SUPPORT VECTOR MACHINE FOR BANDWIDTH ANALYSIS OF SLOTTED MICROSTRIP ANTENNA

SUPPORT VECTOR MACHINE FOR BANDWIDTH

ANALYSIS OF SLOTTED MICROSTRIP ANTENNA

Venmathi A.R.

1

_{& Vanitha L.}

2

1_{Department of Electronics Communication Engineering, Kings Engineering College, Irungattukottai, Chennai, India,} ([email protected])

2_{Department of Electronics Communication Engineering, Sri Venkateswara College of Engineering, Sriperumbadur, Chennai, India,} ([email protected])

Abstract: In this work, the Support Vector machine (SVM) is adopted for analysis of slotted microstrip rectangular patch antenna, which is a technique based on the rigorous mathematical fundamentals. SVM is one of the most competitive techniques to the popular artificial neural networks. In this design process, accuracy, computational efficiency of Support Vector Machine is compared to artificial neural network performance. It can be concluded that the artificial neural network may be replaced by the Support Vector Machines due to its high approximation capability and much faster convergence rate.

Keywords: Support vector machine, slotted microstrip antenna, EM Simulator

1. INTRODUCTION

In high-performance spacecraft, aircraft, missile and satellite applications, where size, weight, cost, performance, ease of installation, and aerodynamic profiles, are constraints, low profile antennas are required. To meet these requirements, microstrip antennas are used [1–4].

In the literature, artificial neural network (ANN) models have been built for the design and analysis of microstrip antennas [5]. Many new tools for microwave CAD are developed during the past decades. Models are generally developed using analytical, electromagnetic simulation, and/or measurement based methods. Accurate and efficient models for circuit components are essential for cost-effective circuit design. The advantage of neuro computing is that, after proper training, a neural network completely bypasses the repeated use of complex iterative processes for new cases presented to it. The single network structure can predict the results for patch antenna provided that input values are in the domain of training values.

Support Vector Machines (SVM) and kernel methods, which enable to generalize ‘discrete’ data into the ‘continuous’ domain have become one of the most popular learning machines in the last few years. In particular, support vector machines are based on a judicious and rigorous mathematics combining the generalization and optimization theories together and verified to be computationally very efficient

(the so-called Vapnik- Chervonenkis theory [6]–[7]). This learning machine has found many fruitful applications in science and engineering.

Inset feed microstrip antenna with slots in it improves the antenna bandwidth compared to antenna without slots of the same physical dimensions [8]. In the present work SVM model is developed to analyze the bandwidth of the example antenna. The Method of Moments (MOM) based IE3D software has been used to generate training and test data for the SVM.

The paper is organized as follows: Definition of the problem in section II, the design and data generation in section III. Section IV gives the theory of the SVM. Section V includes results and conclusions. The results are compared with two Neural Network architecture, Back propagation(BP) and Radial Basis Function(RBF).

2. DEFINITION OF THE PROBLEM

The frequency of a patch antenna can be increased by a capacitive or inductive load. In this paper, a two slot microstrip antenna has been designed to achieve a wide bandwidth. The proposed SVM model calculates the cut off frequencies f1 and f2 and hence the bandwidths (f2 - f1) of the example antenna for different coordinates of the slots i.e., X1, Y1 and X2, Y2. The patch dimensions of the antenna as well as that of slot are kept constant for this specific model [8].

3. DESIGN AND DATA GENERATION

The rectangular microstrip antennas are made up of a rectangular patch with dimensions width (W) and length (L) over a ground plane with a substrate thickness h having dielectric constant∈_r[8], [9]

As an example an inset feed microstrip antennas is designed to resonate at 10 GHz frequency with dielectric constant∈_r= 4:7, substrate thickness h = 1:588 mm, L = 6 mm, W = 8:88mm on a ground plane. All dimensions of the antenna are in mm. The length and the width of the patch are calculated initially by the relationships in equations (1)-(6). 2 2or r 1 W f ν = ∈ + (1) 2 2 r o reff L L f ν = = ∆ ∈ (2)

(

)

(

)

0.3 0.264 0.412 – 0.258 0.8 reff reff W L h W h h   ∈ + _ + _ ∆ _{=  }   ∈ _ + _   (3) –1/2 1 _{–1 1 12} 2 2 r r reff ∈ + ∈  _Wh  ∈ = + _ + _   (4) L_g= 6(h) + L = 6 (1.588) + 6 = 5.528mm (5) W_g= 6(h) W = 6 (1.588) + 8.88 = 18.40mm (6) Figure 1 shows the geometry of inset feed microstrip antenna with two slots in the patch. The patch is energized electromagnetically using 50 ohm

microstrip feed line. The length of the current path is increased due to the slot which leads to additional inductance in series. Hence wide bandwidth is generated as the resonant circuits become coupled. The slots aggregate the currents, which give additional inductance controlled by the patch width.

The software used to model and simulate the proposed microstrip patch antenna is Mentor Graphics IE3D software. IE3D is a full wave electromagnetic simulator based on the method of moments.

IE3D software has been used to calculate the return loss (S11) and hence the cut off frequencies f1 and f2 of the antenna. First the example antenna is designed without slots in IE3D EM Simulator with patch dimensions L = 6, W = 8:88 for resonating frequency of 10 GHz. The bandwidth for the antenna is around 450 MHz. The present work signifies that by introduction of two slots in the same design, the bandwidth gets enhanced about 25%-45%, i.e., from 450 to 650 MHz. The same antenna is designed in IE3D Simulator for different coordinate values of both the slots X1, Y1 and X2, Y2 in the specified range (X1 = 0; 0.5 < Y1 < 4 mm, X2 =0;-0.5 < Y2 < -4mm) keeping patch dimensions, slot dimensions, and h constant and the corresponding cut off frequencies are recorded and this data has been used as a training data and test data for SVM.

4. SVM, ARTIFICIAL NEURAL NETWORK AND RADIAL BASIS FUNCTION ARCHITECTURE 4.1. Support Vector Machine (SVM)

SVM has proven its efficiency over neural networks and RBF classifiers. Unlike neural networks, this model builds does not need hypothesizing number of neurons in the middle layer or defining the centre of Gaussian functions in RBF [14]. SVM uses an optimum linear separating hyperplane to separate two set of data in a feature space. This optimum hyperplane is produced by maximizing minimum margin between the two sets [12]. Therefore the resulting hyperplane will only be depended on border training patterns called support vectors.

The support vector machine operates on two mathematical operations: (1) Nonlinear mapping of an input vector into a high-dimensional feature space that is hidden from both the input and output. (2) Construction of an optimal hyperplane for separating the features discovered in step 1. Support vectors are determined by using the equations 7 -12. [10, 11]

4.2.Variable Definition

1. Let x denote a vector drawn from the input space, assumed to be of dimension m_o.

Figure 1: Inset Feed Microstrip Antenna with Two Slots. All Dimensions are in mm

2. Let {ϕ_j(x)} for j=1 to m_1, denote a set of nonlinear transformations from the input space to the feature space.

3. m₁is the dimension of the feature space.

4. {w_j} for j=1 to m₁denotes a set of linear weights connecting the feature space to the output space. 5. {ϕ_j(x)} represent the input supplied to the weight

w_jvia the feature space. 6. b is the bias

7. α_i is the Lagrange coefficient 8. d_icorresponding target output.

4.3. Steps Involved in the Design of SVM

1. Hyperplane acting as the decision surface is defined as 1 ( , ) 0 N i i i i d K x x = α =

Σ

(7) Where K(x,x_i) =ϕT_(x)ϕ(x

i) represents the inner product of two vectors induced in the feature space by the input vector x and input pattern xi pertaining to the ith example. This term is referred to as inner-product kernel. where 1 ( ) N i i i i w d x = =

Σ

α ϕ ₍₈₎

[

0 1 1

]

( )x ( ), ( ),...,x x m x( )T ϕ = ϕ ϕ ϕ (9) ϕ₀ (x) = 1 for all x w_odenotes the bias b

2. The requirement of the kernel K(x,x_i) is to satisfy Mercer’s theorem. The kernel function is selected as a polynomial learning machine.

K(x,x_i) = (1 + xT_x

i)2 (10) 3. The Lagrange multipliers {α_i} for i = 1 to N that maximize the objective function Q(α), denoted byα_0,i

is determined.

(

)

1 1 1 1 ( ) – , 2 N N N i i j i i j i i j Q d K x x = = = α

Σ

= α

ΣΣ

α α (11)

Subject to the following constraints:

1 0 N i i i d = α =

Σ

(12) 0≤ α ≤i C for i=1,2, ...,N (13) 4. The linear weight vector w₀ corresponding to the optimum values of the Lagrange multipliers are determined using the following formula:

0 0, 1 ( ) N i i i i w d x = α ϕ

Σ

(14)

ϕ(x_i) is the image induced in the feature space due to x_i. w₀ represents the optimum bias b₀.

4.4. Back Propagation Algorithm

These are supervised networks, and also they require a desired response to be trained. The weights of the network are computed by training the network. Artificial neural network with one hidden layer and trained by back propagation training algorithms is used to design microstrip antenna. [12],[13]

ANN structure (number of layers, number of neurons in each layer, neurons activation function, learning algorithm and training parameters) is not known in advance. Hence the network model is analyzed with different number of hidden layers in the structure and also the numbers of processing elements are also varied to acquire the accuracy. Hence it is concluded that three layers one hidden layer and 25 processing elements in the hidden layer is the optimum network structure for the proposed problem

4.5. RBF Algorithm

Radial basis function(RBF) network is a feed forward neural network with a single hidden layer that uses radial basis activation functions for hidden neurons.

It consists of three layers of neurons, Input, hidden and output. The hidden layer neurons represent a series of centres in the input data space. Each of these centres has an activation function, typically Gaussian. The generation of the centres and their widths is done using unsupervised k-means clustering algorithm.The centres and widths created by this algorithm then form the weights and biases of the hidden layer, which remain unchanged once the clustering has been done.

5. SIMULATION RESULTS

The various inputs to the network are coordinates of both slots, i.e., X1, X2, Y1 and Y2. The output of the network is bandwidth, i.e., f1 and f2 cut off frequencies. All the networks were trained with 250 samples and tested with 50 samples. For Back Propagation Network, the training time is 50 seconds and training performed in 290 epochs. For Radial Basis Function (RBF) network, the training time is 40 seconds and training performed in 175 epochs. SVM

Classifier takes less time for training compared to other ANN networks Thus SVM performs training in 31 seconds.

The results simulated using IE3D, Back Propagation Network, RBF and SVM are tabulated in Table 1. Figure 3 shows the comparison between the bandwidth obtained by IE3D, Back propagation, RBF and SVM. Table 2 shows the error calculated in percentage for BP, RBF and SVM compared with IE3D Simulated results.

Table 1

Comparison of Band Width Results of IE3d, ANN, RBF and SVM BAND WIDTH IN GHz

L of slot W of X₁, Y₁ X₂,Y₂ IE3D BP RBF SVM BW without

mm slot mm coordinate coordinate slots in GHZ

1st _{slot 2}nd _slot 5.2 0.2 0,0.5 0,-0.5 0.5915 0.5912 0.5914 0.5913 0.45 5.2 0.2 0,0.7 0,-0.7 0.5902 0.5875 0.5881 0.5901 0.45 5.2 0.2 0,1.0 0,-1.0 0.5894 0.5811 0.5896 0.5713 0.45 5.2 0.2 0,1.2 0,-1.2 0.5922 0.594 0.591 0.592 0.45 5.2 0.2 0,1.5 0,-1.5 0.5981 0.5874 0.5882 0.5980 0.45 5.2 0.2 0,2.0 0,-0.5 0.5849 0.5884 0.5835 0.5800 0.45 5.2 0.2 0,2.5 0,-2.5 0.6032 0.6029 0.6030 0.6029 0.45 5.2 0.2 0,3.5 0,-3.5 0.6491 0.6496 0.6490 0.6492 0.45 5.2 0.2 0,4 0,-4 0.6672 0.6508 0.6549 0.6675 0.45

(X, Y) CO-ORDINATES OF SLOT 2 & SLOT 1

conventional patches. The paper concludes that results obtained using SVM techniques are quite satisfactory.

REFERENCES

[1] D.M. Pozar, Microstrip antennas, Proc IEEE 80, 79-8, 1992.

[2] Vandana Vikas Thakare, 1 Pramod Singhal, Microstrip Antenna Design Using Artificial NeuralNetworks, Wiley Periodicals, 2009.

[3] C. A. Balanis, Antenna theory, 3rd Edition, John Wiley and Sons, Hoboken, NJ, 1997.

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[9] Herscovici, N., New consideration in the design of microstripantenna,” IEEE Trans. on Antennas Propagat., Vol. 46, No. 6, 807-812, Jun. 1998. [10] G. Angiulli, M. Cacciola, and M. Versaci,

Microwave Devices and Antennas Modelling by Support Vector Regression Machines, IEEE Transactions on Magnetics, Vol. 43, April 2007. [11] Nurhan Turker Tokan, Filiz Gunes, Analysis and

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[12] Simon Haykin, ]Neural Networks – A comprehensive foundation, Pearson Education Asia, 2001. [13] Zhang, Q. J. and K. C. Gupta, Neural Networks for

RF and Microwave Design, Artech House Publishers, 2000.

Figure 4: Graph Plotted Between the Percentage of Error in the y-axis and Slot Co-ordinates in the x-axis

Table 2

% of Error Compared with IE3d Output ERROR IN % X1, Y1 X2,Y2 BP RBF SVM coordinate coordinate 1st _{slot 2}nd _slot 0,0.5 0,-0.5 0.05 0.02 0.03 0,0.7 0,-0.7 0.46 0.36 0.02 0,1.0 0,-1.0 1.41 -0.03 3.07 0,1.2 0,-1.2 -0.30 0.20 0.03 0,1.5 0,-1.5 1.79 1.66 0.02 0,2.0 0,-2.0 -0.60 0.24 0.84 0,2.5 0,-2.5 0.05 0.03 0.05 0,3.5 0,-3.5 -0.08 0.02 -0.02 0,4 0,-4 2.46 1.84 -0.04 6. CONCLUSION

The inset fed microstrip patch antenna is a versatile structure which can be modified by the addition of simple slots in the design structure to overcome selected limitations of conventional patch antennas. The antenna can provide improved bandwidth enhancement, under certain conditions, while maintaining many of the desirable features of