IEEE TRANSACTIONS ON MAGNETICS, VOL. 50, NO. 2, FEBRUARY 2014 7015004

Parallel Implementation of a Combined Moment Expansion andSpherical-Multipole Time-Domain Near-Field to

Far-Field TransformationGlaucio L. Ramos1,4, Cássio G. Rego2, and Alexandre R. Fonseca3

1Department of Telecommunications and Mechatronics Engineering, Federal University, São João del-Rei 36307, Brazil2Department of Electronic Engineering, Federal University, Minas Gerais 30000, Brazil

3Computer Department, Federal University, Vales do Jequitinhonha e Mucuri 39100, Brazil4Graduate Program in Electrical Engineering, Federal University, Minas Gerais 30000, Brazil

A time-domain spherical-multipole near-to-far-field algorithm running in a parallelized code version is introduced in this paper.Such an approach is employed to obtain UWB antenna radiated fields and radiation patterns directly in time domain, which is moreconvenient to perform a unified characterization in time and frequency domains. We propose the use of the OpenMP, an applicationprogram interface that permits the code parallelization with almost no intervention. The results show that the proposed techniqueallows a code running 28 times faster when using a computer with 24 processors, each one with two threads, when compared withthe sequential one. It also suggested a moment expansion technique to improve the accuracy and computational efficiency whenusing Gaussian pulse excitation when compared with the Fourier transform.

Index Terms— OpenMP, parallel processing, time-domain spherical-multipole expansion, UWB antenna characterization.

I. INTRODUCTION

THE near-field components, which can be calculated byan FDTD solver, over a Huygens surface can be replaced

by equivalent electric and magnetic dipoles, and the far-fieldgenerated by a UWB antenna can be obtained in the timedomain by a spherical-mutipole near-to-far-field transforma-tion and expressed as [1], [2]

�e∞(�r , t) = −∑

n,m

an,m

(t − r

vc

)�nn,m(θ, φ)

+Z0

∑

n,m

bn,m

(t − r

vc

)�mn,m(θ, φ) (1)

where Z0 and vc = 1/√μ0ε0 are, respectively, the intrinsic

impedance and the velocity of light in vacuum, �nn,m(θ, φ) and�mn,m(θ, φ) are the set of vector functions with respect to thetransverse fields on a spherical surface [2]

�mn,m(θ, φ) = − 1

sin θ

∂Yn,m(θ, φ)

∂φθ̂ + ∂Yn,m(θ, φ)

∂θφ̂ (2)

�nn,m(θ, φ) = ∂Yn,m(θ, φ)

∂θθ̂ + 1

sin θ

∂Yn,m(θ, φ)

∂φφ̂. (3)

an,m(t) and bn,m(t) are the electric and magnetic time-domain spherical-multipole amplitudes [2]

Manuscript received June 21, 2013; revised August 12, 2013; acceptedSeptember 1, 2013. Date of current version February 21, 2014. Correspondingauthor: G. L. Ramos (email: glopesr@gmail.com).

Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TMAG.2013.2280796

akn,m = (−1)m

rn (n + 1)

×⎧⎨

⎩ZLel∑

i=1

k−1∑

l=0

[((1 + l) �ck−l,[i]

el − l�ck−(l+1),[i]el

)· �ζ l,[i]

n,m

+ 1

�t

(�ck−(l+1),[i]

el − �ck−l,[i]el

)· �ξ l,[i]

n,m

]

+Lmag∑

i=1

k−1∑

l=0

[((1 + l) �ck−l,[i]

mag − l�ck−(l+1),[i]mag

)· �χ l,[i]

n,m

+ 1

�t

(�ck−(l+1),[i]

mag −�ck−l,[i]mag

)· �ψ l,[i]

n,m

]}(4)

bkn,m = (−1)m

rn (n + 1)

×⎧⎨

⎩1

Z

Lmag∑

i=1

k−1∑

l=0

[((1 + l) �ck−l,[i]

mag − l�ck−(l+1),[i]mag

)· �ζ l,[i]

n,m

+ 1

�t

(�ck−(l+1),[i]

mag − �ck−l,[i]mag

)· �ξ l,[i]

n,m

]

−Lel∑

i=1

k−1∑

l=0

[((1 + l) �ck−l,[i]

el − l�ck−(l+1),[i]el

)· �χ l,[i]

n,m

+ 1

�t

(�ck−(l+1),[i]

el −�ck−l,[i]el

)· �ψ l,[i]

n,m

]}. (5)

cel and cmag are, respectively, the electrical and magneticcurrent moments, Lel and Lmag are, respectively, the numberof the equivalent Hertzian electrical and magnetic dipoles and�ζ , �ξ , �χ and �ψ as defined in [2].

The main advantage of time-domain multipole approach isthat the radiated fields can be obtained for an antenna excitedby any sources with pulsed temporal behavior and thus itallows the characterization of its radiation pattern in time and

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7015004 IEEE TRANSACTIONS ON MAGNETICS, VOL. 50, NO. 2, FEBRUARY 2014

Fig. 1. Strategy to allocate the node in a random manner.

frequency domains and such an approach is very useful forperforming a unified characterization of UWB antennas.

Another advantage of this technique relies on the fact thatthe representation expressed in (1) is not based on the retardedpotentials, which eliminates the need for source integration foreach observation point. Such a feature makes this techniqueinteresting when compared with traditional ones, as once mul-tipole amplitudes were obtained, the time-domain multipoleexpansion is valid at any arbitrary point in the far field.However, it has the issue of involving a high computationalburden due to the process of an intrinsic time convolution [2].

II. PARALLEL VERSION OF THE TIME-DOMAIN

SPHERICAL-MULTIPOLE CODE

An analysis of the approach developed in [2] shows thateach set of multipoles an,m(t) and bn,m(t) within the sametime interval could be independently calculated. An analysisof (1) also shows that the electrical far field could also be time-independently calculated. Thus, an alternative to speed up thecalculation of these multipoles amplitudes and the electricalfar field is the use of a parallel processing technique, whichhas been widely used for speeding up several electromagneticsolver techniques [3]–[6]. Adopting such parallel processtechnique, the multipole and the time-domain electrical far-field computation can be divided into n threads of executionand thus be calculated in a parallel implementation.

Among the alternatives for constructing parallel imple-mentations we can highlight the OpenMP [7], which is anapplication program interface (API). The OpenMP directiveuses a pre-compiler to indicate where the code should beparallelized, allowing parallelization of routines with almostno extra code.

We propose herein the allocation of each node in a randommanner among the threads [8]. The code implementationof the node distribution is shown in Fig. 1. Therefore, theconvolutions are randomly allocated through the availablethreads and the usage time of each processor is almostequalized and the time-domain spherical-multipole near-to-far-field transformation can be implemented in a more efficientway. The OpenMP parallelism applied to the code is shownin Fig. 2.

When performing antenna far field and radiation patternanalysis, the conventional spherical-multipole near-to-far-fieldtransformation algorithm suggests the use of an FFT to per-form a time to frequency domain conversion [2]. This papersuggests the use of a moment expansion technique [9] together

Fig. 2. OpenMP parallelism implementation applied to the time-domainspherical-multipole amplitudes.

Fig. 3. Validation of the parallel code: vertical radiation pattern of a halfwavelength dipole.

with parallel version of spherical-multipole code which allowsfor obtaining the radiation pattern of the UWB antenna directlyin time domain. The numerical complexity of the momentexpansion and FFT techniques can be compared in terms ofmultiplications. For each frequency to be computed, the totalnumber of real multiplications for the FFT algorithm is aboutOFFT = N log2 N + N [9], where N is the number of FDTDfar field samples. For the fourth-order deconvolution momentexpansion technique, the number of multiplications is aboutOME = 3N [9]. Therefore, the use of the moment expansiontechnique together with the spherical-multipole near-to-far-field transformation algorithm can also improve the compu-tation efficiency of the proposed algorithm.

III. RESULTS AND COMMENTS

The code parallelization was applied to calculate the verticalradiation pattern of a half wavelength dipole at a frequency of10 GHz. It was used a computer equipped with an Intel(R)Xeon(R) CPU E7530@1.87GHz processor, with 128 GBRAM and 24 cores, each core with two threads, with a totalof 48 threads. The results obtained for time and frequency-domain radiation patterns are shown in Fig. 3 and we can seea good agreement between the proposed parallel technique andthe analytical solution [10], validating the spherical-multipoleexpansion parallelized version.

The results of the implementation of the proposed paralleltechnique on the simulation time can be seen in Fig. 4. It canbe observed that the use of the proposed parallel processingtechnique with up to 48 cores permits that the radiation patternbe obtained with a processing time 28 times lower. In Fig. 4,we can observe that the processing gain initially shows a linear

RAMOS et al.: PARALLEL IMPLEMENTATION OF A COMBINED MOMENT EXPANSION 7015004

Fig. 4. Processing gain of spherical-multipole technique with parallelism.

Fig. 5. Geometry of the printed UWB antenna under analysis.

Fig. 6. Simulated time-domain a1,1 multipole amplitude.

behavior (described by the red line) when running the codewith approximately up to six threads. After this point theprocessing gain does not tend to follow this linear behaviorand when using all of 48 available threads, the code is about28 times faster than its unparalleled version. This behaviorcan be explained by the loading distribution process. As thisload distribution is randomly performed, the processing timechanges for each iteration and thus some cores will be usedby the processing for a longer time than others. In addition tothe last iterations, it is possible that some cores can be free ofiteration when other cores can be still being used to processthe last iterations. The computer used for this simulation has atotal of 24 cores, each core with two threads. When the processis started with up to 24 parallel threads, only a single thread

Fig. 7. Eθ component of the electrical far field of the printed UWB antennaat θ = 0◦ and φ = 90°.

Fig. 8. Normalized radiation pattern of the UWB antenna at 3 and 7 GHzfor the x–y plane.

is used for each core. When the code is processed with theuse of 25–48 threads, the second thread of each core startedto be used, causing a decrease in the processing gain.

Once the code was validated, it was employed to charac-terize the time and frequency-domain radiation pattern of aUWB antenna [11], as the proposed technique can improvethe simulation time of radiation patterns of such antennas. Theantenna geometry can be seen in Fig. 5 and its dimensions areL = 33 mm, Wg = 30 mm, L1 = 16.5 mm, W1 = 12 mm,L2 = 4 mm, W2 = 3 mm, Gw1 = 1.5 mm, Lg = 15 mm,Lsc = 2.25 mm, Ls1 = 4 mm, Ws1 = 2.5 mm, Ls2 = 4 mm,and Ws2 = 0.3 mm. With the beginning of antenna operationwith pulsed signals, which requires a wider bandwidth, it isuseful to define a time-domain radiation pattern, based on theenergy of radiated fields. The norm of a time-dependent signalis a way to quantify the energy contained in a signal. The time-domain radiation pattern can be defined as

Gt (θ, ϕ) = 4πU tE (θ, ϕ)∫ 2π

0

∫ π0 U t

E (θ, ϕ) senθdθdϕ(6)

where

U tE (θ, ϕ) = r2

Z0‖�e∞(�r , t)‖2 (7)

7015004 IEEE TRANSACTIONS ON MAGNETICS, VOL. 50, NO. 2, FEBRUARY 2014

Fig. 9. Normalized radiation pattern of the UWB antenna at 3 and 7 GHzfor the y–z plane.

and ‖�e∞(�r , t)‖2 is a norm of a time-dependent signal [12],which can be evaluated for energy and power signals.

To characterize the time-domain radiation pattern at dif-ferent frequencies, a moment expansion technique [9] wasapplied altogether with the spherical-multipole transformation.Once the time-domain far fields were obtained, the antennatransfer function is estimated by the moment expansion andthrough a convolution process the far field is estimated for aspecific frequency.

The results for the simulated multipole amplitude a1,1 canbe seen in Fig. 6. A time-domain electric far field, Eθ wasalso obtained and can be seen in Fig. 7. The results for theradiation pattern of the UWB antenna at 3 and 7 GHz on x–yand y–z plane can be seen in Figs. 8 and 9.

IV. CONCLUSION

In this paper, a parallelized time-domain spherical-multipoletogether with a moment expansion of time signals was pro-posed to characterize the radiation pattern of a UWB antennaat two different frequency bands. The parallel version ofthe code is very important as it significantly reduces thesimulation time for obtaining time and frequency domain farfields of antennas at different frequency bands. A momentexpansion technique was also proposed to obtain the radiationpattern directly in the time domain. This technique was shownto be more computationally efficient than the deconvolutionprocess, which involves the Fourier transform [9]. The impor-tance of time-domain radiation patterns relies on the factthat energy and power norms of temporal signals can beused to obtain the antenna radiation patterns for transientand steady-state responses, from any arbitrary source feedingthe antenna, which is very important especially for UWBantennas.

APPENDIX

MOMENT EXPANSION TECHNIQUE

Using the FDTD time increment t = n�t , n = 0, 1, ... andfor a Gaussian pulse excitation in the form of

vi (t) = exp

[− (t − τ0)

2

2T 20

](8)

we can use a fourth-order approximation for the input i andoutput port k leading to an impulse response between theseports hik as [9]

hik(n) = a4

24�t4 [vk(n + 2 + n0)+ vk(n − 2 − n0)]+

( a2

2�t2 − a4

6�t4

)[vk(n + 1 + n0)

+ vk(n − 1 − n0)] +(

a0 − a2

�t2 + a4

4�t4

)vk(n + n0)

(9)

where the parameters a0 = 1/√

2πT0, a2 = −a0T 20 , a4 =

3a0T 40 and n0 = int (t0/�t) and the shift parameter t0 =

j V (1)(0)/V (0), with V (p)i (0) = d pVi (ω)/dωp|ω=0, is chosen

to put the center of gravity of the input signal at t = 0 [9].The moment expansion technique with fourth-order trunca-

tion has been shown to improve accuracy and computationalefficiency when using Gaussian pulse excitation when com-pared with the Fourier transform, especially when the upperfrequency of the impulse response is greater than its upper(10% amplitude) frequency [9].

ACKNOWLEDGMENT

This work was supported in part by the Brazilian agenciesCAPES, CNPq, Fapemig and in part by Instituto Nacional deCiência e Tecnologia em Comunicações sem Fio (INCT-CSF).

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