Overview of Novel Post-Processing Techniques to Reduce Uncertainty in Antenna Measurements <s> Iterative algorithms to reduce the truncation error in planar, cylindrical and not complete spherical acquisitions <s> A new view of the problem of continuing a given segment of the spectrum of a finite object is presented. Based on this, the problem is restated in terms of reducing a defined ‘error energy’ which is implicit in the truncated spectrum. A computational procedure, which is readily implemented on general purpose computers, is devised which must reduce this error. It is demonstrated that by so doing, resolution well beyond the diffraction limit is attained. The procedure is shown to be very effective against noisy data. <s> BIB001 </s> Overview of Novel Post-Processing Techniques to Reduce Uncertainty in Antenna Measurements <s> Iterative algorithms to reduce the truncation error in planar, cylindrical and not complete spherical acquisitions <s> An alternative method is presented for computing far-field antenna patterns from near-field measurements. The method utilizes the near-field data to determine equivalent magnetic current sources over a fictitious planar surface that encompasses the antenna, and these currents are used to ascertain the far fields. Under certain approximations, the currents should produce the correct far fields in all regions in front of the antenna regardless of the geometry over which the near-field measurements are made. An electric field integral equation (EFIE) is developed to relate the near fields to the equivalent magnetic currents. The method of moments is used to transform the integral equation into a matrix one. The matrix equation is solved with the conjugate gradient method, and in the case of a rectangular matrix, a least-squares solution for the currents is found without explicitly computing the normal form of the equation. Near-field to far-field transformation for planar scanning may be efficiently performed under certain conditions. Numerical results are presented for several antenna configurations. > <s> BIB002 </s> Overview of Novel Post-Processing Techniques to Reduce Uncertainty in Antenna Measurements <s> Iterative algorithms to reduce the truncation error in planar, cylindrical and not complete spherical acquisitions <s> In this paper, experimental results of the application of a new method for the reduction of the truncation error in near-field measurement are presented. It is shown that the use of the proposed algorithm allows the test of larger antennas obtaining a significant upgrading of near-field system possibilities. <s> BIB003 </s> Overview of Novel Post-Processing Techniques to Reduce Uncertainty in Antenna Measurements <s> Iterative algorithms to reduce the truncation error in planar, cylindrical and not complete spherical acquisitions <s> The authors demonstrate through numerical simulation and experimental measurement that a polyplanar scan geometry consisting of a flat-topped pyramid provides a possible solution to the measurement of high-gain antenna patterns in the forward hemisphere using a planar scanner of size of order 1.5 times the size of the radiating aperture. The importance of correctly determining the normal field component for each partial scan data set is demonstrated and a new auxiliary rotation near-field to far-field transform algorithm is proposed. Additionally it is shown experimentally that by enclosing a medium-gain antenna (e.g. a corrugated horn) within a scan geometry formed by an imaginary box and measuring the near field on all six sides of the box using a suitable rotation of the AUT, a prediction of the full spherical radiation pattern of the antenna can be obtained. <s> BIB004 </s> Overview of Novel Post-Processing Techniques to Reduce Uncertainty in Antenna Measurements <s> Iterative algorithms to reduce the truncation error in planar, cylindrical and not complete spherical acquisitions <s> A simple and effective procedure for the reduction of truncation errors in planar near-field measurements of aperture antennas is presented. The procedure relies on the consideration that, due to the scan plane truncation, the calculated plane wave spectrum of the field radiated by the antenna is reliable only within a certain portion of the visible region. Accordingly, the truncation error is reduced by extrapolating the remaining portion of the visible region by the Gerchberg-Papoulis iterative algorithm, exploiting a condition of spatial concentration of the fields on the antenna aperture plane. The proposed procedure is simple and computationally efficient; it does not require any modification of the measurement procedure and it allows for the usual probe correction. Far-field patterns reconstructed from both simulated and measured truncated near-field data demonstrate its effectiveness and stability against measurement inaccuracies. <s> BIB005 </s> Overview of Novel Post-Processing Techniques to Reduce Uncertainty in Antenna Measurements <s> Iterative algorithms to reduce the truncation error in planar, cylindrical and not complete spherical acquisitions <s> This paper presents a technique for the determination of the full sphere pattern of an antenna from truncated spherical near field measurements. The approach is based on an iterative procedure which exploits the band-limitedness properties of the radiated field to correctly reconstruct the full sphere data. Tests performed on synthetic as well as measured near-field data demonstrate the effectiveness of the procedure and its stability against measurement errors. <s> BIB006
One important requirement to determine exact far-field patterns from near-field acquisitions is the ability to measure the electric or magnetic field that is tangential to an arbitrary surface that encloses the AUT. The planar and cylindrical scanning geometries are mechanically simpler than in the case of the spherical near-field (SNF). Moreover, the spherical near-field to far-field transformation is more complex, requiring more calculations to obtain the far-field pattern from the acquired data. However, the most accurate antenna patterns are obtained using this last type of acquisition because it is the only measurement set-up where the AUT is fully enclosed by the acquisition surface. Therefore, there are no truncation errors in the calculated far-field pattern. In the planar near-field (PNF) and cylindrical near-field (CNF) measurements, because of the finite size of the scan surface, the closed surface condition is never fulfilled, and, consequently, the true far-field pattern is never known in the whole sphere, i.e., the pattern is only valid within the called reliable region. There are some studies that attempt to obtain a good estimation of the true pattern outside of the reliable region. One of these approaches, called the equivalent magnetic current approach BIB002 , presents a method of computing far-fields in the whole forward hemisphere from planar near-field measurements. Another strategy to reduce truncation errors is to rotate the AUT about one or more axes BIB004 , measuring in different planes and combining them to increase the maximum validity angle. In  and BIB003 , the problem of truncation is addressed using a priori information about the AUT. The main idea of this approach is to estimate the near-field data outside of the scanning area by extrapolating the measured data before calculating the far-field pattern. A recent publication BIB005 ) also uses a priori information about the AUT in an iterative algorithm to extrapolate the reliable portion of the calculated far-field pattern. As commented before, truncation errors are always present in PNF and CNF measurements but not in SNF measurements. However, there are special cases, e.g., when measuring electrically large antennas, in which the measurement time may be prohibitively long and an acquisition over the whole sphere is not practical. A solution to reduce the data acquisition time is to measure over a partial sphere. Nevertheless, the new acquisition surface does not fully enclose the AUT, and, as in the PNF and CNF cases, a truncation error appears in the far-field pattern. In  , the truncated spherical near-field data are used to the spherical wave coefficients using forward hemisphere data only, based on a least-squares technique. This method can significantly reduce truncation errors, but it is not efficient for large antennas. This problem is also addressed in BIB006 where the band-limited property of the spherical wave coefficients is exploited in an iterative algorithm that substitutes the unreliable portion of the measurement sphere with new samples at each iteration. In the present work, a method to reduce truncation errors when measuring the field over truncated surfaces is developed. The method is based on the iterative algorithm that was proposed in BIB001 and (Papoulis, 1974) , and it has already been applied to the PNF case in BIB005 applied to any scanning geometry by taking certain considerations into account. Therefore, this work can be viewed as a generalization of that method, although in the proposed method it is not necessary to take samples in surfaces external to the actual scanning area. Therefore the reduction of truncation errors is achieved without increasing the measurement time. A bottleneck of the method presented in BIB006 may be the time required to find the optimum termination point in the iterative procedure. In this work, a faster procedure based on the Gradient Descent algorithm, with which is possible to obtain the iteration number where the error is minimum, is proposed. Before describing all of the steps of the method, the two orthogonal projection operators that play an important role in the method are presented. The first operator is applied in the spectral domain and defines the reliable portion of the PWS. This first operator is given by where A is the spectral-truncation operator and Ω R is the reliable region. The second operator is called band-limited operator and is applied to the field distribution on the AUT plane. where h stands for the band-limited operator and w AUT is the region where the AUT is located. In the spectral domain, this last operator can be expressed as follows being B the band-limited operator defined in the spectral domain, P(k x ,k y ) represents the PWS and H is the inverse Fourier transform of the operator h. The schematic diagram of the method is shown in Fig. 16 . Fig. 16 . Diagram of the method to reduce truncation errors.