1. Introduction

To accommodate large number of communication channels, a frequency independent antenna design is preferred. One of the powerful methods to address a requirement is designing of Log Periodic Dipole Array (LPDA). The initial work of LPDA with its mathematical description was carried out by Isbell and Carrel[1-2]. An LPDA is a wideband antenna whose characteristics are not affected by changing the spectrum of frequencies. Therefore, it is known as frequency independent antenna[3]. It consists of N number of dipole elements having unequal length, spacing and diameter. The main task in the designing of LPDA is the evaluation of current behaviour across the dipole elements[1]. The initial work to determine the current across dipole elements is carried out by King in 1957[4].

The design of an LPDA can be optimized by employing an optimization technique. There are numerous optimization techniques such as swarm optimization, genetic algorithm, artificial neural networks, etc. are frequently applied in electromagnetic related problems. Out of those techniques, swarm optimization and its different variants are considered powerful optimization tools which offer almost guaranteed convergence for complex problems[5-6]. PSO was successfully applied in various applications such as array design[7], device modelling[8], communications[9], etc.

Previously, the design of an LPDA was presented with particle swarm optimization and genetic algorithm[10-11]. The purpose of the presented work was to increase the gain of an LPDA with the reduction in size. In this paper, the variant of swarm optimization which is Quantum Particle Swarm Optimization (QPSO) is used to optimize the physical variables of an LPDA for its potential applications in mobile communication. From the results, it is noted that the QPSO based design offers at least 12.69% improvement in gain and small size than the previously presented designs [10-11].

2. Swarm Optimization

In swarm optimization, a particle tries to get a best position which is called its personal best p_best. There is also a best position with respect to the entire swarm called global best g_best. The number of parameters to be optimized for a given problem is a dimension of a swarm. A swarm is initialized with a random positions and velocities. To avoid explosion[12], sometimes, a search space is also define to ensure the confinement of particles within the defined space. For this purpose, one can define v_max by using following equation

(1)

Let us consider a j^th-dimensioanl problem with N number of particles for k^th-iteration. The position vector is represented by

(2)

The associated velocity of these particles is represented as

(3)

Optimization is achieved by updating position and velocity vectors by defining an appropriate objective function. The new position and velocity is referred as p_best and v_pbest. Optimization process compares the error associated with each particle and the lowest error is referred as g_best. The velocity of the particles is then updated by using equation which is given below[13-15].

(4)

where

(5)

(6)

The update position vector is represented by

(7)

In swarm optimization, convergence depends on controlling the parameters and changing them to appropriate values. But, this is sometimes difficult especially when multiple parameters are involved in an optimization process [16]. To resolve this problem, a quantum formula for swarm optimization is suggested, where the behaviour of particles follow the quantum mechanics principle[17-18]. In QPSO, particles update their positions by using following expression

(8)

where

(9)

(10)

3. LPDA Fundamental

The successive dipole elements of an LPDA are connected alternatively with a balanced transmission line called feeder as shown in Figure 1. It is a self-complementary structure because of its periodicity[6]. The length of shortest and longest dipole element is determined by the upper and lower frequency. The ratio between length and diameter of two consecutive elements of an LPDA is given by[19]

(11)

The scaling factor and the geometry of an LPDA is given by[19]

(12)

(13)

The scaling factor τ has a value in the range 0.76 ≤ τ ≤ 1. By choosing an appropriate value of τ and the length of first element, the length of rest of the elements can be determined which is also defined in Equation (11). In order to get an acceptable gain, there must be an appropriate distance of dipole elements from the source and it can be calculated as

(14)

Thus, the total length of an LPDA is given by

(15)

To obtain proper radiation pattern, we need to know the behaviour of current across each dipole element which can be expressed as[20]

(16)

where,[Z_E] is the impedance of dipole elements and[Y_F] is the admittance of feed line. The impedance matrix[Z_E] of dipole elements can be calculated by adopting different methods. The self and mutual impedance method is a very warm subject for the researchers and is given by[4]

(17)

The admittance matrix[YF] of feed line is determined by the following method.

(18)

To determine accurate results, the method of moment and finite element method is used. The method of moment is well defined in[21]. For the case of parallel connected dipole elements with unequal lengths, spacing and diameter, the resistance and reactance is determined by using equations as defined in[19]. The impedance of each dipole element is determined by combining the resistive and reactive components of dipole elements. As we know the impedance of dipole elements and admittance of feed line, we can evaluate current by using Equation (16). After that, it is easy to determine far-field E-plane, H-plane radiation patterns and gain of an LPDA which is as follows[20]:

(19)

(20)

and

(21)

where, P_in = 0.5Re {[V_E]},[V_E] =[Z_E][I_E] and β = 2π/λ. Figure 2 shows electric (E-plane) and magnetic (H-plane) field patterns which are drawn using Equation (19) and (20), respectively.

Figure 1. Design and geometry of Log-Periodic Dipole Array

Figure 2. Radiation patterns of LPDA (a) E-plane and (b) H-plane

4. Log Periodic Dipole Array Design

Designing an LPDA for different wireless applications is a cumbersome process. The spectral band of interest in the proposed design is 400-2800 MHz. For the desired band, the conventional design parameters are listed in Table 1. The length, spacing and diameter of each dipole element are calculated by using Equations (11), (12) and (14). We choose 13 dipole elements for optimization and the initial/ conventional values of13diole elements are given in Table 2. After designing LPDA, we optimize it by using two optimization techniques which are PSO and QPSO. In order to optimize an LPDA, a fitness function is developed which based on the gain of an antenna and defined as

(22)

where

(23)

Table 1. Conventional design parameters Serial No. Parameters Values 1 τ 0.9 2 σ 0.16 3 L1 λref×0.5 4 N 13 5 Ln/an 125 6 Zo 100

Table 2. Parameters used in LPDA design Element # Ln (m) dn (m) Dn (m) hn (m) Xn (m) 1 0.375 0.003 0.12 0.1875 0.0437 2 0.3375 0.0027 0.108 0.1688 0.0394 3 0.3038 0.0024 0.0972 0.1519 0.0354 4 0.2734 0.0022 0.0875 0.1367 0.0319 5 0.246 0.002 0.0787 0.123 0.0287 6 0.2214 0.0018 0.0709 0.1107 0.0258 7 0.1993 0.0016 0.0638 0.0996 0.0232 8 0.1794 0.0014 0.0574 0.0897 0.0209 9 0.1614 0.0013 0.0517 0.0807 0.0188 10 0.1453 0.0012 0.0465 0.0726 0.0169 11 0.1308 0.001 0.0418 0.0654 0.0153 12 0.1177 0.0009 0.0377 0.0588 0.0137 13 0.1059 0.0008 0.0355 0.053 0.0124

The optimizing parameters of LPDA are length, spacing and diameter. The swarm particles are initialized randomly. The initial velocity of the particles is zero and the maximum velocity is calculated by using Equation (1). The next step is to update the position and velocity of swarm particles. The two constant parameters which define the movement dynamics of the particles are c₁ and c₂. The value of c1 and c2 is 1.4 for the proposed design. The required optimized design is achieved after 100 iterations and is given in Table 3. The optimum response of gain against these parameters is shown in Figure 3 and 4.

The quantum particle swarm optimization (QPSO) is applied is applied on conventional design to analyse the behaviour of LPDA. In case of QPSO, particle position is important instead of velocity as discussed in Equation (8). QPSO algorithm is followed to observe the impact of optimization on conventional design[16]. The gain comparison with conventional and QPSO based design is shown in Figure 3 and 4.

Table 3. Optimized parameters of LPDA using PSO Element # Ln (m) dn (m) Dn (m) hn (m) Xn (m) 1 0.235 0.0027 0.108 0.1175 0.0274 2 0.2115 0.0024 0.0972 0.1058 0.0247 3 0.1903 0.0022 0.0875 0.0952 0.0222 4 0.1713 0.0020 0.0787 0.0857 0.02 5 0.1542 0.0018 0.0709 0.0771 0.0180 6 0.1388 0.0016 0.0638 0.0694 0.0162 7 0.1249 0.0014 0.0574 0.0624 0.0146 8 0.1124 0.0013 0.0517 0.0562 0.0131 9 0.1012 0.0012 0.0465 0.0506 0.0118 10 0.0910 0.0010 0.0418 0.0455 0.0106 11 0.0819 0.0009 0.0377 0.0410 0.0096 12 0.0737 0.0008 0.0300 0.0369 0.0086 13 0.00664 0.0008 0.0300 0.0332 0.0077

Table 4. Optimized parameters of LPDA using QPSO Element # Ln (m) dn (m) Dn (m) hn (m) Xn (m) 1 0.24 0.0019 0.0786 0.12 0.028 2 0.216 0.0017 0.0691 0.108 0.0252 3 0.1944 0.0016 0.0622 0.0972 0.0227 4 0.175 0.0014 0.0560 0.0875 0.0204 5 0.1575 0.0013 0.0504 0.0787 0.0184 6 0.1417 0.0011 0.0453 0.0709 0.0165 7 0.1275 0.0010 0.0408 0.0638 0.0149 8 0.1148 0.0009 0.0367 0.0574 0.0134 9 0.1033 0.0008 0.0331 0.0517 0.0121 10 0.093 0.0007 0.0298 0.0465 0.0108 11 0.0837 0.0007 0.0268 0.0418 0.0098 12 0.0753 0.0006 0.0241 0.0377 0.0088 13 0.0678 0.0005 0.0235 0.0339 0.0079

It is clear from the figures that the gain is improved by using PSO and QPSO optimization techniques. The average gain values of the conventional design, PSO and QPSO based designs are given in Table 4 and the percentage improvement in gain is given in Table 5. One thing is noted from Figure 4(a) and (b) that at higher frequencies PSO optimized gain is better than the QPSO optimized gain. Another important factor considered the designing of LPDA is its physical size. It is observed that PSO and QPSO techniques reduced the physical size of LPDA and the optimized values are given in Table 3 and 4. The reduction in size tends to the ease of implementation and provides cost effective solution.

Table 5. Comparison between PSO and QPSO based designs Frequency Bands Frequency (MHz) Percentage Improvement PSO Percentage Improvement QPSO WiMAX 500-700 20.93 39.45 GSM 800-900 21.44 32.85 DCS 1800-1900 15.85 14.26 3G, Wi-Fi and Bluetooth 2200-2600 12.69 11.11

Figure 3. Conventional and optimized gain for (a) WiMAX and (b) GSM frequency bands

Figure 4. Conventional and optimized gain for (a) DCS and (b) Wi-Fi,, 3G mobile and Bluetooth frequency bands

5. Conclusions

A Log Periodic Dipole Antenna (LPDA) is designed by using conventional technique for microwave communication bands. The response as a gain is observed against the spectral band. After that the design is optimized by using stochastic techniques called PSO and QPSO. It is observed from the comparative optimized results that optimized design improved the gain for each frequency band compared to the conventional design. It is also noted that the optimized design is much smaller as compared to the conventional design.

ACKNOWLEDGEMENTS

The authors are thankful to Dr. Muhammad Mansoor Ahmed, Executive Vice President of Mohammad Ali Jinnah University, Islamabad, Pakistan and Mr. Qamar-ud-Din Memon, Lecturer at NFC Institute of Engineering, Fertilizer and Research, Faisalabad, Pakistan, to giving us understanding about swarm optimization.

Nomenclature

x_i i^th Particle position

v_i i^th Particle velocity

x_max Maximum position

x_min Minimum position

v_max Maximum velocity

c₁ Cognitive component

c₂ Social component

ϕ₁, ϕ₂ Random numbers

χ Random selected spacing factor

ω Inertia weight

k Number of iterations

M Number of particles in swarm

v_pbest Personal best velocity

v_gbest Global best velocity

j dimension of problem

τ Scaling factor

σ Spacing factor

α Geometry of LPDA

N Number of dipole elements

L_n Length of n^th dipole element

X_n Distance from source to nth dipole element

h_n Half-length of n^th dipole element

D_n Spacing between elements

d_n Diameter of n^th dipole element

I_E Current across dipole elements

Z_E Impedance of dipole element

Y_F Admittance of feeder line