In this paper, an odd-order probe for spherical near-field antenna measurements is defined. A probe correction technique for odd-order probes is then formulated and tested by computer simulations. The probe correction for odd-order probes is important, since a wide range of realistic antennas belongs to this class. To the authors' knowledge, the proposed technique is the first practical high-order probe correction technique that has been formulated in detail, has been tested, and has been shown to work.
 The spherical near-field measurement is an extensively used method for accurate characterization of the radiated fields of antennas [Hansen, 1988]. The near field of the antenna under test (AUT) is measured in a number of points on a spherical surface enclosing the AUT by a probe. In the near field of the AUT, the ratio between the signal received by the probe and the incident tangential field at the probe is not constant but a function of the measurement direction. This ratio, that is necessary to know for an accurate far-field determination [Yaghjian, 1986], is inherently found by applying a probe correction.
 The probe correction for spherical near-field measurements was first formulated by Wacker [1974, 1975] and Jensen . Since then this probe correction technique has become a generally accepted technique [Hansen, 1988]. Several suggestions for other probe correction techniques have been later presented by Larsen , but to the authors' knowledge none of these techniques have been formulated in detail, reportedly tested nor reportedly shown to work. It is noted that, according to Hansen [1988, chapter 4], one of the suggested techniques for the probe correction, the straightforward method of forming a system of linear equations and numerically inverting the matrix, is inefficient and not practical.
 The well-known probe correction [Hansen, 1988] is practical for first-order (μ = ±1) probes for which the azimuthal variation of the receiving pattern of the probe is restricted to a simple sine-cosine variation [Wacker, 1974, 1975; Jensen, 1975]. In this case, the full-sphere measurement of the AUT must be performed only twice, with different orientations of the probe. Moreover, if the probe is dual polarized, the probe rotation may be fully avoided, and one full-sphere measurement of the AUT is then sufficient. This probe correction technique is in principle also applicable to probes which are not just first-order probes and which thus possess patterns with more complicated azimuthal variations. This requires, however, that the full-sphere measurement of the AUT be performed for more than two orientations of the probe, and it thus leads to an overall longer measurement time. Since the measurement time is generally the limiting factor in practical antenna measurement projects, this is highly undesirable.
 A typical example of a first-order probe is a conical horn fed by a circular-cylindrical waveguide [Larsen and Hansen, 1979]. These probes have been used extensively for spherical near-field antenna measurements for years [Hansen, 1988]. Because of a relatively narrow bandwidth of the circular-cylindrical waveguide and the need for antenna measurements over a wide frequency range, a large set of probes is required. Conical horns may also be impractically heavy and large at low frequencies.
 The purpose of this paper is to introduce a specific high-order probe correction technique, namely an odd-order probe correction technique, for spherical near-field antenna measurements. An odd-order (μ odd) probe is defined and the probe correction technique for odd-order probes is formulated. As is the case with the existing first-order probe correction technique [Hansen, 1988], the odd-order probe correction technique presented here requires only two probe orientations during the measurement.
 Odd-order probes, in particular, are important for antenna measurements, because a wide and significant range of realistic antennas belongs to this class. A good example of an odd-order probe is an open-ended rectangular waveguide operating with the TE10 waveguide mode.
 In section 2 the background theory is presented and an odd-order probe is defined. In section 3 the probe correction technique for odd-order probes is derived. Section 4 documents the validation of the odd-order probe correction technique. Finally, conclusions are given in section 5.
2. Background Theory
2.1. Spherical Measurement Configuration
 For the purpose of this work, it is not necessary to describe the full physical configuration of a spherical near-field measurement [Hansen, 1988, chapter 3]; we need only to consider the two coordinate systems involved and the two minimum spheres enclosing the AUT and the probe, respectively, as shown in Figure 1.
 The coordinate systems of the AUT and the probe are the unprimed and the primed coordinate systems, respectively. The origin of the primed coordinate system in the unprimed coordinate system is defined by the (r, θ, ϕ) coordinates of the standard spherical coordinate system [Institute of Electrical and Electronics Engineers, 1979]. The positive z′ axis coincides with the r unit vector of the unprimed spherical coordinate system. The χ angle is the probe rotation angle. For χ = 0° and χ = 90° the x′ axis coincides with the θ and ϕ unit vectors of the unprimed coordinate system, respectively. The radii of the minimum spheres of the AUT and probe are r0 and r′0, respectively. For this work, it is further required that these two spheres do not intersect, thus r > r0 + r′0 [Hansen, 1988, chapter 3].
2.2. Spherical Wave Expansion
 The radiated field of an AUT outside its minimum sphere may be expressed in terms of a series of orthogonal spherical vector wave functions [Hansen, 1935, 1988, chapter 2]. The truncated form of the power-normalized spherical vector wave expansion of the radiated electric field of an AUT is
where E is the radiated electric field, k is the wave number, η is the intrinsic admittance, v is the input signal at the port of the AUT, Tsmn are the transmission coefficients of the AUT, Fsmn(3)(r, θ, ϕ) are the power-normalized spherical vector wave functions, and N is the truncation number for the spherical wave series [Hansen, 1988, chapter 2]. The assumed time dependence of the electromagnetic field is e−iωt, where ω is the angular frequency, and t is time. These notations apply for the AUT coordinate system.
 Correspondingly, the radiated electric field of the probe outside its minimum sphere in the primed coordinate system may be expressed as
The symbols of equation (2) are analogous to equation (1); νmax is the truncation number, (r′, θ′, ϕ′) are the spherical coordinates of the primed coordinate system, E (r′, θ′, ϕ′) and Fσμν(3)(r′, θ′, ϕ′) are the radiated electric field and the spherical vector wave functions in the primed coordinate system, respectively, and Tσμν are the transmission coefficients of the probe. The notation of equations (1)–(2) follows Hansen .
 It is a practical rule that the spherical wave series for the AUT field may be truncated at [Hansen, 1988, chapter 2]
where [kr0] is the smallest integer greater than or equal to kr0. Naturally, the same practical rule applies for the probe as follows:
 The power normalization of the spherical vector wave expansion means that the total radiated power (of the AUT) is
where Qsmn are referred to as the Q coefficients of the AUT [Hansen, 1988, chapter 2]. The Q coefficients of the AUT are defined as
 For later use we now define two types of power spectra for the spherical wave expansion of the field radiated by the AUT [Hansen, 1988, chapter 6]. The n mode power spectrum, Prad(n), expresses the power of all the spherical wave modes with the same index n for n = 1…N. From equation (5) the n mode power spectrum thus is
 The m mode power spectrum, Prad(m), expresses the power of all the spherical wave modes with the same ∣m∣ for m = 0…N, respectively. The m mode power spectrum thus is
where a = 4 for m = 0, and a = 2 for m > 0. The maximum values of the corresponding normalized power spectra, Prad,norm(n) and Prad,norm(m), are set to 0 dB.
Equations (5)–(8) become valid for the probe by replacing the indices (s, m, n) by indices (σ, μ, ν), and by replacing N by νmax. The power spectra for the probe are referred to as the ν mode power spectrum, Prad(ν), and the μ mode power spectrum, Prad(μ), respectively. The corresponding normalized power spectra for the probe are Prad,norm(ν) and Prad,norm(μ).
 For later use, it is also necessary to define the receiving coefficients (R coefficients) of the probe. For a reciprocal probe, which is assumed for simplicity in this paper, the relationship between the R coefficients (Rσμν) and the transmission coefficients is Rσμν = (−1)μTσ−μν [Hansen, 1988, chapter 2].
2.3. Transmission Formula
 The probe-corrected transmission formula, henceforth the transmission formula, is the starting point for performing the probe correction in spherical near-field antenna measurements. The transmission formula for spherical coordinates, that was first derived by Jensen  and later presented in a modified form by Wacker  and Jensen , expresses the signal received by the probe as a function of the Q coefficients of the AUT, the probe location and orientation relative to the AUT, and the R coefficients of the probe. An alternative transmission formula has been presented later by Yaghjian and Wittmann . In this paper we have, however, chosen to apply the Jensen-Wacker transmission formula [Wacker, 1974; Jensen, 1975], because of its complete formulation by Larsen  and Hansen .
 The Jensen-Wacker transmission formula for the signal received by the probe is
where w is the complex valued signal received at the port of the probe, Psμn (kr) are the probe response constants, and eiμχ, dμ mn (θ), and eimϕ are the rotation coefficients [Hansen, 1988, chapter 3]. The summation is carried out for s = 1, 2, n = 1…N, m = −n…n, and μ = −μ0…μ0, where μ0 = min(n, νmax). The formulas for dμmn (θ) are given in [Hansen, 1988, Appendix A2]. The probe response constants are
where Cσμνsn(3)(kr) are the translation coefficients of the spherical vector wave functions, and Rσμν are the R coefficients of the probe. The formulas for the translation coefficients are given by Hansen [1988, Appendix A3].
 In the spherical near-field antenna measurement, the only unknowns of the transmission formula are the Q coefficients of the AUT. The R coefficients of the probe are known from a separate probe calibration.
2.4. Odd-Order (μ Odd) Probes
 Probes for spherical near-field antenna measurements may be classified according to their R coefficients defined in section 2.2. First-order (μ = ±1) probes constitute a well-known class of probes [Hansen, 1988]. For a first-order probe the R coefficients for μ ≠ ±1 are zero [Hansen, 1988, chapter 3]. In this paper we define a high-order probe to be a probe that possesses nonzero R coefficients for ∣μ∣ > 1. Odd-order probes constitute an important subclass of high-order probes. We define an odd-order probe to be a probe that possesses nonzero R coefficients only for odd values of μ.
 Practically, odd-order probes are very significant, because many typical antennas, that are used as probes in antenna measurements, belong to the class of odd-order probes. For example, a current distribution, which is symmetric about a symmetry plane and antisymmetric about a perpendicular plane, so that the intersection of the two planes coincides with the z′ axis, creates a radiated field with nonzero Q coefficients only for odd μ [Hansen, 1988, Appendix A1]. Thus reciprocal probes exciting this type of a current distribution are odd-order probes.
 An open-ended rectangular waveguide, operating with the TE10 waveguide mode, is a widely used probe and a good example of an odd-order probe. The symmetry and antisymmetry planes of the current distribution of a rectangular waveguide, operating with the TE10 mode, are illustrated in Figure 2a. The radiated field of a standard X band rectangular waveguide, operating with the TE10 mode, has been calculated with the Ansoft HFSSTM software by using perfect magnetic and electric surfaces in place of the symmetry and antisymmetry planes of the current distribution. In Figure 2b the normalized μ mode power spectrum of the simulated radiated field is presented. It is seen that the spectrum consists of several modes with odd μ, while the power of the even μ modes is zero.
3. Odd-Order Probe Correction Technique
 The probe correction technique for odd-order probes will now be presented. The aim is to solve the transmission formula (9) for the Q coefficients assuming an odd-order probe. In doing so, the probe correction is also accomplished since the R coefficients of the probe are taken into account in the transmission formula.
 In section 3.1 we first explain how the work presented in this paper is related to earlier reported work. Also a brief introduction to the odd-order probe correction is given. Section 3.2 describes the sampling scheme assumed in this paper for the odd-order probe correction. Sections 3.3 and 3.4 present the odd-order probe correction equations.
3.1. Brief Introduction
Larsen  presents various ideas for solving the transmission formula, but an actual formulation of the probe correction is presented only for first-order probes. It is clear that many of the suggestions have severe practical or computational limitations, and it is not the purpose of this paper to make a full comparison of the techniques presented by Larsen  with the technique introduced in this paper. However, some issues relevant for this paper are discussed in the following.
 For computational reasons, the application of the discrete Fourier transform (DFT), or the fast Fourier transform, of the sampled field to the greatest possible extent is useful [Jensen, 1975; Hansen, 1988]. The obvious first step, without making any assumptions about the probe, is to exploit the orthogonality of the eimϕ term of the transmission formula (9) and to use a DFT of the sampled field in ϕ [Hansen, 1988, chapter 4]. In doing so, the transmission formula is divided into a number of equations independent of ϕ. In the second step, provided that the field is sampled only for two χ values of the probe, either the matrix method or a DFT of the sampled field in θ is applied for the remaining equations [James and Longdon, 1969; Jensen, 1975; Larsen, 1980]. The application of the matrix method allows the use of almost an arbitrary high-order probe. This method, that applies a DFT of the sampled field in ϕ and a matrix method of the sampled field in θ, the so-called DFT-matrix method, may be shown to lead to inversions of full matrices of dimension of the order 2N × 2N at maximum. This method has not been shown to work. On the other hand, if the probe is restricted to a first-order probe, the exploitation of the orthogonality of the dμmn(θ) term of the transmission formula (9) and the subsequent use of a DFT of the sampled field in θ is possible [Hansen, 1988, chapter 4]. This method leads to 2 × 2 matrix inversions for the Q coefficients and constitutes a stable and a computationally efficient method for the probe correction [Hansen, 1988, chapter 4].
 In this paper, instead of applying the orthogonality of the dμmn(θ) term in the second step, the dμmn(θ) is expanded in a Fourier series. Performing the DFT of the sampled field in θ is now possible for odd-order probes. It turns out that after the second DFT the problem is reduced to inversion of matrices of dimension of the order 2N × 2N at maximum. However, the difference from the DFT-matrix method discussed above is that now the largest matrices to be inverted are not full matrices but of block upper triangular type, which is likely to offer computational advantages for solving the corresponding matrix equation. A comparison of the technique introduced in this paper with the above mentioned DFT-matrix method is the subject of a future publication.
 The necessary equations for the probe correction are different for different sampling schemes. In general, for each value of θ the samples should be taken with a constant ϕ step, δϕ, so that 2π/δϕ is a positive integer. Similarly the θ step, δθ, must be constant so that 2π/δθ is a positive integer. These requirements are due to the application of the DFT for the measured signal. It should also be noted that samples are taken for different χ values as illustrated in Figure 1.
 Although it may be possible to choose either an even or odd number of samples in θ as well as in ϕ, it is assumed in this paper that δϕ = δθ, and that 2π/δϕ is a positive, odd-valued integer. Therefore the first and the last θ and ϕ angles, where the samples are taken, are θ = 0 and θ = π − δθ/2, and ϕ = 0 and ϕ = 2π − δϕ, respectively. For a fixed θ the number of samples in ϕ is 2N + 1, and for a fixed ϕ the number of samples in θ is N + 1. This sampling grid for δϕ = δθ = 40° is illustrated in Figure 3.
 The formulas are derived for a case where a dual-polarized probe with identical ports is used in the measurements. This choice also corresponds to the case where a single-port probe is used, and the field is sampled for both χ = 0° and χ = 90°, thus with two full-sphere measurements of the AUT. Yet, it is clear that a similar formulation for the probe correction could be carried out for a dual-polarized probe with nonidentical ports. The measurement distance will be considered later.
3.3. Fourier Transforms
 We will now derive a two-dimensional Fourier spectrum of the received signal of the probe. By interchanging the n and m summations of the transmission formula (9), this can be expressed as
where the summation is now for m = −N…N, s = 1, 2, n = n0…N, and μ = −μ0…μ0, where
Equation (11) may be expressed in terms of a finite Fourier series,
where the Fourier coefficients for −N ≤ m ≤ N are
 Since the function w(r, χ, θ, ϕ) of equation (13) is bandlimited and periodic in ϕ, with the period of 2π, the Fourier coefficients can be determined by the inverse DFT of the equidistantly sampled signals in ϕ. For each value of θ, the number of samples in ϕ is 2N + 1. Thus wm(r, χ, θ) has been now found.
 By inserting the Fourier expansion of the rotation coefficient [Hansen, 1988],
 Δmμn in equation (15) is defined by Hansen [1988, Appendix A2]. According to the sampling scheme presented in section 3.2, the values of wm(r, χ, θ) are available only for certain discrete values of θ for 0 ≤ θ < π. The number of samples in θ is N + 1. In order to apply a DFT here, the data series must be made periodic, with the period of 2π, by a proper data extension for π < θ < 2π [Hansen, 1988, chapter 4]. We carry out the data extension in a similar manner as in chapter 4 of Hansen , and thus the extended wm(r, χ, θ) is
 Noticeably, the extended data is dependent on μ, and thus not readily available. However, as we have confined ourselves to using an odd-order probe, which means that (−1)μ = −1 on the right-hand side of equation (18), the extended data becomes independent on μ and is thus known. The extended data, m(r, χ, θ), now contains 2N + 1 equidistant samples in θ in the interval 0 ≤ θ < 2π. Equation (16) may now be written, by using m(r, χ, θ) instead of wm(r, χ, θ), as follows:
where the Fourier coefficients for −N ≤ m ≤ N and for −N ≤ m′ ≤ N are
The coefficients m′m(r, χ) may be found by the inverse DFT. The useful inverse DFTs have been now performed.
3.4. Solution for Q Coefficients
 For each fixed m (−N ≤ m ≤ N), equation (20) is solved for a set of Q coefficients (Q,Q…Q1mN, Q2mN), where n0 is given by equation (12), as follows: A matrix equation
is formed from equation (20), where = [Q, Q…Q1mN, Q2mN]T and = [1m(r, 0°), 1m(r, 90°)…Nm(r, 0°), Nm(r, 90°)]T. A block matrix is formed separately for each m as follows:
where N′ = N − n0 + 1. The matrices m′n′, for n′ = 1…N′ and for m′ = 1…n′ + n0 − 1, are 2 × 2 matrices as follows:
where l = n′ + n0 − 1. m′n′ are zero matrices for m′ > n′ + n0 − 1. Note that in the calculation of equation (17) the μ summation is now from −min(l, νmax) to min(l, νmax). The Q coefficients of the same index m are now found by a pseudoinverse operation [Campbell and Meyer, 1991] from
where T denotes the transpose of a matrix. In this paper the pseudoinverse command, pinv, of MatLAB™ is used.
Equations (21)–(24) provide just one possibility for solving equation (20) for the Q coefficients, and because of the fact that the matrices to be inverted are not full matrices more efficient solutions may also exist. The method described here involves inversions and pseudoinversions of matrices with dimensions ranging from 2 × 2N to 2N × 2N. For ∣m∣ > 1 the matrix equation is overdetermined and may be solved by a pseudoinverse operation. For ∣m∣ ≤ 1 the matrices to be inverted are of block upper triangular type with dimension 2N × 2N. The method provides an exact solution, only limited by the numerical accuracy of the computation, for the Q coefficients in the noise-free case, where the radiated fields of the AUT and the probe can be expressed by spherical wave series truncated at N and νmax, respectively, and an ideal odd-order probe is used.
 Furthermore, it should be noted that we make use of the Fourier coefficients m′m for 1 ≤ m′ ≤ N, but the values of m′m for m′ = 0 could be used as well. Since the Fourier spectrum m′m is symmetric along m′ = 0, the use of negative values of m′ does not provide any useful additional information.
 This section describes a simulated measurement which is performed to validate the odd-order probe correction technique. First, it is necessary to calculate the signal received by a probe model from the transmission formula (9). The required inputs to the transmission formula are the Q coefficients of an AUT model and the R coefficients of the probe model. Once the received signal is known, the inverse operation, namely the probe correction, may be carried out to validate the new probe correction.
 In sections 4.1 and 4.2 the AUT model and the probe model are described, respectively. In section 4.3 the simulated measurement and the probe correction are described. In section 4.4 results of the simulation are presented.
4.1. AUT Model
 The required set of Q coefficients for the AUT model (AUT) will now be determined on basis of the AUT model illustrated in Figure 4. The AUT model is an array of four linearly x-polarized Huygens sources [Hansen, 1988, chapter 2]. The (x, y, z) coordinates of the four Huygens sources are (1.75λ, −0.25λ, 0), (2.25λ, −0.25λ, 0), (1.75λ, 0.25λ, 0), and (2.25λ, 0.25λ, 0), respectively, where λ is the wavelength. Each Huygens source consists of an x-oriented electric Hertzian dipole and a y-oriented magnetic Hertzian dipole situated at the same location. The excitations of these two dipoles ensure that their electric fields are in phase in the +z axis direction while they cancel each other in the −z axis direction. Again, the electric fields of the four Huygens sources are in phase in the +z axis direction.
 A simulated measurement for this AUT model represents a situation where, for example, an element pattern of an antenna array is measured. Normally, it is desirable to mount an AUT in such a way that the AUT is located near the origin of the unprimed coordinate system to minimize the minimum sphere of the AUT. However, since it is not desirable to mount and dismount an array between each element pattern measurement, the actual radiating element may be located with a significant offset from the origin of the unprimed coordinate system [Pivnenko et al., 2003].
 The radiated field of the AUT model is now calculated using the well-known formulas for electric and magnetic Hertzian dipoles [Hansen, 1988, chapter 2], and it is expanded into spherical wave modes using the truncation number N = 25 obtained from equation (3). The expansion is performed by first assuming an x′ oriented electric Hertzian dipole sampling the radiated electric field on the surface of a sphere. The signal received by the electric dipole for χ = 0° and χ = 90°, that is known to be proportional to the θ and ϕ component, respectively, of the electric field by a constant factor [Hansen, 1988, chapter 3], is calculated from the known radiated field. The probe correction technique of Hansen [1988, chapter 4] is then applied to solve for the Q coefficients from the transmission formula using the electric Hertzian dipole as a probe. The obtained Q coefficients constitute the desired set AUT. The radiated field related to AUT by equations (1) and (6) is designated by EAUT. It should be noted at this point, that since the spherical wave expansion for the AUT model is truncated at N = 25, the radiated field is not equal to the exact field of the AUT model but it constitutes a very accurate approximation. In other words, the AUT model has changed from the configuration of Figure 4 to the truncated spherical wave expansion of that configuration.
 The co-polar and cross-polar directivities, according to definition 3 of Ludvig , of the field EAUT are calculated from AUT [Hansen, 1988, chapter 2]. The dBi values of the co-polar and cross-polar directivities in the ϕ = 0° (x-z plane), ϕ = 45° and ϕ = 90° (y-z plane) planes are presented in Figure 5. Ideally, the cross-polar directivity of the AUT model presented in Figure 4 should be zero. The nonzero cross-polar directivity of EAUT in the ϕ = 0° plane is likely due to numerical inaccuracy in the calculations. Instead, the nonzero cross-polar directivity of EAUT in the ϕ = 45° and ϕ = 90° planes are dominantly caused by the truncation.
 The normalized n mode and m mode power spectra of EAUT, calculated from equations (7)–(8), are presented in Figures 6 and 7, respectively. It is seen from Figures 6 and 7 that the radiated power of EAUT in the modes for n = 25 and m = 25, respectively, is less than −80 dB relative to the maximum power of the corresponding spectrum.
4.2. Probe Model
 The required set of R coefficients for a probe model will now be determined. The probe model, shown in Figure 8, is a dipole array consisting of 12 electric Hertzian dipoles both for the vertical and the horizontal polarization of the probe. The dipoles that are parallel with the x′ axis form the first port (port I) of the probe and those parallel with the y′ axis form the second port (port II) of the probe. The excitations of the dipoles with coordinates z′ = 0 and z′ = λ/4 are 1 and −i, respectively. The fields radiated by both ports of the probe model have the maximum directivities in the −z′ axis direction.
 It is only necessary to consider the radiated field of port I of the probe, because the two ports of the probe are identical and the derivations presented in section 3 are based on the assumption of identical ports. The radiated electric field of port I of the probe is first determined from the conventional equations for the electric Hertzian dipole [Hansen, 1988, chapter 2]. The field is expanded into spherical wave modes in a similar manner as described for the AUT model in section 4.1. The truncation number νmax = 19 is obtained from equation (4). Thus a set of Q coefficients (probe) for the port I of the probe is obtained.
 The normalized ν mode and μ mode power spectra of the radiated field of the probe port I are now determined from probe as explained in section 2.2, and presented in Figures 9 and 10, respectively. It is seen from Figure 9 that the power for ν > 15 is less than −75 dB relative to the maximum value of the corresponding spectrum. The power for μ > 15, shown in Figure 10, is less than −80 dB relative to the maximum value of the corresponding spectrum. The power in modes with even μ is zero. As explained in section 2.4, this is, because the currents of the probe port I are symmetric about the symmetry plane (x′-z′ plane) and antisymmetric about the antisymmetry plane (y′-z′ plane). The probe is thus an odd-order probe.
 The necessary set of R coefficients (probe) for port I of the probe may now be determined from the known probe. We have already determined probe for νmax = 19. However, for the calculation of R coefficients, we further truncate the spherical wave series at ν = 15, since spherical wave modes for ν > 15 are insignificant for the purposes of this study. Thus σμν are now calculated up to ν = 15 from probe according to section 2.2 by using ν = 1 for the input signal of the probe. The desired set of R coefficients is denoted probe.
4.3. Simulated Measurement and Probe Correction
 The signal received by the probe model is now determined from the transmission formula (9) with N = 25. The set of Q coefficients of the AUT and R coefficients of the probe required in equation (9) are AUT and probe, respectively. The (θ, ϕ, χ) angles are determined by the sampling scheme described in section 3.2, with δϕ = δθ = 2π/(2N + 1). The simulated measurement distance is 20λ.
 With the known received signal, the probe correction presented in sections 3.3 and 3.4, starting from equation (11), is applied. In the probe correction the truncation numbers N = 25 and νmax = 15 are used. First, equations (11)–(20) presented in section 3.3 are applied to form the two-dimensional Fourier spectrum from the measured signal. Equations (21)–(24) in section 3.4 are then used to solve for the set of Q coefficients of the AUT model. The Q coefficients obtained from equation (24) are referred to as the predicted Q coefficients, and this set is denoted pred.
 We now compare the predicted set of Q coefficients, pred, with the correct set of Q coefficients, AUT, by introducing the following relative error:
where Qj,pred and Qj,AUT are the elements of pred and AUT, respectively. The index j = 2(n(n + 1) + m − 1) + s. For n = 1…N, for m = −n…n, and for s = 1, 2, with N = 25, the index j assumes the values j = 1…1350. The result of the comparison is that the maximum value of δj = 1…1350 is less than −280 dB.
 In the following we will show how this error affects the directivity of the AUT model. The predicted (Epred) and the correct (EAUT) electric far fields are first calculated from pred and AUT, respectively using equation (1). The co-polar and cross-polar directivities of Epred and EAUT are then calculated [Hansen, 1988, chapter 2] and denoted Dco,pred, Dco,AUT, Dcross,pred and Dcross,AUT. Here we use again the definition 3 of Ludvig  for the co-polar and cross-polar directivities.
 The absolute values of the differences between the predicted and correct co-polar directivities in the ϕ = 0°, ϕ = 45° and ϕ = 90° planes are presented in Figure 11. For comparison the correct co-polar directivities in the ϕ = 0°, ϕ = 45° and ϕ = 90° planes are also presented in Figure 11. The presented values are Dco [dBi] = 10 log10(Dco,AUT) and Δco[dB] = 10 log10(∣Dco,AUT − Dco,pred∣). It is seen that the absolute values of the differences are at a level of approximately −140 dB or less relative to the maximum directivity.
 The absolute values of the differences between the predicted and correct cross-polar directivities in the ϕ = 0°, ϕ = 45° and ϕ = 90° planes are presented in Figure 12. The correct cross-polar directivities in the ϕ = 0°, ϕ = 45° and ϕ = 90° planes are presented for comparison. The presented values are Dcross [dBi] = 10 log10(Dcross,AUT) and Δcross [dB] = 10log10(∣Dcross,AUT − Dcross,pred∣). It is seen that the differences are at a level of approximately −100 dB at maximum relative to the maximum value of the corresponding cross-polar directivity in the ϕ = 45° and ϕ = 90° planes. The difference between the correct and predicted cross-polar directivity in the ϕ = 0° plane is at a level of approximately −300 dB.
 In general, Figures 11 and 12 show that the errors between the predicted and correct directivities are insignificant for most practical purposes. The proposed method was also tested for several other AUTs and probe models and this showed similar excellent results in all cases.
 The probe correction in spherical near-field measurements has been mainly, if not exclusively, restricted to the well-known probe correction presented by Hansen . This probe correction is practical only for a first-order probe, since for more general probes it would require a probe rotation into more than two orientations at each measurement direction. This probe correction technique has been known and used for first-order probes for about 30 years now [Wacker, 1974, 1975; Jensen, 1975]. Several suggestions for probe correction of more general probes have been presented by Larsen  but, to the authors' knowledge, none of these techniques have been fully formulated, reportedly tested nor shown to work in practice.
 In this paper we have defined an odd-order probe and formulated a probe correction technique for odd-order probes for spherical near-field antenna measurements. This technique is different from any of the suggestions of Larsen . To the authors' knowledge the proposed technique is the first practical high-order probe correction technique that has been formulated in detail, tested and shown to work. Preliminary investigations have shown that the method provides sufficient accuracy also in the presence of noise. The described technique is practical, because it requires only two orientations for the probe during the measurement. Moreover, if the probe is dual polarized, the probe rotation is avoided.
 Odd-order probes are very important for antenna measurements in general because a wide range of practical probes belong to this class. The introduction of a practical probe correction technique for odd-order probes offers new possibilities for probe design for accurate spherical near-field antenna measurements. Rotationally symmetric geometrical structures are typically utilized to create a field that fulfils the requirements of a first-order probe. A greater number of geometrical structures may be considered for the design of an odd-order probe compared to the design of a first-order probe. This naturally provides wider range of possibilities for wideband probe designs. In general, the odd-order probe correction technique may have a significant impact on the applied measurement procedures at spherical near-field antenna measurement facilities.
 We want to acknowledge J. M. Nielsen for providing help for the verification of the correctness of the computer codes used for this paper and F. Jensen and A. Frandsen for their useful comments. We want to acknowledge TICRA Foundation and H. C. Ørsted Foundation for their financial support.