Design of time modulated concentric circular and concentric

www.ietdl.org Published in IET Microwaves, Antennas & Propagation Received on 8th March 2013 Revised on 13th January 2014 Accepted on 14th January 2014 doi: 10.1049/iet-map.2013.0132

ISSN 1751-8725

Design of time modulated concentric circular and concentric hexagonal antenna array using hybrid enhanced particle swarm optimisation and differential evolution algorithm Mohab A. Mangoud1, Hassan M. Elragal1,2, Mohamed T. Alshara1 1

Department of Electrical and Electronics Engineering, University of Bahrain, P.O. Box 32038, Isa Town, Kingdom of Bahrain 2 Department of Electrical Engineering, Faculty of Engineering, Alexandria University, Alexandria, Egypt E-mail: [email protected]

Abstract: Design of circular arrays (CAs) and hexagonal arrays (HAs) with low sidelobe level (SLL) and high directivity is usually achieved by increasing the number of array elements, which leads to a high undesired mutual coupling. Therefore this study presents an efficient optimisation method and a framework to show how to design multi-ring concentric CAs (CCA) and concentric HAs (CHAs) configurations using a hybrid enhanced particle swarm optimisation and differential evolution (hybrid EPSO/DE) optimisation technique. The presented optimum CCA and CHA have perfect invariant SLL and high directivity with low mutual coupling by keeping the inter-element spacing not less than half a wavelength which is not possible to be achieved in CA and HA arrangements. Different configurations with two-rings, three-rings and four-rings are presented. The rotation angle of outer rings and the complex excitations of array elements are first optimised. Then, this design is further optimised using a time modulation technique by controlling the switch-on times and the phases of elements excitations of the best CCA and CHA array designs. The presented time modulated (TM) concentric CA and TM concentric HA designs attain ultra-low SLL, reduced sideband level and maximised directivity besides reducing the dynamic range of the array excitations.

1

Introduction

A very low sidelobe level (SLL) radiation pattern for a linear array can be synthesised with a high gain main lobe and narrow beamwidth (BW) in any given direction for the elevation plane for beamforming applications. However, these arrays do not work equally well in the azimuth plane. Two-dimensional (2D) antenna arrays have the capability to transmit and receive data in elevation and azimuth angles as well. Thus 2D antenna arrays configurations have been widely used in the field of 3D beamforming. One of the best configurations of 2D antenna arrays is the circular antenna array, since its main lobe can be steered in all azimuth angles without changing its BW [1]. The circular array (CA) has high SLLs that can be minimised by decreasing the inter-element spacing but this leads to increasing the undesired mutual coupling effect. A solution to reduce the CA high SLLs is to use a concentric multi-ring array with adequate inter-element distance. The multi-ring array configuration can be arranged as concentric CAs (CCAs) [2–4], hexagonal array (HA) [5, 6] or concentric HAs (CHAs) which is also referred as the planar uniform HA [7]. In [7], it is proved that the hexagonal and CHAs have better steerability and higher gain properties IET Microw. Antennas Propag., pp. 1–9 doi: 10.1049/iet-map.2013.0132

than CAs. In order to effectively design such complex structure arrays, evolution algorithms have been widely used. Recently, particle swarm optimisation (PSO) algorithm and differential evolution (DE) algorithm have been used as efficient optimisation tools. They are easy to implement with a better convergence, less trapping in local minima and relatively shorter computation time compared with genetic algorithm (GA) and simulated annealing. A PSO algorithm was used in [8] to design non-uniform circular antenna arrays with optimum SLL reduction. PSO was used to determine an optimised set of complex weights and antenna element separations that provide a radiation pattern with lowest SLL with the constraint of a fixed major lobe BW. The results show that the design of non-uniform circular antenna arrays using the PSO method provides a SLL reduction better than that obtained using GA. In [9], the authors used a modified type of PSO to design three-ring concentric circular antenna arrays (CCA) with and without central element feeding. Optimal concentric CA (CCA) designs were determined by optimising both current excitation weights and the radii of the rings to obtain low SLL. Another promising optimisation technique for antenna design is the DE algorithm which was used as an optimisation tool in [10] to synthesise radiation patterns 1

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www.ietdl.org in conformal antenna arrays in the presence of a platform. DE was applied to optimise the amplitude weights of elements to reduce the sidelobe with the desired phase weights found by the scan angle and array geometry. In [11], an adaptive DE algorithm has been used to optimise the distance between the elements of the linear array to synthesise a radiation pattern with minimum SLL and null placement control. As shown in most of these papers, PSO and DE have proved to be the most efficient evolutionary optimisation methods. Therefore, in [12], a modified optimisation algorithm EPSO was introduced for pattern synthesis applications and including wide null in the array patterns. This work was followed by Elragal et al. [13], with the EPSO technique being hybridised with DE to beamform the pattern of phased antenna arrays. Examples for designing reconfigurable linear [11] and circular antenna arrays [14] with prescribed null directions have been presented. Null steering is achieved by position perturbation of array elements in arbitrary directions with a minimum SSL change constraint. The numerical results showed that the hybrid DE/EPSO outperforms the classical DE and the EPSO in obtaining optimum desired patterns. An alternative approach to realise low/ultra-low side lobes is to present the time as another degree of freedom in the design. Antenna arrays with the time parameter are called time modulated arrays (TMAs). The time modulation can be implemented by simple ‘on–off’ switching of antenna elements. The collateral disadvantage of TMAs is that there are many harmonics separated at multiples of the modulation frequency, which means that portion of the radiated or received power is moved to the sidebands. These harmonics appear in SBLs considered as power wastage that affect the directivity and gain of the antenna array at the centre frequency. Therefore the SBLs need to be reduced or eliminated if possible when applying this time modulation technique. Also another drawback of the position-time modulation technique is the notable decrease of the directivity, because of the power losses of sidebands. Moreover, to obtain ultra-low SLL in the azimuth plane, the dynamic-range ratio of the amplitude excitation of the array’s elements will increase. Therefore lower dynamic range ratios are needed to have flexible hardware realisation of the arrays [15]. Most of previous works consider optimising excitations of linear arrays with time modulation technique [16, 17]. However, in [18], DE was used to optimise the excitation phases and the switch on time intervals of time modulated (TM) CA (TMCA) elements. In [19], time modulation is applied to the uniformly excited concentric circular ring array to suppress the SSL with fixed BW. The main objective of this paper is application of hybrid DE/EPSO [13] to design multi-ring CCA or CHA to achieve optimum radiation patterns. We present design procedures of different configurations of CCA and CHA, having the same number of element with different number of rings, to achieve the lowest SSL radiation pattern while keeping the BW as close as possible to the initial CA. This is achieved by optimising both the rotation angles of the outer rings and the excitations of each element in the array. In addition, TM excitations are used to further enhance the design. The rest of this paper is organised as follows: Section 2 describes the hybrid EPSO/DE optimisation technique. In Section 3, the design of CA, HA, CCA and CHA are presented. Design of multi-ring arrays is introduced in Section 4 by optimising the complex current excitation 2

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(amplitudes and phases) and the positions, (rotating angles) of the array elements for best selected CCA and CHA initial configurations. In Section 5, further sidelobe suppression is performed through optimising switching times and phases of complex currents of array excitations. Finally, conclusion remarks are presented in Section 6.

2 Hybrid EPSO/DE algorithm optimisation techniques Evolutionary algorithms such as GAs, SA, PSO and DE have been widely used in solving global optimisation problems. A meta-heuristic optimiser usually converges to the global optimum. The most commonly used method is to run the same optimisation scheme several times. Recent results show that collaborative and successful usage of population based algorithms can lead to performance improvement. Other successful approaches include the combination of several different meta-heuristic optimisers. In [13], two meta-heuristic global optimisers are combined, the first one is an enhanced version of EPSO and the second one is DE. The combination method, hybrid EPSO/DE that is illustrated in Fig 1, starts by initialising a generation of 2P individuals of N-dimensional trial vectors. EPSO is applied to the first P individuals and DE is applied to the remaining P individuals to obtain a new updated generation. The updated generations from EPSO and DE are merged together. The fitness function for each individual is calculated and the individual with the minimum fitness function value (best individual) is located. EPSO is applied again to a randomly selected P individual from the 2P updated generation (containing best individuals) and DE is applied to the remaining P individuals to obtain the new updated generation. This operation of updating individuals using EPSO and DE will continue until a convergence criterion is met. Interchanging individuals between the two parallel running optimisers can provide alternative better solutions to the ones being explored from each optimiser and decrease the chance of trapping in local minima. This hybrid DE/EPSO was introduced and applied successfully for interference suppression by position-only control using a minimum number of mobilised elements for linear and circular phased arrays in [13]. It was demonstrated that the hybrid DE/EPSO is an effective algorithm with good accuracy to be applied for solving global optimisation problems and designing reconfigurable antenna arrays. Therefore this technique is applied in this paper to design multi-ring CCA and CHA arrays as will be shown in the following sections.

Fig. 1 Flowchart of hybrid EPSO/DE optimisation technique IET Microw. Antennas Propag., pp. 1–9 doi: 10.1049/iet-map.2013.0132

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Design of concentric CCA and CHA

The design starts with a CA consisting of equally spaced 2N isotropic elements around a circle of radius r. The radiation pattern of this array [3] is given by the array factor AF(u, w) =

2N !

In ej(kr sin u cos (w−wn )+an )

(1)

n=1

where In is the amplitude excitation of nth element, k is the wave number and r is the radius of the CA (in λ) where (r = 2Ndo/2π) with do being the inter element distance. fn is the angular position of the nth element on the x−y plane and it is given as fn = 2π(n−1)/2N for n = 1, 2, …, 2N. Assuming the CA has an even number of elements 2N with equal spacing between elements, therefore fn + N = fn + π. αn is the phase excitation of nth element. The HA is considered a multi-ring CA, since its elements are placed at different distances (radii) from the centre of the array. The radii of adjacent elements are either identical or slightly different. To form a hexagonal geometry, the number of elements should be a multiple of six. The array factor of the HA will be AF(w) =

Nm M ! !

Imn exp

m=1 n=1

(2)

{krm cos(w − wmn ) + amn } where M is the total number of rings in the HA, Nm is the total number of elements in the mth ring, Imn and αmn are the amplitude and phase excitation of the nth element in the mth ring and rm is the radius of the mth ring. fmn is the predetermined angular position of the nth element in the mth ring considering the first element of the mth ring is placed on the x axis for all rings. Similar to [5], the radii and the angular positions of hexagonal elements in this paper will be calculated according to the number of elements and inter-element distance in the geometry. The maximum radius (rm) in the hexagonal ring as measured from the array centre to the vertices will be used to define the geometry size. If several ring arrays share the same centre point but with have different radii, then the formed configuration is a CCA [5] or CHA that has various advantages like steerablity, high gain and their compact size compared to the CAs. Fig. 2 shows how the elements are placed to produce two different multi-ring CCA and CHA configurations. As shown, dom is the inter-element distance in the mth ring and rm is the radius of the mth circular ring or the maximum radius of the mth hexagonal ring. The AF for both CCA and CHA will be the same as the AF of HA in (2), with the

angular position (fmn) for the nth element in the mth ring defined as follows

wmn

" # n−1 = 2p Nm

(3)

where m = 1, 2, …, M and n = 1, 2, …, Nm. It is assumed that the amplitude excitations are taken to be Imn = Im(n + Nm/2) to have symmetrical radiation patterns. To direct the peak of the main beam in the f0 direction, the phase excitation of the mnth element can be chosen to be

amn = −amn+Nm = −krm cos (w0 − wmn )

(4)

3.1 Design examples: 36 isotropic elements, uniform CCA and CHA Starting with the design of 36 element uniform amplitude CA and HA, it is found that setting inter-element distance do to 0.25λ, (CA radius = 1.43λ and HA maximum radius = 1.5λ) produces radiation patterns with SLLs equal −7.9 and −8.15 dB and BWs equal 31° and 32° for the CA and HA arrays, respectively. The inter-element distance of 0.25λ is small and can produce a high mutual coupling. Therefore CCA and CHA with a larger distance of at least 0.5λ between elements are designed to achieve a similar pattern to that of the 36 elements CA and HA. This can be done in different configurations depending on the number of rings (M ) and the number of elements in each ring (Nm). Defining each configuration by the following notation, Array{N1, N2, …, NM}, two-rings, three-rings and four-rings are assumed. Therefore, the following six arrays are designed for both CCA and CHA, namely Array {18, 18}, Array{12, 24}, Array{6, 6, 24}, Array {6, 12, 18}, Array {6, 6, 12, 12}and Array{6, 6, 6, 18}. Table 1 depicts the details of the design parameters for the six cases considered in two, three and four circular and hexagonal rings. Also it shows the SLLs and the BW obtained for each configuration. As shown, the BW of some CCA and CHA configurations varies slightly from that obtained before for CA and HA, for example, Array {6, 6, 12, 12} has 31° and 32° for CCA and CHA, respectively. Also, SLLs of most configurations are found to be lower than that of the 36 elements CA (−7.9 dB) and HA (−8.15 dB). Note that this is achieved with larger distances between array elements which are set to be at least 0.5λ. The best case according to the SLL lowest values, was found to be the Array {6, 12, 18} with SLL of −11.8 and −14.3 dB for CCA and CHA, respectively. For this configuration a BW increase of 4° has been noted compared to the BW of the initial CA and HA. However, if BW of the initial array needs to be unchanged, one can use CCA Array {6, 6, 12, 12}. Also, it can be seen that in general the CHA configurations have better SLL than CCA configurations. Fig. 3 illustrates the radiation pattern of Array {6, 12, 18} for both CHA and CCA in the azimuth plane at θ = 90°.

4 Optimisation of element positions and complex weights for CCA and CHA configurations Fig. 2 Geometries of CCA and CHA for different configurations IET Microw. Antennas Propag., pp. 1–9 doi: 10.1049/iet-map.2013.0132

In this section, the objective is to design multi-ring CCA and CHA with minimum SLL and maximum directivity without 3

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www.ietdl.org Table 1 Number of elements, dimensions of each ring, BW and SLL for different CCA and CHA configurations Configuration

array {18, 18} array {12, 24} array {6, 6, 24} array {6, 12, 18} array {6,6,12,12} array {6, 6, 6, 18}

First ring, λ

CCA CHA CCA CHA CCA CHA CCA CHA CCA CHA CCA CHA

Second ring, λ

Third ring, λ

r1

do1

r2

do2

r3

do3

r4

do4

1.43 1.5 0.95 1.0 0.48 0.5 0.48 0.5 0.48 0.5 0.48 0.5

0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5

1.91 2 1.91 2 0.95 1 0.95 1 0.95 1 0.95 1

0.67 0.67 0.5 0.5 1 1 0.5 0.5 1 1 1 1

— — — — 1.91 2 1.43 1.5 1.43 1.5 1.43 1.5

— — — — 0.5 0.5 0.5 0.5 0.75 0.75 1.5 1.5

— — — — — — — — 1.91 2 1.91 2

— — — — — — — — 1 1 0.67 0.67

Fig. 4 Fig. 3 Radiation pattern of Array {6, 12, 18} UCHA and UCCA in the azimuth plane at θ = 90°

changing the BW. The optimisation is performed by rotating the outer circular or hexagonal rings and setting the complex weights of element excitations rather than using uniform amplitude. In this case, the optimisation parameters are the angles of the outer rings and the complex excitations of the CCA and CHA array elements. 4.1 Objective function formulation of the design problem We define the rotation angle (σm) as the angle by which every element in the mth ring is rotated anticlockwise which equals the rotation angle of the first element of the mth ring about the x axis. Therefore σm is the rotation angle that is required to be optimised for the mth ring starting from the second ring with the inner ring kept fixed with rotation angle (σ1 = 0). The angular position of the nth element in the mth ring is defined as fmn = σm + 2π (n−1)/Nm. Fig. 4 illustrates the optimised rotated angles of the outer rings for the three ring CCA and CHA cases. Also, for each nth element in the mth ring, defining the excitation amplitude as (Imn) and the phase as (βmn). Therefore it is needed to optimise the following objective function of the problem under the small bandwidth variation constraint min [ C sm ,Imn ,bmn

%

$ $ max $AF(w, sm , Imn , bmn )$

w!DwBW

Fourth ring, λ

&

subject to dBW ≤ 1 4

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(5)

BW, deg.

SLL, dB

27 28 29 30 31 32 35 36 31 32 30 30

−8.5 −10.1 −10 −9.5 −11.5 −9.5 −11.8 −14.3 −10.9 −7 −7.9 −9.3

Geometries with rotated outer rings and fixed inner ring for

a CCA and b CHA

where AF(f, σm, Imn, βmn) is the array factor of the multi-ring CCA and CHA arrays, f is the azimuth angle, C is the set of all vectors of {σm, Imn, βmn}, ΔfBW and δBW are the main lobe BW and the change in the main lobe BW, respectively, and ɛ is the maximum allowable change in the main lobe BW. This problem is optimised using the three algorithms EPSO, DE and the hybrid EPSO/DE techniques according to the following fitness function

f =

◦ w=180 !

w=−180◦

$ $w

BW (w)

$ · AF(w)$ + D

(6)

where wBW(f) is the shaping function that is defined as follows wBW (w) =

%

0 1

if w [ DwBW otherwise

(7)

The directivity of the array (D) is $ $ $E(u , w )$2 0 0 D= $2 '2p 'p $ (1/4p) 0 0 $E(u, w)$ · sin u · du · dw

(8)

where |E(θ, f)| is the far-field radiation of the array with the main beam directed towards the elevation and azimuth angles {θ0, f0}, respectively IET Microw. Antennas Propag., pp. 1–9 doi: 10.1049/iet-map.2013.0132

www.ietdl.org 4.2 Design example: optimised 36 isotropic elements, complex excited and rotated CCA and CHA

Figs. 5 and 6. In EPSO, DE and hybrid EPSO/DE, (1)–(4) are used to implement the fitness function with the following constraints. The boundaries of the rotation angles σm were set to {0–60} degrees. The boundaries of the optimised parameters are: the amplitude Imn = {0.1–1}, for the phase βmn = {−π–π}. Table 2 illustrates the optimisation technique used for each case, the optimised values of rotated angles in each ring, the SLLs, the BW of the main beam, the directivity, the dynamic ratio for each case of the three ring and four ring CCA and CHA. Also the table includes the SLL and directivity that are calculated for the CA with Chebyshev-like pattern [14]. It is clear from the obtained numerical results that the hybrid DE/EPSO outperforms the classical DE and EPSO in achieving the optimum designs. Compared to the initial designs shown in Table 1, it is found that for all cases, optimising the rotation angles and element excitations leads to a significant reduction in the SLL values. For example, the SLL drops in CCA Array{6, 12, 18} from −11.8 to −25.4 dB, in CHA Array {6, 12, 18} from −14.3 to −35 dB, in CCA Array{6, 6, 12, 12} −10.9 to −27.8 dB and in CHA Array {6, 6, 12, 12}, the SLL dropped from −7 to −33.4 dB. From the obtained results shown in Table 2, and by comparing the design with Chebyshev-like CA design. It can be concluded that the optimised CHA Array {6, 6, 12, 12} using hybrid EPSO/DE has best design in terms of SLL and directivity. This optimum design is obtained by rotating the outer rings with the values 42°, 16.6° and 57.3° and setting the

Three optimisation algorithms EPSO, DE and hybrid EPSO/ DE are used to optimize the 36-element CCA and CHA Array {6, 12, 18} and Array {6, 6, 12, 12} by rotating the outer rings and setting the element excitations. These arrays are chosen as an initial design for the optimization process since they have the best SLL and BW as illustrated in Table 1. In this case, the three rings Array {6, 12, 18} with the radiation pattern is assumed to be symmetric, therefore 2 angles for the outer rings and 18 amplitudes and 18 phases for the element excitations are optimised. Likewise, the optimisation parameters for the four rings Array {6, 6, 12, 12} configuration, there are 3 rotation angels and 18 element complex excitations. The SLL, BW and directivity of the resultant optimised arrays are compared with optimum Chebyshev-like radiation pattern of a 36 elements CA [14]. It should be noted, that the optimum CA design in [14] has 0.25λ inter-element distance. However, the optimum CCA and CHA with rotated outer rings preserve the inter-element distances to at least 0.5λ. 4.3

Numerical results

This section presents the results obtained from the optimisation process and these are shown in Table 2,

Table 2 Optimisation technique, the values of rotated angles in each ring, SLL, BW and directivity, dynamic ratio for each case of the three ring and four ring CCA and CHA in compare with CA with Chebyshev-like [14] Configuration

Optimisation technique

Rotated angle, deg. σ2

36-elements CA with Chebyshev-like [14] CCA array {6,12 ,18} DE EPSO HYBRID CHA array {6,12 ,18} DE EPSO HYBRID CCA array {6, 6, 12, 12} DE EPSO HYBRID CHA array {6, 6, 12, 12} DE EPSO HYBRID

σ3

σ4

13.4

10.7

—

30

10

—

15.9

12.5

13.7

42

16.6

57.3

SLL, dB

δBW, deg.

Directivity, dBi

Imax/Imin

−31 −22.2 −24.2 −25.4 −15.2 −23.1 −35 −19.2 −23.5 −27.8 −22 −27.4 −33.4

0 2 5.8 6 −2.2 −4.4 1.8 −12.2 −4.2 −2.4 −9.1 −1.2 1.6

7.9 7.47 7.22 8.03 6.67 8.9 8.85 10.95 10.77 10.48 10.87 10.38 10.2

2 5 6.2 8.4 5.4 6.3 8.2 3.3 6.6 7.9 3 7.8 6.5

Fig. 5 Optimised amplitude and phase for the elements of CHA Array {6, 6, 12, 12} using hybrid EPSO/DE algorithm IET Microw. Antennas Propag., pp. 1–9 doi: 10.1049/iet-map.2013.0132

5

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Fig. 6 Radiation pattern of CHA {6, 6, 12, 12} using the three optimisation algorithms compared with Chebyshev-like pattern of CA

amplitude and phase of elements excitations to the values shown in Fig. 5. It should be noted that this optimum four hexagonal ring array achieves a SLL of −33.4 and directivity of 10.2 dBi with an increase in the BW of 1.6° with large inter-element distances of 0.5λ, λ, 0.75λ and λ in rings 1, 2, 3 and 4 unlike the Chebyshev-like CA that has a 0.25λ. Fig. 5 shows the values of the optimised 18 amplitude and phase excitations for the array elements using hybrid EPSO/DE algorithm, for the symmetric 36 elements CHA Array {6, 6, 12, 12}. As shown, the excitations are not uniform with a dynamic range ratio Imax/ Imin = 6.5. Fig. 6 shows the optimised radiation pattern of CHA Array {6, 6, 12, 12} using the three optimisation algorithms. As seen, the hybrid technique produces the design with the lowest SLL of −33.4 dB compared with other optimisation techniques and to the reference CA with Chebyshev-like pattern.

TMHA complex excitations (amplitude and phase) the switching times will be optimised with gain and dynamic range taken into considerations. 5.1

Objective function formulation

Consider the CHA Array {6, 12, 18} and CCA Array {6, 6, 12, 12} that are optimised in Section 4. Considering the same dimensions for the rings diameter and the isotropic elements number and each element is connected to a high-speed RF switch then the AF is given by AF(u, w, t) = ej2pf0 t ×

Nm M ! ! m=1 n=1

5 Optimisation of switching times and phases of excitations of TM CCA and CHA In the previous section, a low SLL up to −35 dB was achieved in the azimuth plane by optimising the complex weights and rotation angles of the 36-elements with different configurations. In order to achieve a lower value of SLL and synthesise ultra-low side lobes this may be achieved by using a very high excitation dynamic-range ratio as in [15] making them difficult for the practical implementation. A TM technique, besides controlling the complex weights similar to the approach used in [18] could be used to synthesise the ultra-low SLL in the azimuth plane for further SLL reduction. In [18], the capability to lower the equal-ripple SLLs of 36 isotropic elements λ∕4-spaced– TMCA with a uniform amplitude excitation is considered by optimising switching times and phases of the excitations. This section presents a continuation to the optimisation process of the best CCA and CHA geometries obtained in Section 4 in order to achieve an ultra-low SLL radiation by applying TM technique. These arrays will be referred as time modulated concentric CA (TMCCA) and TMCHA. Therefore, ultra-low SLL can be achieved without increasing the number of elements while preserving low mutual coupling by separating the elements to at least 0.5λ unlike the λ∕4-spaced TMCA. Also, for TMCCA and 6

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Imn Umn (t) · ej[krm sin u cos (w−wmn) +amn ] (9) " # n−1 = 2p Nm

wmn

(10)

where M is the total number of rings in the array, Nm is the total number of elements in the mth ring, Imn is the absolute value of amplitude excitation and rm is the radius of the mth ring. αmn is the phase excitation. The time switching function Umn(t) in a TM period Tp is given as Umn (t) =

%

1 0 ≤ t ≤ tn 0 otherwise

(11)

where tn is the pulse width of the nth element. Owing to the periodicity of Umn(t), the array far-field pattern in (9) can be expanded into a Fourier series, and given by AF(u, w, t) = ×

1 !

i=−1

( Nm M ! ! m=1 n=1

j[krm sin u cos (w−wmn) +amn ]

amni (t) · e

)

· ej2p(f0 +ifp )t (12)

where fo is the centre frequency with harmonics at ifp, the dynamic amplitude of the Fourier component amn0 can be IET Microw. Antennas Propag., pp. 1–9 doi: 10.1049/iet-map.2013.0132

www.ietdl.org follows

simplified as (13)

amn0 = imn. · tmn /Tp

GTM = DTM ·

To facilitate the optimisation process it will be divided into two-step fitness function using the hybrid EPSO/DE algorithm. In the first step of the fitness function, it is aimed to find the desired fundamental radiation having a prescribed main-lobe BW and the lowest SLL, and in the second step of the fitness function, the SBL is tried to be suppressed as low as possible in according with the results of the first step. In the first step of the optimisation, a single optimisation vector v1 = {amn0, αmn} is used, and the so-called dynamic amplitude amn0 here is treated as one variable according to the following fitness function f1 (v1 ) = w1 · SLLmax (v1 ) $ $ $ $ × $ f0 + w2 · wBW (v1 )$ f0

The objective of this example is to synthesise a concentric geometries with ultra-low SLL patterns and trying to achieve dynamic-range ratios as low as possible. A 36-elements arrays will be considered in order to be able to compare with the available published designs as [18] and to lower the SLL of the previous section optimum design. From the previous section, it was found that, the best optimised CCA configuration was Array {6, 6, 12, 12} with SLL = −27.8 dB and Imax/Imin = 7.9, and the best optimised CHA configuration Array {6, 12, 18} with SLL = −35 and Imax/Imin = 8.2. It can be noticed that both of these dynamic-range ratios are high, and the use of the time-modulation intend to solve this problem. The aim of this section is to optimise the switch-on instants and the complex weights of the 36-elements TMCHA and TMCCA to minimise SLL and sideband levels (SBLs) with minimum change in main-lobe BW at centre frequency f0 simultaneously using hybrid EPSO/DE algorithm. A comparison between numerical results of this section and results obtained in [18] will be discussed. The reference pattern in [18] uses time modulation beside phase excitation optimisation with non-uniform fixed amplitude excitation to minimise SLL and SBLs for 36-elements CA with 0.25λ spacing between elements

(14)

(15)

5.3

SBL(1), SBL(2) and SBL(3) are the maximum SBL at the first sideband, the second sideband and the third sideband, respectively. By extending (8), to include the centre frequency and all the sideband components, the directivity of TMCCA and TMCHA is given by DTM =

(1/4p)

,1

$ $ $E (u , w )$2 0 0 0 $ '2p 'p $ $E (u, w)$2 sin u du dw

i=−1 0

0

(17)

5.2 Design example: optimised 36 isotropic elements, TMCCA and TMCHA

SLLmax is the optimised maximum SLL at the centre frequency, fBW is the BW of the main-lobe at the centre frequency, w1 and w2 are the corresponding weighting factors of each term in the cost function. For the second step of the optimisation, the related vector will be v2 = {tmn/ Tp}, where tmn/Tp is the normalised switch-on time for each element to be optimised to suppress different SBLs of the TMCCA and TMCHA, according to the following fitness function $ * $ f2 (v2 ) = min SBL(1) (v2 )$ f0+ fp , $ $ + $ $ SBL(2) (v2 )$ f0+ 2fp , SBL(3) (v2 )$ f0+ 3fp

. $ $2 $I $ · t /T n p n=−N n $ $2 ,N $I $ n=−N n

,N

Numerical results

The search ranges for the first vector will be: the dynamic amplitude ahn0 = {0.1, 2}, and the excitation phase αmn = {−π, π}. The proper values of w1 and w2 were found by trial and error to be 1 and 2, respectively. For the second vector the normalised switch-on times range will be tmn/Tp = {0, 1}. It is obvious from Table 3 that the time modulation technique in the case of TMCHA using hybrid EPSO/DE succeeded to achieve ultra-low SLL up to −43.1 dB in the azimuth plane, with a considerable gain up to 9 dBi compared to the pattern in [18] which has lower SLL = −45 dB, but low gain = 6.2 dBi and high SBLs reached −20 dB in the first SBL, while in the case of TMCHA the highest SBL was −39.7 dB as shown in Fig. 7. The first null BW, FNBW = 56.4°, which is thinner than the BW of the reference pattern FNBW = 60° and with acceptable dynamic-range ratio Imax/Imin = 5.55. For the case of TMCCA, the SLL could not be less than the –35.6 dB with

(16)

i

where Eo and Ei are the radiation patterns of the centre frequency fo and the ith sideband components (harmonic frequency components fo + ifp, i ≠ 0), respectively. (θ0, f0) is the direction of the main beam. Here, the total radiated power as expressed in the denominator includes the centre frequency and all the sideband components. The gain of TM arrays (GTM) is calculated as in [18] by numerically multiplying the directivity by the feed-network efficiency as

Table 3 Radiation parameters of optimised 36-elements TMCHA and TMCCA in compare with CA Configuration

Algorithm

SLL, dB

BW, deg.

SBL(1), dB

SBL(2), dB

Gain, dBi

Imax/Imin

circular [18] array {6, 12, 18} TMCHA

DE DE EPSO Hybrid DE EPSO Hybrid

−45 −33.2 −39.83 −43.1 −30.3 −34.6 −35.6

60 50.7 55.4 56.4 50.2 51.4 51.6

−20 −26.4 −29.7 −39.7 −31.0 −32.4 −33.2

−23.5 −27.2 −32.8 −45 −27.8 −32.8 −39.6

6.2 9.6 9.0 9.0 13 12.2 11.59

2 4.2 4.9 5.55 3.2 3.35 4.15

array {6,6,12,12} TMCCA

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Fig. 7 Radiation pattern of optimised TMCHA at f0, f0 + fp, f0 + 2fp and f0 + 3fp using hybrid EPSO/DE algorithm

Fig. 8 3D space and frequency response of optimised TMCHA

the use of time modulation technique, however, it achieves better gain values. This proved the role of the geometry in synthesising the radiation pattern for the same number of elements in the antenna array. Fig. 8 illustrates the 3D space and frequency response of TMCHA that shows how hybrid EPSO/DE algorithm is used to achieve optimum radiation patterns in both space and frequency domains.

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Conclusions

An approach for the design of optimum TM multi-ring arrays such as CCA and CHA has been presented. The optimisation has been performed for complex current excitations, the rotation angles and the switching times of the elements of TM multi-rings circular and hexagonal antenna arrays. The objective was to find the best design configurations with ultra-low SLL, reduced mutual coupling, high directivity and low dynamic-range ratio. The presented CCA and CHA are compared with CA and HA having the same number of elements with and without applying time modulation. Three evolutionary optimisation methods EPSO, DE and hybrid EPSO/DE have been used to achieve the target designs. The obtained results confirm that hybrid EPSO/DE algorithm [13] performs better than meta-heuristics PSO and DE individually. The numerical results show that hybrid EPSO/ 8

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DE optimisation efficiently produce optimum designs for such complex geometries. A numerical design example of 36 elements two, three and four rings CCA and CHA have been synthesised to reduce SLL, increase the directivity and reduce the mutual coupling. Compared with 36 element CA with equal-ripple Chebyshev-like radiation design which has −31 dB SLL, the hybrid EPSO/DE has produced a better CHA designs with the same number of elements distributed over three and four rings with SLL of −35 and −33.4 dB, respectively, and with higher directivity and minor BW change. The mutual coupling has been fully taken into consideration by preserving the inter-element spacing to at least 0.5λ. To further enhance the achieved radiation pattern, a time modulation technique has been applied to the optimum CCA and CHA. By optimising switching time and phases of excitations, an optimum TMCHA design has been produced that has an enhanced radiation pattern with SLL equals −43.1 dB with suppressed SBL, mutual coupling and lower dynamic range compared to the non-TM case.

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References

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