REDUCTION OF TRUNCATION ERROR IN THE NEAR-FIELD FAR-FIELD TRANSFORMATION WITH PLANAR SPIRAL SCANNING

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2 REDUCTION OF TRUNCATION ERROR IN TE NEAR-FIELD FAR-FIELD TRANSFORMATION WIT PLANAR SPIRAL SCANNING F. D Agostino (), F. Ferrara (), C. Gennarelli (), R. Guerriero (), G. Riccio (), C. Rizzo () () D.I.I.I.E. University of Salerno, via Ponte Don Melillo, 8484 Fisciano (SA), Italy. () MI Technologies Euroe, 3 ither Green, Southbourne Emsworth, PO 8JA, UK. ABSTRACT An elaborate and effective strategy for estimating the samles external to the measurement region in the lanar siral scanning is develoed in this aer. It relies on the nonredundant samling reresentations of the electromagnetic field and on the otimal samling interolation exansions of central tye and uses the singular value decomosition method for extraolating the outside samles. It is so ossible to reduce the inevitable truncation error affecting the near-field reconstruction, thus giving rise to a more accurate far-field rediction. Numerical examles assess the effectiveness of the roosed technique. Keywords: Extraolation, Planar siral scan, NF FF transformations, Nonredundant reresentations of electromagnetic fields, Truncation error reduction.. Introduction Each member of the antenna measurement techniques community can rofit today by about fifty years of research activity on near-field data acquisition and related near-field to far-field (NF FF) transformations. Over these years, many solutions have been roosed to meet the demands of the various alications [, ]. The measured NF data are usually transformed into FF atterns by using an exansion of the antenna field in terms of modes, i.e., a comlete set of solutions of the vector wave equation in the region outside the source. Plane, cylindrical or sherical waves are generally used for reresenting the field, thus determining the geometry of the NF scanning surface. Each of the NF FF transformation techniques has its own articular advantages and disadvantages deending on the tye of radiating structure under consideration and the measurement requirements. Among the NF FF transformation techniques, those emloying the lanar scannings are articularly suitable for high gain antennas radiating encil beam atterns well within the solid angle secified by the edges of the antenna under test (AUT) and those of the scanning area. As suggested by Rahmat-Samii et alii in [3], the comlexity and cost of the measurement set-u, as well as the time needed for data acquisition, can be reduced by means of continuous and synchronized movements of the ositioning systems of the robe and of the AUT. In order to satisfy these requirements, NF FF transformation techniques using innovative siral scannings [4] have been recently develoed. In articular, an effective NF FF transformation using a lanar siral arrangement of samles has been roosed in [5]. To this end, a nonredundant reresentation of the voltage data acquired by the measurement robe on the siral has been develoed by alying the theoretical results on the nonredundant samling reresentations of electromagnetic (EM) fields [6]. Then, the choice of the siral ste equal to the samle sacing needed to interolate the data along a radial line allows one to obtain the required two-dimensional otimal samling interolation (OSI) formula. It is so ossible to reconstruct the NF data needed by the classical robe comensated NF FF transformation with lanerectangular scanning [7]. owever, since the measurement region is always finite, an inevitable truncation error affects the NF reconstruction in the zone close to the boundary of such a region. As a consequence, the NF recovery results to be accurate in a zone smaller than the measurement one and this imlies a decrease of the angular region wherein an accurate FF reconstruction is attained. Accordingly, to obtain an accurate field reconstruction in the whole measurement region, it becomes very imortant to estimate a roer number of samles external to it. The goal of this aer is just the estimation of these outside samles, otherwise equal to zero in the alication of the OSI algorithm. Such an estimation allows one to reduce the truncation error, thus giving rise to a more accurate FF reconstruction, also in the far out side lobe region. The extraolation rocess relies on the knowledge of extra data acquired on very few eriheral rings and makes use of the singular value decomosition (SVD) algorithm [8] for determining the outside data.

3 . Efficient samling reresentation of the robe voltage Let us consider the field radiated by an antenna and observed on a siral lying on a lane located at distance d from AUT the center. Since the voltage V measured by a non directive robe has the same effective satial bandwidth of the field, the theoretical results relevant to the nonredundant reresentation of EM fields [6] can be alied to such a voltage. Accordingly, if the AUT is enclosed in a shere of radius a and the lanar siral is described by an analytical arameterization r = r() x, it is ossible to consider the robe reduced voltage ( ) = ( ) e j () g( x) V x V x where g (x) is a hase function to be determined. The error, occurring when Ṽ is aroximated by a bandlimited function, becomes negligible as the bandwidth exceeds a critical value W x [6]. Therefore, such an error can be controlled by choosing a bandwidth equal to c 'W, c' > being an excess bandwidth factor. x According to the theoretical results in [6], a nonredundant samling reresentation of the voltage on a lanar siral with constant angular ste Dq (see Fig. ) can be develoed by roerly using the following exressions for the otimal hase function and arameterization: g s b ( ) = Ú x = x( s) = s È max ˆ ˆ min ˆ ˆ Rt + Rt Î Í r' r' ds () s b È max ˆ W ˆ min ˆ ˆ Rt - Rt x r' r' Ú Î Í ds (3) where r' denotes the source oint, s is the curvilinear abscissa along the siral, ˆt is the unit vector tangent to it at the observation oint P, b is the wavenumber and ˆR is the unit vector from the source oint to P. d x P( j) O z a S r max Figure - Planar siral scanning y The coordinates of P are given by: Ïx = d tanj cosj = r cosj Ô Ìy = d tanj sinj = r sin j (4) Ô Óz = d wherein j is the angular arameter describing the siral. It is worthy to note that the siral angle J = k j, unlike the zenithal angle q, can assume negative values. Moreover, the siral angle j is always continuos, whereas, according to (4), the azimuthal angle f exhibits a discontinuity jum of when the siral crosses the ole. Such a siral can be obtained as intersection of the lane z = d with the line from the origin to the oint which moves on a siral wraing a shere of unit radius. In order to allow the two-dimensional interolation, the angular ste Dq of the siral is chosen equal to the samle sacing required to interolate the reduced voltage along a radial line. Then, the arameter k is such that Dq, fixed by two consecutive intersections P( j ) and P(j + ) of the siral with the consider radial line, is Dq = ( M + ), with M = Int[ c M'] + and M' = Int[ c' ba] +. Accordingly, being Dq = k, it follows that k = ( M + ). The function Int[ x] gives the integer art of x and c > is an oversamling factor. P R tˆ ˆ e R r d d O a P P R ˆr Rˆ Figure - Geometry in the lane ˆr, ˆt It can be verified that the extreme values of Rt ˆ ˆ in () and (3) are determined by considering the intersection of the lane defined by ˆt and the unit vector ˆr (ointing from the origin to P) with the cone with the vertex at P and the generatrices coincident with the tangents to the AUT ball (see Fig. ). As shown in [5] for the lanar siral, and in [4] with reference to the more general case of a siral wraing a quite arbitrary rotational surface, it results: r -Ê a ˆ g = b - a r dr = b r -a - ba cos Á (5) Ú Ë r x = b a Wx j Ú k + sin kj' d j' (6)

4 Namely, x is roortional to the arclength of the siral wraing the shere of unit radius. Since such a siral is a closed curve, it is convenient to choose the bandwidth W x such that x covers a range when the whole curve on the shere is drawn. Therefore, ( M + ) a Wx = b k + sin kj' dj' Ú According to these results, the OSI formula of central tye [6] to reconstruct the field at any oint of the siral is: n + q V ( x) = V Â ( xn) WN'' ( x - xn) DN ( x - xn ) (8) n = n - q + ( ) where n = Int ( x - x( j i )) D x is the index of the samle nearest, on the left, to the outut oint, q is the number of retained samles, and (7) x = x( j ) + ndx = x( j ) + n ( N + ) (9) n i i N = Int( c N' ) + ; N' = Int( c' W x ) + () Moreover, DN x W N " ( x) = sin (( N + ) x ) N + sin x ( ) = ( ) ( ) [ ( ) - ] TN" cos( x ) cos( x ) TN" cos ( x ) - [ ] () () are the Dirichlet and Tschebyscheff Samling functions, T N" () being the Tschebyscheff olynomial of degree N" = N - N' and x = qd x. The OSI formula (8) allows one to evaluate the voltage at any oint P on the scanning lane. As a matter of fact, it can be used for the evaluation of the intermediate samles, namely the voltage values at the intersection oints of the siral with the radial line assing through P. Once these samles have been determined, due to the articular choice of Dq, the voltage at P can be recovered by means of the following OSI exansion: m + V ( q, f) Â = V ( qm ) D M ( q -qm) W M "( q -qm ) (3) m = m - + where m = Int [( q -q ) Dq] qm = md q + k f = md q + q () f ; M" = M - M' (4) V ( q m ) are the intermediate samles, and the other symbols have the same meaning as in (8). 3. Extraolation rocedure Due to the finite sizes of the measurement lane, the reconstruction of voltage data is affected by an inevitable error in the zone close to the erihery of the scanning region. This will reflect in a decrease of the angular region wherein an accurate FF recovery is attained. In order to overcome this drawback, we now tackle the roblem of estimating the voltage samles external to the scanning region r r max on the measurement lane (see Fig. ). To this end, let us assume, besides the regular samles acquired via the siral scanning, the knowledge of the voltage data on the J rings saced at a fixed ste Dr, from the end of the measurement circle. On each of these rings, the extra samles are known at the oints secified by fnj, = ndfj = n ( Nj + ) (5) where Nj = Int( cn' j) + ; Nj ' = Int ( c * Wf ) + (6) Wf = W a j f( qj) = b sin qj (7) ( )[ ] - 3 / c* = + c' - sinq j (8) - [ ] qj = tan ( rmax - ( j -) D r) d j =, K, J (9) On each radial line fixed by f, the reduced voltage at the intersection oints P( q j, f ) with the extra rings can be evaluated via the OSI exansion [6] ( ) = ( ) = V q V j q j, f n + Â q n = n - q+ = V ( q j, fn, j) W N " ( f - fnj, ) DN ( f - f j n, j) j j () where q is the number of retained samles along f, n = Int( f/ D f j ), and N " j = N j - N ' j. When emloying the exansion (3) for reconstructing Ṽ at each of the oints P( q j, f ), just unknown outside samles V ( q m ) are always involved, since the other ones can be reconstructed via (8). Accordingly, by denoting with m the index of the last intermediate samle inside the measurement lane on the considered radial line, for each j =, K, J we obtain: m V ( q V j ) - Â ( qm ) DM ( qj - qm ) WM" ( qj - qm ) = m = m - + Âm + = V ( qm ) DM ( qj - qm ) W M" ( qj - qm ) () m = m +

5 where is the number of external samles to be estimated. These J equations can be rewritten in matrix form as Ax = b, where b is the sequence of the known terms, A is a J matrix, whose elements Ajm = DM( qj - qm) W M" ( qj - qm) are given by the weight functions in the considered OSI exansion and x is the sequence of the unknown outside samles V ( q m), with m = m +,..., m +. A solution, which is the best aroximation in the least squares sense of the linear system (), can be obtained by using the SVD technique. Once the outside samles relevant to the considered radial line have been estimated, the voltage values at any oint on it can be evaluated via (3). Note that the matrix A deends on the samles osition on the radial line. In fact, the r-coordinate of the regular samles (those obtained as intersection between the radial line and the siral) increases on increasing the angle f. A Tikhonov regularization aroach can be usefully alied [8] to obtain a more accurate solution of the linear system (). Such a solution corresonds to minimize the functional: Ax - b + a x () a being the regularization arameter. Thus, we can write the regularized solution x reg and the corresonding residual vector b - Axreg in term of the SVD of A in the generic form ui xreg = Â fi v i (3) s i i= J b - Axreg = Â( - fi) ui b ui + Â ui b ui (4) i= i= + In (3) and (4), the symbol denotes the conjugate transosition oerator, s i, (i =,..., ) are the singular values of A, ordered from the maximum to the minimum, f i = s i ( s i + a ) are the corresonding filter factors, and u i, v i are the left and right singular vectors of A, resectively [8]. The choice of the otimal arameter a to be used can be made by means of the L-curve, which is simly a lot of the norm of the regularized solution x reg versus the corresonding residual norm of b - Axreg drawn in log-log scale for a set of admissible regularization arameters. In this way, the L-curve dislays the comromise between the minimization of these two quantities, which is the heart of any regularization method. With reference to the Tikhonov regularization, the best comromise is reresented by the so-called corner, i.e., the distinct oint searating the vertical and the horizontal arts of the curve. 4. NF FF transformation The basic theory of robe comensated NF measurements on a lane [7] is based on the alication of the Lorentz recirocity theorem. In the here used reference system, the key relations, for evaluating the FF comonents E Q, E F of the AUT, are: ( V V ) EQ( Q, F) = I E' F ( Q, -F) -I E' D F ( Q, -F) ( V V ) EF( Q, F) = I EQ' Q F I E D (, - ) - Q' ( Q, -F) where D = E ' ( Q, -F ) E ' ( Q, - F ) + Q FV (5) (6) -EQ ' ( Q, -F) E' F ( Q, -F) (7) + + d cosq I V, = A cosq e jb VV, ( x, y Ú Ú ) - - jbxsinq cosf jbysinq sinf e e dxdy V (8) A being a constant. Namely, the AUT far field is related to: a) the two-dimensional Fourier transforms I V and I of the voltages V V and V for two indeendent sets of measurements (the robe is rotated by 9 in the second set); b) the FF comonents E ' Q V, E ' F V and E ' Q, E ' F radiated by the robe and the rotated robe, when used as transmitting antennas. Equations (5) and (6) are valid whenever the robe maintains its orientation with resect to the AUT and this requires that it rotates together with the AUT. Probes exhibiting only a first-order azimuthal deendence in their radiated far field can be used without corotation, since V V and V can be evaluated from the measured voltages V f and V r, via the relations VV = Vf cosf - Vr sin f, V = Vf sin f + Vr cos f. 5. Numerical tests The reorted numerical tests refer to a uniform lanar circular array (Fig. ) having diameter a = 3. 4 l (l being the wavelength). Its elements are elementary uygens sources linearly olarized along the y axis and are radially and azimuthally saced of 95. l. An oen-ended circular waveguide having radius a' equal to.338 l is chosen as robe. The measurement lane is 5 l away from the AUT center and the samles of the robe voltages V f and V r are collected on a siral that covers a circular zone of radius 6 l. According to the described samling reresentation, J = 8 extra rings have been acquired in the eriheral zone. They are saced at Dr = 95. l, starting from the end of the measurement circle. It is

6 worthy to note that, on a given radial line, the number of outside samles is = 5. A Tikhonov regularization aroach has been alied to obtain the best estimation of the outside samles for all the considered radial lines. We have assumed = 7 in the extraolation rocess of the outside samles, whereas q = 6 has been adoted both in (8) and in () to obtain the involved known samles and the extra data, resectively. Note that the SVD is alied to a small matrix with a negligible comutational effort. Figures 3 and 4 show the amlitude of the outut voltage V V on the radial lines at f = and f = 9, resectively. They have been reconstructed without using the extraolation rocess and utting the outside samles equal to zero. As can be seen in Figs. 5 and 6, by using the roosed estimation rocedure, the reconstruction is very accurate not only in the whole measurement region, but also in a zone outside it. It is worthy to note that, in all the cases, = 6 has been adoted when alying (3) for the reconstruction. In order to assess more quantitatively the effectiveness of the roosed aroach, the maximum and mean-square reconstruction errors have been eval = q = 7-5 c =. c' = Figure 3 - Amlitude of the robe voltage V V on the radial line at f =. Solid line: exact. Crosses: reconstructed without estimated outside samles. uated by comaring in the measurement zone the exact voltage values and those reconstructed with and without the estimated outside samles. Figure 7 shows such errors, normalized to the voltage maximum value on the lane, for c = c' =., and = q ranging from to. As can be seen, the errors evaluated by taking into account the estimated samles decrease until very low levels are reached. On the contrary, those obtained without considering them saturate to constant values, due to the truncation error resent near to the boundary of the measurement region. The algorithm stability has been assessed (see Fig. 8) by adding random errors to the exact samles. These errors simulate a background noise, bounded to Da (db) in amlitude and with arbitrary hase, and an uncertainty on the data of ±Da r (db) in amlitude and ±Da (degrees) in hase. The described aroach has been alied to recover the lane-rectangular data needed for the NF FF transformation and lying in a 36l 36l square grid. Figures 9 and reort the AUT attern in the E- lane, reconstructed via the NF FF transformation without and with estimated outside samles. As can = q = 7-5 c =. c' = Figure 5 - Amlitude of the robe voltage V on V the radial line at f =. Solid line: exact. Crosses: reconstructed with estimated outside samles = q = 7-5 c =. c' = Figure 4 - Amlitude of the robe voltage V on V the radial line at f = 9. Solid line: exact. Crosses: reconstructed without estimated outside samles = q = 7-5 c =. c' = Figure 6 - Amlitude of the robe voltage V on V the radial line at f = 9. Solid line: exact. Crosses: reconstructed with estimated outside samles.

7 Normalized errors (db) c' = c =. -7 maximum error -8 mean-square error = q Figure 7 - Normalized reconstruction errors. Dashed lines: without estimated outside samles. Solid lines: with estimated outside samles Da = -5 db Da r =.5 db Da = 5-4 = q = 7-5 c =. c' = Figure 8 - Amlitude of the robe voltage V V on the radial line at f =. Solid line: exact. Crosses: reconstructed with estimated outside samles from error affected data. be seen, the FF reconstruction obtained by considering the estimated samles is accurate in a significantly wider angular range, thus assessing the effectiveness of the technique. Note that the number of emloyed samles is 6. In articular, the number of extra samles on the eriheral rings is REFERENCES [] A.D.Yaghjian, An overview of near-field antenna measurements, IEEE Trans. Antennas Proagat., vol. AP-34,. 3-45, 986. [] C.Gennarelli, G.Riccio, F.D Agostino, and F.Ferrara, Near-Field Far-Field Transformation Techniques, CUES, Salerno, Italy, 4. [3] R.G.Yaccarino, L.I.,Williams, and Y.Rahmat- Samii, Linear siral samling for the biolar lanar antenna measurement technique, IEEE Trans. Antennas Proagat., vol. AP-44,. 49-5, 996. [4] F.D Agostino, C.Gennarelli, G.Riccio, and C.Sa- Relative field amlitude (db) = q = 7 c =. c' = ba sin J Figure 9 - E-lane attern. Solid line: exact. Dots: reconstructed via the NF FF transformation without estimated outside samles. Relative field amlitude (db) = q = 7 c =. c' = b a sin J Figure - E-lane attern. Solid line: exact. Dots: reconstructed via the NF FF transformation with estimated outside samles. varese, Theoretical foundations of near-field far-field transformations with siral scannings, Progress in Electromagn. Res., PIER 6,. 93-4, 6. [5] O.M.Bucci, F.D Agostino, C.Gennarelli, G.Riccio, and C.Savarese, Probe comensated FF reconstruction by NF lanar siral scanning, IEE Proc. - Microw., Antennas Proagat., vol. 49,. 9-3,. [6] O.M.Bucci, C.Gennarelli, and C.Savarese, Reresentation of electromagnetic fields over arbitrary surfaces by a finite and nonredundant number of samles, IEEE Trans. Antennas Proagat., vol. 46, , 998. [7] D.T.Paris, W.M.Leach, Jr., and E.B.Joy, Basic theory of robe-comensated near-field measurements, IEEE Trans. Antennas Proagat., vol. AP- 6, , 978. [8] P.C.ansen, Rank-deficient and discrete illosed eoblems, SIAM, Philadelhia, 998.