International Journal of Antennas and Propagation

Volume 2017 (2017), Article ID 2563901, 7 pages

https://doi.org/10.1155/2017/2563901

## Synthesizing Sum and Difference Patterns with Low Complexity Feeding Network by Sharing Element Excitations

College of Electronics Engineering, Ninevah University, Mosul 41001, Iraq

Correspondence should be addressed to Jafar Ramadhan Mohammed; moc.oohay@marrafaj

Received 12 January 2017; Revised 6 April 2017; Accepted 11 April 2017; Published 20 April 2017

Academic Editor: Herve Aubert

Copyright © 2017 Jafar Ramadhan Mohammed. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In monopulse radar antennas, the synthesizing process of the sum and difference patterns must be fast enough to achieve good tracking of the targets. At the same time, the feed networks of such antennas must be as simple as possible for efficient implementation. To achieve these two goals, an iterative fast Fourier transform (FFT) algorithm is used to synthesize sum and difference patterns with the main focus on obtaining a maximum allowable sharing percentage in the element excitations. The synthesizing process involves iterative calculations of FFT and its inverse transformations; that is, starting from an initial excitation, the successive improved radiation pattern and its corresponding modified element excitations can be found repeatedly until the required radiation pattern is reached. Here, the constraints are incorporated in both the array factor domain and the element excitation domain. By enforcing some constraints on the element excitations during the synthesizing process, the described method provides a significant reduction in the complexity of the feeding network while achieving the required sum and difference patterns. Unlike the standard optimization approaches such as genetic algorithm (GA), the described algorithm performs repeatedly deterministic transformations on the initial field until the prescribed requirements are satisfied. This property makes the proposed synthesizing method converge much faster than GA.

#### 1. Introduction

Conventional approaches for synthesizing sum and difference patterns require the use of two separate element excitations for one monopulse radar antenna, for example, Taylor excitation [1] for sum pattern formation and Bayliss excitation [2] for difference pattern formation. Thus, these approaches require a feed network of considerable complexity [3–5].

Currently, this drawback of the conventional approaches can be overcome by using subarrays [6, 7] or another method known as common element excitations [8, 9]. Although these approaches allow a significant reduction in the complexity of the feeding network, they rely on the use of an optimization procedure, by either simulated annealing or genetic algorithm. Generally, the optimization algorithms depend on a successive improvement of randomly initialized patterns by applying more or less random variations, which are necessary to overcome local optima. Therefore, the optimization algorithms can be considered as trial-and-error methods and show a rather slow convergence due to the large number of unsuccessful trials [10].

In [11, 12], the complexity of the feeding network was simplified to only one attenuator and two phase shifters by controlling the amplitude and phase excitations of the side elements only, while keeping the excitation of the rest of array elements uniform.

This paper introduces a simple and fast algorithm for synthesizing sum and difference patterns with a number of common element excitations for the purpose of reducing the feeding network. For active antenna arrays, the element excitations can be implemented by using a number of digital attenuators and phase shifters. More number of elements results in a complex feeding network. Thus, higher sharing percentage in the element excitations results in lower complexity and cost. By using an iterative fast Fourier transform (FFT) with a specific constraints on the required element excitations and their corresponding radiation patterns, it is possible to reach a good compromise among the sum/difference patterns quality and the complexity of the required feeding networks. Unlike the current optimization approaches, the proposed method may be regarded as a deterministic method. Therefore, it delivers results much faster as compared to standard optimization approaches.

#### 2. Description of the Method

Consider an array of an even number of isotropic elements that are equally spaced by . For sum beam pattern, the relation between antenna’s radiation and its corresponding element excitation can be written asand, for difference beam pattern, this relation can be given bywhere , representing the wavelength in free space, , representing the angle with respect to the direction normal to the array axis, and and are two separate sets of the excitation coefficients belonging to the corresponding sum and difference beam patterns, respectively.

It can be seen from these two equations that the element excitations and and their array patterns and are related together through FFT algorithm. In order to apply FFT, the antenna array is assumed to be linear with equally spaced elements. By using these relationships in an iterative manner with some constraints enforcing on the element excitation and sidelobe level, the required goal can be achieved. Specifically, we wish to calculate the coefficients, and , so that the corresponding sum and difference beam patterns have a prescribed sidelobe requirements and, at the same time, these two sets have a number of common element excitations. To design such antenna patterns we adopted an iterative fast Fourier transform method [13, 14]. First, we started to calculate the sum and difference patterns assuming some initial values for the coefficients, and (i.e., for we used uniform excitation and for we used odd linear excitation). Since we are using amplitude-only synthesis, the phase of the element excitation is unchanged and thus its value is made to be equal to that of the initial excitation.

As we mentioned earlier, for linear arrays with equally spaced elements, the radiation pattern and its corresponding element excitation are related together by an inverse Fourier transformation. The algorithm starts by updating the sum and difference patterns of the corresponding initial excitations in an iterative manner using 4096-point inverse FFT. During each update, only the sidelobe values that exceed the prescribed sidelobe are modified, and the other sidelobe values are left unchanged. After this modification, a direct 4096-point FFT is performed on the adapted sum and difference patterns to get two new sets of the excitation coefficients and . Note that the length of vectors and is extended using zero padding. Then, from those 4096 excitation coefficients, only the samples belonging to the array are retained.

The constraints on the element excitations are performed by replacing the values of a number of the element excitations of the synthesized sum pattern to be equal to that of the synthesized difference pattern.

It is worth mentioning that the constraints are applied to a certain number of element excitations of the corresponding sum pattern (this means that the rest of the element excitations of the synthesized difference pattern remains unchanged). Finally and after applying these element excitation constraints, the fast Fourier transform is performed to get new sum and difference patterns.

#### 3. Simulation Results

To validate the effectiveness and the convergence speed of the described method, a number of numerical experiments have been performed on a 2.4 GHz Laptop equipped with a 4 GB of RAM. In the following examples, the synthesis of equally spaced linear arrays composed of and elements is considered.

As a first example, the iterative FFT algorithm is used to synthesize sum and difference patterns to reach a prescribed sidelobe requirement. In this case, the required sidelobe level of both patterns is chosen to be −24 dB. Note that, in this example, none of the constraints have been applied to the element excitation, that is, and have no common excitation. The results are shown in Figure 1. This figure also shows the corresponding element excitations for both patterns. Clearly, implementing such antenna with these two separate excitations (i.e., no common excitations) requires a considerable complexity in the feeding network. More importantly, these two excitations tend to be approximately equal at the side elements of the array. This feature can be highly exploited in the proposed method as can be seen in the following examples.