Uniform amplitude distribution for N = 20 with different spacings.

## Abstract

Antenna array thinning is the tuning of same antenna elements with uniform spacing or periodic antenna array to generate the desired amplitude density across the aperture area. Large antenna arrays are difficult to build and have increased fabrication cost. The process of eliminating the radiating elements from the array would be desirable if arrays performance is not significantly degraded. One method of achieving this goal is array thinning by systematically removing elements without change in the performance. This chapter presents and explores machine learning and its applications in the design of antenna array. This chapter also gives the characteristics of machine learning, deep learning, different learning algorithms and its usage in the design of an antenna array with thinning. This chapter presents the performance of an array with and without thinning and the radiation characteristics are observed for both the cases with different spacings. The major advantage of the present work is the reduction of number of elements to achieve better and specified Radiation patterns.

### Keywords

- Thinning
- Spacing
- linear array
- machine learning algorithm

## 1. Introduction

Artificial Intelligence (AI) is an approach to make a robot, or a product to think how smartly humans can think. It is the study of how the human brain thinks, learns, decides, and works, when it attempts to solve the problems. And finally this study outputs intelligent software systems. By using algorithms and large amount of data sets, iterative processing permits AI software to learn automatically. Artificial Intelligence (AI) and Machine Learning(ML) are similar and can be used interchangeably. Figure 1 shows the relationship between ML, AI, and Deep Learning. ML is a huge subset of AI. In fact, ML can be used as an approach to achieve applications of AI. These ML algorithms are very powerful in optimization, but the performance of these algorithms depends upon the size of the data collected [1, 2, 3]. Therefore, ML is often accompanying with data statistics and data analysis. Deep Neural Networks (DNN) are neural networks which contain more than one hidden layer. DNN consist of layers of interconnected nodes/neurons where each and every node/neuron produces a nonlinear function of its input given. All these algorithms are considered as the types of machine learning algorithms [4].

Single-element antennas have less directivity i.e. low gain in the particular direction as the radiation pattern of single element is relatively wide. Usually, high directivity i.e. more gain in the desired direction antennas required for long distance communications. High directivity antennas are possible by broadening the size of the radiating aperture which has maximum size larger than wavelength λ. But, this approach leads to the presence of multiple side lobes near the main lobe. And also, the antenna fabrication becomes difficult when the size of the antenna is large. There is an another method to increase the electrical size of an antenna is, an assembly of radiating elements in a suitable electrical and geometrical alignment is called an antenna array. Generally, all the elements in an array are identical. The design and fabrication of antenna arrays becomes simple by using identical elements in an array. The different discrete antenna elements are wire dipoles, loops, apertures, and etc. The vector combination of all the field patterns radiated by the individual antenna elements is termed as the radiation pattern. The more directive radiation pattern of an antenna array can be obtained by the fields produced by the individual antenna elements which are interfered constructively in one desired direction and then interfered destructively in the other directions [5].

A group of antenna elements whose current distributions are with various amplitudes and phases is referred as array antenna. To enhance the radiated signals in the particular direction and lessen it in the non-desired direction, the phenomena of electromagnetic wave is used. Antenna arrays are used to solve the limitations using single antenna. For example, a dipole antenna provides better radiation pattern compared to that of an isotropic antenna which radiates uniformly in all directions. But, the control of field pattern in a particular direction reduces when the length of the dipole increases. Therefore, controlling of radiation pattern by varying the lengths of single antenna is restricted. Hence, more flexibility and control over the gain can be attained for directing the antenna beam using multiple antennas/radiators arrangement (Figure 2).

### 1.1 Radiation pattern, antenna arrays and array factor

Antenna arrays of two types, i.e. one dimension and two dimension based on the arrangement of the antenna elements. Due to the simple construction of an array, one-dimension antenna array is used in this chapter. These arrays provide a specific radiation/field pattern. The total radiation pattern of the array varies when different antenna elements with different radiation pattern are combined [6]. This occurs due to array factor which measures the effect of joining radiating elements in an array without which the specific radiation pattern of each element is considered. Therefore, the complete radiation pattern of an array is calculated by this array factor and each antenna element radiation pattern. The whole radiation pattern results in a certain direction. Thus the efficiency and the directivity are calculated. These both are same if the overall efficiency is 100% [7].

Depending on the orientation of field pattern in a particular direction, arrays can be divided into broadside i.e. the radiation pattern is normal to the array axis and end fire i.e. the field pattern is parallel to the array axis. In this chapter, broadside arrays are taken into consideration and also radiation pattern in z direction is considered.

### 1.2 Defining array factor

Array factor is based on the number of elements in an antenna, spacing between the radiating elements, and also amplitude, phase of the signal applied to individual element. The overall surface area of radiating structure depends on the number of elements in an array and the spacing between elements. This area is termed as aperture. Large aperture area results high gain. The aperture efficiency indicates that how efficiently the aperture area is used. In this chapter, the influence of all these parameters are explained by considering linear array of isotropic radiating antenna elements [8]. An isotropic antenna is one which radiates power equally in all directions, i.e. isotropic antenna has unity directivity (0dB), unity gain (0dB) and efficiency of 100%.

The Array Factor is a function of the antenna element’s position and the weights respectively [9]. By fitting these position and weight parameters, the performance of an array is optimized to get required properties. By changing the weights of elements, the antenna arrays can be steered in direction of maximum radiation pattern and the directivity can be increased with the increase in the number of antenna elements [10]. But due to the increase in number of elements increases the side lobes which are appeared next to the main lobe. From the definition of array factor there are two main lobes. One lobe is at θ = 0° and an another main lobe at theta = 180° in the directions of positive and negative x-axis respectively.

### 1.3 Thinned antenna arrays

One of the methods for optimizing antenna arrays around 1960 is known as array thinning. Thinning an array is simple and used in a uniformly spaced linear or planar array. Generally, building of large arrays with more number of elements are complex and also increased the fabrication cost. Therefore, removing some of radiating elements in an array would be desirable at which the performance is not significantly reduced.

The process of removing elements systematically from an array without significant degradation in the array performance i.e. the elements can then be disturbed from their exact positions if necessary. The main advantage of thinned array is the elimination of large side lobes easily. From the definition, the actual radiation pattern of an array is the multiplication of the Array Factor and the field pattern of individual antenna elements which make the array. So, by choosing an antenna with less gain and high angle, side lobes are removed instantaneously [11, 12]. Another advantage of thinning is that it does not need any mathematical computations. The array spacing uses non-uniform and sparse spacing. Thinning an antenna array comprises of the deduction (turning off) of radiating elements from an array [13] with reduced cost and weight of individual elements. This method of thinning permits closely the same narrow beam width which is equal to the filled array with total elements of equal size. In array thinning, lower side lobes can be acquired during turned ON process, the antenna elements operate with equal amplitude similar to the same filled array with uniform weighting.

One of the successful process of lowering the side lobe levels in an array is to reduce the magnitude of the weights of the corresponding elements from the center of the array [11]. This process is similar to the method of windowing in digital signal processing. Higher side lobes will appear when the array elements with uniform weights which are set across the array than the weights of individual elements are taper down. The density tapering requires uniform weights for all type of antennas. The elements in antenna array away from the center have less radiated energy compared to the remaining elements which are far away from the center. So, by removing these elements from the array will not decrease the performance of array.

As the computers are so computationally fast in these days, there is an another way of thinning and placement optimization has done using machine learning algorithms. All of these computerized methods employ statistical optimization techniques remove the antenna elements from an array without degrading the performance of an array [14]. The different machine learning algorithms like Support Vector Machine (SVM), Logistic Regression (LR) provide the enhanced thinned antenna array with high accuracy levels, reduced error and time while maintaining a feasible prediction of the antenna behavior, good computing efficiency with lesser number of iterations.

The steps to apply ML algorithms in the design of antenna arrays are explained here.

Step 1: Find the electromagnetic characteristics of an array Antenna using multiple simulations

Step 2: These simulated features are kept in the database and can be used as reference data for training using ML algorithms.

Step 3: Using these algorithms and making predictions array antenna provides the nearest results.

Support Vector Machines (SVMs) algorithm is used for the design of thinned arrays. This algorithm is used for regression, so it is termed as Support Vector Regressors (SVRs). This learning algorithm is trained with the dataset stored which includes computed values of the operational bandwidth, input impedance and the resonant frequency of the antenna array. It is shown that thinning of linear antenna array design using ML algorithms provides better and more accurate characteristics compared to theoretical. This section analyses the concept of thinning an array for lower side lobe sum patterns and also introduces these thinned arrays for difference patterns with low side levels. The Amplitude weights taken as either ‘1’or ’0’. The antenna element is connected to matched feed input when the amplitude weight is ‘1’. When the amplitude weight is ‘0’ the antenna element is connected matched load [11].

To get the improved performance of thinned arrays with the reduced side lobe levels, Mailloux and Cohen [12] proposed the 3-level amplitude weighting structure. This structure is used to approximate side lobe levels with Taylor amplitude distribution and statistical thinning process of the radiating elements. Hence, the thinning process in antenna array having been used more than 40 years and get good performance in military phased array radars. These phased array radars provide different operations like instantaneous wideband operation, mono pulse tracking and they are especially used for the recognition, tracking, and detection of intercontinental ballistic missiles [14]. Finally, thinned arrays in phased array radar systems provide high performance with low side lobe levels [15, 16].

## 2. Formulation

The amplitude is calculated for antenna arrays by considering different number of elements and these results are presented in Tables 1 and 2.

Number of elements N | Amplitude distributions A(X_{n}) | Amplitude distributions A(X_{n}) after Thinning for 2 &19 elements | λ/2 spacing | Ishimaru spacing X_{n} |
---|---|---|---|---|

1 | 1.00 | 1.00 | 0.5 λ | −0.95 |

2 | 1.00 | 0.00 | 0.5 λ | −0.85 |

3 | 1.00 | 1.00 | 0.5 λ | −0.75 |

4 | 1.00 | 1.00 | 0.5 λ | −0.65 |

5 | 1.00 | 1.00 | 0.5 λ | −0.55 |

6 | 1.00 | 1.00 | 0.5 λ | −0.45 |

7 | 1.00 | 1.00 | 0.5 λ | −0.35 |

8 | 1.00 | 1.00 | 0.5 λ | −0.25 |

9 | 1.00 | 1.00 | 0.5 λ | −0.15 |

10 | 1.00 | 1.00 | 0.5 λ | −0.05 |

11 | 1.00 | 1.00 | 0.5 λ | 0.05 |

12 | 1.00 | 1.00 | 0.5 λ | 0.15 |

13 | 1.00 | 1.00 | 0.5 λ | 0.25 |

14 | 1.00 | 1.00 | 0.5 λ | 0.35 |

15 | 1.00 | 1.00 | 0.5 λ | 0.45 |

16 | 1.00 | 1.00 | 0.5 λ | 0.55 |

17 | 1.00 | 1.00 | 0.5 λ | 0.65 |

18 | 1.00 | 1.00 | 0.5 λ | 0.75 |

19 | 1.00 | 0.00 | 0.5 λ | 0.85 |

20 | 1.00 | 1.00 | 0.5 λ | 0.95 |

Number of elements N | Gaussian amplitude distributions A(X_{n}) | Gaussian amplitude distributions A(X_{n}) with Thinning of 2 & 19 elements | Ishimaru spacing X_{n} | λ/2 spacing |
---|---|---|---|---|

1 | 0.650 | 0.650 | −0.95 | 0.5 λ |

2 | 0.599 | 0 | −0.85 | 0.5 λ |

3 | 0.606 | 0.606 | −0.75 | 0.5 λ |

4 | 0.652 | 0.652 | −0.65 | 0.5 λ |

5 | 0.718 | 0.718 | −0.55 | 0.5 λ |

6 | 0.792 | 0.792 | −0.45 | 0.5 λ |

7 | 0.862 | 0.862 | −0.35 | 0.5 λ |

8 | 0.921 | 0.921 | −0.25 | 0.5 λ |

9 | 0.963 | 0.963 | −0.15 | 0.5 λ |

10 | 0.985 | 0.985 | −0.05 | 0.5 λ |

11 | 0.985 | 0.985 | 0.05 | 0.5 λ |

12 | 0.963 | 0.963 | 0.15 | 0.5 λ |

13 | 0.921 | 0.921 | 0.25 | 0.5 λ |

14 | 0.862 | 0.862 | 0.35 | 0.5 λ |

15 | 0.792 | 0.792 | 0.45 | 0.5 λ |

16 | 0.718 | 0.718 | 0.55 | 0.5 λ |

17 | 0.652 | 0.652 | 0.65 | 0.5 λ |

18 | 0.606 | 0.606 | 0.75 | 0.5 λ |

19 | 0.599 | 0 | 0.85 | 0.5 λ |

20 | 0.650 | 0.650 | 0.95 | 0.5 λ |

d = resonant spacing, (* λ*/2), k = 2π/

λ

A(n) = Amplitude distribution

The amplitude is also measured by considering different number of elements with Ishimaru spacings and these results are presented in Tables 1 and 2.

Using the above expressions, the radiation pattern of arrays are calculated

Here, A(x_{n}) is the Amplitude distribution

u = sinθ, θ is the angle in the bore sight direction

_{xn} = Ishimaru spacing function

Where ϕ(x_{n}) is the excitation of phase distribution.

Thinning an array comprises of the deduction (turning off) of radiating elements from an array [13] with reduced cost and weight of individual elements. This method of thinning permits closely the same narrow beam width which is equal to the filled array with total elements of equal size. In array thinning, lower side lobes can be acquired during turned ON process, the antenna elements operate with equal amplitude similar to the same filled array with uniform weighting.

Tables 1 and 2 gives Gaussian distribution and uniform amplitude distribution for N-20 elements using different spacing respectively.

Figure 3 presents the radiation pattern for N = 20 elements with resonant spacing and Ishimaru spacing for uniform amplitude distribution.

Figure 4 gives the radiation pattern for N = 20 elements with Resonant spacing and Ishimaru spacing for uniform amplitude distribution with thinning concept where the antenna elements i.e. ‘9’ and ‘19’ are eliminated from an array antenna.

Figure 5 gives radiation pattern for N = 20 elements with Resonant spacing and Ishimaru spacing for Gaussian amplitude distribution.

Figure 6 represents the Radiation pattern for N = 20 elements with Resonant spacing and Ishimaru spacing for Gaussian amplitude distribution with thinning concept where the antenna elements i.e. ‘9’ and ‘19’ are eliminated from an array antenna.

## 3. Conclusions

The major advantage of present work is reduction of number of elements for getting specific patterns for the given array length by thinning. It is also more useful for obtaining the desired excitation levels at uniform levels while some of the elements are turned off. Here thinning concept is applied for resonant spacing and for Ishimaru spacing and observed the response for 2^{nd} and 19^{th} element turned off. Hence, the percentage of Thinning is 10% or percentage of Filling is 90%. From the results the beam width for Ishimaru spacing is more whereas the beam width for Resonant spacing is less and side lobe levels for Resonant spacing is more and the same response is observed for Gaussian distribution for array of 20 elements using machine learning algorithms. Such patterns are useful in radar applications to reduce EMI and creating nulls in unwanted directions.

## References

- 1.
Razavi A, Forooraghi K. Thinned arrays using pattern search algorithms. In: Progress in Electromagnetics Research. PIER; 2008. pp. 61-71 - 2.
Russell SJ, Norvig P. Artificial Intelligence: A Modern Approach. 3rd ed. Upper Saddle River, New Jersey: Pearson Education, Inc.; 2016 - 3.
Mohri M, Rostamizadeh A, Talwalkar A. Foundations of Machine Learning. The MIT Press; 2012 - 4.
Harrington P. Machine Learning in Action. Manning Publications; 2012 - 5.
Skolnik M, Sherman J III, Ogg F Jr. Statistically designed density- tapered arrays. IEEE Transactions on Antennas and Propagation. 1964; AP-12 (4):408-417 - 6.
Lo Y, Lee S. A study of space-tapered arrays. IEEE Transactions on Antennas and Propagation. 1966; AP-14 (1):22-30 - 7.
Raju GSN, Reddy RR, Satyanarayana M. Sum patterns from arrays of longitudinal slots. In: International Conference on Electromagnetic Compatibility. Bangalore, India: INCEMIC; 2010. pp. 335-339 - 8.
Linear arrays with arbitrarily distributed elements. IEEE Transactions on Antennas and Propagation. 1960; AP-8 :222-223 - 9.
Unz H. Linear Arrays with Arbitrarily Distributed Elements. California, Berkeley: Electron. Res. Lab., Univ; 1956 - 10.
Chakradhar KS. Three different compact elliptical slot UWB antennas for wireless applications. Journal of Communications. 2021; 16 :52-59 - 11.
Keizer WPMN. Amplitude-only low Sidelobe synthesis for large thinned planar array antennas. IEEE Transactions on Antennas and Propagation. 2012; 60 :1157-1161 - 12.
Tomiyasu K. Combined equal and unequal element spacings for low sidelobe pattern of a symmetrical array with equal-amplitudeelements. IEEE Transactions on Antennas and Propagation. 1991; 39 (2):265-266 - 13.
Randy I, Haupt L. Fellow interleaved thinned linear arrays. IEEE Transactions on Antennas and Propagation. 2005; 53 :9 - 14.
Jiang R, Wang X, Cao S, Zhao J, Li X. Deep neural networks for channel estimation in underwater acoustic OFDM systems. IEEE Access. 2019; 7 :23579-23594 - 15.
Rocca LP. Large array thinning by means of deterministic binary sequences. IEEE Antennas and Wireless Propagation Letters. 2011; 10 :9 - 16.
Randy L. Thinned arrays using genetic algorithms. IEEE Antennas and Wireless Propagation Letters. 1994; 42 :7