NEAR-FIELD TO FAR-FIELD TRANSFORMATION WITH PLANAR SPIRAL SCANNING

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1 Progress In Electromagnetics Research, PIER 73, 49 59, 27 NEAR-FIELD TO FAR-FIELD TRANSFORMATION WITH PLANAR SPIRAL SCANNING S. Costanzo an G. Di Massa Dipartimento i Elettronica Informatica e Sistemistica Universitá ella Calabria 8736 Rene (CS), Italy Abstract A transformation proceure irectly computing the antenna far-fiel pattern from near-fiel samples acquire on a planar spiral is propose in this paper. The convolution property of the raiation integral is exploite to efficiently perform the evaluation by taking avantages of the Fast Fourier Transform, without the nee of any intermeiate interpolation process. Valiations on circular arrays of elementary ipoles are presente to show the effectiveness of the metho. 1. INTRODUCTION Measuring techniques in the raiating near-fiel are establishe as compact an controlle environments to perform accurate antenna test an iagnostics. They require a processing of the probe near-fiel istribution to recover the corresponing far-fiel pattern, with a variety of existing transformation techniques base on moal expansions an equivalent source reconstruction, or employing traine neural networks [1]. Stanar near-fiel setup are base on planar, cylinrical an spherical geometries, but improve scanning configurations, in terms of complexity an cost, have been introuce. In the framework of planar near-fiel measurements, a strong improvement with respect to scanner compactness is given by the bipolar configuration [2], base on exclusive rotational motions of the Antenna Uner Test (AUT)an the measuring probe. Near-fiel ata are collecte at the intersections between concentric rings an raial arcs, by imposing a full revolution of the AUT, followe by an incremental rotation of the probe arm. A further improvement of

2 5 Costanzo an Di Massa the bi-polar configuration has been recently achieve by imposing a simultaneous an continuous motion of the AUT an the measuring probe, so reucing the acquisition time as well as the overall system complexity. This measurement strategy results in a sampling arrangement on a planar spiral [3], which however strongly complicates the near-fiel to far-fiel transformation process, as a conversion to a rectangular format is neee to take avantage of the Fast Fourier Transform (FFT)algorithm. An Optimal Sampling Interpolation (OSI)technique [4] is then applie [3] to express the raiating fiel in a carinal series form by employing appropriate sampling functions. An approximate interpolation formula is also aopte [3] to map the non-uniform behavior of spiral samples in the raial coorinate into a sequence of uniformly space ata. In orer to avoi the near-fiel oversampling inherent to the spiral sampling arrangement, a fast an accurate interpolation algorithm is propose in [5, 6] to reconstruct the raiate electromagnetic fiel on a rotational surface from the knowlege of a non reunant number of its samples on a spiral wrapping the scanning surface. In this paper, a fast ata processing algorithm is propose to compute the AUT far-fiel irectly from near-fiel samples acquire on the planar spiral. The convolution property of the raiation integral is exploite to evelop an efficient proceure computing the far-fiel pattern in terms of FFT algorithm. This avois the application of intermeiate interpolations usually aopte in literature to enable the use of planar near-fiel to far-fiel transformation. Numerical simulations on circular arrays of Huyghens sources are iscusse to valiate the propose technique. 2. NEAR-FIELD SAMPLING ON PLANAR SPIRAL The planar spiral geometry is obtaine by imposing a simultaneous rotation of the AUT an the probe arm in terms of angles α an β, respectively (Fig. 1). This gives a samples istribution on raial arcs at points P (s,α )(Fig. 2), where s is a surrogate for the raial coorinate ρ, efine as [3]: s = ρ (1) The parameter in relation (1)gives the istance between the AUT an the near-fiel measuring plane (fig.1), while α is the angle escribing the AUT rotation, which is relate to the azimuthal coorinate φ by the equation [2]:

3 Progress In Electromagnetics Research, PIER 73, α = φ + β 2 β being the probe arm rotation angle. (2) z Far-fielpoint Probearm θ β α= L α= α x y φ AUT x y Figure 1. Planar spiral scanning geometry. The archimeean spiral scanning is mathematically escribe by the equation: ρ = aα +2πaγ, a >, γ =, 1, 2,... (3) which ientifies all points lying on the raial arc associate to a specific value of the azimuthal angle α (Fig. 2). The raial spacing between two ajacent points on the same arc is erive from (3)as: ρ =2πa (4) It must be coherent with the sample spacing neee for the planepolar geometry, that is: ρ = λ (5) 2 So, the correct value for the parameter a into expression (3)is erive from sampling consierations, by equating relations (4)an (5)to

4 52 Costanzo an Di Massa y α= α ρ P α = x L β Figure 2. Sampling arrangement on planar spiral. obtain: a = λ 4π (6) 3. FAR-FIELD COMPUTATION FROM NEAR-FIELD ON PLANAR SPIRAL Let us consier a near-fiel ata set collecte on a polar scan plane having raius ρ max. A raiation-type integral for the equivalent aperture current on the acquisition plane can be erive as [7]: T (θ, φ) = ρmax 2π q(ρ,φ ) e jkρ sinθcos(φ φ ) ρ ρ φ (7) where the scalar form is consiere, for the sake of simplicity. Uner the assumption of a omniirectional probe, the left han sie of equation (7)gives the far-fiel at coorinates (θ,φ)(fig. 1), while the term q(ρ,φ )represents the near-fiel istribution on the measurement plane x -y (Fig. 1), k being the free-space propagation constant. In the presence of a near-fiel spiral trajectory with maximum raial extension ρ max, the coorinate transformations (1), (2) from polar

5 Progress In Electromagnetics Research, PIER 73, variables (ρ,φ )to spiral variables (s,α )moifies the raiation integral (7)as follows: T (θ, φ) = ρmax β 2 +2π β 2 q(s,α ) e jks sinθcos(φ α + β 2 ) 2 s s α (8) A compact form of equation (8)can be easily erive as: T (θ, φ) = ρmax β 2 +2π β 2 q 1 (s,α ) r(θ, φ, s,α )s α (9) where the following efinitions are aopte: q 1 (s,α )=s 2 q(s,α )(1) r(θ, φ, s,α )=e jks sinθcos(φ α + β 2 ) (11) A convolution form in the variable α can be recognize for the inner integral appearing in (8), so leaing to apply the Fourier transform for its computation, by invoking the convolution theorem as [8]: where: T (θ, φ) = ρmax F 1 { q 1 (s,w) r(θ, φ, s,w) } s (12) q 1 (s,w)=f { q 1 (s,α) } (13) r(θ, φ, s,w)=f { r(θ, φ, s,α ) } (14) an the symbols F an are use to enote the Fourier transform operator. Let us consier a near-fiel spiral trajectory with samples locations at coorinates: α m = m α, m =, 1, 2,..., M 1 (15) s nm = ρ nm, n =, 1, 2,..., N 1 (16) where: ρ nm = a(α m +2πn)(17) N being the number of loops in the spiral arrangement an M the number of samples for each loop. After inserting relation (17)into equation (16)an making use of expressions (5)an (6), a pair of iscrete mathematical relationships

6 54 Costanzo an Di Massa are obtaine which uniquely escribe the near-fiel spiral trajectory in terms of spacings coherent with the plane-polar sampling requirements, namely: α = φ = λ (18) 2r o s = ρ = λ (19) 2 where r o is the raius of the smallest sphere completely enclosing the AUT. With the above assumptions on spiral samples istribution, the numerical computation of the raiation integral (12)can be performe as: T (θ, φ) = N 1 M 1 n= m = In this latter relation, the terms: q 1 (s nm,w)= 1 M r(θ, φ, s nm,w)= 1 M [ q 1 (s nm,w) r(θ, φ, s nm,w)] e j 2πm w M (2) M 1 m= M 1 m= 2πmw j q 1 (s nm,α m ) e M (21) 2πmw j r(θ, φ, s nm,α m ) e M (22) represents the Discrete Fourier Transform (DFT)of the sequences q 1 (...)an r(...), respectively, which can be efficiently performe by aopting the FFT algorithm [8]. An overview of the ata processing metho for far-fiel computation from near-fiel samples on planar spiral is reporte uner Fig NUMERICAL RESULTS Numerical tests on ipole arrays are performe to show the effectiveness of the propose far-fiel transformation process from nearfiel samples on planar spiral. As a first case, a circular array of 18 y-oriente Huyghens sources λ/2 space is consiere, with excitation coefficients chosen to have a main lobe in the irection θ =1 in the H-plane. Near-fiel acquisition is simulate on a plane at a istance =1λ from the array, with samples lying on a planar spiral having N = 2 loops an M=136 points along each loop. Sampling spacings

7 Progress In Electromagnetics Research, PIER 73, Near-fiel ata on planar spiral Multiply by coorinate s Perform FFT on exp function Perform FFT over coorinate α X Perform FFT -1 Perform sum over coorinate s Far-Fiel at coorinates θ, φ Figure 3. Data processing scheme for the planar spiral configuration y [ ] x [ ] Figure 4. Normalize near-fiel amplitue on planar spiral for a circular array of 18 elements.

8 56Costanzo an Di Massa Intensity Pattern [B] Reference Reconstructe from irect transformation [eg] Figure 5. Co-polarize H-plane pattern for circular array of 18 elements y [ ] x [ ] Figure 6. Normalize near-fiel amplitue on planar spiral for a planar circular array.

9 Progress In Electromagnetics Research, PIER 73, α an s coherent with relations (18)an (19)are consiere, by assuming r o =1.85λ. The contour plot of the normalize intensity pattern on the near-fiel spiral trajectory is shown in Fig. 4. The irect transformation algorithm is then applie to near-fiel spiral samples for recovering the co-polarize H-plane pattern reporte uner Fig. 5 an successfully compare with the exact array solution. As a further valiation, a planar circular array of iameter equal to 14λ is consiere, with elements given by y-oriente ipoles raially an azimuthally space of λ/2. Simulations are performe on a near-fiel plane at a istance = 15λ from the array, with samples locate on a spiral arrangement with N = 3 loops an M = 133 points along each loop. An azimuthal spacing α =2.72 is assume, with r o =7.5λ, an a normalize raial step s as given by equation (19) is again consiere. The normalize amplitue of the simulate nearfiel on the planar spiral is shown in Fig. 6, while the co-polarize H plane pattern as obtaine from the irect transformation algorithm is reporte uner Fig. 7. A high accuracy is prove again by comparison with the exact analytical solution. Reference Reconstructe from irect transformation Intensity Pattern [B] [eg] Figure 7. Co-polarize H-plane pattern for planar circular array.

10 58 Costanzo an Di Massa 5. CONCLUSIONS AND FUTURE DEVELOPMENTS A far-fiel transformation proceure irectly performe on near-fiel samples coming from a planar spiral arrangement is evelope in this paper. The convolution property of the raiation integral is exploite to efficiently perform its computation in terms of FFT. This avois the use of interpolation techniques usually aopte in literature to obtain a rectangularly regularize format of the near-fiel ata which enables the application of the well known planar near-fiel to farfiel transformation. The propose ata processing is numerically valiate on circular arrays of elementary ipoles. Concerning future evelopments, two open points will be consiere. First of all, the proceure will be extene to take into account the irective effect of a non-ieal probe, by incluing a correct probe compensation. Furthermore, the application of a two-probes base metho [9] will be consiere for recovering the far-fiel pattern from the knowlege of intensity-only ata on a single near-fiel spiral surface. REFERENCES 1. Ayestaran, R. G. an F. Las-Heras, Near fiel to far fiel transformation using neural networks an source reconstruction, Journal of Electromagnetic Waves an Applications, Vol. 2, No. 15, , Williams, L. I., Y. Rahmat-Samii, an R. G. Yaccarino, The bi-polar planar near-fiel measurement technique, Part I: Implementation an measurement comparison, IEEE Trans. Antennas Propag., Vol. 42, No. 2, , Yaccarino, R. G., L., I. Williams, an Y. Rahmat-Samii, Linear spiral sampling for the bipolar planar near-fiel antenna measurement technique, IEEE Trans. Antennas Propag., Vol. 44, No. 7, , Bucci, O. M., C. Gennarelli, an C. Savarese, Fast an accurate near-fiel far-fiel transformation by sampling interpolation of plane-polar measurements, IEEE Trans. Antennas Propag., Vol. 39, No. 1, 48 55, D Agostino F., C. Gennarelli, an G. Riccio, Theoretical founations of near-fiel far-fiel transformations with spiral scannings, Progress In Electromagnetics Research, PIER 61, , D Agostino, F., F. Ferrara, C. Gennarelli, an G. Riccio, Directivity computation by spherical spiral scanning in near-

11 Progress In Electromagnetics Research, PIER 73, fiel region, Journal of Electromagnetic Waves an Applications, Vol. 19, No. 1, , Costanzo, S. an G. Di Massa, Direct far-fiel computation from bi-polar near-fiel samples, Journal of Electromagnetic Waves an Applications, Vol. 2, No. 9, , Weaver, H. J., Theory of Discrete an Continuous Fourier Analysis, John Wiley an Sons, New York, Costanzo, S. an G. Di Massa, Far-fiel reconstruction from phaseless near-fiel ata on a cylinrical helix, Journal of Electromagnetic Waves an Applications, Vol. 18, No. 8, , 24.