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Synthesis of randomness in the radiated fields of antenna array

Published online by Cambridge University Press:  06 October 2011

Praveen Kumar Malik  and
Harish Parthasarthy
Praveen Kumar Malik*
Affiliation:
Department of Electronics and Communication, Radha Govind Engineering College, Meerut, UP, India. Phone: 9719437711.
Harish Parthasarthy
Affiliation:
Department of Electronics and Communication, N.S.I.T, New Delhi, India.
*
Corresponding author: P.K. Malik Email: malikbareilly@gmail.com
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Abstract

An alternative approach is based on statistically computed signal analysis technique for the design of antenna array exhibiting lower side lobes in their radiation pattern. New and generalized expressions for the array factor of all physically realizable linear antenna arrays are introduced. An algorithm based on the statistically computed signal analysis is designed. By considering the random elements, distance between the sensors, their mean, variance, average amplitude pattern and correlation of the amplitude between the two angles, some mathematical formulations have been done and shown with the help of MATLAB. Based on these generalized expressions, a new way of synthesizing arrays with reduced side lobes is available. It applies to end fire antenna array, which may have either even or odd numbers of sensors with restricted elements spacing. Final expressions shown are a clear relationship between elements excitation and null location in the radiation patterns.

Keywords

DipoleArray factorCorrelationAmplitudeStatisticsSignal analysis
Type
Research Papers
Information
International Journal of Microwave and Wireless Technologies , Volume 3 , Issue 6 , December 2011 , pp. 701 - 705
Copyright
Copyright © Cambridge University Press and the European Microwave Association 2011

I. INTRODUCTION

The study on the radiation pattern of linear dipole antenna arrays, especially the far-field pattern, is discussed in this note. To design a linear array, the far-field pattern and its formulation should be synthesized. The pattern of an array can be expressed as

E_{total} = E_{single\, element} \times Array\, factor.

This is a general rule in array patterns. As per this rule, simple dipoles placed in some geometry can form an arbitrary pattern. Hence, a desired pattern can be achieved by placing these elements in a proper geometry and feeding them with required current. The array factor depends on the current feed of elements and also the way in which these dipoles have been placed together. To achieve the desired pattern with a phased array, the array factor should be properly chosen. Tapering the excitation amplitude in antenna arrays is known to reduce the lobes' level especially those adjacent to the main beam. Suppressing the side lobes saves power in undesired directions on the expense of broadening the main beam, i.e., less directive radiation. A formulation, which makes no direct use of these polynomials, was given in [Reference Kida and Kida1]. During the design it should be taken into account that the main lobe should have maximum power and there must be a minimum number of side lobes with minimum radiation power. The relation between the array factor and designing parameters like spacing, geometry is discussed in the following sections and it will be shown how these parameters can affect the entire pattern.

II. SIGNAL ANALYSIS AND ITS APPLICATION TO ANTENNA ARRAYS

A) Current source in array elements

Let us, for a linear array, assume that the array elements are located on the x − y plane (at the points md x, nd y), m = 0, 1, 2…, M − 1 and n = 0, 1, 2…, N − 1. Let the current source in the array elements have the separable form I(m, n) = I 1(m), I 2(n). The array amplitude pattern is given by (m and n are points of x − y plane)

\eqalign{A\lpar \theta\comma \; \Phi \rpar &=\sum_{m = 0}^{M - 1} \sum_{n = 0}^{N - 1} I\lpar m\comma \; \, n\rpar \exp \lpar\, jk\lpar md_x \cos \lpar \Phi\rpar \sin \lpar \theta \rpar \rpar \rpar \cr &\quad + nd_y \sin \lpar \theta \rpar \sin \lpar \Phi \rpar \cr &=A_1 \lpar \theta\comma \; \Phi \rpar A_2 \lpar \theta\comma \; \Phi \rpar \comma \; }

where

\eqalign{&A_{1 } \lpar \theta\comma \; \Phi \rpar = \sum I_1 \lpar m\rpar \exp\lpar\, jk\lpar md_x \, {\rm cos}\lpar \Phi\rpar \sin \lpar \theta \rpar \rpar) \comma \; \cr &\qquad m=0\; {\rm to}\; M - 1\comma \; \cr &A_2 \lpar \theta\comma \; \Phi \rpar = \sum I_2 \lpar n\rpar \exp \lpar\, jk\lpar md_y \sin \lpar \Phi\rpar \sin \lpar \theta \rpar \rpar) \comma \; \cr &\qquad n=0\; {\rm to}\; N - 1.}

We assume that

\eqalign{I_1 \lpar m\rpar &= \vert I_1 \lpar m\rpar \vert \exp \lpar\, jm\psi_x \rpar \quad {\rm and}\; \cr I_2 \lpar n\rpar &= \vert I_2 \lpar n\rpar \vert \exp \lpar\, jm\psi_y\rpar .}

Then we conclude that max occurs when

\eqalign{kd_x \cos \lpar \Phi \rpar \sin \lpar \theta \rpar + \psi_x &= 0 \quad {\rm and} \cr kd_y \sin \lpar \Phi \rpar \sin \lpar \theta \rpar +\psi_y &= 0.}

These two equations can be solved for θ and Φ. Now assume that |I 1(n)| and |I 2(m)| are constants, say unity; then

\eqalign{A_1 \lpar \theta\comma \; \Phi \rpar &=\sum \exp \lpar\, j\, k\, m\, d_x \, \cos \lpar \Phi \rpar \sin \lpar \theta \rpar) \cr &= \sin \lpar k\, m\, d_x \, \cos \lpar \Phi \rpar \sin \lpar \theta \rpar /2\rpar \comma \cr & \quad {\rm for}\, m=0\, {\rm to}\, M - 1.}

B) Array synthesis

Consider an array with an odd number of elements. The amplitude pattern is given by

\eqalign{AF\lpar \theta \rpar &=\sum a\lpar m\rpar \exp \lpar \,jm \,\psi \rpar \comma \cr \psi&=kd\cos \lpar \theta \rpar +\beta\comma \quad m=- M\ {\rm to }\ M.}

From this equation we obtain

a\lpar m\rpar = {1 \over 2 }\pi \vint \, AF\lpar \Psi \rpar \exp \lpar - jm \Psi\rpar d\Psi\comma \; \quad {\rm limits} - \pi \ {\rm to}\ \pi

Let the amplitude pattern be specified over the range θ €[θ 1, θ 2]. We assume that kd(cos(θ 1) − cos(θ 2)) ≤ 2π.

Then setting Ψ i = kd cos(θ i) + β, i = 1, 2, 3 … .

For the current weights,

a\lpar m\rpar = {1 \over 2 }\pi \vint AF\lpar \Psi \rpar \exp \lpar - jm\, \Psi\rpar d\, \Psi\comma \; \quad {\rm limits}\, \Psi_1 \ {\rm to}\ \Psi_2\comma \;

where AF(Ψ) is obtained by replacing θ in AF with

\cos ^{ - 1} \lpar \lpar \Psi - \beta \rpar /kd\rpar .

C) Width of the main lobe of a uniform linear array

The width of the main lobe of a uniform linear array that has N elements can be determined using

\eqalign{A\lpar \theta \rpar &= \sum \exp \lpar\, jknd\, \cos \lpar \theta \rpar \rpar \cr &= \lpar \exp \lpar \,jkNd\, \cos \lpar \theta \rpar \rpar - 1\rpar /\lpar \exp \lpar \,jkd\, \cos \lpar \theta \rpar \rpar - 1\rpar \cr &\qquad {\rm for}\, n=0\ {\rm to}\ N - 1\comma \; \cr \vert {A\lpar \theta \rpar } \vert &= \vert {\sin \lpar kNd\, \cos \lpar \theta \rpar /2\rpar } \vert /\vert {\sin \lpar kd\, \cos \lpar \theta \rpar /2\rpar } \vert .}

The maximum occurs when θ = π/2, and the first minimum occurs when

kNd\, \cos \lpar \theta \rpar /2=\pi

or equivalent, when cos(θ) = λ/Nd or

\theta=\cos^{ - 1} \lpar \lambda /Nd\rpar

or

\theta = {\pi \over2} - \sin^{ - 1} \lpar \lambda /Nd\rpar .

Width of the main lobe, i.e. angle between the first two nulls on either side of the max is given by

2\sin^{ - 1} \lpar \lambda /Nd \rpar.

To design a three-element linear array with non-uniform excitation, so that the nulls of the array occur at θ = θ i, i = 1, 2, 3, … assume that inter-element space is d.

The array factor can be written as

\eqalign{A\lpar \theta \rpar &= I_0 + I_1 \exp \lpar \,jkd\cos \lpar \theta \rpar \rpar + I_2 \exp \lpar\, j2kd\cos \lpar \theta \rpar \rpar \comma \; \cr z_i &= \exp \lpar\, jkd\cos \lpar \theta _i \rpar \rpar \comma \; \quad i = 1\comma \; 2\comma \; 3\comma \; \ldots.}

Setting u = I 1/I 0 and v = I 2/I 0.

Then the following equation can be solved for the above design:

1 + uz_i + vz_i^2 = 0\comma \; \; i = 1\comma \; 2.

D) Correlation between the amplitude patterns

Consider a linear array with a sensor located on the z-axis at a distance d n, n = 1, 2, 3, … , N from the origin. Let d n be a random with mean d no and variance σ n2 small. If we determine the average amplitude pattern and power pattern as well as the correlation between the amplitude patterns along two different directions. And, if this theory is applied to the analysis of dipole antenna. It has been observed that, if we make small variance and the correlation is expected to be nearly 1. The side lobes of the radiation pattern will be reduced to a great extent:

A\lpar \theta\rpar = \sum I_n \exp \lpar jkd_n \cos \lpar \theta \rpar \rpar \comma \; \quad n = 1\; {\rm to}\; N.

Let

\Phi_n \lpar \xi \rpar = \dot{E} \exp \lpar\, j \xi d_n \rpar\comma \;

the characteristic function of d n. Note that

\Phi_n \lpar \xi \rpar \approx 1 + j\, \xi \, d_n o - \xi^2 \lpar \sigma\,_{n} ^{2} + d_n o^2\rpar /2.

The average amplitude pattern is given by

\langle A\lpar \theta\rpar \rangle = \sum I_n \Phi_n \lpar k.\cos \lpar \theta\rpar \rpar \comma \; \quad n = 1\; {\rm to}\; N.

The amplitude correlation is

\eqalign{\langle A\lpar \theta_1\rpar A^{\ast} \lpar \theta_2 \rpar \rangle &= \sum I_n^2 \Phi _n \lpar k\lpar \cos \lpar \theta_1 \rpar - \cos \lpar \theta_2\rpar \rpar \rpar \cr &\quad+ \sum I_n I_m \Phi_n \lpar k\cos \lpar \theta_1\rpar \rpar \Phi_m^{\ast} \lpar k\cos \lpar \theta_2\rpar \rpar .}

The multiple correlations may also be evaluated. In particular, the average power pattern is given by

\langle \vert A\lpar \theta\rpar \vert^2 \rangle = \sum \, I_n^2 + \sum \, I_n \, I_m \, \Phi_n \lpar k\cos \lpar \theta\rpar \rpar \Phi_m^{\ast} \lpar k\cos \lpar \theta\rpar \rpar .

III. EXPERIMENTS

To examine the effects of the randomness in an antenna array, two programs in MATLAB have been written, one for using the conventional method and other in which the effect of the randomness has been studied. Few assumptions are considered as:

  1. 1) Uniform current distribution is assumed to be 1 A.

  2. 2) There is no mutual coupling effect.

  3. 3) Operating frequency for antenna is about 300 MHz.

  4. 4) η = 120π.

Conventional plots are represented using red and random plots are represented using blue.

A) Experiment 1

During the first experiment a program is written in MATLAB to verify the characteristics (current distribution, linear, and polar plot of dipole antenna array) of the antenna, such that the effect of randomness can be added further and can be visualized.

For value of length of antenna (in terms of wavelength) 0.5, 0.75, 1.00, 1.25, 1.50, 1.75, current distribution is as shown below (Figs. 1–3).

Fig. 1. Current Distribution (in terms of wavelength) 0.5, 0.75, 1.00, 1.25, 1.50, 1.75

Fig. 2. Array factor of conventional dipole antenna (wavelength): 0.5, value of d (dipole spacing in meters): 0.25 and value of beta (phase shift, 0 or 180). N = 4.

Fig. 3. Array factor of conventional dipole antenna (wavelength): 0.5, value of d (dipole spacing in meters): 0.25 and value of beta (phase shift, ±90 ). 90, N = 4.

As per the [2] for array factor, different plot will satisfy the equation with MATALB program.

B) Experiment 2

Some experimental results are shown for the conventional dipole antenna design.

Finally in the synthesis, it has been assumed that the sensors are placed randomly over the distance d n (distance 0.20–0.40, in terms of wavelength). With their mean d no and variance σ n2, it has to be taken care that the value of variance σ n2 is very small. As per the amplitude correlation [Reference Harish3] equation, if we KEEP THE value of correlation coefficient is always near to unity, then following results are obtained (Figs 4 and 5).

Fig. 4. Conventional and random array factor with N = 4.

Fig. 5. Conventional and random array factor with N = 12.

IV. RESULTS

A procedure for reducing the level of the side lobes of large array antennas made of equal contiguous sub-arrays has been presented. Statistically computed theory of the variation of side-lobe level with range has been designed for long-distance end fire antenna arrays. The purpose of the note is to show that the main characteristics of polar diagrams of long-distance end fire arrays may be inferred from amplitude radiation patterns and the correlation coefficient taken in the Fresnel region.

V. CONCLUSION

The objective of the note is to provide an explanation of the differences of the conventional and random data input behavior of a dipole antenna array. An algorithm that efficiently determines the node coordination of sensors topology has been successfully designed with the help of statistically computed signal analysis. The array geometry that has been considered is a linear node array. Using statistically computed signal analysis in synthesizing array factors results in a lower side lobe level. In order to prove this hypothesis, graphical results are presented. Results for randomly synthesized array factors having suppressed alternate side lobes are compared with those of the conventional array (in number of elements, frequency, and spacing) [Reference Stiles and Jenshak4]. An approximate theory of the variation of the side lobe level was developed for end fire antenna arrays.

ACKNOWLEDGEMENTS

The author would like to thank Prof. Harish Parthsarthy, N.S.I.T, New Delhi for all his support. I am always grateful for the encouragement of Prof. Seema Verma, Banasthali Vidyapeth, Banasthali that made me write and publish paper. The author will also like to express sincere appreciation and gratitude to R.G.E.C, Meerut which has provided tremendous assistance throughout the project.

Mr. Praveen Kumar Malik A professional qualified and experienced TECHNICAL and CREATIVE person with extensive knowledge and skills in Microcontroller, Embedded System and Antenna. Praveen Kumar Malik is working in the department of Electronics and Communication, Radha Govind Engineering College, Meerut, UP INDIA. He is B Tech and M Tech from Electronics and Communication system resp. He is having 2 papers in international referred journals. Having more then 10 papers in international and national conference. E mail:

Prof. Harish Parthasarthy is working in department of Electronics and Communication, Netaji Subhash Institute of Technology, New Delhi, INDIA. He is B Tech from IIT Kanpur, Ph D from IIT Delhi and Post-doctoral fellowship from IIA Bangalore. His major area of interest is, DSP, statistically signal processing, Antenna. He is having more then 10 papers in international referred journals and IEEE transaction. Having more then 20 papers in international conference and has written more then 6 technical and sound books also. e-mail:

References

REFERENCES

[1]Kida, Y.; Kida, T.: The Optimum Approximation of Signals That Have Small Side-Lobes Bounded by a Measure Like Kullback–Leibler Divergent, IEICE Technical Report, RCS2008-35, 2008.Google Scholar
[2]Constantine, A.B.: Antenna Theory: Analysis and Design, John Wiley & Sons, New Delhi, 2008.Google Scholar
[3]Harish, P.: Application of advance signal analysis, in Electromagnetism and Relativity Theory, I K International publication, New Delhi 2008, 441–507.Google Scholar
[4]Stiles, J.; Jenshak, J.: Sparse array construction using marginal Fisher's information, in IEEE Waveform Diversity Conf., February 2009.Google Scholar
[5]Harish, P.: Application of advance signal analysis, in Probability Theory and Statistics, I K International publication, New Delhi 2008, 657–705.Google Scholar
[6]Harish, P.: Advanced signal analysis and its applications to mathematical physics, in Signal Analysis, I K International publication, New Delhi, 2009, 73–147.Google Scholar
[7]Harish, P.: Advanced signal analysis and its applications to mathematical physics, in Electromagnetism, I K International publication, New Delhi 2009, 265–321.Google Scholar
[8]Harish, P.: Advanced signal analysis and its applications to mathematical physics, in Antenna Theory, Analysis and Synthesis, I K International publication, New Delhi 2009, 560–571.Google Scholar
[9]Jenshak, J.; Stiles, J.: A fast method for designing optimal codes for side lobes, in IEEE Radar Conf., May 2008.Google Scholar
[10]Haupt, R. L.: Optimized weighting of uniform sub arrays of unequal sizes. IEEE Trans. Antennas Propag., 55 (4) (2007), 1207–1210.CrossRefGoogle Scholar
Figure 0

Fig. 1. Current Distribution (in terms of wavelength) 0.5, 0.75, 1.00, 1.25, 1.50, 1.75

Figure 1

Fig. 2. Array factor of conventional dipole antenna (wavelength): 0.5, value of d (dipole spacing in meters): 0.25 and value of beta (phase shift, 0 or 180). N = 4.

Figure 2

Fig. 3. Array factor of conventional dipole antenna (wavelength): 0.5, value of d (dipole spacing in meters): 0.25 and value of beta (phase shift, ±90 ). 90, N = 4.

Figure 3

Fig. 4. Conventional and random array factor with N = 4.

Figure 4

Fig. 5. Conventional and random array factor with N = 12.