimum.

Some of the significant Stochastic algorithms useful for solving optimization problems

are Simulated annealing [2.24], Tabu search [2.25], Neural networks[2.26], Particle

Swarm Optimization [2.27], Ant Colony Optimization [2.28] and Evolutionary

algorithms.

Out of these, Ant Colony Organization (ACO), Particle Swarm Optimization (PSO) and

Evolutionary algorithms have been used for solving the antenna array thinning problem.

All the three algorithms derive their inspiration from nature and can be classified under

NBSA.

Óscar Quevedo-Teruel, et al [2.29] have applied ACO technique to successful thinning

of 100 element linear and 200 element planar arrays. In their approach they use the array

with all elements “on” as the initial value and take advantage of the entire structure in

which there is only one nest from which all ants depart. For a 100 element linear array

they obtained a Sidelobe level (SLL) of -20.52 dB with 20 elements optimally selected to

be “off”. In case of planar array the SLL obtained was -25.76 and -25.67 dB in the two

orthogonal planes.

S. Mosca and M. Ciattaglia [2.30] has applied ACO for array thinning in combination

with the classical combinatorial Knapsack problem. By this technique they achieved

sidelobe levels of -18 dB for a 30 x 30 element planar array with Taylor distribution with

a thinning factor of 0.556.

In a detailed article dealing with antenna design, Nanbo Jin and Yahya Rahmat-Samii has

analyzed various derivatives of PSO including real-number, binary, single-objective and

Multi-objective optimizations for antenna design[2.31,2.32]. Their simulation study

involved various linear arrays from 10 elements to 200 elements and they were able to


establish a pareto-front to trade off between peak SL and antenna elements “turned on”.

Results obtained from simulation were also compared with some experimental antenna

arrays.

Using Simple Genetic Algorithm (SGA) as a tool for optimisation in electromagnetic

problems was studied initially by Rahmat-Samii, Johnson, Weile, Haupt, DH Werner and

others [2.33-2.37]. Typically for a 400 element planar array of isotropic sources, a

thinning factor of 0.4 having a sidelobe level of -22.6 db could be achieved. An alternate

way of obtaining element positions in a non-uniform array is to perturb the element

positions within a uniformly spaced array [2.38]. GA has been used to find the new

positions of a linear array with 2N +1 elements by using a side lobe level cost function.

The same principle has been extended to scanning radar arrays as well as other

applications.

Synthesis of thinned antenna arrays using Real-Coded GA [2.39-2.41] and Hybrid GA

i.e., combining both Boolean and real valued variables [2.42, 2.43] have been explored

for specific cases. When the thinned array has variable amplitude as a parameter for

element excitation, SGA with binary coding shall not be adequate. Using a binary coding

technique may not only give rise to quantization noise but also shall require time

consuming coding / decoding procedures. In order to effectively address this problem by

means of a GA-based procedure, a hybrid coding is used. The chromosome assumes this

structure

; … . . … . ; … . . … . (2.1)

where M is the number of active elements, is a Boolean value indicating the state

(turned on or off) of the kth array element, ωk is the corresponding excitation coefficient,

and N is the number of intervals in which the array length has been discretized.

According to the adopted representation, suitable genetic operators have to be defined in

order to obtain admissible solutions and possibly enhance the convergence process.

In spite of all the good features of GA in solving a multi-objective problem, a GA-based

procedure is fairly slow to “fine tune” the optimum solution after locating an appropriate


region (called attraction basin) in the solution space. According to [2.44], SGA must be

customized for each application in order to give optimal results.

Other techniques such as Iterative Fourier technique [2.45], alternating projection

technique [2.46], Fractal technique [2.47,2.48], Matrix pencil method [2.49,2.50] have

also been attempted for thinning antenna arrays in some form. Wang, Fang, and Chow

[2.51] propose a combined approach including sub array amplitude weights, random sub

array and the random displacing rows to reduce the Grating lobes in a thinned phased

array.

2.1.5 CONCLUSIONS BASED ON THE LITERATURE REVIEW:

Following conclusions are drawn from the literature review conducted on the subject

• The topic of thinned array synthesis is a current research topic and many

researchers are working in newer areas.

• Synthesis of thinned arrays is a multi-dimensional, multi-objective problem

involving nonlinear functions. Though classical analytical methods were being

used in earlier years, better results are possible with the advent of fast computing

machines and newer techniques.

• Stochastic optimization techniques are better suited for solving this

problem.However these stochastic techniques are fairly slow in converging to

optimum results. There is also a need for fine tuning the parameters and

customize the approach for each application individually.

• Most of the studies on thinning of antenna arrays are restricted to smaller arrays,

involving a maximum of a few hundreds of elements. There appears to be a

problem in scaling the approach for handling larger arrays.

2.1.6 ADDITIONAL OBSERVATIONS ABOUT NBSA

As mentioned earlier, NBSA draw inspiration from nature’s ability to optimise. They are

not deterministic and thus form a subgroup under stochastic algorithms.


Based on the literature study reported above, it was observed that NBSA have the

following special characteristics, which are useful for solving optimization problems

related to thinning of EAA,

• Gradient free (do not use derivative information).

• Effective for multi-objective functions.

• Robust leading to practical acceptable solution rather than the best solution.

• Largely independent of initial design/solution domain.

• Useful when dealing with problems, where the solution space is vast and unknown.

2.2 PROBLEM FOR INVESTIGATION

2.2.1 HYPOTHESES FOR INVESTIGATION Based on the above study and the conclusions drawn as above, it was proposed to take up

the following hypotheses for investigation for the present research work

1. The NBSA are well suited for tackling the optimization problem related to

thinning in antenna arrays.

2. It is possible to formulate a search algorithm by which NBSA can be used for

solving the thinning problems related to large arrays.

3. The approach can be extended so that the concept can be used for building

practical systems with large number of elements.

2.2.2 NEED TO INVESTIGATE OPTIMAL SEARCH STRATEGY

It was observed from the literature review that in spite of possessing many good features

for solving a problem of the nature of thinned array, NBSA needs to be fine tuned and

customised for an application. In fact Eiben [2.52] in his review paper observes optimal

parameters of these techniques vary not only from problem to problem but also from

stage to stage during the optimization process for a given problem. The paper sites nearly

140 references to support his view and suggests the need for optimal search strategies

based on appropriate choice of algorithm parameters for a given problem. To quote


another researcher [2.53], “cost of determining the exact optimal GA parameters usually

exceeds the cost of solving the original problem!”

2.2.3 APPROACH FOR PRESENT INVESTIGATION

In view of this, the investigation was taken up in three phases.

Phase I: Technique for establishing Optimal Search Strategy

Phase II: Study on Scaling and Dynamic thinning

Phase III: Simulation Results and Discussions

These are discussed in the following chapters. Figure 2.1 shows the flow of the total

investigative procedure.

In Chapter 3 an optimal search strategy using NBSA is established for array thinning

problems. This is augmented with the simulated results presented for a benchmark

problem.

Chapter 4 discusses the methodology for scaling up the above strategy for large practical

systems involving few thousands of antenna elements. Search technique for exploring

large solution space and implications of thinning for dynamic cases are also discussed

here. In this connection zoning, geometric and dynamic symmetry, Bulk Array

Computation (BAC) and Thinning Algorithm based on Genetic Search (TAGS) have

been evolved as a part of the present investigation.

Corresponding simulation results are presented in Chapter 5.

Lastly Chapter 6 concludes with the summary of current research work and directions for

future work.

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