Mohammed Serhir
email: mohammed.serhir@supelec.fr

ON THE NEAR-FIELD SAMPLING AND TRUNCATION ERRORS IN PLANAR TIME-DOMAIN NEAR-FIELD TO FAR-FIELD TRANSFORMATION

and research institutions in France

INTRODUCTION

The Ultra Wide Band (UWB) antennas operating over large frequency bandwidth have attracted a lot of interest over wide telecommunication applications [START_REF] Jin | Super-wideband printed asymmetrical dipole antenna[END_REF][START_REF] Tuovinen | Impedance dependency on planar broadband dipole dimensions: an examination with antenna equivalent circuits[END_REF][START_REF] Akhoondzadeh-Asl | Frequency and time domain characteristic of a novel notch frequency UWB antenna[END_REF][START_REF] Tuovinen | Analysis of the impedance behaviour for broadband dipoles in proximity of a body tissue: approach by using antenna equivalent circuits[END_REF][START_REF] Tsai | A coplanar-strip dipole antenna for broadband circular polarization operation[END_REF]. The measurement of the radiation pattern of these antennas has to be carried in a wide frequency band, which is time consuming. For these antennas, the time-domain (TD) techniques seem to be more adapted than frequency-domain (FD) techniques. Indeed, using a short pulse, one can measure the antenna transient response covering the frequency band of interest. This measurement is fulfilled using near-field (NF) or far-field (FF) techniques [START_REF] Yaghjian | An overview of near-field antenna measurements[END_REF].

In TD FF measurements we record the antenna under test (AUT) transient response when it is excited by a specific pulse. The excitation pulse has to cover the frequency band of interest and supply the AUT with enough power that allows the measurement with acceptable signal to noise ratio at farfield distances. In TD NF techniques the tangential components of radiated field are collected in the antenna vicinity over a scan surface (planar, cylindrical or spherical). Then, the measured data are transformed to calculate the asymptotic behavior of the antenna far-field. For NF measurement, we use pulse generators providing a trade-off between the output voltage magnitude and a short pulse rise-time for wide frequency band characterization.

For accuracy improvement, the post processing tools such as TD gating technique are used to filter out the multiple reflections occurring during the measurement [START_REF] Blech | A time domain spherical near-field measurement facility for UWB antennas employing a hardware gating technique[END_REF]. The TD techniques allow the radiation pattern measurement in non-anechoic environment as presented in [START_REF] Jinhwan | Free Space Radiation Pattern Reconstruction from Non-Anechoic Measurements Using an Impulse Response of the Environment[END_REF]. These are well adapted for the parasitic electromagnetic radiation emitted from electronic devices for electromagnetic compatibility purpose [START_REF] Liu | Fully time-domain scanning of EM near-field radiated by RF circuits[END_REF]. Also, using TD techniques one can measure the radiation pattern of radar structures fed by non-sinusoidal excitation signals. Particularly, when the feeding system is integrated in the radiating antennas for radar applications.

The main difficulty related to the practical use of NF TD techniques concerns the ability to manage the quantity of measured data. The required computation time and memory storage became problematic for large measurement surface and long-time window (long transient response). The quantity of NF data depends on the used time and spatial sampling criteria. These are determined based on the AUT excitation signal maximum frequency as presented in [START_REF] Hansen | Plane-wave theory of time-domain fields, near-field scanning applications[END_REF]. Authors in [START_REF] Bucci | Near-field far-field transformation in time domain from optimal plane-polar samples[END_REF][START_REF] Bucci | Optimal time-domain field interpolation from plane-polar samples[END_REF][START_REF] Bucci | Representation of electromagnetic fields over arbitrary surfaces by a finite and nonredundant number of samples[END_REF] present a different NF sampling strategy that aims at representing the NF measurement data using plane polar samples allowing a minimum redundancy in NF sampling. In the FD, Wang in [START_REF] Wang | An examination of the theory and practices of planar near-field measurement[END_REF] presents a detailed analysis on the theory and practices for planar near-field measurement. The sampling strategies, the NF probe characteristics, the filtering of the NF data and the evanescent modes effect have been addressed. Authors of [START_REF] Levitas | UWB system for time-domain near-field antenna measurement[END_REF][START_REF] Serhir | Development of pulsed antennas measurement facility: Near field antennas measurement in time domain[END_REF][START_REF] De Jough | Antenna time-domain measurement techniques[END_REF][START_REF] Serhir | An efficient near field to near or far field transformation in time domain[END_REF] have presented some experimental setups for NF or FF antenna measurement. These papers did not present the sampling strategies used in the NF or FF measurements.

In planar TD near-field antenna measurement, the sampling approach is based on the assumption that the time signal is band-limited to a maximum angular frequency ω max = 2πc/λ min . This leads to the NF sampling criterion ∆y = ∆z = λ min /2 and the AUT excitation pulse defines approximately the frequency band of interest (minimum and maximum frequencies). This assumption gives a reduced view of the reality of TD NF measurement practices. Considering the spatial sampling criterion based on the highest frequency of the AUT excitation signal (λ min /2), as presented in [START_REF] Hansen | Plane-wave theory of time-domain fields, near-field scanning applications[END_REF], does not guarantee the accuracy of the TD far-field calculation. Indeed, the AUT directivity depends on the frequency and using the same measurement surface for time-domain FF calculation leads to a frequency dependent truncation error. Also, the NF measurement has to be carried in the radiating NF zone (Fresnel zone). This Fresnel zone distance depends on the AUT dimensions and the operating frequency. For NF TD technique, the measurement distance has to be adapted to avoid the reactive NF zone for low frequencies. These three parameters are responsible of the TD FF errors.

In this paper we propose to determine the AUT TD FF using different sampling criteria and for several NF measurement distances. By comparing the far-field results, we show that the TD FF errors are resulting from NF evanescent modes with low measurement distances, under-sampling and scan surface truncation errors. It is difficult to distinguish between these errors in the TD FF directly. The frequency-domain (FD) comparisons are presented to make out the effect of each error.

In the NF to FF transformation scheme we use the reconstruction formula to rigorously interpolate the NF in the time-domain. We propose a simple and rigorous formulation to calculate the E-field time derivative from the measured E-field as complementary development of the analysis presented in [START_REF] Hansen | Plane-wave theory of time-domain fields, near-field scanning applications[END_REF]. The computation scheme for NF to FF field transformation is validated using analytic radiating array composed of infinitesimal dipoles. The paper is structured as follows. In Section II, the NF to FF formalism based on the Green's function is described and the different formulation needed for our analysis are commented. In Section III, we present the transformation results and a comparison of the calculated far-field with the actual one in the time and the frequency domains. In Section IV, recommendations and concluding remarks are presented.

TIME-DOMAIN NEAR-FIELD TO FAR-FIELD TRANSFORMATION

In this section, we first recall the equations expressing the NF to FF transformation based on the Green's formulation. The detailed developments can be found in [START_REF] Hansen | Plane-wave theory of time-domain fields, near-field scanning applications[END_REF]. Then, we present how to calculate rigorously the E-field time derivative using the reconstruction formula. Finally, we present the analytic expressions of infinitesimal dipole radiated field in the TD when it is excited by modulated Gaussian pulse. This infinitesimal dipole is used to generate the TD radiation pattern of our AUT composed of 40 infinitesimal dipoles.

Time-domain near-field to far-field formalism

In this paper we use the time-domain near-field to far-field transformation algorithm based on the Green's function representation [START_REF] Hansen | Plane-wave theory of time-domain fields, near-field scanning applications[END_REF]. The antenna NF tangential components E y and E z are collected over the plane x = x meas . Equation (1) expresses the antenna far-field F (θ, φ, t) in the direction (θ, φ) (spherical coordinates associated with the Cartesian coordinates) as a function of the time derivative of the tangential components of the measured NF E meas .

F (θ, φ, t) = - 1 2πc +∞ -∞ +∞ -∞ e r × e x × ∂ E meas ∂t ( r 0 , t + e r . r 0 /c) dy 0 dz 0 with F (θ, φ, t) = lim r→∞ r E( r, t + r/c) (1)
The measurement surface is defined by -D max ≤ y, z ≤ D max , and (1) is expressed in a truncated form as

F (θ, φ, t) = - 1 2πc +Dmax -Dmax +Dmax -Dmax e r × e x × ∂ E meas ∂t ( r 0 , t + e r . r 0 /c) dy 0 dz 0 (2)
The tangential components of the measured field vector E meas are collected at a regular grid described by r 0 = x meas e x + y e y + z e z with -D max ≤ y, z ≤ +D max . Using the sampling theorem [START_REF] Hansen | Plane-wave theory of time-domain fields, near-field scanning applications[END_REF], we transform (2) to the following double summation

F (θ, φ, t) = - 1 2πc
We have developed a Matlab routine expressing (3) to calculate the FF using NF data calculated at several measurement distances x = x meas with different sampling criterion ∆y, ∆z. In (3), the first step in NF to FF transformation is the calculation of the time derivative of the measured NF at the instant t+ er. r 0mn c

. The accuracy of the result depends on the chosen interpolation method. In our analysis, we will use the reconstruction formula to determine the time derivative of the measured field and to interpolate the field at the required instants.

The reconstruction formula

Let us define a function f (t) where f ω is its Fourier transform. f (t) is a band limited function if for ω ≥ ω max

f ω = - 1 2π +∞ -∞ f (t) exp(jωt)dt = 0 (4)
The reconstruction formula allows the exact interpolation of the band limited function f (t) using the discrete points f (t 0 + l∆t). f (t) is expressed for every t ∈ [t 0 , t 0 + (N t -1)∆t] using the cardinal series as

f (t) = Nt-1 l=0 sinc π t -(t 0 + l∆t) ∆t f (t 0 + l∆t) (5)
Using the discrete measurement samples E meas (t 0 + l∆t) 1≤l≤Nt-1 , one can express accurately E meas (t) for every t ∈ [t 0 , t 0 + (N t -1)∆t]. To do this, the Nyquist sampling criterion ∆t = π/ω max has to be respected. The reconstruction formula is used to interpolate the measured E-field as:

E meas ( r 0mn , t) = Nt-1 l=0 h(t) E meas ( r 0mn , t 0 + l∆t) with h(t) = sin π t-(t 0 +l∆t) ∆t π t-(t 0 +l∆t) ∆t (6)
For t 0 ≤ t ≤ t 0 + (N t -1)∆t, the time derivative of the E-field is expressed rigorously as

∂ E meas ( r 0mn , t) ∂t = Nt-1 l=0 ∂h(t) ∂t E meas ( r 0mn , t 0 + l∆t) where ∂h(t) ∂t = 1 t -(t 0 + l∆t) cos(π t -t 0 ∆t -l) -h(t) (7)
If the sampling Nyquist criterion is respected, ( 6) and ( 7) interpolate the E-field and its time derivative. Here, we consider the antenna under test (AUT) composed of infinitesimal dipoles distributed over the plane x = 0 as shown in Fig. 1. Each dipole is excited by a pulse and the TD NF is collected over the plane x = x meas at regularly spaced positions. In the next paragraph, we express the analytical expression of the field radiated by an elemental dipole.

Transient radiation of an infinitesimal dipole

Let us consider a z-polarized elemental dipole placed at the origin of the coordinate system. This dipole is excited by e(t), a sinusoidal current modulated by a Gaussian pulse written as

e(t) = e 0 sin(βt) exp(-αt 2 ) whith α = 2/τ 2 (8)
The transient radiated fields E r and E θ are written in the spherical coordinates system

E r (r, θ, φ, t) = 2η 4πr 2 cos(θ) ∂e(t -r/c) ∂t + c r e(t -r/c) E θ (r, θ, φ, t) = η 4πr sin(θ) ∂ 2 e(t -r/c) c∂t 2 + ∂e(t -r/c) r∂t + e(t -r/c) r 2 (9)
with

∂e(t) ∂t = e 0 [β cos(βt) -2αt sin(βt)] exp(-αt 2 ) ∂ 2 e(t) ∂t 2 = e 0 4α 2 t 2 -β 2 -2α sin(βt) -4αβt cos(βt) exp(-αt 2 ) ( 10
)
We have used ( 8)- [START_REF] Hansen | Plane-wave theory of time-domain fields, near-field scanning applications[END_REF] to generate the tangential components E z and E y radiated by 40 elementary dipoles placed at the plane x = 0. The distance between dipoles is 5cm, β = 4π10 9 rad/s and τ = 0.8.10 -9 s. The dipole distribution is described in Fig. 1. The measurement surface is limited by D max as shown in Fig. 2 and the TD radiation pattern in the cut planes y meas = 0 and z meas = 0 are presented in Fig. 3.

NUMERICAL RESULTS

In this section we are interested in evaluating the accuracy of the time-domain (TD) FF as a function of the spatial sampling criterion ∆y and ∆z. The antenna under test (AUT) is composed of 40 elementary dipoles excited by a sinusoidal current modulated by a Gaussian pulse. The NF measured at the distance x meas = 50cm = 5λ 3GHz (x meas = 5λ 1GHz /3) is presented in the TD in Fig. 3 and in the FD in Fig. 4.

3.1. The near-field measurement surface truncation and the sampling criterion

The time-domain far-field results comparison

In the planar rectangular frequency-domain measurement technique, the NF is spatially sampled using the Nyquist rule ∆y = ∆z = λ/2 with λ being the working wavelength. As presented in [START_REF] Hansen | Plane-wave theory of time-domain fields, near-field scanning applications[END_REF] the TD NF is spatially sampled based on the AUT excitation signal maximum frequency ∆y = ∆z = λ min /2 = cπ/ω max . For a short pulse (UWB antenna), the maximum frequency can be F -3dB , F -10dB , F -20dB , F -30dB . F -XdB is the frequency for which the magnitude of the excitation pulse spectrum is lower by X value compared with the normalized maximum level (0dB) of the central frequency. For our case of study, as it is seen in Fig. 1,

F -3dB ≈ 2.3GHz , F -6dB ≈ 2.4GHz, F -10dB ≈ 2.6GHz, F -20dB ≈ 2.
8GHz and F -30dB ≈ 3GHz. The maximum magnitude of the excitation signal spectrum corresponds to the central frequency

F 0dB = f 0 = 2GHz. We propose to express F -3dB ≈ 1.15f 0 , F -6dB ≈ 1.2f 0 , F -10dB ≈ 1.3f 0 , F -20dB ≈ 1.4f 0 , F -30dB = 1.5f 0 .
In order to study the effect of NF spatial sampling criterion over the calculated FF, we use χ to parameterize the spatial sampling ∆y χ = ∆z χ = λ 0 /2χ, where χ = 1.1, 1.2, 1.3, 1.4, 1.5 and λ 0 = c/f 0 . By comparing the far-field calculated using different χ, we identify the influence of the chosen maximum frequency on the accuracy of the calculated AUT FF. The NF time-axis is sampled using ∆t = Over the planar surface at the distance x meas = 50cm from the AUT, we calculate the near-field at regularly spaced points with ∆y χ = ∆z χ = λ 0 /2χ and D max = 150cm (10λ 0 ). Then, using (3) and ( 7), we calculate the FF at the directions (θ = π/2, φ = 0), (θ = π/3, φ = 0), and (θ = π/2, φ = π/6) for χ = 1.1, 1.2, 1.3, 1.4, 1.5. The E z component of the FF is presented in Fig. 5 (a-c-e) and compared with the reference FF determined directly from ( 9) and ( 10) for r going to infinity. This actual far-field is labeled Ref in Fig. 5. The differences between the calculated FFs and the actual one are quantified using the error expressed in [START_REF] Bucci | Near-field far-field transformation in time domain from optimal plane-polar samples[END_REF]. E N F toF F (θ, φ, t) is the field calculated using NF to FF transformation and E Ref (θ, φ, t) is the actual far-field.

Error(θ, φ, t) = 100 × E N F toF F (θ, φ, t) -E Ref (θ, φ, t) max(E Ref (θ, φ, t)) t 0 ≤t≤t 0 +(Nt-1)∆t (11)
The errors of the curves in Fig. 5(a) are presented in Fig. 5(b) for different χ. The errors of the comparisons presented in Fig. 5(c) are plotted in Fig. 5(d). Fig. 5(f) presents the errors of Fig. 5(e). From Figs. 5 (b-d-f) the early-time far-field introduced in [START_REF] Hansen | Plane-wave theory of time-domain fields, near-field scanning applications[END_REF] as being the part of the radiated FF without the truncation error is visually observed in Fig. 5(e). The measurement surface truncation effect is visible at t = 7ns and 10ns ≤ t ≤ 13ns. As explained in [START_REF] Hansen | Plane-wave theory of time-domain fields, near-field scanning applications[END_REF], the effect of the truncation error arrives chronologically after the actual FF signal. This is presented in [START_REF] Hansen | Plane-wave theory of time-domain fields, near-field scanning applications[END_REF] for a point source. Nevertheless, in Fig. 5 (b-d-f), we notice quantitative differences between the calculated far-fields (FFs) and the actual one even in the time portion of the radiation pattern attributed to early-time far-field (4ns ≤ t ≤ 6ns).

In order to understand the origin of the errors for 4ns ≤ t ≤ 6ns, we have used a larger measurement surface to calculate the FF (D max = 300cm) with the same sampling conditions. The obtained results have shown the same error level for 4ns ≤ t ≤ 6ns. These errors are not attributed to the size of the measurement surface. However, we have applied the same analysis (same excitation pulse, same sampling conditions, same measurement distance) for a smaller antenna (4x4) composed of 16 dipoles with 5cm between dipoles and D max = 150cm, the error level is smaller for 4ns ≤ t ≤ 6ns. Moreover, this decreasing tendency is confirmed for 4 dipoles (2x2). The errors for 4ns ≤ t ≤ 6ns is directly linked to the size of the AUT.

As presented in Fig. 5(b-d-f), the error levels depend on the FF observation point (θ, φ). The angular FF directions free from truncation error depends on three parameters, namely the measurement distance (x meas = 50cm), the AUT size (Ly = 45cm and Lz = 15cm) and the measurement surface dimensions (D max = 150cm). Here, the critical angle is defined by α θ = 70deg and α φ = 68deg as expressed in Fig. 2. This allows an angular FF area free from truncation errors for 30deg ≤ θ ≤ 150deg and -68deg ≤ φ ≤ 68deg. This provides an indication and not the exact angular area, especially when the measured field level in the edge of the measurement surface is of comparable levels with the maximum measured field.

In Fig. 6 we show the FF errors calculated at the principal plane cuts θ = π/2 with -π/2 ≤ φ ≤ π/2 and φ = 0 with 0 ≤ θ ≤ π. The presented errors are for χ = 1.2, 1.3 and 1.5. It is seen from Figs. 6 (c) and (f) that the NF sampled using χ = 1.2 is under-sampled since the observable oscillations are due to the aliasing errors. In Figs. 6 (a-b-d-e) we have less oscillations. The errors behavior do not reveal early-time far-field region where the truncation error happens later in time compared with the actual FF. The errors level for 4ns ≤ t ≤ 6ns (actual far-field) is of comparable level with the errors for 7ns ≤ t ≤ 10ns. The errors for 0 ≤ θ ≤ 30deg, 150deg ≤ θ ≤ 180deg, -90deg ≤ φ ≤ -68deg, and 68deg ≤ φ ≤ 90deg are due to the truncation of the measurement surface as predicted by α θ = 70deg and α φ = 68deg. In addition, the errors for χ = 1.5 and χ = 1.3 have comparable behavior as it is seen from Figs. 6 (a-b-d-e). Consequently, comparing the TD FF results for this AUT, the weighting factor χ = 1.3 (F -10dB ≈ 1.3f 0 ) allows a comparable accuracy as χ = 1.5 (F -30dB ≈ 1.5f 0 ).

The frequency-domain far-field results comparison

Using the NF sampling criterion ∆y χ = ∆z χ = λ 0 /2χ x meas = 50cm and D max = 150cm we compare the calculated far-fields in the frequency-domain (FD). These FFs are set by Fourier transforming the previously calculated TD far-fields. The comparisons are carried out at the principal plane cuts θ = π/2 and φ = 0 at F -10dB (1.4GHz and 2.6GHz), F -20dB (1.2GHz and 2.8GHz), F -30dB (1GHz and 3GHz) for different χ.

It is seen in Fig. 4 that the AUT NF truncation level depends on the frequency and the field truncation is more visible in the plane cut z meas = 0 than the plane cut y meas = 0. Hence, the calculated far-field (from NF to FF transformation) is damaged by the truncation error in the cut plane θ = π/2 more than the cut plane φ = 0. Let us recall that the angular area of the free error FF is for 30deg ≤ θ ≤ 150deg and -68deg ≤ φ ≤ 68deg.

In Figs. 7, we present the FD FFs comparison at the plane cut φ = 0. The FF for high and low frequencies of F -XdB fits well the actual far-field when the sampling criterion reaches the Nyquist limit (λ/2) for 30deg ≤ θ ≤ 150deg. Also, the FF results for 30deg ≤ θ ≤ 150deg are identical using χ = 1.3 and χ = 1.5 even for the frequency 2.8GHz. In Figs. 8, we compare the calculated FFs with the actual one in the plane cut θ = π/2. As shown, the results are more sensitive to the truncation error compared with the results of Figs. 7. The FFs for χ = 1.3, 1.4 and 1.5 are similar for -40deg ≤ φ ≤ 40deg. The measured NF amplitude at the edge of the measurement surface is only -22.6dB lower than the maximum NF amplitude in the plane cut z meas = 0. Consequently, the FF for -68deg ≤ φ ≤ -40 and 40deg ≤ φ ≤ 68 presents important errors. When the truncation level of the AUT NF measurement data changes significantly as a function of the frequency, the measurement surface has to be adapted to satisfy the same truncation level for every frequency. Otherwise, if the measurement surface stays unchanged, the AUT valuable measurement frequency band is chosen as a function of the truncation level of the measured field. In particular, when measuring the NF of the AUT in the time-domain, we can primarily verify by measuring in the principal cut planes y meas = 0 and z meas = 0 the level of the E-field truncation level for every measured frequency by Fourier transforming the time-domain NF. Then, we calculate the NF truncation level by determining the level difference between the maximum E-field and the field level at the measurement surface limit for each frequency. The AUT presents at the cut plane z meas = 0 the truncation levels of (22.58dB, 26, 18dB), (33.84dB, 24.40dB), (28.78dB, 33.35dB) and (27dB, 26.45dB) for F -30dB (1GHz and 3GHz), F -20dB (1.2GHz and 2.8GHz), F -10dB (1.4GHz and 2.6GHz) and F -3dB (1.7GHz and 2.3GHz), respectively as presented in Fig. 9. The AUT NF measurement data guarantee truncation levels less than -30dB in the plane cut y meas = 0. As a consequence, the far-field is accurately calculated at the plane cut φ = 0 for 30deg ≤ θ ≤ 150deg. The NF plane cut y meas = 0 presented in Fig. 9 (b) shows truncation levels of -22.5dB for 1GHz (χ = 1.5) and 24.4dB for 2.8GHz (χ = 1.4) which are responsible of the FF discrepancies in frequency and time domains. From Fig. 9 (b), the frequency 2.6GHz (χ = 1.3) is the highest frequency allowing the lowest truncation error level. For this reason the errors presented in the time-domain in Fig. 5-6-7 for χ = 1.3 were equivalent to the error associated with χ = 1.5. For this AUT and this measurement surface (D max = 150cm), the maximum frequency to consider for the time-domain NF sampling criterion is 2.6GHz (χ = 1.3). The FD FF results confirm these conclusions. 3.2. The near-field measurement distance and the sampling criterion

F -30dB 1GHz F -30dB 3GHz F -20dB 1.2GHz F -20dB 2.8GHz F -10dB 1.4GHz F -10dB 2.6GHz F -3dB 1.7GHz F -3dB 2.3GHz (a)
F -30dB 1GHz F -30dB 3GHz F -20dB 1.2GHz F -20dB 2.8GHz F -10dB 1.4GHz F -10dB 2.6GHz F -3dB 1.7GHz F -3dB 2.3GHz (b)

The time-domain far-field results comparison

Let us consider the NF measurement surface with D max = 150cm = 10λ 0 . We are interested in studying the effect of the NF measurement distances x meas where the TD NF data are collected with ∆y χ = ∆z χ = λ 0 /3 (χ = 1.5). The actual TD FF is compared with the ones calculated using the TD NF to FF transformation for different measurement distances x meas = 10cm (2λ 0 /3), 20cm(4λ 0 /3), 30cm(2λ 0 ), 40cm(8λ 0 /3) and 50cm(10λ 0 /3). The errors resulting from these comparisons are presented in Fig. 10 (a) and (b) for the directions (θ = π/2, φ = 0) and (θ = π/3, φ = 0) respectively. As it is seen in Fig. 10(a-b), the error depends on the measurement distance, especially for 4ns ≤ t ≤ 6ns. The error values decrease as the measurement distance increases. Namely, it is difficult to set the limit of the radiating NF zone (Fresnel zone) in the TD NF measurement. The Fresnel zone is defined based on the AUT size and the operating frequency. In TD NF measurement we deal with a frequency band and the NF data measured below the Fresnel zone contain evanescent modes that are probably responsible of the errors for 4ns ≤ t ≤ 6ns. Considering smaller antennas, the Fresnel zone is rapidly reached and the errors for 4ns ≤ t ≤ 6ns are smaller because less evanescent NF modes are taken into account. In [START_REF] Hansen | Plane-wave theory of time-domain fields, near-field scanning applications[END_REF] authors have considered a point source and the Fresnel zone is rapidly reached (low evanescent mode) for that case. This explains the early-time far-field free from truncation error as presented in [START_REF] Hansen | Plane-wave theory of time-domain fields, near-field scanning applications[END_REF]. x meas =10cm

x meas =20cm

x meas =30cm

x meas =40cm

x meas =50cm

(a) In addition, from Fig. 10(a), we can isolate the effect of the measurement distance error (4ns ≤ t ≤ 6ns) since the truncation error happens later in time and the same error level is observed for every measurement distance for 7ns ≤ t ≤ 11ns. If the measurement distance is larger than 2λ 0 (30cm), the error values for x meas = 2λ 0 are equivalent to the error values for x meas = 50cm. In contrast with Fig. 10(b), it is difficult to distinguish between the measurement distance error and the truncation error.

The frequency-domain far-field results comparison

We are interested in studying in frequency-domain the effect of the measurement distance x meas . The FF results presented in Fig. 11 are determined by Fourier transforming the time-domain results of the NF to FF transformation of NF data collected at x meas = 10cm, 20cm, 30cm, 40cm and 50cm with the sampling criterion ∆y χ = ∆y χ = λ 0 /3 (χ = 1.5). The comparisons presented in Fig. 10 show that TD FF errors decrease as the measurement distance increases. The time-domain errors are helpless to identify which frequency is mostly sensitive to the measurement distance. In contrast, the FD comparisons presented in Fig. 11 (a-d) show that the frequency (1GHz) is sensitive to the measurement distance with a maximum difference of 1.47dB at φ = 0 and 1.45dB at θ = π/2. The frequency 1.2GHz presented in Fig. 11 (b-e) shows a maximum difference of 0.4dB. The other frequencies (greater than 1.4GHz) stay unchanged (error ≤ 0.2dB) since the NF measurement distance is greater than a wavelength.

As a conclusion, using D max = 150cm and x meas ≥ 2λ 0 the error values are of the same order for χ = 1.3 and χ = 1.5. Hence, F -10dB ≈ 1.3f 0 can be considered as the maximum frequency for NF sampling of this AUT. We have tried to isolate the effect of three important measurement parameters in the planar TD NF technique. These parameters are : the measurement distance (reactive NF), the surface truncation (truncation error) and the NF sampling criterion (aliasing error). As it is presented in Figs. 5-6-7 it is difficult to predict from TD FF errors the consequences over the AUT radiation pattern in the frequency-domain. For this reason, the frequency-domain (FD) comparisons have been presented to make out the effect of each parameter over the AUT FD FF. x meas =10cm

x meas =20cm

x meas =30cm

x meas =40cm

x meas =50cm

Ref x meas =10cm

x meas =20cm

x meas =30cm

x meas =40cm |Ez| dB(V/m) at the frequency= 1.0 (GHz) , φ= 0

x meas =10cm

x meas =20cm

x meas =30cm

x meas =40cm

x meas =50cm

Ref x meas =20cm

x meas =30cm

x meas =40cm

x meas =50cm

Ref (f)

CONCLUSION

The effect of three parameters in planar time-domain near-field to far-field transformation have been presented. The followed approach aims to optimize the computation time and memory requirements by studying the near-field sampling measurement criterion. For antennas characterization using the time-domain near-field technique, we have shown that multiple conditions have to be met to correctly calculate the far-field. The size of the measurement surface decides predominantly on the frequency band to consider. The NF spatial truncation is responsible for the frequency limitation. The maximum frequency taken into account depends on the antenna excitation pulse and the behaviour of the AUT near-field directivity as a function of the frequency. Once the maximum frequency is defined, the sampling criterion is based on the Nyquist rate. Specific care has to be applied in choosing the measurement distance which determine the minimum frequency to be considered. Comparisons in TD and FD have been carried out to confirm these assumptions.

Figure 1 :

 1 Figure 1: The time-domain excitation signal (b) applied to each of the 40 dipoles composing the AUT (a) and the magnitude of the excitation signal spectrum (c). The maximum of the spectrum is at 2GHz.

Figure 2 :

 2 Figure 2: Near-field measurement grid at the distance x meas from the AUT. The tangential components E y and E z are regularly recorded over a square surface in the zy plane.

1 3F

 1 -30dB to prevent the time-domain aliasing error.

Figure 3 :Frequency

 3 Figure 3: The Ez (V/m) near-field radiated by the AUT (40 dipoles) as a function of the time. (a) The Ez(V /m) at the plane cut z meas = 0. (b) The Ez(V /m) at the plane cut y meas = 0.

Figure 4 :

 4 Figure 4: The normalized Ez(dB V/m) NF measured at (x meas = 50cm) radiated by the AUT (40 dipoles) as a function of the frequency. (a) The Ez(dB(V /m)) at the plane cut z meas = 0. (b) The Ez(dB(V /m)) at the plane cut y meas = 0.

Figure 5 :

 5 Figure 5: Comparison of the NF to FF transformation results for different sampling criteria ∆y = ∆z = λ 0 /2χ, D max = 150cm and x meas = 50cm. The TD FF Ez component at (a) θ = π/2 and φ = 0, (c) θ = π/3 and φ = 0 (e) θ = π/2 and φ = π/6. The errors of Figs. (a-c-e) are respectively (b) θ = π/2 and φ = 0, (d) θ = π/3 and φ = 0 (f) θ = π/2 and φ = π/6.

2 Time 2 Time 2 Time

 222 Ez error (%) χ =1.5 θ = π/2 , -π/2< φ <π/Ez error (%) χ =1.3 θ = π/2 , -π/2< φ <π/Ez error (%) χ =1.2 θ = π/2 , -π/2< φ <π/Ez error (%) χ =1.5 φ = 0deg , 0< θ <π Time(ns) Ez error (%) χ =1.3 φ = 0deg , 0< θ <π Time(ns) Ez error (%) χ =1.2 φ = 0deg , 0< θ <π Time(ns)

Figure 6 :

 6 Figure 6: The FF Ez component error as a function of time for the plane cut θ = π/2. (a) for χ = 1.5, (b) for χ = 1.3, (c) for χ = 1.2. The Ez error as a function of time for the plane cut φ = 0, (d) for χ = 1.5, (e) for χ = 1.3, (f) for χ = 1.2.

Figure 7 :

 7 Figure 7: Comparison of far-field Ez-component at the plane cut φ = 0 with D max = 150cm, x meas = 50cm and ∆y = ∆z = λ 0 /2χ for χ = 1.1, 1.2, 1.3, 1.4, 1.5. (a-d) For 1GHz and 3GHz (F -30dB ) respectively. (b-e) For 1.2GHz and 2.8GHz (F -20dB ) respectively. (c-f) For 1.4GHz and 2.6GHz (F -10dB ) respectively.

Figure 8 :

 8 Figure 8: Comparison of far-field Ez-component at the plane cut θ = π/2 with D max = 150cm, x meas = 50cm and ∆y = ∆z = λ 0 /2χ for χ = 1.1, 1.2, 1.3, 1.4, 1.5. (a-d) For 1GHz and 3GHz (F -30dB ) respectively. (b-e) For 1.2GHz and 2.8GHz (F -20dB ) respectively. (c-f) For 1.4GHz and 2.6GHz (F -10dB ) respectively.

Figure 9 :

 9 Figure 9: The normalized Ez NF component for F -30dB (3GHz, 1GHz), F -20dB (2.8GHz, 1.2GHz), F -10dB (2.6GHz, 1.4GHz), F -3dB (2.3GHz, 1.7GHz). (a) At the plane cut y meas = 0. (b) At the plane cut z meas = 0.

  error , θ = π/2 , φ = 0

Figure 10 :

 10 Figure 10: The Ez error as a function of time and as a function of the NF measurement distance (x meas ) for the FF observation points (a) θ = π/2 and φ = 0, (b) θ = π/3 and φ = 0.

  |Ez| dB(V/m) at the frequency= 1.0GHz , θ= π/2

  |Ez| dB(V/m) at the frequency= 1.2GHz , θ= π/2

  |Ez| dB(V/m) at the frequency= 1.4GHz , θ= π/2

  |Ez| dB(V/m) at the frequency= 1.4 (GHz) , φ= 0 x meas =10cm

Figure 11 :

 11 Figure 11: Comparison of the FF radiation pattern for x meas = 10cm, 20cm, 30cm, 40cm and 50cm at the plane cut θ = π/2, (a) at 1GHz, (b) at 1.2GHz, (c) at 1.4GHz. Comparison of the FF radiation pattern at the plane cut φ = 0, (d) at 1GHz, (e) at 1.2GHz, (f) at 1.4GHz.

  + e r . r 0mn /c) ∆y 0 ∆z 0 where r 0mn = x meas e x + m∆y e y + n∆z e z , e r = cos(φ) sin(θ) e x + sin(φ) sin(θ) e y + cos(θ) e z

	Ny n=-Ny	N z m=-N z	e r × e x ×	∂ E meas ∂t	( r 0mn , t


Mohamed Farouq
email: mohamed.farouq@lss.supelec.fr

Mohammed Serhir

Dominique Picard

Matrix Method for Antenna Plane Wave Spectrum Calculation Using Irregularly Distributed Near-Field Data: Application to Far-Field Assessment

The matrix method for the calculation of antenna far-field using irregularly distributed nearfield measurement data is presented. The matrix method is based on the determination of the plane wave expansion (PWE) coefficients from the irregular near-field samples using a matrix form that connects the radiated field with the corresponding plane wave spectrum. The plane wave spectrum is used to determine the far-field of the antenna under test (AUT). The matrix method has been implemented, and its potentialities are presented. The validations using analytical radiating model (dipoles array) and experimental measurement (X band standard gain horn antenna) results have demonstrated the efficiency and stability of the proposed method.

INTRODUCTION

The characterization of antenna far-field pattern is an important task of antenna measurement community. Among the existing methods for far-field assessment, we can find direct and indirect measurement techniques. The direct technique measures the antenna response when illuminated by a plane wave. This is reached directly at the far-field distance or using a compact range. The indirect technique is based on the near-field measurement over canonical surfaces in an anechoic chamber from which the radiation pattern is calculated through near-field to far-field transformation (NFFFT) algorithms. An overview of these algorithms is presented in [START_REF] Yaghjian | An overview of near-field antenna measurements[END_REF]. The NFFFT are based on an expansion of the electromagnetic field into plane wave (PWE), cylindrical wave (CWE) or spherical wave (SWE). The geometry of the used scanning near-field surface imposes which wave expansion to use [START_REF] Joy | Spatial sampling and filtering in near-field measurements[END_REF][START_REF] Leach | Probe compensated near-field measurements on a cylinder[END_REF][START_REF] Bucci | Use of sampling expansions in near-field-far-field transformation: The cylindrical case[END_REF][START_REF] Wacker | Non-planar near-field measurements: Spherical scanning[END_REF][START_REF] Hansen | Spherical Near-field Antenna Measurements[END_REF][START_REF] Wang | An examination of the theory and practices of planar near-field measurement[END_REF]. The planar near-field measurement technique is used to characterize directive antennas and have enjoyed a great popularity due to the relative simplicity involved in the data acquisition and processing. For the plane wave expansion, Wang [START_REF] Wang | An examination of the theory and practices of planar near-field measurement[END_REF] has presented a detailed examination of the theory and practice of this technique in 1988.

As an alternative to the wave expansion methods, it is possible to model the AUT through a set of equivalent currents distribution ( [START_REF] Petre | Planar near-field to far-field transformation using an equivalent magnetic current approach[END_REF][START_REF] Sarkar | Near-field to near/far-field transformation for arbitrary near-field geometry utilizing an equivalent electric current and MoM[END_REF][START_REF] Taaghol | Near-field to near/far-field transformation for arbitrary near-field geometry, utilizing an equivalent magnetic current[END_REF][START_REF] Las-Heras | Evaluating near-field radiation patterns of commercial antennas[END_REF][START_REF] Alvarez | Reconstruction of equivalent currents distribution over arbitrary three-dimensional surfaces based on integral equation algorithms[END_REF][START_REF] Persson | Reconstruction of equivalent currents using a near-field data transformation with radome application[END_REF]). These currents are set from the near-field measurement data, by solving the inverse radiation problem using the method of moments (MoM) [START_REF] Harrington | Field Computation by Moment Methods[END_REF]. This is the solution of the integral equations relating the equivalent currents and their radiated field that matches the measured one. From these equivalent currents, it is possible to calculate the AUT radiated field at any point of the space outside the equivalent sources domain. This method needs, however, the AUT a priori information: AUT location, dimensions, type.... The quality of the field reproduced (also the far-field) by the equivalent currents depends on the chosen distribution and spatial positioning of these equivalent electric or magnetic sources.

Newell [START_REF] Newell | Error analysis techniques for planar near-field measurements[END_REF] identified 18 error sources in the antenna planar near field measurements procedure and the impact of each near-field error on the far-field pattern was estimated. In [START_REF] Muth | Displacement errors in antenna near-field measurements and their effect on the far field[END_REF], Muth evaluated the effect of probe displacement errors in the planar near-field measurement procedure on the farfield pattern. In [START_REF] Muth | Displacement errors in antenna near-field measurements and their effect on the far field[END_REF] not only the maximum far-field errors caused by systematic position errors or the dependence of this maximum on the wave-number were studied, but also it deduced the exact error that contaminated the far-field when the probe displacements were known. Moreover, the theoretical study was carried out with enough generality so that the analysis can be extended to study position errors in cylindrical and spherical scanning geometries. In [START_REF] Hojo | Error analysis for bi-polar near-field measurement technique[END_REF] authors analyzed the same kind of errors but in bi-polar measurements. The NF measured with position errors may be consider either as an erroneous NF or as a NF measured on an irregular meshing. Considering this second situation the position errors may be random but have to be known (laser interferometry is often used for this purpose).

Dealing with irregular measurement data, the near-field plane wave expansion (PWE) based on 2-D Discret Fourier Transform (DFT) allowing the far-field calculation is not possible. Using the fact that the measured field is a band limited function [START_REF] Yen | On nonuniform sampling of bandwidth-limited signals[END_REF], the irregular near-field is interpolated to reconstruct a regular near-field grid, then, the data are processed using classical 2-D DFT algorithm to determine the far-field radiation pattern. This solution based on interpolating the near-field has been presented by many authors and exists in a variety of coordinate systems [START_REF] Rahmat-Samii | Nonuniform sampling techniques for antenna applications[END_REF][START_REF] Dehghanian | A new interpolation method for reconstructing non-uniformly spaced samples into uniform ones in planar near-field antenna measurements[END_REF][START_REF] Dehghanian | A new interpolation technique for the reconstruction of uniformly spaced samples from non-uniformly spaced ones in plane-rectangular near-field antenna measurements[END_REF][START_REF] Bucci | Interpolation of electromagnetic radiated fields over a plane by nonuniform samples[END_REF]. Wittmann in [START_REF] Wittmann | Near-field antenna measurements using nonideal measurement locations[END_REF] have proposed an approach to deal with these irregularities by combining unequally spaced DFT [START_REF] Marvasti | Nonuniform Sampling, Theory and Practice[END_REF], interpolation, and the iterative conjugate gradient algorithm. More recently an interesting approach has been presented in [START_REF] Schmidt | Fully probe-corrected near-field far-field transformation employing plane wave expansion and diagonal translation operators[END_REF][START_REF] Schmidt | Assessment of irregular sampling near-field far-field transformation employing plane-wave field representation[END_REF][START_REF] Qureshi | Adaptive sampling in multilevel plane wave based near-field far-field transformed planar near-field measurements[END_REF] in which a plane wave expansion is used to calculate near-field to far-field transformation with similar paradigm of equivalent-currents method. This method permits the far-field calculation of the antenna from its near-field sampled over irregular grid. To overcome the computational complexity of this method the Multilevel Fast Multipole Method has been used [START_REF] Schmidt | Multilevel plane wave based near-field far-field transformation for electrically large antennas in free-space or above material halfspace[END_REF] to perform matrix-vector products and accelerate the resolution of the linear system. The implementation of the fast multipole method proved to be a rather difficult task in part because of its complexity and because of the need to optimize all the steps of this method.

In this paper we present the matrix method to calculate the antenna PWE and consequently the farfield using non-uniformly spaced samples distributed randomly in 3-D rectangular surface (irregularities in x, y and z directions), where the Shannon criterion may be broken in limited area of the measurement surface [START_REF] Yen | On nonuniform sampling of bandwidth-limited signals[END_REF]. The principle of the matrix method proposed here is based on the matrix formulation of the PWE that expresses the linear relationship between the irregularly distributed near-field data and the plane wave spectrum. Solving this linear system, we calculate the plane wave spectrum and the far-field in the half space in front of the AUT.

We aim at introducing the matrix technique for PWE calculation where no interpolation is used and no a priori information are needed to calculate the AUT far-field. The equivalent currents technique, where the a priori information is essential, is popular and we are interested in comparing the two techniques using synthetic and experimental data.

The paper is structured in the following sections. In Section 2 the mathematical developments of the matrix method are presented. Results issued from this method are presented in Section 3 for different antennas. The first studied AUT is an array of 10 × 10 infinitesimal dipoles to generate the irregular near-field data analytically. Then, the far-field results of the proposed method are presented for an X band standard gain horn antenna measured in a planar near-field range at 12 GHz. Finally, a conclusion is outlined in Section 4.

THE MATRIX METHOD FOR PLANE WAVE EXPANSION

In this section we present the mathematical basis of the matrix method to calculate the PWE from a rectangular irregular near-field samples distributed randomly in a 3-D domain.

We start by recalling the theory of classical plane wave spectrum for regularly distributed near field data. Then we introduce the matrix method for such case. Finally we extend the matrix method to the case where the samples are irregularly distributed. Throughout this paper, exp(jωt) time dependence is suppressed in all the E-field expressions.

The plane wave expansion (PWE) expresses the electric field vector E at the point (x meas , y, z) of the measurement surface as a function of the plane wave spectrum vector F . In our analysis we suppose that only the tangential components E y and E z of the electric field are measured at the plane x meas where the antenna boresight is in the x-direction. The field E is expressed as a function of the plane wave spectrum F following

E (x meas , y, z) = 1 4π 2 +∞ -∞ +∞ -∞ F (k y , k z ) e -j(kxxmeas+kyy+kzz) dk y dk z (1)
F (k y , k z ) = +∞ -∞ +∞ -∞ E (x meas , y, z) e j(kxxmeas+kyy+kzz) dydz (2)
In practice, the measurement surface can not be infinite and y min ≤ y ≤ y max and z min ≤ z ≤ z max . Accordingly we present the truncated form of (2) as

F (k y , k z ) = ymax y min zmax z min E (x meas , y, z) e j(kxxmeas+kyy+kzz) dydz (3)
The near-field is discretized using the sampling steps Δy and Δz over y and z directions.

F k n y , k m z = P p=1 Q q=1 E (x meas , y p , z q ) e j(k n,m x xmeas+k n y yp+k m z zq) ΔyΔz (4) E (x meas , y p , z q ) = 1 4π 2 N n=1 M m=1 F k n y , k m z e -j(k n,m x xmeas+k n y yp+k m z zq) Δk y Δk z (5)
where,

k n,m x = k 2 -(k n y ) 2 -(k m z ) 2 , k is the wave-number in free space, Δk y = 2π/(y max -y min ), Δk z = 2π/(z max -z min ), P = (y max -y min )/Δy+1, Q = (z max -z min )/Δz+1, N = (k ymax -k y min )/Δk y +1 and M = (k zmax -k z min )/Δk z + 1 with k y min ≤ k y ≤ k ymax and k z min ≤ k z ≤ k zmax .
The field and the plane wave spectrum can be expressed as follow:

E = E x e x + E y e y + E z e z and F = F x e x + F y e y + F z e z . E y (x meas , y p , z q ) = 1 4π 2 N n=1 M m=1 F y k n y , k m z e -j(k n,m x xmeas+k n y yp+k m z zq) Δk y Δk z (6) E z (x meas , y p , z q ) = 1 4π 2 N n=1 M m=1 F z k n y , k m z e -j(k n,m x xmeas+k n y yp+k m z zq) Δk y Δk z ( 7
)
We express the matrix form of ( 6) and ( 7) as

E y (x meas , y p , z q ) = A regular F y k n y , k m z E z (x meas , y p , z q ) = A regular F z k n y , k m z ( 8
)
with

A regular = Δk y Δk z 4π 2 ⎛ ⎜ ⎜ ⎝ e -j(k 1,1 x xmeas+k 1 y y 1 +k 1 z z 1) . . . e -j(k N,M x xmeas+k N y y 1 +k M z z 1) . . . . . . . . . e -j(k 1,1 x xmeas+k 1 y y P +k 1 z z Q) . . . e -j(k N,M x xmeas+k N y y P +k M z z Q) ⎞ ⎟ ⎟ ⎠ (9)
The matrix method consists in the inversion of the equation system [START_REF] Petre | Planar near-field to far-field transformation using an equivalent magnetic current approach[END_REF] where F y and F z are the unknowns.

As an extension, we generalize the matrix method to deal with the irregular distributed data over a known grid, which are due to the errors caused by the probe displacement. In other terms, the irregular grid provides from a slightly modified regular grid. The Near-Field (NF) data are collected over a 3-D grid defined by (

x irreg min ≤ x irreg l ≤ x irreg max , y irreg min ≤ y irreg l ≤ y irreg max and z irreg min ≤ z irreg l ≤ z irreg max ), for 1 ≤ l ≤ L, with L is the number of measured points.
In this situation (irregular distributed data), ( 8) can be rewritten as follow:

⎧ ⎨ ⎩ E irreg y x irreg l , y irreg l , z irreg l = A irreg F y k n y , k m z E irreg z x irreg l , y irreg l , z irreg l = A irreg F z k n y , k m z ( 10
)
where,

A irreg = Δk y Δk z 4π 2 ⎛ ⎜ ⎜ ⎝ e -j(k 1,1 x x irreg 1 +k 1 y y irreg 1 +k 1 z z irreg 1 ) . . . e -j(k N,M x x irreg 1 +k N y y irreg 1 +k M z z irreg 1 ) . . . . . . . . . e -j(k 1,1 x x irreg L +k 1 y y irreg L +k 1 z z irreg L ) . . . e -j(k N,M x x irreg L +k N y y irreg L +k M z z irreg L ) ⎞ ⎟ ⎟ ⎠ (11) E irreg y = ⎛ ⎜ ⎜ ⎜ ⎝ E irreg y x irreg 1 , y irreg 1 , z irreg 1 . . . E irreg y x irreg L , y irreg L , z irreg L ⎞ ⎟ ⎟ ⎟ ⎠ , E irreg z = ⎛ ⎜ ⎜ ⎜ ⎝ E irreg z x irreg 1 , y irreg 1 , z irreg 1 . . . E irreg z x irreg L , y irreg L , z irreg L ⎞ ⎟ ⎟ ⎟ ⎠ (12)
F y = ⎛ ⎜ ⎝ F y k 1 y , k 1 z . . . F y k N y , k M z ⎞ ⎟ ⎠ , F z = ⎛ ⎜ ⎝ F z k 1 y , k 1 z . . . F z k N y , k M z ⎞ ⎟ ⎠ (13)
In our problem the near-field measurement is devoted to antenna far-field determination. Indeed, we take into account only the propagating plane waves (k

n,m x = k 2 -k n y 2 -(k m z ) 2 real
) for the calculation of the plane wave spectrum F . This stay true, since the distance between the probe and the antenna under test is beyond the evanescent near-field zone. As a consequence, the spatial sampling steps are chosen at a value slightly lower than half a wavelength. Equation ( 10) is over determined since the number of equations exceeds the number of unknowns, we look for the least square solution (LSQR) of [START_REF] Taaghol | Near-field to near/far-field transformation for arbitrary near-field geometry, utilizing an equivalent magnetic current[END_REF]. The LSQR solution passes by the solution of [START_REF] Harrington | Field Computation by Moment Methods[END_REF].

A H irreg E irreg y = A H irreg A irreg F y A H irreg E irreg z = A H irreg A irreg F z (14)
The operator A H irreg is the Hermitian (conjugate) transpose of A irreg . The least square solution F y and F z of ( 14) minimize

||A H irreg E irreg y -A H irreg A irreg F y || and ||A H irreg E irreg z -A H irreg A irreg F z ||.
In the given example, different direct and iterative methods, the pseudo inverse, conjugate gradient (CG) and LSQR method [START_REF] Paige | LSQR: An algorithm for sparse linear equations and sparse least squares[END_REF] have been applied. LSQR and CG achieve a small residual of the same order, whereas LSQR shows a slightly faster convergence. The matrix inversion using the pseudo inverse method reveals, as expected, very long computation time, even for rather small matrices. The LSQR method is based on the bidiagonalization procedure of Golub and Kahan [START_REF] Paige | LSQR: An algorithm for sparse linear equations and sparse least squares[END_REF]. It is analytically equivalent to the standard method of conjugate gradients, but possesses more favorable numerical properties.

Once F y and F z are determined, the normal component F x of the vector spectrum F is calculated using the property that the region contains no free charges (div( E) = 0))

k • F = 0 k x F x + k y F y + k z F z = 0 F x = - k y F y + k z F z k x ( 15
)
Once the measured near-field plane wave spectrum is calculated the far-field in the spherical coordinates system (r, θ, φ) is directly determined based on the following expression

E far-field (r, θ, φ) = jk sin θ cos φe -jkr r F (k sin θ sin φ, k cos θ) ( 16
)
As it can be seen the mathematical development of the matrix method for the plane wave expansion from irregular near-field data is easy to implement.

RESULTS

The matrix method is tested in two situations. In the first situation a numerical radiating antenna is considered. It is composed of 10 × 10 infinitesimal electric dipoles. In this case, the NF irregular positions are modeled considering the following procedure. First, we create a regular grid in both y and z directions using respectively the sampling steps Δy and Δz. Then, we add random functions (Random y , Random z defined in Matlab) varying between -1 and +1. These functions allow us to create an irregular grid in two dimensions. The irregularities are controlled using the weighting factors χ y and χ z . Finally, to generate a three dimensions irregularities we consider a given x meas to which we add a random function Random x varying between 0 and 1 and multiplied by a controlled irregularity factor χ x . This procedure is formulated in ( 17)

y irreg l(p,q) = pΔy + Random y χ y z irreg l(p,q) = qΔz + Random z χ z x irreg l(p,q) = x meas + Random x χ x ( 17
)
In the second situation, we measure an X-band standard gain horn antenna in SUPELEC planar near-field range. This measurement setup allows only the near-field measurement over regular grid at certain distance from the AUT. In order to simplify the experimental validation of the matrix method, we just consider the irregularities created in x direction, because the majority of the existing methods are not able to take into account this irregularity. This is done by measuring the near field in different planes x = x meas . Then, experimental irregular near field data are constructed by choosing randomly the samples from the different measurement planes for each (y p , z q ).

In both situations, the far-field pattern is calculated using the matrix method, equivalent currents method and by DFT while ignoring irregularities and the different obtained far-field patterns are compared with the reference far-field (Fig. 1). In the numerical study the reference far-field is calculated directly. The reference far-field for the experimental study is obtained by DFT of the measured regular near-field at x meas = 12 cm.

Numerical Study

The 10 × 10 z-dipoles array under consideration is composed of equally excited dipoles distributed regularly over the plane yOz. The dipoles are λ/2 spaced along the y and z directions (λ is the working wavelength). The near-field measurement surface is centered on the axis of the AUT and the field is collected over a square surface of side 20λ (y max = -y min = z max = -z min = 10λ). The near field is collected at a distance of x irreg with x meas = λ from the antenna at the frequency 2 GHz. The finite size of the measurement surface limits the reliable region of the calculated far-field. The reliable far-field angular region (θ valid and φ valid ) is computed as

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ θ valid = tan -1 (z max -z min ) -d 2x meas φ valid = tan -1 (y max -y min ) -d 2x meas (18)
where, d is the diameter of the AUT and x meas is the distance between the AUT and the measurement plane. In this case of study θ valid = 80 • and φ valid = 80 • . In order to evaluate the accuracy of the proposed method, we quantify the deviation between the co-polar far-field E calc issued from the different methods (matrix method and equivalent currents method) and the co-polar of the exact one E exact calculated directly from the dipoles radiation pattern, by means of the root mean square difference of the two far-fields. This quantity is evaluated either on the 2D or 1D far-field pattern. For example, the 2D far-field pattern deviation, i.e., error, is defined as below:

Error(%) = 100 θ,φ |E calc (θ, φ) -E exact (θ, φ)| 2 θ,φ |E exact (θ, φ)| 2 , 10 • ≤ θ ≤ 170 • and -80 • ≤ φ ≤ 80 • (19)
The results are presented and compared with the reference field (E exact ) in Figs. 2(a)-(b)-(c)-(d). Fig. 2 shows the far-field pattern calculated from a near field measured in irregular positions using χ y,z = χ x = λ/10. This means that the irregularity magnitude is λ/5 peak to peak. It is seen that the use of the matrix method based on the PWE gives a comparable accuracy with the equivalent currents method. Both methods fit very well the actual radiated far-field. In the case where we don't take into account the irregularities, we use directly the discrete Fourier transform to calculate the plane wave spectrum and the far-field radiation pattern. This is presented in Fig. 2(d). The differences between this far-field and the actual one are important. The principal plane cuts θ = 90 • and φ = 0 • of the co-polar component of the E-field pattern are presented in Fig. 3 for irregularities associated with χ y,z = χ x = λ/10. This shows the comparison between the reference and the far-field pattern obtained using the matrix method, equivalent currents method and when the irregularities are ignored. We note that both methods present a good agreement with the reference radiation pattern with an error of order 1.1% for θ = 90 • and 1.6% for φ = 0 • for the matrix method. The use of the equivalent currents method leads to errors of 1.6% for θ = 90 • and 1.4% for φ = 0 • . In the situation where the irregularities are not taken into account, significant errors in the side lobes of the radiation pattern are induced.

We study the influence of the weighting factor χ y,z , χ x . Fig. 4 shows the far-field pattern determined using irregular near-field with χ y,z = χ x = λ/5. In this figure we present the results of the calculated far-field using the matrix method, the equivalent currents method and when ignoring the irregularities.

As it can be seen, increasing the weighting factor influences directly the far-field accuracy especially in the low level far-field areas. Moreover, in the case where we ignore the irregularities the radiation pattern is strongly modified.

In Fig. 5 we present the plane cuts θ = 90 • and φ = 0 • for χ y,z = χ x = λ/5. In these plane cuts the error between the reference far-field and the far-field pattern obtained using the matrix method is of order 2.3% for θ = 90 • and 1.4% for φ = 0 • . The equivalent currents method presents the errors 1.9% for θ = 90 • and 1.5% for φ = 0 • plane cuts. Considering χ y,z,x = λ/5 makes the determination of the far-field pattern in the case where irregularities are not taken into account very critical (green curve). This leads to high errors in the main and side lobes of order 31.4% for θ = 90 • and 31.1% for φ = 0 • . Finally, we are interested in studying the effect of the signal to noise ratio (SNR) over the stability of the matrix and the equivalent currents methods. In fact, this can influence the accuracy of the matrix inversion when calculating the plane wave spectrum or for the equivalent currents determination. To do so, the near-field calculated at irregular positions (x irreg l , y irreg l , z irreg l ) with χ = λ/10 and χ = λ/5 is contaminated with a controlled white Gaussian noise to reduce the SNR.

Figure 6 presents the error values for χ = λ/10 and λ/5 as a function of the near-field data SNR for the far-field calculated over a hemi-sphere (2π steradian). It is seen from Fig. 6 that the error takes high values for low SNR near-field data. However, beyond the threshold of 30 dB the error level stays stable. Moreover, we can see that increasing χ changes slightly the far-field results for both methods (+3%). These increasing values influence the low level far-field areas.

Experimental Study

Our purpose is to illustrate the capability of the matrix method for the determination of the far-field of a standard gain pyramidal horn antenna (Fig. 7) measured in SUPELEC planar near-field setup. We consider the operating frequency of 12 GHz. The horn aperture dimensions are 13 cm × 9 cm and the measurement has been performed using an open-ended rectangular waveguide.

Using the planar translation axes set in the anechoic chamber we measure the tangential components of the near-field E y and E z radiated by the horn antenna over a regular grid (-60 cm ≤ y meas ≤ 60 cm and -30 cm ≤ z meas ≤ 30 cm) with Δy = Δz = λ/2.5 sampling criterion in each dimension over 5 different planes (x meas = 10 cm, 11 cm, 12 cm, 13 cm and 14 cm). Then, the irregular near-field is constructed by choosing randomly the near-field samples from the 5 measurement grids for each (y, z) positions.

Figure 8 presents the near-field magnitude for the case of regularly distributed near-field data ((a) ). The comparison includes: reference radiation pattern, the matrix method, the equivalent currents method and when ignoring irregularities. The far-field calculated using the matrix method is compared with the one calculated directly using DFT of regular near-field data measured at x meas = 12 cm. This is considered as the reference far-field pattern. The green curve is associated with irregular near-field data to which a classical near-field to far field transformation is applied while neglecting the irregularities and by considering x meas = 12 cm (medium value). For the equivalent currents assessment (electric and magnetic) we have defined a planar surface at the horn antenna aperture. The dimensions of this source surface are -9 cm ≤ y s ≤ 9 cm and -6 cm ≤ z s ≤ 6 cm and the number of equivalent currents to assess is 20 × 30 = 600.

The comparison concerns the principal plane cuts as shown in Fig. 9. The far-field radiation patterns when neglecting irregularities are strongly affected. Using the proposed matrix method and the equivalent currents method the far-field results fits very well the reference one for the co-polarization plane cuts θ = 90 • and φ = 0 • .

To get more insight into the efficiency of the proposed matrix method, we have presented in Table 1 the computation times and the condition numbers of the proposed method and the equivalent currents method. As it can be seen the matrix method converges rapidly to the desired solution and presents a low condition number which means that the problem is well posed.

CONCLUSION

The matrix method allowing the plane wave expansion of near-field over 3-D irregular grid has been proposed. The technique has shown a good accuracy and high calculation stability. The numerical validation examples have shown that for the near-field data measured at known irregular positions, the far-field can be accurately calculated. Experimental results have been presented using a standard gain horn antenna measured in the SUPELEC planar near-field facility. It has been shown that the radiation pattern of the antenna under test can be reconstructed accurately using the proposed method. However, in the case where we do not take into account the irregularities, the radiation pattern is strongly modified. The matrix method has shown interesting features: a low computational complexity, implementation simplicity, fast convergence to the solution, low condition number and also no a priori information about the AUT is needed in comparison with the well-known equivalent currents method.

APPENDIX A. THE EQUIVALENT CURRENTS METHOD

According to the equivalence theorem, the field radiated by an antenna can be reproduced from an equivalent set of electric and magnetic equivalent currents over the virtual surface S placed in the aperture of the radiating antenna (Fig. A1). The electric field integral equation [START_REF] Balanis | Advanced Engineering Electromagnetics[END_REF] expresses the AUT radiated field, in the half space in front of the AUT (x > 0), as the superposition of the contribution of electric (J) and magnetic (M ) currents.

E meas ( r) = E J ( r) + E M ( r) ( A 1 )

E J ( r) = -jη 4πk S k 2 J( r ) e -jkR R + ∇ ∇ J( r ) e -jkR R dS (A2) E M ( r) = -1 4π ∇ × S M ( r ) e -jkR R dS (A3)
In (A2) and (A3) r(x, y, z) is the position of the observation point and r (x , y , z ) is the position of the source point (equivalent currents). R = | rr | is the distance between these two points. The equivalent currents are expressed as M = M y e y + M z e z and J = J y e y + J z e z .

The Cartesian components of the electric field can be expressed according to [START_REF] Balanis | Advanced Engineering Electromagnetics[END_REF] as with

E
G 1 = -1-jkR+k 2 R 2 R 3
and

G 2 = 3+3jkR-k 2 R 2 R 5
. where, Δy and Δz are the dimensions of the source rectangular patches. The solution of (A8) is performed as (10) using LSQR method. As presented in this paper, the mathematical formulation of the equivalent current method seems to be more complicated to handel than the matrix method presented above. Indeed, the accuracy of the equivalent currents method depends on the definition of the currents surface (S ) where these sources are uniformly distributed. This surface is defined based on the AUT a priori information and the number of considered equivalent sources is of crucial importance. The matrix method is easy and systematic in its application where no a priori information are needed.

Figure 1 .

 1 Figure 1. Validation procedure of the matrix method.

Figure 2 .

 2 Figure 2. Amplitude of the radiation pattern (co-polar) as a function of θ and φ. (a) Reference pattern, (b) matrix method, (c) equivalent currents method, (d) ignoring irregularities.

Figure 3 .Figure 4 .

 34 Figure 3. The far-field (co-polar) comparison at the principal plane cuts. The comparison includes: reference radiation pattern, matrix method, equivalent currents methods and when ignoring irregularities.

Figure 5 .

 5 Figure 5. Principal far-field (co-polar) plane cuts. The comparison includes: reference radiation pattern, matrix method, equivalent currents methods and when ignoring irregularities.

Figure 6 .

 6 Figure 6. 2D error values for χ y,z = χ x = λ/10 and χ y,z = χ x = λ/5 versus the SNR for the matrix and the equivalent currents methods.

Figure 7 .

 7 Figure 7. The horn antenna measured in the planar near-field facility.

Figure 8 .Figure 9 .

 89 Figure 8. Near-field magnitude measured (a), (b) over a regular planar surface and (c), (d) over irregular surface.

  co and (b) cross polarization) and for the case of irregularly distributed data ((c) co and (d) cross polarization).

  My (x, y, z) = -1 4π y z xx M z + jkR R 3 e -jkR dy dz (A4) E Jy (x, y, z) = -jη 4πk y z G 1 J y + yy G 2 yy J y + zz J z e -jkR dy dz (A5) E Mz (x, y, z) = -1 4π y z xx M y 1 + jkR R 3 e -jkR dy dz (A6) E Jz (x, y, z) = -jη 4πk y z G 1 J z + zz G 2 yy J y + zz J z e -jkR dy dz (A7)

Figure A1 .

 A1 Figure A1. The equivalent currents method for the Far-Field calculation from irregular near-field measured samples.

Table 1 .

 1 Computation time (s) and condition number for matrix method and equivalent currents method.

		Computation time	Condition number
		Matrix method Equivalent currents Matrix method Equivalent currents
	Numerical validation	6.8	30	2.2	3 10 11
	Experimental validation	572	1628	2.7	1.4 10 15