STAIR-STEP PATTERNS FROM ARRAYS

STAIR-STEP PATTERNS FROM

ARRAYS

M.Chandrasekhar

Research Scholar, Dept. of Electronics and Communication Engineering, College of Engineering (A), Andhra University, Visakhapatnam, Andhra Pradesh, India-530 003.

Email: [email protected]

Dr. G.S.N.Raju

Professor, Dept. of Electronics and Communication Engineering, College of Engineering (A), Andhra University, Visakhapatnam, Andhra Pradesh, India-530 003.

Email: [email protected]

Abstract:

In array antennas, the shapedbeams are produced either by amplitude control or phase control. In the present work, an attempt is made to produce them with the introduction of new aperture distribution without additional phase. Stair-step patterns are produced using Fourier transform method after introducing the proposed aperture distribution. The computed patterns are presented. These patterns are intensively used to identify more than one target moving in different altitudes and different angular regions.

Key words: Luneburg Lens antenna, Geometrical optics, Stair- step patterns, spill over, and Fourier Transform

method.

1. Introduction

For the past few decades, various antenna designs have been developed that are capable of either significantly increasing the directivity, lowering the sidelobe levels, or steering an antenna beam (C.A.Balanis). For satellite antennas, one of the attractive, feasible ways to realize a shaped beam with a fixed or variable pattern is by using array fed parabolic reflector or lens antennas (Yi-Xi Zhang and Jun-Mei Fu). The parabolic reflector based antenna is most commonly used for this purpose. The parabolic reflector has some disadvantages when put into use, they are: 1.Difficult to realize large aperture due to mechanical constraints like back up structure and gears.2.Blockage by the feed structure.3.Slow in response as related to scanning in comparison to electronics scanning(Tse-Tong Chia). In order to avoid the above limitations, an attempt has been made to generate shaped beams from optical Luneburg lens antenna with optimum performance.

Most of the wireless communication and direct broadcast satellite antennas require shaped and reconfigurable radiation pattern to illuminate a well defined irregularly shaped angular regions. The antennas associated with the multi target search radars are required to produce patterns with different stair steps which are considered as shaped beam. Shaped beam patterns are generated from array antennas. These patterns are basically symmetric with one main lobe in bore sight and sidelobe structure around the main beam. In the present work, these patterns are generated from optical antenna system using Luneburg lens and intensively used to identify more than one target moving in different altitudes and different angular regions.

Desired beam shapes can be generated from optical antenna system using Luneburg lens ( E.C.Dufort). It is convenient to explain the principle involved in beam shaping in terms of geometrical optics. A radiating structure is described (Giorgio.V.Borgiotti) to generate shaped beams, which consists of a bootlace lens with linear outer and circular inner profiles resulting excellent scan performance over moderate frequency band. The aim of these studies has been the reduction of the number of costly control elements, phase shifters or variable power dividers as compared to the number in a conventional phased array designed.

Antenna beam shaping can be obtained using the amplitude only control or phase only controls (M.T.Ma). In non-scan applications, amplitude only control method is used. Phase only control method is used for scan applications. For the amplitude only control method, aperture phase distribution is fixed. On the other hand, aperture amplitude distribution is fixed in phase only control method.

Shaped beams are generated using wood ward Lawson method. This method is based on the decomposition of the aperture excitation function into a sum of uniform amplitude and linear phase sources (A.Sudhakar and G.S.N.Raju). Stairstep patterns are realized from continuous line source using iterative sampling method(G.R.L.V.N.Srinivasa Raju).These patterns have more ripple components in the desired angular sectors and also in the trade off region.

element spacing and zero additional phase to obtain a desired array pattern performance, especially for larger arrays.

Stair step patterns are generated from amplitude only control technique. Moreover, the patterns of very small arrays have some ripple components in the desired angular sectors and also in the trade off region. On the other hand they are found to be smoothened out in larger arrays.

The rest of this paper is organized as follows: Analysis of an optical antenna system is given in Section 2, Section 3 describes the mathematical formulation, the numerical simulation results are reported in Section 4 and finally the conclusion is given in Section 5.

2. Analysis

In this paper, the geometry is described in terms of geometric optics where it is shown that a generalized Luneburg Lens is a necessary element in the optical system (E.C.Dufort). Performance can be analyzed for large optical elements using simplified corrections for diffraction effects.

The Luneburg Lens is a spherically symmetric delay-type lens formed of a dielectric with index of refraction which varies as a function of radius. A Luneburg lens is fabricated using the dielectric material such as polystyrene or polyethylene. These dielectric materials have dielectric losses. At higher frequencies, the dielectric loss will be greater (Kang Wook Kim). The effect of dielectric loss is less than effect of air gaps if low loss dielectric material is used for lens fabrication. However, the gain loss cannot be ignored at higher operating frequencies.

Consider a circular lens shown in fig.1. Its dielectric constant depends on radial distance. Assume that this lens acts as corrective lens of the system. Also assume that the lens bends the rays coming from a point source placed on the lens surface at radial distance ‘d’ to another point at distance D>d. The source, center of the lens and focal point lie along same straight line. If the above focal condition is true for line pair of points then all points on the circular lens surface will image to unique points on the image surface, radius D. The distribution of these image sources will be a stretched replica of the source distribution

These sources on the image circle are again mapped to a linear aperture without distortion by means of equal line lengths connecting all point pairs whose arc lengths measured from the line of symmetry are the same.

Fig.1. Major Components of Optical Antenna System.

The Bootlace aperture lens used in this system is not the commonly used Abbe or Rotman lens (W.Rotman and R. F.Turner). Only broadside incidence will focus all the rays to a single point. At maximum scan all incoming rays are tangent to the lens. This gives maximum scan angle and usable D/L ratio. The portion of the lens which will be illuminated on receive must not overlap the feed array.

of radius (D-d) centered on the receiving element located on the geometrical focus is obtained using geometrical optics. It is used to generate stair step type of beams.

3. Formulation

In the present work, it is of interest to generate stair-step of beam shapes. They are expressed in the following form over desired angular regions.

For three Stair –steps,

D 3 kSd 3 D 3 kSd 2 , cos d 6 D D 3 kSd 2 D 3 kSd , cos d 6 D 2 D 3 kSd D 3 kSd , cos d 2 D D 3 kSd D 3 kSd 2 , cos d 6 D 2 D 3 kSd 2 D 3 kSd 3 , cos d 6 D ) ( F 2 1 2 1 2 1 2 1 2 1          _          _           _            _            _   (1)

For four stair steps,

Five stair steps, D 5 kSd 5 D 5 kSd 4 , cos d 10 D D 5 kSd 4 D 5 kSd 3 , cos d 10 D 2 D 5 kSd 3 D 5 kSd 2 , cos d 10 D 3 D 5 kSd 2 D 5 kSd , cos d 10 D 4 D 5 kSd D 5 kSd , cos d 10 D 5 D 5 kSd D 5 kSd 2 , cos d 10 D 4 D 5 kSd 2 D 5 kSd 3 , cos d 10 D 3 D 5 kSd 3 D 5 kSd 4 , cos d 10 D 2 D 5 kSd 4 D 5 kSd 5 , cos d 10 D ) ( F 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1          _           _           _           _            _             _             _             _             _   (3)

The transmit pattern E(θ) of a distribution An on the inner array of the aperture lens is given by

          _    N N n sin jnkS n 2 1 e A cos 2 kS ) (

E (4) It can be written in u-domain and considered for pattern computation.

              N N n jnkSu n 2 1 2 e A 2 u 1 kS ) u (

E (5) Here,

S=element spacing

n=zero for a element centered on the geometric focus 

sin

u ,



For an odd number of elements M=2N+1, the elements are n0,1,2....N.

If E (θ) is a conjugate match to F (θ), entire power is delivered to the aperture lens, An can be calculated by using Fourier inversion.

For three stair steps

For four stair steps,                                                                                                                                             D 4 nksd D 4 nksd sin D 16 ksd D 4 nksd D 4 nksd sin D 64 ksd 3 D 2 nksd D 2 nksd sin D 16 ksd 3 D 2 nksd D 2 nksd sin D 8 ksd D 4 nksd 3 D 4 nksd 3 sin D 64 ksd D 4 nksd 3 D 4 nksd 3 sin D 64 ksd 9 D nksd D nksd sin D 4 ksd A 2 1 2 1 2 1 2 1 2 1 2 1 2 1 n (7)

For five stair steps,

                                                                                                                                                                                    D 5 d nkS D 5 d nkS sin D 25 kSd D 5 d nkS D 5 d nkS sin D 125 kSd 4 D 5 d nkS 2 D 5 d nkS 2 sin D 125 kSd 16 D 5 d nkS 2 D 5 d nkS 2 sin D 500 kSd 48 D 5 d nkS 3 D 5 d nkS 3 sin D 500 kSd 108 D 5 d nkS 3 D 5 d nkS 3 sin D 250 kSd 36 D 5 d nkS 4 D 5 d nkS 4 sin D 250 kSd 64 D 5 d nkS 4 D 5 d nkS 4 sin D 500 kSd 64 D d nkS D nkSd sin D 5 kSd A 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 n (8) Here,

D=Distance between center of the circular lens and aperture lens, and d=Radius of the circular lens.

4. Simulation Results

-10 -8 -6 -4 -2 0 2 4 6 8 10 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Element number

A

m

pl

it

ude

Fig.2. Aperture distribution of central subarray for N=21

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 -50

-45 -40 -35 -30 -25 -20 -15 -10 -5 0

u

E

(u)

in dB

Fig.3. Radiation pattern of central subarray for N=21 and three stair steps

-40 -30 -20 -10 0 10 20 30 40

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Element number

A

m

p

lit

u

d

e

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 -50

-45 -40 -35 -30 -25 -20 -15 -10 -5 0

u

E

(u)

in dB

Fig.5. Radiation pattern of central subarray for N=81 and three stair steps

-10 -8 -6 -4 -2 0 2 4 6 8 10

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Element number

A

m

p

lit

u

d

e

Fig.6. Aperture distribution of central subarray for N=21

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 -50

-45 -40 -35 -30 -25 -20 -15 -10 -5 0

u

E

(u)

in dB

-40 -30 -20 -10 0 10 20 30 40 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Element number

A

m

p

lit

u

d

e

Fig.8. Aperture distribution of central subarray for N=81

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 -50

-45 -40 -35 -30 -25 -20 -15 -10 -5 0

u

E

(u)

in dB

Fig.9. Radiation pattern of central subarray for N=81 and four stair steps

-10 -8 -6 -4 -2 0 2 4 6 8 10

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Element number

A

m

p

lit

u

d

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 -50

-45 -40 -35 -30 -25 -20 -15 -10 -5 0

u

E

(u)

in dB

Fig.11. Radiation pattern of central subarray for N=21 and five stair steps

-40 -30 -20 -10 0 10 20 30 40

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Element number

A

m

p

lit

u

d

e

Fig.12. Aperture distribution of central subarray for N=81

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 -50

-45 -40 -35 -30 -25 -20 -15 -10 -5 0

u

E

(u)

in dB

Fig.13. Radiation pattern of central subarray for N=81 and five stair steps

5. Conclusion

one target moving in different altitudes and different angular regions. Moreover, the patterns of very small arrays have some ripple components in the desired angular sectors and also in the trade off region. On the other hand they are found to be smoothened out in larger arrays.

References

[1] C.A.Balanis(1982) , “Antenna Theory Analysis and Design”, John Wiley and sons, Inc., United States of America.

[2] Yi-Xi Zhang and Jun-Mei Fu(2004), “Frequency-domain principle of pattern multiplication for array fed reflector or lens antennas”, IEEE Microwave and wireless components letters, vol.14, no.7.

[3] Tse-Tong chia(2014 ), “Design of low profile cylindrical Luneburg lens antenna” ,The 8th _{European conference on Antennas and} Propagation.

[4] E.C.Dufort( Sept. 1986), “Optimum Optical limited Scan Antenna”, IEEE Trans. Antennas and Propagat. vol. AP-34, no.9.

[5] G.V.Borgiotti(June 1975), “Degrees of freedom of an antenna scanned in a limited sector”, in IEEE Antennas Propagat. Symp. Dig., p. 319.

[6] M.T. Ma(1973), “Theory and Application of Antenna Arrays”, Johh Wiley and sons, Inc, United States of America. [7] A.Sudhakar and G.S.N.Raju, “Generation of Stair-Step Radiation Patterns from an array antenna”, AMSE Journal France.

[8] G.R.L.V.N.Srinivasa Raju (August 2013), “Generation of shaped beam radiation patterns from a line source using iterative sampling method”, International Journal of Engineering Science and Technology (IJEST), Vol. 5, No.08.

[9] Kang Wook Kim(May 2001), “Characterizations of Spherical Luneburg Lens Antennas with Air-gaps and Dielectric Losses”, Journal of the Korea Electromagnetic Engineering Society, vol.1.No.1.

[10] Rotman and R.F.Turner(Nov. 1963), “Wide angle microwave lens for line source applications”, IEEE Trans. Antennas Propagat. Vol. AP-11, p. 623.