JOURNAL OF TELECOMMUNICATIONS, VOLUME 3, ISSUE 1, JUNE 2010 22

Modelling and Design of Microwave Structures Using Neural Networks and Particle Swarm Optimization Mahdieh Zeinadini, Saeed Tavakoli, and Amir Banookh Abstract— Optimization of design parameters based on electromagnetic simulation of microwave circuits is a time-consuming and iterative procedure. To provide a fast and fairly accurate frequency response for a given case-study, this paper employs a neural network modelling approach. First, the real and imaginary parts of case-study's output, S11 are predicted using two neural network models. Then, particle swarm optimization is employed to optimize the case-study's design parameters so that a design objective is optimized. Index Terms— Design of microwave structures, neural netwotk modelling, particle swarm optimization.

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1 INTRODUCTION eural networks are information processing systems which are inspired by the human brain’s ability to learn from observations [1]. They are efficient alternatives to conventional methods such as numerical modelling methods which could be computationally expensive, or analytical methods which could be difficult to obtain, or empirical models whose range and accuracy could be limited. Due to their ability and adaptability to learn, generalizability, fast operation, and ease of implementation, they have been recently applied to microwave computer-aided design problems [2]. First, a neural network is trained to model the electrical behavior of a given microwave circuit. This trained neural network is often referred to as the neural network model. Then, it can be used in simulation and design to provide fast answers to the task it has learned [3, 4]. Moreover, neural networks can be employed along with evolutionary algorithms such as genetic algorithm [5] and particle swarm optimization [6] for optimization purposes. For a given case-study, a branch-line coupler, we employ a neural network to accelerate the calculation of its S-parameters. The neural network is trained to approximate the S-parameters for given design parameters within the region of interest. Once trained, the neural network model will be capable of performing the frequency sweep in a fraction of a second. Using particle swarm optimization, then, the optimal design parameters are determined so that a given objective function is optimized.

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2 STRUCTURE OF NEURAL NETWORK

To understand why neural networks have the ability to model the behavior of microwave components, the structure of neural networks should be described. A typical neural network has two types of basic components. They are neurons which are processing elements and links or synapses which are interconnections between neurons. Each link has a weighting parameter. Each neuron receives stimulus from other neurons connected to it, processes the information, and produces an output. Neurons are categorized to input, output and hidden neurons. Input neurons receive stimulus from outside the network, while the output of output neurons are externally used. Hidden neurons receive stimulus from some neurons and their outputs are stimulus for some other neurons. Different neural networks can be constructed by using different types of neurons and by connecting them differently [2]. Using MATLAB Neural Networks Toolbox [7], we model the given case-study with a multilayer perceptron (MLP) neural network. MLP is a popularly used neural network structure, in which neurons are grouped into layers. The first and the last layers are called input and output layers, respectively, and the remaining layers are called hidden layers. An MLP neural network with an input layer, one hidden layer, and an output layer, is referred to as a three-layer MLP. Considering an L -layer MLP, the first layer is the input layer, whilst the Lth layer is the output layer. Layers 2 to ( L − 1) are hidden layers. Consider N k as the number of neurons in the k th layer. ———————————————— Let w ijk represent the weight of the link between the jth • M. Zeinadini is with the Faculty of Electrical and Computer Eng., The University of Sistan and Baluchestan, Iran. neuron of the (k − 1)th layer and the i th neuron of the k th • S. Tavakoli is with the Faculty of Electrical and Computer Eng., The Uni- layer. There is an additional weighting parameter for each versity of Sistan and Baluchestan, Iran. k th neuron of • A. Banookh is with the Faculty of Electrical and Computer Eng., The Uni- neuron ( w i 0 ) representing the bias for the i versity of Sistan and Baluchestan, Iran. © 2010 JOT http://sites.google.com/site/journaloftelecommunications/

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the k th layer. The weighting parameters, which are real numbers, are initialized before training the MLP neural network. During training, they are updated iteratively in a systematic manner [8]. Once the neural network training is completed, the weighting parameters remain fixed throughout the usage of the neural network as a model. The neural network requires a suitable number of hidden neurons to accurately model a given case-study. The number of hidden neurons depends on the degree of input-output nonlinearity as well as the number of input and output neurons of the neural network. Highly nonlinear components need more neurons whilst smoother items need fewer neurons. However, the precise number of hidden neurons required for a modelling task may be found either by experience or by trial-and-error [2]. The use of magnitude of S-parameters may result in sharp variations around the resonant frequency [9, 10]. However, the real and imaginary parts of S-parameters show a smoother behavior. Therefore, we train two neural networks, one for the real part of S11 and another for its imaginary part. Then, the magnitude of S11 is calculated from its real and imaginary parts, using (1).

S11 = Re 2 (S11 ) + Im 2 (S11 )

(1)

3 PARTICLE SWARM OPTIMIZATION In this section, a brief description of particle swarm optimization (PSO) is given. Inspired by social behavior of bird flocking or fish schooling, PSO is a population based stochastic optimization technique developed by Kennedy and Eberhart in 1995. Having initialized the optimization system with a random population of individuals, optimal solutions are obtained by updating generations. In an n-dimensional search space, the position and velocity of each particle is given by x i = [ x i1 , x i 2 ,..., x in ]T and vi = [ vi1 , vi 2 ,..., vin ]T , respectively. The position of particle indicates the possible solution in the n-dimensional search space, whereas its velocity indicates the amount of change between the current and next positions. Corresponding to the personal best solution obtained so far at time t, each particle has its own best position, pi = [ pi1 , p i 2 ,..., pin ]T . The global best particle, p g , represents the best particle found so far at time t in the entire swarm [6]. The new velocity of each particle is calculated by (2). v ij ( t + 1) = wv ij ( t ) + c1r1 ( p ij − x ij ( t )) (2) + c 2 r2 ( p gj − x ij ( t )), j = 1, 2, ..., n The position of each particle is updated in each generation according to (3) x ij ( t + 1) = x ij ( t ) + v ij ( t + 1), j = 1, 2,..., n (3) where w is the inertia weight and c1 and c 2 are called acceleration coefficients. r1 and r2 are two independent random numbers.

4 CASE-STUDY: BRANCH-LINE COUPLER The branch-line coupler is shown in Figure 1. It is made of a perfect conductor on the top of a Duroid substrate with a relative dielectric constant of 2.2 and a height of 0.794 mm, backed with a perfect conductor ground plane. The geometrical parameters of the coupler, W1 and W2 , affect the coupler's response, S11 . To represent such input–output relationship, we develop a neural network model which can be used to provide fast estimation of S11 for given geometrical parameters. W1 W2

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Fig. 1. Structure of the branch-line coupler.

Let n and m represent the number of input and output neurons of the neural network. Suppose x and y refer to the neural network input and output, respectively, while w refer to the weighting parameters of the neural network. The structure of the neural network is determined by both the manner in which y is computed from x and w and by the weighting parameters. The neural network input-output relationship is given by (4). y = f ( x, w ) (4) The neural network input is given by (5) x = [w1 w 2 ω]T where W1 and W2 are the coupler design parameters and ω is a range of frequency, in which we intend to model the coupler. We train two neural network models, one for the real part of S11 and another for its imaginary part. The outputs of the neural network models are as follows. y re = Re(S11 ) (6) yim = Im(S11 ) (7) The sample points are generated using an electromagnetic simulator, e.g. HFSS [11]. The region of interest, in and mm, is specified by 1.73395 ≤ W1 ≤ 3.73395 1.19213 ≤ W2 ≤ 2.55213 . For each neural network model, we use 25 uniformly distributed sample points, called training data. Also, 16 sample points are used to test the neural network model. The objective of training is to adjust the weighting parameters so that the neural network output best match the training data output. Generally, MLP neural networks with one or two hidden layers are commonly used for microwave applications [12]. In this work we use an MLP neural network which has one hidden layer with 11 neurons.

© 2010 JOT http://sites.google.com/site/journaloftelecommunications/

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For three test points, real and imaginary parts of S11 as well as S11 obtained from HFSS and the neural network models are depicted in Figures 2-4.

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Fig. 3. Real and imaginary parts of S11 and S11 for 2nd test point.

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Fig. 2. Real and imaginary parts of S11 and S11 for 1 test point.

To find the optimal values of the geometrical parameters, we employ a PSO technique. The objective function is to minimize S11 at the resonant frequency. The optimal parameters are then given to HFSS for validation. Figure 5 shows S11 for the optimal design parameters. The optimal parameters, in mm, obtained by particle swarm optimization technique are given by W1 =2.7883 and W2 =1.6869.

This paper presented a neural network modelling approach for fast and accurate calculation of S11 , for a branch-line coupler. We used two MLP neural network models to predict the real and imaginary parts of S11 . It was shown that the trained neural network models were capable of providing a fast and fairly accurate frequency response. The objective function was to minimize S11 at the frequency of 5.5 GHz. To optimize the design objective, a particle swarm optimization technique was employed to determine the optimal design parameters.

© 2010 JOT http://sites.google.com/site/journaloftelecommunications/

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REFERENCES

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Q. J. Zhang and K. C. Gupta, Neural Networks for RF and Microwave Design. Norwood, MA: Artech House, 2000. [2] Q. J. Zhang and K. C. Gupta, “Neural Networks for RF and Microwave Design- From Theory to Practice,” IEEE Transactions on Microwave Theory and Techniques, Vol. 51, April 2003. [3] K. C. Gupta, “Emerging trends in millimeter-wave CAD,” IEEE Transactions on Microwave Theory and Techniques, Vol. 46, pp. 747–755, June 1998. [4] V. K. Devabhaktuni, M. Yagoub, Y. Fang, J. Xu, and Q. J. Zhang, “Neural networks for microwave modeling: Model development issues and nonlinear modeling techniques,” International Journal of RF Microwave Computer-Aided Engineering, Vol. 11, pp. 4–21, 2001. [5] D. E. Goldberg, Genetic Algorithms and Machine Learning. MA, Addison-Wesley 1989. [6] M. Clerc, Particle Swarm Optimization. Antony Rowe Ltd, 2006. [7] MATLAB, version 7.0, The MathWorks Inc., 2004. [8] F. Wang, V. K. Devabhaktuni, C. Xi, and Q. J. Zhang, “Neural network structures and training algorithms for microwave applications,” International Journal of RF Microwave Computer-Aided Engineering, Vol. 9, pp. 216–240, 1999. [9] E. A. Soliman, M. H. Bakr, and N. K. Nikolova, “Neural Networks — Method of Moments (NN-MoM) for the efficient filling of the coupling matrix,” IEEE Transactions on Antennas and Propagation, Vol. 52, 1521–1529, June 2004. [10] E. A. Soliman, “Rapid frequency sweep technique for MoM planar solvers,” IEE Proceedings Microwaves, Antennas & Propagation, Vol. 151, 277–282, Aug. 2004. [11] Ansoft High Frequency Structure Simulator (HFSS), ver. 10, Ansoft Corporation, Pittsburgh, PA, 2006. [12] J. de Villiers and E. Barnard, “Backpropagation neural nets with one and two hidden layers,” IEEE Transactions on Neural Networks, Vol. 4, pp. 136–141, Jan. 1992. [1]

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Fig. 5. Response of the branch-line coupler for optimal parameres.

Mahdieh Zeinadini received her BSc degree in electrical engineering from Shahid Bahonar University of Kerman, Iran in 2007. Now, she is an MSc student at the University of Sistan and Baluchestan, Iran. Her areas of research are space mapping optimization and microwave circuit design. Saeed Tavakoli received his BSc and MSc degrees in electrical engineering from Ferdowsi University of Mashhad, Iran in 1991 and 1995. In 1995, he joined the University of Sistan and Baluchestan, Iran. He earned his PhD degree in electrical engineering from the University of Sheffield, England in 2005. As an assistant professor at the University of Sistan and Baluchestan, his research interests include space mapping optimization, control of time delay systems, PID controller design, robust control, and jet engine control. Dr. Tavakoli has published more than 40 papers in refereed journals and conferences. He has served as a reviewer for a number of well-known journals including IEEE Transactions on Automatic Control, IEEE Transactions on Control Systems Technology, IET Control Theory & Applications, and several international conferences. Amir Banookh received his BSc degree in electrical engineering from the University of Sistan and Baluchestan, Iran in 2009. Currently, he is an MSc student at the same university. His research interests are particle swarm optimization and microwave circuit design.

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