NEAR-FIELD measurement techniques are a fundamental

Transcription

1 2940 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 54, NO. 10, OCTOBER 2006 A New Method for Avoiding the Truncation Error in Near-Field Antennas Measurements Ovidio Mario Bucci, Fellow, IEEE, and Marco Donald Migliore, Member, IEEE Abstract A new method to avoid the truncation error in antenna near-field measurements is presented. The approach relies on the concept of information content of the field. According to this point of view, the truncation problem is solved by picking up the information that is lost due to the finite size of scanning area, in points of the space reachable by the measurement system. The method can be applied to any scanning geometry, including the planar and cylindrical ones, whenever the set-up allows to vary the distance between the antenna under test and the probe during the scanning procedure. Application of the method to cylindrical near-field scanning is numerically investigated, assessing the effectiveness of the proposed technique. Index Terms Antenna measurement, near-field far-field transformation, truncation error. I. INTRODUCTION NEAR-FIELD measurement techniques are a fundamental tool for characterizing microwave radiating systems [1]. A fundamental step in the development of near-field measurements has been the identification and minimization of the error sources. Among them, the so called truncation error [2] due to the finite size of the scanning surface, is one of the most important. Its relevance in near-field measurements is testified by the fact that it can force the choice of the scanning surface. In fact, the standard approach to reduce the truncation error is to extend the measurement surface up to include all the region the field radiated by the antenna under test (AUT) is significant. As a consequence, the use of planar scanning geometry is restricted to the characterization of focusing antennas, the cylindrical scanning is adopted for fan-beam antennas. For the measurement of almost omnidirectional antennas or if a complete characterization of the far field pattern is required, the spherical scanning geometry must be applied, as it is the only one which does not suffer from the truncation error. However, it requires mechanical set-ups and near-field far-field transformation algorithms much more complex than those involved in planar and cylindrical scanning systems. On the other side, for a given near-field measurement set-up, it is again the truncation error that fixes the kind and the max- Manuscript received November 5, 2004; revised May 31, O. M. Bucci is with the Dipartimento di Ingegneria Elettronica e delle Telecomunicazioni, Università di Napoli Federico II, Napoli, Italy ( bucci@unina.it). M. D. Migliore is with the Dipartimento di Automazione, Elettromagnetismo, Informazione, Matematica Applicata, Università di Cassino, Cassino, Italy ( md.migliore@unicas.it). Digital Object Identifier /TAP imum size of the antennas which can be tested, limiting in practice a full exploitation of the site. It is worth noting that the choice usually adopted in near-field community of assuming zero the field outside the measuring area introduces a further error, i.e., a spurious ripple in far-field. This ripple can be reduced by a proper windowing of the near-field data [3], [4], but this reduces the angular range the far-field can be accurately evaluated. Consequently, it is of great interest to investigate strategies that allow to reduce or, if possible, completely eliminate the truncation error also in planar and cylindrical scanning systems. The approach followed in this paper to achieve this goal is based on the concept of degrees of freedom and information content of the field radiated by the AUT [5]. Broadly speaking, given a physical system about which we have some a priori information, the number of degrees of freedom of the system is the minimum number of parameters required to definite the state of the system within a given accuracy. In our case, the physical system consists of electromagnetic sources lying inside a given surface and the set of interest is the set of the corresponding fields observed on a surface external to the sources. Of course, being the set infinite dimensional, the number of parameters required to represent the field with infinite accuracy is infinite. However, in the true world any physical quantity can be known only within a finite accuracy, due to the unavoidable presence of noise and the fact that only a finite number of inaccurate measurements can be performed. Even if a finite precision is accepted, this does not ensure, in the case of a generic infinite dimensional set, that a finite number of parameters can represent the elements of the set within the given accuracy. However, due to the mathematical properties of the radiation operator [5] [7], under the only hypothesis that the amplitude of the sources is uniformly bounded, the set of all possible fields radiated on a surface is precompact [8]. Precompact sets can be uniformly approximated by elements of finite-dimensional subspaces with arbitrary precision [8]. The number of degrees of freedom (NDF) of the field is defined as the minimum number of linearly independent functions required to represent the field over within a given error. The evaluation of the NDF can be obtained using the function approximation theory, and the interested reader can refer to [5], [8]. For our purpose, it is useful to recall that in practical cases the NDF of the field turns out to be scarcely dependent on the error level, so that we can refer to the NDF as without explicitly indicating the level of accuracy. The NDF clearly depends on the observation manifold. An observation surface enclosing all the sources collects the maximum number of degrees of field of the field, let be. Since X/$ IEEE

2 BUCCI AND MIGLIORE: A NEW METHOD FOR AVOIDING THE TRUNCATION ERROR 2941 the knowledge of (two tangential components of) the field on a close surface enclosing the sources allows the evaluation of the field in the whole space outside the surface, in the following will be called the NDF of the field, without an explicit indication to the surface. If we consider only a part of a surface enclosing the sources, the NDF of the field on this truncated surface turns out to be lower than. This observation is crucial for the method presented in this paper. An other important observation is that the NDF depends on the a priori information on the physical system. For example, the spherical expansion [9] turns out to be the optimal representation in the case of a priori information consisting in the overall dimension of the source and its position in the space. The number of terms of the expansion to approximate the field within a fixed approximation turns out to be the NDF at the fixed level of accuracy. However, if we have more detailed information on the shape of the source, we can represent the field within the same accuracy with a smaller number of terms using different basis. The a priori information used in this paper consists in the overall dimension of the antenna, in its position in the space and in a further information on the shape of the antenna. The concept of NDF allows to state the truncation error problem in a very simple and general way: The truncation error is caused by a lack of information due to the fact that the number of degrees of freedom of the field on the observation surface is lower than the number,, of degrees of freedom of the field. Accordingly the elimination of the truncation error requires the evaluation of the remaining degrees of freedom.however, such evaluation is not straightforward. First of all, the evaluation of and the corresponding optimal (i.e., nonredundant) representation is by no means trivial. As discussed above, depends on the geometry of the smallest surface enclosing all the sources. Consequently, effective nonredundant representation can be obtained by using geometrical a priori information on the AUT. A simple and effective nonredundant representation of the field for substantially arbitrary source and scanning surface geometries is based on the concepts of reduced field and field effective bandwidth [10], [11]. This representation is a sampling representations, involving the field values in correspondence of a proper Nyquist lattice over the observation surface. In order to introduce the nonredundant sampling representation, let us recall that the NDF represents a lower bound for the number of functions required to represent the field (within a fixed accuracy) using linear approximation. Consequently the effectiveness of a linear representation of the field can be quantified by evaluating how many functions are required to represent the field within a given accuracy, or equivalently evaluating the dimension of the basis required to represent the field within a given accuracy. The closer such a dimension to the NDF is, the better the basis is. The reduced field requires a sinc basis (e.g., it is a bandlimited representation) whose dimension is only slightly larger than the NDF, or equivalently, the number of Nyquist samples required to represent the reduced field is practically equal to the number of degrees of freedom of the field on the surface [5]. This means that the sampling representation adopted in this work is an almost optimal representation. Consequently, the sentence regarding the significance of the truncation error can be expressed in terms of Nyquist samples (for the sake of simplicity, in the following discussion we will neglect the slight difference between the number of Nyquist samples and the number of degrees of freedom of the field) as follows. The truncation error is caused by the missing of field samples due to the fact that the number of Nyquist samples,, falling inside the observation surface is smaller than the number of Nyquist samples,, required to represent the field on a complete surface, i.e., a surface surrounding the source. Consequently, the elimination of the truncation error requires the estimation of the samples falling outside the observation surface. In the case of spherical scanning geometry, the partial recovery of the missing information can be obtained by estimating the spherical harmonics coefficients by a (regularized) mean square fitting of the data, oversampled over the truncated scanning region [12]. A more general approach, valid for arbitrary source and scanning geometry, has been proposed in [13], and experimentally validate in the case of aperture antennas in [14], and in the case of elongated antennas (in particular, mobile communication base station antennas) in [15]. This method gives satisfactory results with a significant reduction of the truncation error, but is not able to obtain a complete elimination of the truncation error, because of the ill posedness of the extrapolation problem. In fact, in spite of the use of sophisticated regularization procedures [16], only few samples outside the measurement zone can be reliably recovered. The good performance of the procedure is strictly related to the adoption of an optimal representation used in [13] [15], that allows a significant enlargement of the surface the field is accurately evaluated even if few samples are recovered. The method proposed in this paper to practically eliminate the truncation exploits a different idea, first proposed in [17], which is based on the following two key points. a) In planar and cylindrical scanning systems the probe can usually move not only on the scanning surface, but also along an axis perpendicular to it; b) The nonredundant sampling theory allows the identification of how the information content of the field radiated by the AUT is distributed in the space around the AUT. These observations suggest that it is possible to recover the information associated with the samples falling in the part of the (ideal, infinite) scanning surface external to the actual scanning area by measuring the field at other points around the AUT that are reachable by the scanning set-up. The algorithm to recover the lost information is implemented in the following way. First, a nonredundant sampling representation on the ideal scanning area is adopted. The samples of the representation

3 2942 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 54, NO. 10, OCTOBER 2006 falling inside the actual scanning area are directly measured. To recover the external samples, measurements having the same information content of the lost samples are performed in an area reachable by the measurement set-up, usually in a frame around the available scanning area. The positions of these latter measurement points are obtained by the nonredundant sampling theory. Then, a linear system relating the samples on the ideal scanning surface and the data measured on the frame is implemented by means of near-field near-field transformation. The solution of the linear system gives the value of the samples falling outside the actual scanning surface. It must be noted that, thanks to the choice of the unknowns, the dimension of the linear system is quite small, and its inversion has a low computationally cost. Finally, the field is interpolated on a very large (theoretically unbounded) portion of the ideal scanning surface, and standard near-field far-field transformation is applied to obtain the farfield pattern. The paper is organized as follows. The sampling strategy is presented in Section II. In the same section, some problems concerning the truncation of the series when standard sinc base functions are adopted to represent the field on open curves are discussed. The algorithm is described in Section III. Section IV is devoted to the clarification of the practical use of the method. Some examples of an extensive numerical investigation performed in the cylindrical scanning geometry are reported. Even if the examples are focused on a cylindrical scanning geometry, the method can be applied also to planar scanning. The stability of the method is investigated taking into account noisy data and nonideal probe. Finally, conclusions are collected in Section V. II. THE SAMPLING STRATEGY As pointed out in the introduction, the method proposed in this paper to practically eliminate the truncation error exploits the movement of the probe antenna along an axis perpendicular to the scanning surface in order to modify the standard planar or cylindrical measurement procedure and to recover the information contained in the part of the (ideal) scanning surface external to the actual scanning area. To devise such a strategy, we must answer two key questions. The first one regards how many measurements are required to recover the lost information. The second one regards where these measurements must be carried out, in order to allow a stable recovery of this information. Both questions can be answered by exploiting the above mentioned nonredundant sampling representations of the radiated field [10], [11]. Let us consider (Fig. 1) the field radiated by an AUT enclosed in the convex domain bounded by. Let us denote by an observation curve (supposed external to ), a curvilinear abscissa along and a component of the field radiated by the AUT on. A monochromatic field of angular frequency is considered, the time dependence being assumed and dropped in the following. Let us introduce the reduced field [10], [11] (1) Fig. 1. Relevant to nonredundant sampling of the field along an observation curve. where is a phase function. Provided that the parameterization and the phase functions do not introduce spurious singularities, the reduced field is an analytical function, which can be closely approximated [10] by a function bandlimited to, being is the generic source point,, and is an bandwidth enlargement factor. In the case of electrically large antennas the band limitation error decreases very rapidly with and becomes negligible for values of slightly larger than one. Accordingly, can be identified as the effective bandwidth of the field corresponding to the chosen parameterization and phase function, and the field along can be represented by a Shannon-Whittaker sampling series with Nyquist interval. As discussed in details in [11], a proper choice of the phase function and of the curve parameterization allows the minimization of the number of required samples, thus obtaining a non- redundant sampling representation. The corresponding optimal functions depend only on the source and observation curve, and can be explicitly determined for a large number of practically relevant geometries [11]. It turns out that the number of samples required to represent the field on the observation domain encircling the source is practically equal to the number of degrees of freedom of the field on the observation domain. As discussed in the Introduction this is the key-point of the technique proposed in this paper. When the observation curve does not encircle the sources, as in our case, retaining only the samples within the interval of interest introduces a truncation error whose quantitative evaluation is discussed in [18]. This could be avoided using, on the truncated domain, the spheroidal prolate functions (SPF) as basis function [19]. However, the expansion coefficients should be numerically evaluated from the measured field, and this would require a number of measurements larger than the Nyquist one. A much simpler approach is to include some extra samples outside the domain of interest. In the case of large sources and observation domains, the increase in the number of required samples is negligible [18]. (2)

4 BUCCI AND MIGLIORE: A NEW METHOD FOR AVOIDING THE TRUNCATION ERROR 2943 Fig. 2. Sampling position in a 2-D geometry; filled circles: measured samples on C; empty cirles: lost samples on C; filled squares: samples on C and C added to recover the information associated to the lost samples on C; empty squares: samples on C and C added to partially recover the information associated to samples falling outside the visible domain. By making reference to a proper set of coordinate curves, the theory can be applied to a generic observation surface. However, for sake of clarity, as first step in the following we consider the simple example in 2-D geometry shown in Fig. 2, consisting of an AUT limited by a convex curve, and an unbounded observation line parallel to the -axis. In the same Fig. 2 the curves corresponding to constant values of and are shown. The parameter ranges in when the observation point moves on the whole unbounded observation line. A fundamental property of the functions is that their value in the space around the AUT depends only on the dimension, shape and position of the surface including the AUT. Furthermore, they define an orthogonal coordinate system that is the natural one for the representation of the field by sampling series [11]. For example, the adoption of this coordinate system allows us to immediately find the sampling positions, that are simply the intersections between the observation curve and the curves, (circles in Fig. 2; note that the sampling positions associated to the samples 4 and 4 fall outside the figure). Consequently, we can identify the number of degrees of freedom of on the curve by simply counting the number of intersections between and the curves (f.i. in the example of Fig. 2 we have degrees of freedom on C). Let us suppose now that the actual scanning line is limited to a segment having finite length (see Fig. 2 again). This segment will be denoted by. In this case the parameter on ranges in, and does not intercept all the curves. Consequently only samples can be measured (points drawn as black circles in Fig. 2) while those falling outside the available scanning interval (points 3 and 3 drawn as white circles in the figure, and 4 and 4 falling outside the figure) are missed. This makes impossible to represent the field along the whole C line, causing the truncation error. In the following the number of lost samples will be indicated by 2Q. Let us now assume that the AUT-probe distance can be changed. In this case the lost curves can be intercepted by moving the probe on a frame around the AUT (the curve and drawn as dashed lines in Fig. 2). In this case the sampling positions are the intersection points between the curves and the curve given by (points plus the points,,3 and 4 drawn as dark squares in Fig. 2). The introduction of these new points modifies the scanning geometry from an unbounded line to a curve (half) enclosing the AUT. Accordingly, we can state that the adoption of the modified scanning geometry consents the collection of information on all the degrees of freedom of the field required to reconstruct the far field without truncation error. 1 Of course, in order to be practically exploitable we must be able to recover, starting from the measurements made on, the lost samples on, a point which will be addressed in the next section. With reference to this point, a deeper discussion on the representation of the field along the unbounded line C is required. With reference to Fig. 2, our goal is to accurately represent the field by Shannon-Whittaker series along the whole unbounded line, i.e., in the range. This range will be called the visible domain in the following discussion. However, as discussed before, bandlimited function on a domain not enclosing the sources using the Shannon-Whittaker series requires a knowledge of some samples outside the domain on which we want to represent the function, i.e., in the range and, that are associated with the analytic continuation of the field. Summarizing, in order to obtain an accurate representation of the field on the whole line C it is necessary to include some of the nearest samples outside the visible domain. Consequently, the field along the unbounded curve C is represented by the series 2P is the number of samples falling outside the visible domain added to the sampling series, and is the number of samples falling on C. The curves, fall outside the interval. The above example regards a 2-D geometry. The extension to 3-D geometries is straightforward [10], [11]. Let us consider, without significant loss of generity, an AUT included in a surface that has a rotational symmetry. Let be a plane through the axis of rotation, and be the intersection between and the plane. On this plane it is possible to consider the coordinate system, according to the discussion outlined in the first part of this section. The surface is obtained by rotating i.e. changing the azimuthal angle. This leads to the introduction of a coordinate system that represents a natural orthogonal coordinate system for sampling representation [11]. In the case of a 1 Of course in this example we are only interested in the evaluation of the far-field in the z>0 half space. (3)

5 2944 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 54, NO. 10, OCTOBER 2006 Fig. 3. Sampling positions in case of cylindrical scanning geometry. cylindrical observation surface around the source, the field on the surface can be represented by a double Shannon-Whittaker series [10], [11] is the number of samples along the th azimuthal circumference,, is the effective bandwidth along the th azimuthal circumference, is the corresponding enlargment bandwidth factor, is the Dirichlet function of order, while the meaning of the other symbols are the same as (3). As discussed in the Introduction, the number of samples falling inside the visible domain is practically equal to the number of degrees of freedom of the component of the field in (4) [5], and it turns out to be slightly larger than the area of the surface divided by [11]. Since the field outside a surface including the AUT is determined by the two tangential components, the number of degrees of freedom of the field is almost equal to 2. In Fig. 3 the sampling positions chosen according to the strategy outlined in this Section are plotted in the case of a cylindrical scanning. The Figure shows that the unbounded cylindrical surface is substituted by a surface that completely encloses the AUT by adding the two bases of the cylinder (the sampling points lying on these bases are plotted as white circles). This modified surface is equivalent to a spherical surface from the point of view of field information content. A further example is reported in Fig. 4, the generatrix curve is rotated around the axis. In this case we have a modified plane-polar scanning geometry. III. DESCRIPTION OF THE ALGORITHM The sampling strategy described in the previous section allows us to collect all the information required to practically (4) Fig. 4. Sampling positions in case of plane-polar scanning geometry. avoid the truncation error in far-field evaluation from near-field data. In fact, the samples enables us, in principle, to evaluate the field on the modified surface surrounding the source. Consequently, it is possible to evaluate the far-field starting from the knowledge of the near-field on the modified surface. To reach this goal, it is advantageous to pass from the modified surface to a canonical surface, like planar, cylindrical or spherical in order to adopt a standard near-field far-field transformation. For example, we can evaluate the equivalent currents on a planar surface, following the method outlined in [20]. Another possibility consists in calculating the coefficients of the modal expansion relative to the surface chosen from the samples on the modified surface, and then evaluating the far-field directly from these coefficients. However, all these choices require the inversion of a large linear system having a number of unknowns of the order of the electrical area of the antenna. The inversion can be performed using iterative algorithms, like the conjugate gradient method, but care must be taken in the stopping rule in case of ill-conditioned systems. A natural choice to reduce the number of the unknowns is to estimate the unknown samples in the representation (4). In fact, since the field on a large portion of the not truncated scanning area is known, the number of unknowns is significantly reduced. After evaluating the field on the unbounded scanning surface via the sampling representation, the far-field transformation can be performed using the very efficient algorithms available on trade. This two-step procedure to evaluate the far-field from modified scanning surface is the strategy followed in this paper. For sake of clarity, as a first step the algorithm for the evaluation of the field on the canonical surface from data collected on the modified surface is described with reference to a 2-D geometry. Let us reconsider again the simple example shown in Fig. 2. Furthermore, again for sake of clarity let us assume that the electric field is directed along the axis. The field on C is represented by means of the sampling series, with sampling points falling outside the visible domain.

6 BUCCI AND MIGLIORE: A NEW METHOD FOR AVOIDING THE TRUNCATION ERROR 2945 According to the strategy introduced in Section II, we collect the data on the modified scanning geometry. As discussed in Section II, the data collected in the points [ (filled squares + filled circles in Fig. 7)] on have the same information content as the data collected in the points [, 3, 4, (empty + filled circles in Fig. 7)] on C. Regarding the points in the invisible domain, they belong to the analytical continuation of the field, so that their recovery from the information obtained from the measurement on is an ill-posed problem. However, we can recover part of their information by adding measurement points placed on close to the -axis (empty squared in Fig. 2) and adding some kind of regularization. The field on C value of the samples on falling outside and consequently to evaluate the field along the whole line by (5). If in (5), i.e., no samples outside the visible domain are introduced in the representation, the condition number of the linear system is very low thanks to the choice of the sampling positions. When we have an ill-conditioned system that can be solved using regularization procedure. An analysis of the regularization procedures is reported in [14]. In particular, it is shown that if only a couple of samples must be estimated, the procedure using a simple truncation of the singular values of the matrix gives satisfactory results [14], [15]. Consequently, considering the singular value decomposition (SVD) of the matrix [16] (9) is related to the measured data collected on the curve, say,, by a linear relationship, that in practice is a near-field near-field transformation 2 from to. Consequently, adopting a matrix notation, we have is the vector of the samples on, that of the measured field and the matrix that relates the value of the field at the sampling points to the values of the field in the samples on. However, the values of the samples on and of the samples on placed on (in the example the points ) coincide, and only samples (i.e., 5, 4, 3, 3, 4, 5) are unknowns. Accordingly, we introduce two vectors, one collecting the samples falling inside the segment (and consequently having known values), let be, and other collecting the samples falling outside the segment, let be. Furthermore, the values measured on the curves C and C are collected in a vector, while the other measured values are collected in a vector. By partitioning the matrix in the proper way, we obtain The unknowns can be obtained by solving the linear system (5) (6) (7) (8) is a square matrix whose dimension is. The solution of the system allows to obtain the 2 Note that if the near-field far-field transformation is available on C, the nearfield near-field transformation is also available. Of course, because it is exploited inward, it can be applied only up the smallest cylinder enclosing the source. This determines the position of the measurement points respect to =0; ( is a unitary matrix whose columns are the left singular vectors of the matrix, is a unitary matrix whose columns are the right singular vectors,, are the singular values of the matrix, and apex + stands for the transpose conjugate of the matrix), a stable solution is obtained by (10), and can be chosen by the methods outlined in [14], [15]. In the above discussion a simple example has been considered. In more general cases there are two transverse components of the electric field on C. However, the algorithm does not change. The only difference is that we need to represent each of the two tangential components by a sampling series, obtaining samples. Consequently, the near-field near-field transformation in (7) relates the samples of the two tangential component of the field to the values of the two tangential components measured on and so that the linear system in (8) has unknowns. The above outlined procedure can be extended in a straightforward way in the case of 3-D geometry. In this case each of the two tangential components of the electric field on the measurement surface is represented by a sampling series [see (4)]. The steps of the algorithm are the same as those outlined in the 2-D case. For example, by considering Fig. 3, we start from a truncated cylindrical scanning geometry, then we collect the data on a modified cylindrical geometry including the two basis of the truncated cylinder, successively we estimate the samples on the (unbounded) cylindrical geometry and finally we perform the near-field far-field transformation from the data interpolated on this latter surface. It is useful to note that from a theoretical point of view we could estimate the samples also on other surfaces rather than the unbounded cylinder, like f.i. on a sphere. The only difference compared to the choice of a cylinder is that we must use the near-field near-field transformation for spherical geometry. However, in this case we should estimate all the samples belonging to the spherical surface, obtaining a much

7 2946 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 54, NO. 10, OCTOBER 2006 larger linear system, so that from a practical point of view it is not convenient. The same procedure can be applied, f.i., in case of a truncated plane-polar scanning geometry, using the modified geometry shown in Fig. 4. Finally, it is worth noting that even if the above discussion regards the case of an ideal measurement probe, the same discussion is valid also in the case of a real probe. In fact it is shown in [21] that the measured voltage, over the observation surface, also has an effective bandwidth equal to that of the radiated field. Consequently, the same algorithm described above can be adopted, the only variation being the introduction of the probe response in the operator relating the field samples to the measured quantities. Even if accuracy, and not computational efficiency, is our goal, some considerations on the computational complexity are in order. The algorithm consists of two steps. In the first one we fill the matrices and, which requires the evaluation of entries, via a standard near-field near-field transformation. The second step consists in solving the linear system. The computational complexity is proportional to, is the number of unknowns. However, is quite small, since only the value of the samples falling outside the scanning surface are unknowns, so that the inversion of the matrix is very fast. Note that the evaluation of and can be done off-line, since they are functions of the sampling point and of the AUT shape, while the inversion of the matrix is very fast. In the case of the test of antennas with same geometry, the computational time slightly increases with the number of antennas to be tested. IV. DISCUSSION AND NUMERICAL EXAMPLES In order to show the practical application of the approach outlined in previous Sections, let us consider the case of a cylindrical near-field scanning geometry. The AUT consists of a linear array of 13 directed electric dipoles placed along the axis and equispaced. Each dipole is fed by a unit electric current. The near-field component along of the electric field is measured by an ideal probe on a scanning surface placed at a distance from the AUT. In Fig. 5 the exact near-field amplitude is plotted in the range as a continuous line. We suppose that, due to the limited extension of the measurement set-up, the data are collected on a line having length. The extension of this region is indicated by two dotted vertical lines in the same Fig. 5, showing a truncation around 35 db level. Due to the rotational symmetry of the source, the field radiated by the AUT is independent of, and can be obtained from measurement on this single scanning line. The far-field obtained by standard near-field far-field procedure is plotted in Fig. 6 as dotted line. In the same Fig. 6 the exact far-field pattern is reported as continuous line. The plot shows that the reconstruction is erroneous outside the angular range (50, 130 ). Let us consider now the proposed approach. The first step of the procedure is to choose a proper convex surface enclosing the AUT. We choose a prolate spheroid having an eccentricity slightly smaller than 1 (equal to 0.99) in order to avoid Fig. 5 Near-field pattern from noiseless data collected by an ideal probe: continuous line: reference pattern; dashed line: from the truncated data using the proposed technique. Fig. 6. Far-field pattern, from noiseless data collected by an ideal probe: continuous line: reference; dotted line: from truncated data; dashed line: from the truncated data using the proposed technique. the presence of singularities in the representation [11], and the distance between the focuses equal to. The parameterization and phase functions are reported in Appendix I for sake of reader convenience. The transformation maps the entire unbounded observation line onto the interval. Since the field radiated along the axis of the AUT, i.e., and is known, being null, it is advantageous to place two samples respectively at and.a is chosen to obtain a negligible bandlimitation error, obtaining samples in the visible range (sampling step ). The curves are plotted in Fig. 7 as solid curves while the unbounded scanning line placed at is drawn as dashed line. Furthermore, we introduce four samples falling outside the visible domain in the series representation (5), two of them falling in, and the other two falling in. Coming back to Fig. 7, the plot shows that since only 13 curves are intersected by the available scanning line (the intersection points are marked as filled circles

8 BUCCI AND MIGLIORE: A NEW METHOD FOR AVOIDING THE TRUNCATION ERROR 2947 Fig. 8. Normalized singular values. Fig. 7. Relevant to the example regarding the linear array; circles and squares: same meaning than in Fig. 2. in Fig. 7), while the remaining 6 are lost. However, the two samples related to the curves and are known, their value being equal to zero, so that only the 4 samples placed in and are really lost. Let us suppose that the measurement set-up permits also a movement of the probe in the -direction, so that it is possible to collect data along the lines. In this case the information content associated with the lost samples can be obtained by measuring the field in the positions (, ) and (, )(filled squares in Fig. 7). Besides these measurement points, we introduce four other measurements samples in (, ) (along the curves and ) and (, ) (along the curves and ) (these points are plotted as empty squares in Fig. 7) in order to estimate the four samples falling outside the visible domain. Then the linear system (6) is written by using the near-field near-field transformation in cylindrical geometry reported in Appendix II. Since the information related to the samples falling outside the interval can only be partially recovered by the measurements, the linear system is ill-conditioned. Fig. 8 reports the normalized singular values of the related matrix, showing a conditioning number around 25 db. Fig. 8 clearly shows a knee between the sixth and the seventh singular values. A stable and reliable solution of the system can be obtained from (10) by cutting the last two singular values [14], [15]. The solution of the linear system allows us to obtain the unknown samples on. Then, the near-field is reconstructed in the desired interval by interpolation. Fig. 5 reports the interpolated near-field amplitude in the range as dashed line, showing a good match with the exact near-field amplitude (continuous line). Similar good results are obtained Fig. 9. Far-field pattern from noisy truncated data collected by an ideal probe: continuous line: reference; dotted line: 045 db noise level; dashed line: 055 db noise level. for the near-field phase. Finally, the standard near-field far-field transformation is applied to the interpolated data, obtaining the far-field pattern plotted in Fig. 6 as a dashed line, confirming that the procedure allows to practically eliminate the truncation error. Let us consider now the presence of noise in the measurement data. As an example, in Fig. 9 the far-field pattern is reported in the case of the above reported example, but considering the presence of 55 db white Gaussian noise (dashed line), and a 45 db white Gaussian noise (dotted line). The results show a good stability with reference to the noisy data. The above examples regard the case of an ideal probe. However, in practical measurements the accurate evaluation of the far sidelobes requires the correction of the response of the probe also in the case [20] of low directivity probes. In particular, since the probe correction tends to amplify any reconstruction error in the far-sidelobes, it is important to have information on the stability of the solution as a function of the directivity of the probe. In the following we consider a circular probe with radius having a uniform tangential magnetic field distribution on the aperture [23]. The procedure, outlined above in the case of ideal probe, is repeated, the only difference being the substitution of the terms related to near-field near-field transformation in the linear system, that in this case must take into ac-

9 2948 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 54, NO. 10, OCTOBER 2006 Fig. 10. Far-field pattern from noiseless truncated data collected by the nonideal probe: continuous line: reference; dotted line: probe directivity 10 dbi; dashed line: probe directivity 4 dbi. Fig. 12. Planar array: reference far-field in the range 0 # 180, 0180 ' 180 ; gray scale: 0 10 db (white), db, db db, db, db (black). Fig. 11. Far-field pattern from noisy truncated data collected by the nonideal probe: continuous line: reference; dotted line: probe directivity 10 dbi; dashed line: probe directivity 4 dbi. count the response of the probe (reported in Appendix III). In Fig. 10 the far-field pattern obtained from noiseless near-field data collected by a probe having a directivity equal to 4 dbi is plotted as dashed line. The plot shows an almost complete elimination of the truncation error. In the same Fig. 10 the far-field pattern from noiseless data in the case of a probe having a directivity equal to 10 dbi is plotted as dotted line, showing less satisfactory estimation of the farthest side-lobes. Finally, the procedure is repeated including noise on the data. In Fig. 11 the far-field obtained by the proposed procedure is shown in case of a 4 dbi (dashed line) and a 10 dbi (dotted line) probe directivity with 55 db noise level in the near-field data. The results show again a good stability with respect to noisy data. An extensive numerical investigation of the technique proposed in this paper has been carried out, considering different AUT. As an example, we report the case of an AUT simulating a mobile communication base station antenna. It consists of two linear arrays one of them having the same geometry of the AUT adopted in the example reported above. The two arrays are Fig. 13. Planar array: far-field pattern in the range 0 # 180, from truncated data using the proposed technique 0180 ' 180 ; gray scale: 0 10 db (white), db, db db, db, db (black). placed at a distance of. The exact far-field pattern of the AUT is plotted in Fig. 12. Noiseless data is collected by means of an ideal probe on a cylindrical surface having length at a distance from the axis of the AUT. The algorithm to eliminate the truncation error is applied including the AUT in a prolate spheroid having semi-axis and.a value of and is chosen, obtaining 105 samples, 71 falling inside the measurement area, 16 outside the measurement area but inside the visible range, 16 outside the visible range, 1 at and 1 at. Since the value of these two latter samples is equal to zero, there are 32 unknowns. 32 measurement are performed on the planes according to the method outlined in Section III. Then, the linear system is solved by SVD algorithm by cutting 10 singular values. Finally, the field is interpolated on a cylinder high, and the far-field is evaluated by standard near field-far field transformation. The result is shown in Fig. 13, confirming the effectiveness of the technique.

10 BUCCI AND MIGLIORE: A NEW METHOD FOR AVOIDING THE TRUNCATION ERROR 2949 V. CONCLUSION A novel method for avoiding the truncation error in near field measurement has been presented. The method takes advantage of the possibility, present in most of the scanning set-ups, to move the probe not only on the scanning surface, but also along the axis perpendicular to it. This possibility allows us to modify the standard planar or cylindrical measurement procedure in order to recover the information contained in the part of the (ideal) scanning surface external to the actual scanning area. The algorithm proposed in the paper can be applied both to planar and cylindrical scanning geometry. Numerical examples regarding a cylindrical scanning and taking into account the noise and the probe correction confirm the effectiveness of the method. and is the enlargement bandwidth factor along. APPENDIX II CYLINDRICAL NEAR-FIELD NEAR-FIELD TRANSFORMATION FROM NON REDUNDANT SAMPLES IN CASE OF IDEAL PROBE Considering a cylindrical reference system, and taking into account the relationships [1] (II.1) APPENDIX I NON REDUNDANT SAMPLING PARAMETERS IN CASE OF AUT INCLUDED IN A PROLATE ELLIPSOID Let us consider an AUT included in a prolate ellipsoid, having major and minor semiaxis respectively equal to and. With reference to a meridian plane, let us consider an observation point placed on an analytical observation curve. Let and be the distance from the observation point to the foci of the ellipsoid, and,. The phase functions, parameterization functions and effective bandwidth are [11] and their inverse (II.2) (II.3) (II.4) (I.1) (I.2) (I.3) is the eccentricity of the ellipse obtained from the intersection between the prolate ellipsoid and the meridian plane, is the length of the ellipse, the focal distance, denotes the elliptic integral of the second kind ( is the free-space wavenumber, is the Hankel function of the second kind and order, and ), the tangential components of the electric field in a point is related to the tangential components of the electric field collected on a cylinder having radius by the following expressions: (I.4) (II.5) Consequently, the curves and are respectively the family of hyperbolas and ellipses confocal to. The samples on the observation curve are uniformly distributed respect with step is the enlargement bandwidth factor. In the case of azimuthal circle intersecting the meridian plane in the point, the phase function is constant and consequently is chosen equal to. The parameterization coincides with the azimuthal angle, while the effective bandwidth is, The samples on the circumference are placed at uniform angular step, [11], (II.6)

11 2950 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 54, NO. 10, OCTOBER 2006 (II.7) (II.8) Let us represent the tangential components of the field on the cylinder having radius by nonredundant sampling series. In view of the substitution of the series in the expressions (II.5) and (II.6) it is advantageous to express the components of the field as Fourier series respect to, obtaining APPENDIX III CYLINDRICAL NEAR-FIELD NEAR-FIELD TRANSFORMATION FROM NON REDUNDANT SAMPLES IN CASE OF NONIDEAL PROBE Taking into account the relationships reported in [22], the output voltage in a point is related to the voltage at the output of the probe and at the output of the rotated probe collected on a cylindrical surface having radius by the relationships (II.9) (II.10) (III.1) (II.11) (II.12) By substituting the above expressions in (II.5) and (II.6) we have (III.2) (II.3) (III.3) (III.4) (III.5) (II.14) (II.15) (III.6) and and are the coefficients associate to the probe in the two positions.

12 BUCCI AND MIGLIORE: A NEW METHOD FOR AVOIDING THE TRUNCATION ERROR 2951 Let us introduce a nonredundant sampling representation of the output voltage probe paralleling the approach followed in the case of ideal probe: (III.7) (III.8) (III.9) (III.10) Substitution of (III.7) and (III.8) into (III.2) and (III.3) allows to express (III.1) in terms of nonredundant samples (III.11) REFERENCES [1] A. D. Yaghjian, An overview of near-field antenna measurements, IEEE Trans. Antennas Propag., vol. AP-34, no. 1, pp , Jan [2] A. C. Newell, Error analysis techniques for planar near-field measurements, IEEE Trans. Antennas Propag., vol. AP-36, pp , Jun [3] E. B. Joy and C. A. Rose, Windows 96 for planar near-field measurements, in Proc. AMTA, Seattle, WA, Oct. 1996, pp [4] P. R. Rousseau, The development of a near-field data window function for measuring standard gain horns, in Proc. AMTA, Boston, MA, Nov. 1997, pp [5] O. M. Bucci and G. Franceschetti, On the degrees of freedom of scattered fields, IEEE Trans. Antennas Propag., vol. 37, pp , [6] O. M. Bucci and T. Isernia, Electromagnetic inverse scattering: Retrievable information and measurement strategies, Radio Sci., vol. 32, no. 6, pp , Nov. Dec [7] D. S. Jones, Method in Electromagnetic Wave Propagation, 2nd ed. Oxford, U.K.: Clarendon Press, [8] A. Kolmogorov and S. V. Fomine, Elements de la Theorie des Functions et de d Analyse Fonctionelle. Moscow: Editions MIR, [9] R. F. Harrington, Time-harmonic Electromagnetic Fields. New York: McGraw Hill, [10] O. M. Bucci and G. Franceschetti, On the spatial bandwidth of scattered field, IEEE Trans. Antennas Propag., vol. AP-35, pp , Dec [11] O. M. Bucci, C. Gennarelli, and C. Savarese, Representation of electromagnetic fields over arbitrary surfaces by a finite and nonredundant number of samples, IEEE Trans. Antennas Propag., vol. AP-46, pp , [12] R. C. Wittmann, C. F. Stubenrauch, and M. H. Francis, Spherical scanning measurements using truncated data sets, in Proc. AMTA, Cleveland, Nov. 2002, pp [13] O. M. Bucci, G. D Elia, and M. D. Migliore, Experimental validation of a new technique to reduce the truncation error in near-field far-field transformation, in Proc. 20th AMTA Symp., Montreal, Canada, 1998, pp [14], A new strategy to reduce the truncation error in near-field farfield transformations, Radio Sci., vol. 35, no. 1, pp. 3 17, Jan. Feb [15] J. C. Bolomey, O. M. Bucci, L. Casavola, G. D Elia, M. D. Migliore, and A. Ziyyat, Reduction of truncation error in near-field measurement of antennas of base-station mobile communication systems, IEEE Trans. Antennas Propag., vol. AP-52, no. 2, pp , Feb [16] M. Bertero, Linear Inverse and Ill-Posed Problems. Boston, MA: Academic Press, [17] O. M. Bucci and M. D. Migliore, Strategy to avoid truncation error in planar and cylindrical near-field measurement set-ups, Electron. Lett., vol. 39, no. 10, pp , May [18] O. M. Bucci and G. Di Massa, The truncation error in the application of sampling series to electromagnetic problems, IEEE Trans. Antennas Propag., vol. AP36, no. 7, pp , Jul [19] D. Slepian and H. O. Pollak, Prolate spheroidal wave functions, Fourier analysis and uncertainty I, The Bell Syst. Tech. J., vol. 40, no. 1, pp , Jan [20] P. Petre and T. K. Sarkar, Planar near-field to far-field transformation using equivalent magnetic current approach, IEEE Trans. Antennas Propag., vol. AP-40, no. 11, pp , Nov [21] O. M. Bucci, G. D Elia, and M. D. Migliore, Advanced field interpolation from plane-polar samples: Experimental verification, IEEE Trans. Antennas Propag., vol. AP-46, no. 2, pp , Feb [22] W. M. Leach and D. T. Paris, Probe compensated near-field measurements on a cylinder, IEEE Trans. Antennas Propag., vol. AP-21, pp , [23] Z. A. Hussein and Y. Rahmat-Samii, Probe compensation characterization in cylindrical near-field scanning, in Proc. IEEE AP Symp., Jun. 1993, pp Ovidio Mario Bucci (SM 82 F 93) was born in Civitaquana, Italy, on November 18, He received the E.E. degree from the University of Naples Federico II, Naples, Italy, in Since 1976, he has been a Full Professor of electromagnetic fields at the University of Naples Federico II, where he was the Director of the Department of Electronic Engineering from 1984 to 1986 and 1989 to He was the Vice Rector of the University of Naples, from 1994 to He has been the Director of the Interuniversity Research Centre on Mi-

13 2952 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 54, NO. 10, OCTOBER 2006 crowaves and Antennas (CIRMA) since 1997 and of the CNR Institute of Electromagnetic Environmental Sensing (IREA) since His main research interests include scattering from loaded surfaces, reflector and array antennas, efficient representations of electromagnetic fields, near-field far-field measurement techniques, inverse problems and noninvasive diagnostics. He is the author or coauthor of more then 300 scientific papers, mainly on international scientific journals or Proceedings of international Conferences. He is the Principal Investigator or Coordinator of many research programs, granted by national and international research organizations, as well as by leading national companies. Prof. Bucci is a Member of the Associazione Elettrotecnica Italiana (AEI) and the Accademia Pontaniana. He is the Past President of the National Research Group of Electromagnetism and the MTT-AP Chapter of the Centre-South Italy Section of IEEE. He received the International Marconi Prize for the Best Paper in the field of Radiotechnique in 1975, the International Award GUIDO DORSO for Scientific Research in 1996, and the Presidential Gold Medal for Science and Culture in Marco Donald Migliore (M 04) received the Laurea degree (honors) in electronic engineering and the Ph.D. degree in electronics and computer science from the University of Napoli Federico II, Naples, Italy, in 1990 and 1994, respectively. He was a Researcher at the University of Napoli Federico II until He is currently an Associate Professor at the University of Cassino, Cassino, Italy, where he teaches adaptive antennas, radio propagation in urban area and electromagnetic fields. He teaches microwaves at the University of Napoli Federico II. He is also a consultant of industries in the field of advanced antenna measurement systems. His main research interests are antenna measurement techniques, adaptive antennas and medical and industrial applications of microwaves. Dr. Migliore is a Member of the Antenna Measurements Techniques Association (AMTA), the Italian Electromagnetic Society (SIEM), the National Inter-University Consortium for Telecommunication (CNIT) and the Electromagnetics Academy. He is listed in Marquis Who s Who in the World, Who s Who in Science and Engineering, and in Who s Who in Electromagnetics.