Sectorized approach and measurement reduction for mutual coupling calibration of non-omnidirectional antenna arrays

Authors


Corresponding author: T. Aksoy, Department of Electrical and Electronics Engineering, Middle East Technical University, E Building 102, 06800, Çankaya, Ankara, Turkey. (taylanaksoyy@gmail.com)

Abstract

[1] Mutual coupling calibration is an important problem for antenna arrays. There are different methods proposed for omnidirectional antennas in the literature. However, many practical antennas have non-omnidirectional (NOD) characteristics. Hence, the previous mutual coupling calibration methods cannot be applied directly since the mutual coupling matrix of an NOD antenna array has angular dependency. In this paper, a sectorized approach is proposed with a transformation matrix for mutual coupling calibration of NOD antenna arrays. In addition, the symmetry of the array radiation patterns for the symmetric array elements is used to reduce the number of calibration measurements. A novel antenna is used to show the accuracy and performance of the proposed approach in direction finding problem where numerical electromagnetic simulations are used to obtain the simulation data.

1 Introduction

[2] In antenna arrays, mutual coupling between the array elements is an important problem. An extensive review on array mutual coupling analysis is presented in Craeye and Gonzalez-Ovejero [2011] where the relationship between array impedance matrix and embedded element patterns is considered. An antenna array cannot function properly without a proper mutual coupling calibration. In Camps et al. [1998], this is shown for the case of interferometric radiometers through a detailed theoretical analysis of mutual coupling effect which is also supported with experimental results.

[3] In the literature, there are several methods for mutual coupling calibration which consider omnidirectional antenna elements. In Hui [2007], a review of the mutual coupling modeling methods is presented. The open-circuit method presented by Gupta and Ksienski [1983] is one of the earliest methods for mutual coupling modeling. In this method, the N-element antenna array is treated as an N-port network, and the antenna terminal voltages are related to the so-called open-circuit voltages through an impedance matrix consisting of mutual impedances. However, it is insufficient to model the mutual coupling effect completely since the scattering effect due to the presence of other antenna elements is not considered. In Hui [2003; 2004], the receiving mutual impedance concept is proposed to overcome the problems of the open-circuit method. The receiving mutual impedances are calculated by considering the antenna elements in pairs. The calculation method is further developed in Lui and Hui [2010] where all the array elements are considered simultaneously.

[4] Previous works usually use dipole and monopole antenna arrays since mutual coupling characterization is relatively easier. In practice, many antennas are non-omnidirectional (NOD) or at least their patterns change with frequency destroying the omnidirectional pattern. In Mir [2008], multiple calibration matrices are used using a series-like expansion in order to account for directional dependency of array manifold calibration.

[5] In this paper, a new method for mutual coupling calibration of arrays composed of identical NOD antennas is proposed. The proposed approach takes into account the directional dependency of mutual coupling matrix and presents a systematic approach for its treatment. It also presents a way for reducing the number of calibration measurements by introducing the symmetry plane definition which depends on the symmetry of the array radiation patterns for the symmetric array elements.

[6] Mutual coupling matrix for an NOD antenna array has azimuth, elevation, and frequency dependency in general. Therefore, a single mutual coupling matrix cannot characterize the mutual coupling effect adequately. In this paper, a sectorized approach is proposed for mutual coupling calibration of NOD antenna arrays. Azimuth and elevation planes are divided into uniform angular sectors, and a different mutual coupling matrix is found for each sector using a transformation matrix. The size of the angular sectors should be selected appropriately in order to decrease the calibration complexity as well as to avoid ill-conditioned matrices.

[7] The manual labor required for mutual coupling calibration measurements is an important factor for practical implementations. In Aksoy and Tuncer [2012], a measurement reduction method (MRM) is proposed for omnidirectional antennas where the number of calibration measurements are reduced by using the symmetry in the array geometry. In this paper, the definition of symmetry plane is presented, and it is shown that MRM is also valid for NOD antenna arrays. MRM is based on the symmetry planes which are determined by the symmetry of the array radiation patterns for the symmetric array elements. This symmetry depends on the array and antenna geometries as well as the antenna radiation patterns and alignments. The amount of measurement reduction for NOD antennas is usually less than the omnidirectional antennas. However, it is still significant and reduces the cost and complexity of the array calibration process.

[8] The sectorized approach combined with MRM is evaluated through direction-of-arrival (DOA) estimation experiments using Multiple Signal Classification (MUSIC) algorithm [Schmidt,1986]. Calibration measurements are obtained using the numerical electromagnetic simulation tool FEKO [User's Manual,2008]. Several simulations are provided to show that the proposed approach is an effective way of mutual coupling calibration of NOD antenna arrays.

2 Signal Model

[9] In this work, antenna arrays composed of identical NOD elements are considered in a noise-free environment, and narrowband model is used. In this case, the output vector due to an excitation source from (φl,θm) direction for an N-element antenna array is given as

display math(1)

where P is the number of snapshots and Clm is the N × N mutual coupling matrix for the azimuth, φl , and elevation, θm, angles. The N × N diagonal matrix Γlm = diag{γlm,0,γlm,1, … ,γlm,(N − 1)} stands for the gain/phase mismatches due to non-omnidirectional antenna radiation patterns for (φl,θm) direction where any gain/phase mismatch due to cabling and instrumentation is assumed to be perfectly calibrated and ignored. a(φl,θm) = [α0(φl,θm) α1(φl,θm) … αN − 1(φl,θm)]T is the steering vector for (φl,θm) direction where ( · )T denotes transposition. The vector element corresponding to the Nth antenna positioned at (xn,yn,zn) is math formula. s(t) is the complex baseband source signal.

[10] Since the array elements are identical, gain/phase mismatch terms due to antenna radiation patterns from a direction (φl,θm) for all of the array elements are the same, i.e., γlm,0 = γlm,1 = ⋯ = γlm,(N − 1) = γlm. Hence, (1) reduces to

display math(2)

In this case, the MUSIC algorithm estimates the DOA angles as the maxima of the following spectrum,

display math(3)

where ( · )H denotes the Hermitian transpose. G is the matrix whose columns are the noise eigenvectors obtained from the singular value decomposition of the sample covariance matrix, math formula.

[11] Note that if the correct Clm matrix is not supplied, MUSIC algorithm generates large errors in DOA estimation. Moreover, Clm changes with the azimuth and elevation angles for NOD antenna arrays. Fortunately, antenna radiation patterns for most of the antenna types are smooth and slowly changing functions of azimuth and elevation angles. Therefore, Clm can be assumed to remain the same for sufficiently small angular sectors. Hence, a sectorized calibration approach seems to be a natural choice. In this paper, a sectorized approach is proposed for the mutual coupling calibration of NOD antenna arrays. Before presenting the sectorized approach, a summary of the transformation approach for mutual coupling modeling [Aksoy and Tuncer,2012] is presented for completeness.

3 Transformation Approach for Mutual Coupling Calibration

[12] Usually, mutual coupling is modeled by using two types of terminal voltages that are measured from the antenna terminals. The first terminal voltage is measured when all the antennas are residing in the array. Therefore, mutual coupling affects the element voltages. The second terminal voltage is measured when only a single array element exists. In this case, there is no mutual coupling. The first terminal voltage is referred as the coupled voltage and the second as the uncoupled voltage. The target in mutual coupling calibration is to determine the transformation matrix which relates the coupled and uncoupled voltages.

[13] Consider an arbitrary N-element antenna array. Assume that the coupled and uncoupled voltages for the Nth antenna due to an excitation source from (φl,θm) direction are denoted as V n(φl,θm) and Un(φl,θm), respectively. Then the coupled voltage vector, v(φl,θm) = [V 0(φl,θm) V 1(φl,θm)  …  V N − 1(φl,θm)]T, can be obtained with a single measurement when all the antennas are residing in the array. Also, the uncoupled voltage vector, u(φl,θm) = [U0(φl,θm) U1(φl,θm)  …  UN − 1(φl,θm)]T, is obtained with N measurements each of which is taken on a single antenna while all the remaining antennas are removed.

[14] In Aksoy and Tuncer [2012], it is shown that the coupled and uncoupled voltages can be related through a linear transformation as follows:

display math(4)

where T is the N × N transformation matrix. T can be obtained using additional observations. Let V = [v(φ1,θm) v(φ2,θm)  …  v(φL,θm)] and U = [u(φ1,θm) u(φ2,θm)  …  u(φL,θm)] be N × L matrices whose columns are the coupled and un-coupled voltage vectors due to single excitation sources from L distinct directions, respectively. For L ≥ N, T can be found as

display math(5)

where ( · ) †  denotes the Moore-Penrose pseudo inverse. The target in array calibration is to find the mutual coupling matrix C. In this case, it can be found as the inverse of the transformation matrix T, i.e., C = T − 1.

4 Sectorized Approach for Non-Omnidirectional Antenna Arrays

[15] Mutual coupling characteristics do not change with the excitation source direction for omnidirectional antennas [Friedlander and Weiss,1991]. This is a consequence of the perfect angular symmetry present in the antenna radiation pattern. In case of NOD antennas, mutual coupling characteristics change with the source direction since the perfect angular symmetry does not exist in the antenna radiation pattern.

[16] In the sectorized approach, the idea is to divide the azimuth plane into angular sectors and obtain a different mutual coupling matrix for each sector, Cd, d = 1,2, … ,D. While different types of angular sectors can be used, uniform and non-overlapping angular sectors are considered in this work. Therefore, the main issue is to properly determine the number of sectors, d, so that the mutual coupling remains approximately the same within each sector.

[17] Choosing d as large as possible may seem to provide a more accurate mutual coupling characterization for each sector. However, in order to find a Cd matrix for a sector, array data for L ≥ N directions residing in that sector are required. Therefore, choosing an unnecessarily large d will increase the total number of measurements and the manual labor. In addition, measurement directions get closer and the V matrix in (5) may become ill-conditioned [Aksoy and Tuncer,2012]. If V becomes ill-conditioned, the resulting Cd matrix will not correctly characterize the mutual coupling for the corresponding sector. Therefore, d should be selected such that it is large enough to obtain a uniform mutual coupling characteristics in a given sector and small enough to avoid ill-conditioned V matrix.

[18] As it can be seen in (5), T (hence, C) changes with U and V. Therefore, d is an implicit function of U and V. When identical array elements are considered, U and V depend on the array and antenna geometries as well as the antenna radiation patterns and alignments. Considering these factors, it is hard, if not impossible, to obtain a closed form expression for d. Therefore, the following iterative procedure is proposed in order to determine d:

  1. Determine a performance criterion, such as the root mean square error (RMSE) of DOA estimation, and start with D = 2.
  2. Find a mutual coupling matrix, Cd, d = 1, … ,D, for each sector by using measurements for N directions which are uniformly separated in each sector.
  3. Make a performance test to see whether Cd, d = 1, … ,D, satisfy the performance criterion or not.
  4. If the performance criterion is satisfied, use the current value of d as the number of sectors. If the performance criterion is not satisfied, increase d by one and return to Step 2.

[19] Sectorized approach is an effective method for calibrating NOD antenna arrays, but calibration data should be collected for each sector separately. In the following part, MRM for NOD antenna arrays is presented in order to decrease the number of calibration measurements significantly.

5 MRM for NOD Antenna Arrays

[20] The measurement reduction method (MRM) is previously proposed for antenna arrays with identical and omnidirectional elements [Aksoy and Tuncer,2012]. It is shown that measurements from symmetric directions can be generated from each other through simple permutations in data vectors. In this part, MRM is shown to be valid for antenna arrays with identical NOD elements as well.

[21] Consider an array with N identical NOD antennas with the array model given in (2). Then the uncoupled voltage vector due to an excitation source from (φl,θm) is the same as the corresponding steering vector up to a complex scaling factor, γlm, i.e.,

display math(6)

where γlm is the gain/phase mismatch due to the non-omnidirectionality of the antennas. Note that γlm changes with source direction, (φl,θm). However, when a proper angular partitioning is applied, we can assume that γlm is constant within each sector, i.e., γlm = γd. Consider the steering matrix in Sector-d, Ad = [a(φ1,θm) a(φ2,θm)  …  a(φL,θm)]. Then

display math(7)

[22] Therefore, Ad can be used instead of Ud in (5). Thus, a Td matrix (hence, a Cd matrix) can be found for each sector using the following equation:

display math(8)

[23] Note that finding T up to a complex scale factor does not pose a problem since a normalized mutual coupling matrix is usually sufficient for calibration. As a result, we do not need to have measurements to obtain Ud, and it is sufficient to take measurements only for Vd. Below, the approach to further decrease the number of measurements for Vd is presented.

[24] Consider an array with N identical NOD antennas, e0,e1, … ,eN − 1, and S symmetry planes, s1,s2, … ,sS as shown in Figure 1. In this figure, antennas are considered as point elements for simplicity. A symmetry plane is defined as the plane which satisfies the following two properties:

  • Array elements should be symmetric with respect to the symmetry plane.
  • Array radiation patterns for symmetric array elements should be symmetric with respect to the symmetry plane.
Figure 1.

An array with N identical NOD antennas, e0,e1, … ,eN − 1, and S symmetry planes, s1,s2, … ,sS.

[25] Symmetry planes are determined by the array and antenna geometries as well as the antenna radiation patterns and alignments. In Figure 1, s1 is a symmetry plane, and e1 and eN − 1 are symmetric elements with respect to s1. Therefore, array radiation patterns for e1 and eN − 1 should be symmetric with respect to s1 as well.

[26] Array radiation pattern for an element is obtained by exciting that element with a source while the other elements are passive. Depending on which array element is excited, array radiation pattern changes due to mutual coupling. In Figure 2a, a uniform circular array (UCA) composed of eight NOD antennas is shown. The array elements are non-symmetric dipole antennas as shown in Figure 3. In Figure 2b, s1 is one of the symmetry planes. Therefore, array radiation patterns for symmetric elements should be symmetric. The numerical electromagnetic simulation of the array in Figure 2a is performed by using FEKO [User's Manual,2008] in order to obtain the array radiation patterns for e1 and e7. These patterns are shown in Figures 4 and 5, respectively. In Figure 4, the three dimensional array radiation pattern for e1 is shown from different view angles. Similarly, Figure 5 shows the array radiation pattern for e7. It can be easily seen that these radiation patterns are symmetric with respect to the symmetry plane s1.

Figure 2.

(a) Eight-element UCA composed of non-symmetric dipole antennas. (b) s1 is one of the symmetry planes.

Figure 3.

The semi-omnidirectional antenna designed in (T. E Tuncer, Nonsymmetric wideband dipole antenna, patent pending, 2011).

Figure 4.

The three-dimensional array radiation pattern for e1 from top, front, and isometric view angles.

Figure 5.

The three-dimensional array radiation pattern for e7 from top, front, and isometric view angles.

[27] The coupled voltage measured at an array element is determined by the array radiation pattern for that array element. Therefore, the coupled voltages measured at two symmetric array elements due to two sources from symmetric directions are equal. Referring back to the case given in Figure 1, the coupled voltage at eN − 1 due to a source from (φ2,θm) direction is equal to the coupled voltage at e1 due to a source from (φ1,θm), i.e., V N − 1(φ2,θm) = V 1(φ1,θm).

[28] Since each array element has a symmetric element with respect to s1, v(φ2,θm) can be obtained from v(φ1,θm) through data permutations. In Figure 1, a similar fact is true for the remaining S − 1 directions, (φ4,θm),(φ6,θm), … ,(φ2S,θm), that are symmetrical to (φ1,θm) with respect to s2,s3, … ,sS, respectively. In other words, v(φ4,θm),v(φ6,θm), … ,v(φ2S,θ2S), can be generated from v(φ1,θm) through data permutations. Moreover, there are S − 1 directions, (φ3,θm), (φ5,θm), … ,(φ2S − 1,θm), that are symmetrical to (φ2,θm) with respect to s2,s3, … ,sS, respectively. Therefore, v(φ3,θm), v(φ5,θm), … ,v(φ2S − 1,θm), can be generated from v(φ2,θm) through data permutations similarly. As a result, 2S − 1 coupled voltage vectors, v(φ2,θm), v(φ3,θm), … ,v(φ2S,θm), can be generated from v(φ1,θm) through data permutations.

[29] In order to find a Cd matrix for a given sector, coupled voltage vectors for at least L = N directions in that sector are required. Therefore, DN measurements are required in order to find Cd, d = 1,2, … ,D. However, when MRM is used, the coupled voltage vectors for 2S − 1 directions can be generated from a single measured coupled voltage vector. Therefore, it is sufficient to have measurements for math formula directions where ⌈.⌉ denotes the ceiling function. These LMRM measurements should be taken from directions that are uniformly separated in azimuth, and they should not be symmetric with respect to any symmetry plane in the array geometry. For example, the number of symmetry planes for the NOD antenna array given in Figure 2 is equal to 2, i.e., S = 2. Hence, only 2-D measurements are required when compared with 8D measurements for the conventional approach.

[30] The implementation of the sectorized approach combined with MRM for an N-element array with identical NOD antennas and S symmetry planes can be summarized as follows:

  1. Determine the number of uniform and non-overlapping sectors, d, by using the sectorized approach.
  2. Choose math formula directions which are spaced math formula degrees apart and are not symmetric with respect to any symmetry plane.
  3. Obtain the coupled voltage vectors due to single excitation sources from the chosen LMRM directions, v(φl,θm), l = 1,2, … ,LMRM.
  4. Use MRM and generate (2S − 1)LMRM coupled voltage vectors from v(φl,θm), l = 1,2, … ,LMRM, through data permutations. 2SLMRM coupled voltage vectors are obtained.
  5. Consider Sector-d. Select N coupled voltage vectors, from the 2SLMRM set, which belong to Sector-d. Use (8) to find Td and math formula for d = 1,2, … ,D.

In the following part, the proposed approach is evaluated using the NOD antenna array given in Figure 2. The DOA estimation problem is considered, and the MUSIC algorithm is used for evaluation.

6 Simulations

[31] In this paper, full-wave electromagnetic simulation tool FEKO [User's Manual,2008] is used to have measurements. FEKO simulations are implemented by using the method of moments (MOM). The performance of the sectorized approach combined with MRM is evaluated through DOA estimation using the MUSIC algorithm. In the experiments, the non-symmetric dipole antenna (T. E. Tuncer, 2011) given in Figure 3 is used. This antenna has an operating frequency band from 80 MHz to 800 MHz. As seen in Figure 6, approximately omnidirectional characteristic observed at 100 MHz is lost when the operating frequency is increased. This antenna can be used as an NOD antenna at 800 MHz as shown in Figure 6b. Since the antenna is omnidirectional in the lower band and non-omnidirectional in the upper band, we call it as a semi-omnidirectional antenna.

Figure 6.

Radiation pattern of the semi-omnidirectional antenna at (a) 100 MHz and (b) 800 MHz.

[32] In Figure 2, the eight-element UCA model used in the simulations is presented. In the array, the distance between the centers of two adjacent array elements is 18.75 cm, and all the antennas are terminated with a ZL = 50Ω load. The operating frequency is chosen as 800 MHz in order to use the NOD characteristic of the semi-omnidirectional antenna.

[33] It is reasonable to start with the conventional calibration approach where a single C matrix is used. This C matrix is obtained using eight measurements due to single excitation sources from uniformly spaced directions on the azimuth plane. For the performance test, the array is excited with a single source whose azimuth angle is swept with 1° resolution, i.e., φ = 0°,1°, … ,359°, and elevation angle is θ = 90°. Then RMSE in DOA estimation using MUSIC algorithm is calculated. In Figure 7, RMSE performance of the conventional calibration approach is given along with the performance when no calibration is applied, i.e., C is the N × N identity matrix. As seen in Figure 7, a single C matrix is not sufficient to properly model the mutual coupling effect for an NOD antenna array when a performance measure of maximum RMSE of 0.1° in DOA estimation is considered.

Figure 7.

The DOA estimation performance of the conventional calibration approach is compared with the case when no calibration is used, i.e., C is the N × N identity matrix.

[34] When the sectorized approach is used for the array given in Figure 2 with a performance measure of maximum RMSE of 0.1° in DOA estimation, the number of sectors is found as D = 4 following the four-step procedure given in section 4. Then mutual coupling matrices, Cd, d = 1,2,3,4, are found for the sectors by using only eight measurements through the five-step procedure given in section 5. In order to evaluate the azimuth performance of the sectorized approach combined with MRM, the array is excited with a single source whose azimuth angle is swept with 1° resolution, i.e., φ = 0°,1°, … ,359°, and elevation angle is θ = 90°. In Figure 8, RMSE performance of the sectorized approach combined with MRM in DOA estimation using the MUSIC algorithm is given when four sectors are used. As seen in Figure 8, RMSE values get significantly smaller due to the use of the sectorized approach for mutual coupling calibration. RMSE performance of the sectorized approach without MRM is also given in Figure 8. As seen in Figure 8, the performance when MRM is used is close to the performance when MRM is not used. This result shows that MRM is an effective method.

Figure 8.

The DOA estimation performances of the sectorized approach with and without MRM are compared with the case when no calibration is used. D = 4 sectors are used.

[35] In Figure 9, performances for three values of number of sectors, D = 2, 4, and 8, are given. As seen in Figure 9, two sectors fall short in satisfying the selected performance measure of maximum RMSE of 0.1° in DOA estimation. When four sectors are used, RMSE in DOA estimation is below 0.1° which satisfies the selected performance measure. When the number of sectors is increased to D = 8, a slight performance improvement is observed.

Figure 9.

The DOA estimation performances of the sectorized approach combined with MRM for the number of sectors, D = 2, 4, and 8.

[36] In order to evaluate the performance of the sectorized approach for changes in the source elevation angle, the following experiment is performed. The Cd, d = 1,2,3,4, matrices found for θ = 90° are used, while the source elevation angle is varied from 75° to 105° with 3° steps. The source azimuth angle is swept as φ = 0°,1°, … ,359°, and the RMSE values for each elevation angle is calculated as

display math(9)

where e(φ,θ) is the error in DOA estimation found for (φ,θ) direction and E(θ) is the RMSE for the corresponding elevation angle. In Figure 10, the performance of the sectorized approach combined with MRM for changes in the source elevation angle is presented. As seen in Figure 10, the best performance of the calibrated array is at 90° as expected. The calibrated array performance is better than the un-calibrated array performance for about 6° neighborhood of the calibration point. It is possible to improve the performance in elevation angle by using the sectorized approach for the elevation angles as well.

Figure 10.

The DOA estimation performance of the sectorized approach combined with MRM is compared with the case when no calibration is used for changes in the source elevation angle. The Cd, d = 1,2,3,4, matrices found for θ = 90° are used, while the source elevation angle is varied from 75° to 105° with 3° steps.

[37] The sectorized approach combined with MRM is evaluated for an elevation angle θ = 81° in order to show that the technique is applicable for different elevation angles as well. Iterative procedure for determining the number of sectors described in section 4 is used, and the results for D = 4 sectors are presented in Figure 11 for different azimuth angles. Since MRM is based on the symmetry of the array radiation patterns for symmetric array elements, it can be used for different elevation angles as well as shown in Figure 11.

Figure 11.

The DOA estimation performance of the sectorized approach combined with MRM is given for θ = 81° when D = 4 sectors are used.

[38] The performance of the sectorized approach for changes in the source frequency is also examined. The array is excited with a single source with a fixed elevation angle, θ = 90°. The Cd, d = 1,2,3,4, matrices found for 800 MHz are used, while the source frequency is varied from 795 MHz to 805 MHz with 250 kHz steps. The source azimuth angle is swept as φ = 0°,1°, … ,359°, and the RMSE values for each frequency is calculated as

display math(10)

where e(φ,f ) is the error in DOA estimation found for (φ,θ = 90°) direction at f frequency. In Figure 12, the performance of the sectorized approach combined with MRM for changes in the source frequency is presented. As seen in Figure 12, the calibrated array performance is significantly better than the un-calibrated array performance. Also, the best performance for the calibrated array is observed at 800 MHz as expected. As the source frequency is changed, the mutual coupling between the antenna elements changes. However, the performance of the proposed approach is significantly better than the un-calibrated array for a large frequency range even though a single set of mutual coupling matrices are used for all the frequencies.

Figure 12.

The DOA estimation performance of the sectorized approach combined with MRM is compared with the case when no calibration is used for changes in the source frequency. The Cd, d = 1,2,3,4, matrices found for 800 MHz are used, while the source frequency is varied from 795 MHz to 805 MHz with 250 kHz steps.

7 Conclusions

[39] In this paper, mutual coupling calibration of NOD antenna arrays is considered. Mutual coupling matrix for NOD antenna arrays changes with direction. It is shown that a single mutual coupling matrix cannot properly model the mutual coupling effect in an NOD antenna array. A sectorized calibration approach is proposed where the mutual coupling calibration is done in angular sectors. Furthermore, measurement reduction method (MRM) [Aksoy and Tuncer,2012] is used for NOD antenna arrays with identical elements. In this case, the symmetry planes in the antenna array are used, and array data are generated through simple data permutations. A non-symmetric dipole antenna is used in the performance evaluations. This novel antenna is a wideband semi-omnidirectional antenna which has omnidirectional characteristics for the lower frequency band while it exhibits non-omnidirectional characteristics for the upper frequency band. Different experiments are done in order to evaluate the performance of the sectorized approach combined with MRM for the changes in source frequency, azimuth, and elevation angles. The results show that the proposed approach is very effective for calibrating NOD antenna arrays, and it provides significant savings in the number of measurements and labor involved in the calibration process.

Ancillary