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International Journal of Antennas and Propagation
Volume 2011 (2011), Article ID 756153, 11 pages
Wideband Analysis of Mutual Coupling Compensation Methods
Department of Electrical and Computer Engineering, Royal Military College of Canada, Kingston, ON, Canada K7K 7B4
Received 30 April 2011; Accepted 21 June 2011
Academic Editor: Hon Tat Hui
Copyright © 2011 Simon Henault and Yahia M. M. Antar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The performance of various mutual coupling estimation and compensation methods is quantified and compared for wideband operation. Their main limitations and the required procedures to estimate the coupling matrix are outlined. It is determined that only the full-wave method is accurate at all frequencies. The performance tradeoffs associated with the other methods are described. General guidelines based on the electrical size and separation of the array elements are formulated for each method. The results can be valuable for both wideband and narrowband receiving antenna arrays.
Mutual coupling in antenna arrays is a subject that has received considerable attention in recent years. With the increase in signal processing capabilities, smart antennas are a promising technology that seeks the optimization of signal reception through spatial processing and other means. For example, with the knowledge of the relative phase shifts between the elements of an array, it is possible to maximize the received power, minimize interference, and estimate the direction of arrival (DOA) of impinging signals. However, due to the relatively small spacings between the elements, which is generally required to achieve some of the above goals, electromagnetic interactions between the array elements appear which lead to mutual coupling effects.
It is widely acknowledged that mutual coupling may severely impact the performance of smart antennas. To mitigate the effects of mutual coupling, several methods have been proposed in the literature. These methods generally attempt to estimate the coupling matrix describing the mutual coupling. The coupling matrix is subsequently used by the signal processor to compensate for the effects of mutual coupling. A comprehensive review of mutual coupling compensation (MCC) methods available in the scientific literature was done in . These methods typically use different approaches to estimate the coupling matrix. Previously published work that originally introduced the different MCC methods [2–5] and subsequent published work (see ) based on these methods reported significant performance improvements by the implementation of MCC in applications of beamforming and direction of arrival estimation. Although mutual coupling is highly dependent on frequency, the studies performed to this date are limited to the narrowband case where operation for a single frequency was evaluated. Besides the summarized results of , no work has been published on the performance of MCC methods for wideband applications.
Through the wideband performance evaluation of the various available MCC methods, it is expected that one will be able to determine their limitations. The aim of this paper is twofold. First, to show the limitations of each of the MCC methods. Second, to accurately quantify and compare the performance of the MCC methods for both wideband and narrowband applications. The paper is organized as follows. In Section 2, the theoretical development of the MCC methods is reviewed. The performance of the different methods is calculated numerically in Section 3 for a known elevation angle, while it is evaluated in Section 4 for an unknown elevation angle. Finally, conclusions are presented in Section 5.
2. Mutual Coupling Compensation Methods
The four main MCC methods, as described in , are reviewed in this section. They are treated in the chronological order in which they appeared in the scientific literature. In each of the four methods, the coupled signals are related to the uncoupled signals by the following equation: where is a column vector containing the coupled signals, is a column vector containing the uncoupled signals, and is the coupling matrix. With the knowledge of , the uncoupled signals can be retrieved using The coupled signals of are the terminal voltages measured across the load impedances connected to the terminals of the elements, and the uncoupled signals of are the incident signals at each of the elements. Therefore, (2) implies that the incident signals are retrieved by simply multiplying the inverse of the coupling matrix by the terminal voltage vector. In practical situations, the coupling matrix is not necessarily perfectly known, and the incident signals are estimated as follows: where is the estimated coupling matrix, and is the estimated incident signal vector. The accuracy of is thus dependent upon the MCC method used in the calculation of .
2.1. Open-Circuit Voltage Method
The open-circuit voltage method was originally suggested in  in 1983. The method is based on the assumption that the open-circuit equivalent voltage of each array element is independent of mutual coupling. Therefore, (1) is assumed to be equivalent to where is the open-circuit equivalent voltage vector. These voltages are believed to be the voltages that would be measured at the terminals of the elements if the load impedances were removed. The estimated coupling matrix is given by where is a diagonal matrix whose entries are the individual load impedances connected to the elements, and is the mutual impedance matrix. The induced electromotive force (emf) method, introduced in  in 1932, can be used to calculate the entries of . For vertical centrally terminated thin dipoles, centered at , the entries of can, therefore, be calculated using the following equation: where is the current distribution along element when it is excited by a voltage source at its terminals, is the electric field at the surface of element resulting from the excitation of element by a voltage source at its terminals, is the physical length of element , and and are the respective input currents of elements and when separately excited by a voltage source while the terminals of the other are short circuited.
When is approximately equal to half a wavelength (), it is often assumed that the amplitude of is sinusoidally distributed, and the phase is constant over . Based on this assumption, analytical expressions can be obtained for the mutual impedances . However, the assumption is violated for elements that are not infinitely thin, such as practical elements. Moreover, the assumption is narrowband in nature and is not valid for a wideband analysis of mutual coupling. Therefore, alternate calculation techniques are required to solve for the current distributions. Numerical techniques such as the method of moments (MoM)  can be used to calculate the theoretical mutual impedances of the array. To avoid having to first solve for the current distributions, calculate the electric fields, and then use (6) to finally obtain the mutual impedances; there are some alternatives. One of these is to separately excite the elements by a voltage source () and find the induced current at the shorted terminals () of the other elements calculated using the numerical technique. Then, the mutual admittances are simply calculated using The mutual impedance matrix is then calculated with It is noted that the procedure described here lends itself well to experimental implementation where numerical calculations can be avoided completely. Alternatively, a vector network analyzer (VNA) can be used to measure the scattering parameters between the elements, and the mutual impedance matrix can be calculated with where is the scattering matrix containing the scattering parameters, is the reference impedance used in the measurement of , and is the identity matrix.
The deficiencies of the open-circuit voltage method were outlined in [4, 5] and in numerous publications reviewed in . The main problem with the method is that it assumes that the excitation of the array occurs only at the terminals of the elements. While this is usually true for a transmitting array, it is not for a receiving array. The excitation of a receiving array is generally the external illumination of the elements by incident uniform plane waves, which occurs along the entire surfaces of the elements and not only at their terminals. The open-circuit voltage method, therefore, appears to be more suitable for characterizing mutual coupling in transmitting antenna arrays since they are excited by discrete sources at the terminals of the elements . Since the external illumination of a receiving array consisting of electrically small elements can be approximated as the discrete excitation of the terminals of each element, the method is also appropriate for this type of array used in the receiving mode. A proof of this is presented in Section 2.3.
2.2. Calibration Method
The calibration method was first used in  in 1991. It consists in the sequential calculation or measurement of the terminal voltage vector , in response to the external plane wave illumination of the array from a discrete number of calibration points corresponding to different DOAs. For each of the individual calibration points, the terminal voltage vector is governed by (1). The terminal voltage vectors for all the calibration points are combined to form matrix as follows: where is the number of calibration points. Based on the knowledge of the individual DOAs of the calibration points, the matrix is formed as follows: For an infinite number of different calibration points, (10) is related to (11) by the following equation: However, in practical situations, the number of different calibration points is finite, and (12) becomes Therefore, the coupling matrix is estimated by the least-square solution of (13) given by where denotes the Hermitian transpose of . A necessary condition for obtaining a unique solution is that the number of different calibration points must be at least equal to the number of array elements .
In comparison to the open-circuit voltage method, the calibration method is more suitable for the characterization of mutual coupling in receiving arrays since is calculated using vectors obtained in the receiving mode. The accuracy of depends on both the complexity of the antenna array and the number of calibration points. An interesting aspect of the calibration method is that it can completely be implemented experimentally and yield an accurate estimate of the coupling matrix if the number of calibration points is sufficient. However, the procedures involved in the determination of the terminal voltage vectors, either through experimental measurements or numerical calculations, can be tedious.
2.3. Full-Wave Method
The full-wave method was formulated in  in 2000. The method directly uses the interaction matrix numerically calculated by the method of moments (MoM). The MoM requires the array elements to be discretized into electrically small segments in order to solve for the current distribution resulting from an excitation. As an example, Figure 1 shows a single dipole, discretized into segments of length , centrally terminated into the load impedance . The MoM yields a convenient matrix equation that relates the excitations to the currents where is the excitation vector whose entries are the excitation of each of the segments, is the current vector whose entries are the currents on each segment, and is the interaction matrix. To calculate currents based on a known excitation, (15) is inverted to give where .
For clarity, the full-wave method was illustrated in  using an array of two elements comprised of three segments each. Here, the method is generalized to an arbitrary number of vertical wire elements having an arbitrary number of segments. The excitation vector is proportional to the incident signals illuminating the segments and can, therefore, be expressed as where denotes the transpose operation, is the number of array elements, and where This formulation is valid for the illumination of the array by plane waves since the excitation of the segments of a given element, , only varies as a function of , , and the vertical position of the segments given by . The superscript used in the latter expression designates the segment number for element , and its largest value is given by . The common term is a complex scalar value proportional to the external excitation of element based on its horizontal position. Here, it is important to note the following relation with respect to the incident signal vector discussed in Section 2
The entries of relating the incident signals to the terminal currents are given by the rows of whose indexes correspond to the segment indexes of the terminals. These rows are used to form in the new equation for the terminal currents By subdividing matrix into row vectors comprised of entries and substituting (17) into (21), (21) can be expressed as where The terminal voltages are given by where is a diagonal matrix whose entries are the individual load impedances at the terminals of each element. Substituting (22) into (24) gives It follows that the coupling matrix is estimated using
The main advantage of the full-wave method is that the coupling matrix estimate is very accurate provided that the MoM numerical model is also accurate. However, the a priori knowledge of the elevation angle of incoming signals is required to compute . An equivalent method not limited to the MoM has been presented in .
An interesting case occurs for electrically small elements, where a single segment per element is required for the accurate calculation of . When this condition is satisfied, is given by  If is defined as the matrix when all the elements are short circuited, that is, , then using (15), it is seen that For a single segment per element, , and (28) becomes This equation is similar to (4)-(5). Therefore, the open-circuit voltage method is accurate for arrays of electrically small elements.
2.4. Receiving Mutual Impedance Method
The receiving mutual impedance method was proposed in  in 2003. In an attempt to address the main deficiency of the open-circuit voltage method of inaccurately modeling receiving arrays, the method involves a new definition of mutual impedances. Instead of calculating the mutual impedances in the transmission mode, they are calculated in the receiving mode under external plane wave excitation. The mutual impedances are then given by for , where is the current induced in by the external plane wave, and is the voltage induced across , that is solely due to the excitation of element . Hence, the procedure involved for computing in (30) requires the subtraction of the induced voltage across in the absence of element from the induced voltage across in the presence of element . By superposition, the terminal voltages are given by Substituting into (31), the incident signals are given by Equation (32) can be expressed in the following convenient matrix equation: where
The receiving mutual impedance method was demonstrated to be superior to the open-circuit voltage method in applications of direction finding and adaptive nulling . Similar to the open-circuit and calibration methods, it is well suited for practical implementation where the mutual impedances of (30) can be measured instead of being calculated numerically. In comparison with the full-wave method, it claims to offer the additional advantage of not requiring a priori knowledge of the elevation angle of incoming signals. However, this relies on the assumption that the current distributions remain unchanged irrespective of the elevation angle. As acknowledged in , this is not necessarily true. Moreover, the currents in (31) are assumed to be equal to those in (30) despite the presence of only two elements in the latter equation and of all the elements in the former equation. This problem was addressed in the new mutual impedance calculation approach presented in [13, 14]. This approach elegantly reduces the number of coupling parameters that must be estimated in the calibration method described in Section 2.2, reducing the minimum number of calibration points to . Consequently, its performance is more an improvement of the calibration method than it is an improvement of the receiving mutual impedance method, despite its name. It is also assumed in the receiving mutual impedance method that the current distributions used in (30) are independent of the DOA of the external excitation. These assumptions may be violated when the elements are very closely spaced. Therefore, the method cannot be expected to provide an accurate estimate of the coupling matrix for arbitrary frequencies and array configurations.
3. Performance Comparison of Mutual Coupling Compensation Methods
To evaluate the wideband performance of the MCC methods discussed in Section 2, simple dipoles are selected. Although dipoles are generally not used in wideband applications, they are nevertheless a useful tool for the performance analysis at multiple frequencies due to their simplicity. Also, for applications such as wideband direction finding, they are often employed to cover bandwidths exceeding 5 : 1. Their individual performance can be measured by their realized gain , which takes into account both the impedance mismatch and the gain in the direction of interest. It is, therefore, a measure of performance in comparison to a hypothetical perfectly matched isotropic antenna. The realized gain is given by where is the directional gain, and is the reflection coefficient resulting from the impedance mismatch. The latter is given by where is the input impedance of the antenna and is the load impedance connected to its terminals. It is verified that a perfectly matched isotropic antenna yields unity in (35). The realized gain of a dipole antenna having a length of and a radius of , where is the smallest wavelength of interest, and centrally terminated into various values of load impedance , is shown in Figure 2. The minimum acceptable realized gain is dependent upon the intended application. In applications of wideband direction finding, for example, a minimum value of −10 dB is often accepted. The realized gain of Figure 2 is calculated using the MoM-based numerical electromagnetic code (NEC)  for . Since only the performance of a single dipole is calculated, it is independent of azimuth angle. It is seen that using a large value of load impedance raises the realized gain between the first two resonances of the dipole at and . Load impedances of 300 Ω are, therefore, used in the remainder of this paper. An interesting advantage of operating near the second resonance can be observed in Figure 2 since the realized gain is maximum at due to the maximum directional gain achieved at this frequency.
For the wideband analysis, a planar array of nine of these dipoles is used, as illustrated in Figure 3. To avoid grating lobes for any steering direction, the separation between the elements is fixed to . Smart antennas generally aim at maximizing the output signal-to-noise ratio (SNR). A common measure of their effectiveness in achieving this goal is the array gain given by  where SNRin and SNRout are the SNRs at the input and output of the signal processor, respectively. In the presence of receiver noise with no interferers, the terminal voltages are described by where , and the column vector contains the noise at each array element. The SNRin in (37) is obtained using the entries of in (38) where denotes the statistical expectation over time. Assuming ergodicity and uncorrelated noise at the elements, (39) becomes where is the noise power. As illustrated in Figure 4, the signal at the output of the processor is given by where the column vector contains the weights applied to each of the elements. Using (41), the SNRout in (37) is obtained by Since the noise is uncorrelated, , and (42) reduces to Substituting (40) and (43) into (37) yields
Recently, [18, 19] demonstrated that adaptive beamforming based on the minimum-mean-square-error (MMSE) algorithm only gains a slight improvement in convergence rate as a result of MCC. It is found that MCC does not yield a lower mean square error (MSE) between the array output and the reference signal in the steady state. Similarly, the array gain in (44), provided by adaptive array processing, converges to the same value as that in the absence of MCC. However, in certain applications, the signal statistics change too rapidly for adaptive array processing to be effective. For DOA estimation, adaptive processing is also generally not an option. It is, therefore, insightful to evaluate MCC methods in the deterministic case where the signal statistics are known. The optimum weights in the deterministic case are simply given by In the absence of mutual coupling, , and the weights in (45) can be verified to be equal to those generally employed in conventional uniformly weighted phased arrays. If the coupling matrix is perfectly known, , and the use of in (45) ensures that the required signals are added coherently based on the relative phases of the signals at each element contained in . By multiplying the conjugate of each weight with in (41), the phases of the required signals are verified to vanish, and the output SNR is maximized. Substituting (45) into (44), the array gain becomes It is verified that when , (46) reduces to . In other words, in the perfect knowledge of the coupling matrix, the optimum array gain is achieved, and it is equal to the number of elements. When mutual coupling is ignored and no MCC is implemented, the coupling matrix is assumed to be . The weights are then given by , and the array gain is
In the next three subsections, (46) is used to evaluate the performance of the open-circuit voltage, calibration and receiving mutual impedance methods for the nineelement planar array. As discussed in Section 2.3, since the coupling matrix calculated using the full-wave method approaches the actual coupling matrix when the numerical model is accurate, the array gain obtained following MCC based on this method approaches in general, and 9 in the specific case of the nineelement array. Therefore, the full-wave method is used here as a reference to evaluate the other three methods. The array gain obtained using each method is also compared against that obtained in the absence of MCC as given by (47). For all the MCC methods, the estimated coupling matrix is calculated based on currents computed using NEC at 51 different frequencies within the operating frequency range of the dipoles. The array gain is calculated using (46) for a signal known to be arriving from and , that is, from the positive x-axis.
3.1. Open-Circuit Voltage Method Performance
As described in Section 2.1, the coupling matrix can be estimated by successively exciting one of the array elements with a voltage source at its terminals while all the other elements are short circuited. For the nine-dipole array, this technically requires nine different calculations. Since the nine dipoles are identical and are terminated into equal load impedances, symmetry can be used advantageously to reduce the number of calculations. Based on symmetry, the currents resulting from only three different configurations need to be calculated. As shown in Figure 5(a), a voltage source is first applied at the terminals of element 8, and the currents in the segments corresponding to the shorted terminals of the other elements are calculated. These are used to fill the column numbered 8 of the mutual admittance matrix according to (7). Thanks to symmetry, the entries of column 8 can be reused to fill columns 2, 4, and 6. Similarly, by exciting element 7 as illustrated in Figure 5(b), columns 1, 3, 7, and 9 of the mutual admittance matrix are filled. Finally, by exciting element 5 as shown in Figure 5(c), column 5 of the mutual admittance matrix is filled. The mutual admittance matrix is then inverted according to (8) to obtain the mutual impedance matrix, which is then substituted into (5) to calculate . Since the nine elements are to be terminated into 300 Ω, in (5) is given by .
The coupling matrix estimate calculated at the 51 different frequencies is substituted into (46) to calculate the resulting array gain. The result is shown in Figure 6. The array gain using the open-circuit voltage method is seen to provide a significant improvement over the uncompensated case for frequencies below . These include the first resonant frequency at . This result is consistent with most previously published work where the analysis was done at frequencies where the elements were generally half-wave dipoles and, thereby, operating at first resonance. Although the electrical spacing of the elements is smaller at the first resonant frequency for the array being studied in this paper, there is, nevertheless, some performance improvement. However, the performance degrades significantly as frequency increases, reaching a minimum array gain of approximately 4.5 at . Between and , a detrimental effect of implementing MCC using the open-circuit voltage is observed since ignoring mutual coupling would yield better performance. The method is seen to yield results that are highly dependent on frequency. The only frequencies where the array gain approaches that for the full-wave method are the lowest frequencies. As discussed in Section 2.1, this is expected since for small electrical dimensions of the elements, the excitation can be approximated as being only at the terminals. It was shown in Section 2.3 that when this approximation is valid, then the open-circuit voltage method yields accurate results.
3.2. Calibration Method Performance
An important consideration in the implementation of the calibration method is the number of calibration points selected. As mentioned in Section 2.2, a number of calibration points equal to at least the number of elements is required to obtain a unique solution to (13). However, depending on the complexity of the array, this number may not be sufficient to obtain an accurate coupling matrix estimate. For the nine-dipole array, nine calibration points were found to be insufficient when they are equispaced over beginning at 0°. However, the implementation of [13, 14], discussed in Section 2.4, was verified to give satisfactory performance with as little as nine calibration points. The array gain obtained using the implementation of Section 2.2 with 16, 18, and 360 calibration points is shown in Figure 7. In each case, the terminal voltages are calculated under plane wave external excitation over beginning at 0°. The azimuth intervals are given by , , and 1°, respectively.
It is observed that the resulting performance using only 16 calibration points shows important variations with respect to frequency. The number of calibration points is visibly too small. It is interesting to see that adding only two calibration points yields a much better performance where the array gain is seen to reach the optimal value of 9 at and remains constant as frequency increases. Increasing the number of points even more has the effect of reducing the frequency at which the optimal array gain is reached. For 360 calibration points, the array gain is, therefore, very close to 9 at the lowest frequency of interest. When the number of calibration points is too small, in (14) has a high condition number. Consequently, is inaccurate and yields suboptimal performance. This problem is exacerbated at lower frequencies since the phase shifts between the signals in are smaller. The only way to ensure that is accurate is by increasing the number of calibration points. As indicated by the above example, if some performance tradeoffs are acceptable, especially at low frequencies, it is possible to limit the number of points to a reasonable value in order to reduce the amount of computations or measurements.
3.3. Receiving Mutual Impedance Method Performance
Similar to the open-circuit voltage method in Section 3.1, the receiving mutual impedance method necessitates multiple calculations for the determination of the mutual impedances given by (30). Although several variants are found in the literature for determining the receiving mutual impedances, including one where all of the elements are simultaneously excited by a plane wave instead of exciting each element pair separately, the element pairs are treated separately here for better accuracy. This implementation of the method technically first requires calculations where each element is excited separately in the receiving mode. Then more calculations are required to calculate in (30) for the elements excited by the different . Therefore, a total of different calculations are required. By taking advantage of symmetry, this number can be reduced to six for the nine-dipole array to take into consideration all the different configurations of element pairs. The six different configurations used in the calculations are illustrated in Figure 8. Since the nine dipoles are identical and are terminated into equal load impedances, only one element needs to be excited separately. In Figure 8(a), element 9 is arbitrarily selected for the separate excitation. Then, in Figures 8(b)–8(f), , , , , and are calculated, respectively, according to (30). Based on symmetry, these mutual impedances are then reused to completely fill the coupling matrix estimate in (34) where the load impedances, , are replaced with 300 Ω. To verify whether the assumption stated in Section 2.4 that the current distributions are independent of the excitation DOA, is calculated for excitation DOAs of 0°, , , and . The array gain obtained using the four different excitation DOAs is shown in Figure 9.
The performance of the receiving mutual impedance method is observed to be impressively good for frequencies below where the gain approaches the optimal value for any excitation DOA. At higher frequencies, there is a degradation in performance although it remains superior to that when no MCC is implemented. Also, the performance becomes dependent on the excitation DOA used in the calculation of . An excitation DOA of is seen to provide the best results among the four DOAs up to , and an excitation DOA of provides the best results above this frequency. The degradation in performance and the dependence on the excitation DOA are explained by the incorrect assumption that the current distributions are unaffected by the presence of the other elements, as discussed in Section 2.4. Due to the alteration of the current distributions when the elements are in the presence of one another, the symmetry used in filling is also questionable.
4. Performance Comparison of Mutual Coupling Compensation Methods for Arbitrary Elevation Angle
Finally, the performance of the MCC methods can be compared for signals arriving from . As mentioned in Section 2.4, an important advantage of the receiving mutual impedance method over the full-wave method is that no assumption needs to be made with respect to the elevation angle of incoming signals. To verify the relevance of this, the array gain is calculated for the nine-dipole array with an incoming signal arriving from and . Although the coupling matrix can be estimated accurately using the full-wave method in the deterministic case, the coupling matrix calculated for is used to evaluate the impact of relying on the same coupling matrix regardless of the elevation angle. The coupling matrix estimated using 18 points is selected for this example since it provided satisfactory results in Section 3.2. For the receiving mutual impedance method, the excitation DOA of is selected based on its best overall wideband performance in Section 3.3. The results are shown in Figure 10.
For this value of elevation angle, the receiving mutual impedance method is observed to provide nearly optimal gain up to . Compared to the performance obtained for , there is a reduction in the bandwidth over which the method yields optimal results. Beyond , the implementation of the method is seen to be detrimental since it yields a significantly lower gain than without MCC, except for a narrow range of frequencies around . However, it is verified that the receiving mutual impedance method is superior to the full-wave method over all the plotted frequencies. Unfortunately, the improvement in performance is very small, reaching approximately over the range of frequencies where the two methods are superior to all the other methods. Beyond , it is simply better to use the open-circuit voltage method since it provides better performance.
The performance degradation of MCC methods observed in this section is explained as follows. The vertical phase shifts of incident signals along the array elements become more important as the frequency increases, causing square coupling matrices to be less accurate in quantifying mutual coupling for all elevation angles. For array elements having large horizontal dimensions, such as some printed circuit type antennas, horizontal phase shifts of the incident signals are an additional complication. Mutual coupling in such antennas, as well as vertical wire antennas illuminated by signals of arbitrary elevation, may be better described by nonsquare coupling matrices, as discussed in .
Various mutual coupling compensation methods were compared over a wide range of frequencies. Through the accurate quantification of their wideband performance, their limitations were outlined. It was shown that only the full-wave method yields accurate results at all frequencies, provided that the numerical model is accurate. The open-circuit voltage method is only accurate when the elements are electrically small and was shown to yield a performance approaching that of the full-wave method when the array elements can be approximated by a single segment. Although the receiving mutual impedance method is accurate over a larger range of frequencies, its performance degrades significantly when the array elements become electrically larger. For example, it was shown that the method is not accurate for dipoles operating at the second resonant frequency. Due to the resulting instability in the computation of the coupling matrix estimate, the calibration method is less accurate and requires a larger number of calibration points when the distance between the array elements becomes electrically small. The only limitation of the full-wave method with respect to the other three methods is that the coupling matrix can generally only be evaluated theoretically. The other methods can evaluate the matrix experimentally, but the required number of measurements was shown to vary, with the open-circuit voltage method requiring the least. It was demonstrated that the receiving mutual impedance method only provides a marginal advantage over the full-wave method in the sense that it does not require a priori information on the elevation angle of incoming signals. The results presented here can be useful for both wideband and narrowband receiving antenna arrays.
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