Accurate Evaluation of Mutual Coupling for Array Calibration S. Henault and Y. M. M. Antar Royal Military College of Canada, Kingston, K7K 7B4, Canada
[email protected] Abstract—Calibration of receiving antenna arrays generally yields a square coupling matrix characterizing the array response. It is demonstrated in this paper that in the presence of strong structure scattering, the array response is better described by a non-square coupling matrix. Therefore, the calibration procedure must account for these new matrix dimensions. The different approaches providing accurate non-square coupling matrices are described, and important guidelines are given for the determination of the coupling matrix dimensions for obtaining the best performance.
I. I NTRODUCTION Array calibration is commonly used in receiving antenna arrays to eliminate undesired effects caused by mutual coupling and non-ideal components. The calibration procedure is often performed experimentally by illuminating the array using a plane wave source from a finite number of directions of arrival (DOAs). Then, a coupling matrix characterizing the array is estimated based on the measured terminal voltages [1]. Since the number of DOAs is finite, the estimated coupling matrix is a least-square approximation of the true coupling matrix, and is generally a square matrix, i.e., its number of rows is equal to its number of columns. However, a serious problem was pointed out in [2]. In particular, array calibration fails in the presence of structure scattering caused, for example, by a conducting plate located behind the elements to improve their gain in a desired direction. No solution to this problem was proposed, although it is actually possible to take into consideration structure scattering in order to predict the radiation patterns of the array elements. The present paper will demonstrate how this can be made possible. To accurately account for structure scattering, one has to resort to non-square coupling matrices, as defined in [3], where such matrices were used to include the effects of electromagnetic interactions with parasitic wires and plane wave incidence from arbitrary elevation angles. Two approaches are available to estimate a non-square coupling matrix. The first approach is numerical and uses a full-wave analysis of the array including the structure causing scattering [4]-[5]. The second approach is generally experimental and is similar to a conventional calibration, as described previously, but provision is made to take into consideration that the incident signals reach a secondary structure that is external to the array elements. This was done in [6] for yagi elements where additional columns were added in the coupling matrix to account for the parasitic elements. In this reference, the
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parasitic elements were thin wires and the incident signals had constant phases along the individual surfaces of the elements. It will be shown here that by carefully grouping portions of the structure for which common incident signals can be assumed, the radiation patterns of the array elements can be predicted accurately through the use of the non-square coupling matrix estimated by calibration. Results along with the calibration methodology will be presented for coupling matrices of various dimensions in an array of eight dipoles in front of a conducting plate, similar to the arrangement studied in [2]. It will therefore be demonstrated that reliable calibration is possible in practical applications where structure scattering is important. The paper is organized as follows. The known theory of array calibration is reviewed in Section II. Both the numerical and experimental approaches are described. In Section III, it is shown how structure scattering can be accounted for during experimental calibration. Results showing predicted radiation patterns based on different calibration implementations are shown in Section IV, along with the resulting performance of typical array applications. Finally, conclusions are presented in Section V. II. C ALIBRATION T HEORY A receiving antenna array of N elements can be described by the following equation: V = CE
(1)
where V is a column vector containing the N voltages measured at the element ports, E is a column vector containing the incident electric fields along the array structure, and C is the coupling matrix. The goal of calibration is to estimate this C matrix as accurately as possible in order for the array to achieve the desired performance. Note that the dimensions of C are not necessarily N × N , but in general can have dimensions of N × K, where the parameter K will depend on assumptions made about the incident fields and the presence of near-field scatterers [3], [5]. Two different approaches can be employed for the calibration of a receiving array. The first approach is theoretical and involves the numerical solution of Maxwell’s equations. The second approach is better suited for experimental procedures. They will be described in the following two sub-sections.
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A. Numerical Approach The method of moments (MoM) [7] inherently implies that the behavior of an antenna can be completely characterized independently of its excitation, according to the following relation: I = YM oM VMoM
(2)
where I is a column vector containing the currents flowing on the discrete portions of the antenna, VMoM is the excitation vector whose entries are proportional to the electric fields on the discrete portions of the antenna, and YM oM is the MoM admittance matrix. Since currents are generally only collected at the array element ports, only the rows of I and YM oM associated with those ports are of interest in the estimation of a coupling matrix [4]. These rows are used to form the vector Iports and the matrix Yports , and (2) reduces to: Iports = Yports VMoM
(3)
The port voltages are simply given by V = ZL Iports where ZL is a diagonal matrix containing the load impedances connected to each of the elements, and the incident electric fields are given by E = αVMoM where α is a scaling factor. Therefore, the estimated coupling matrix is given by: C˜ = αZL Yports
(4)
This method for the estimation of the coupling matrix is generally referred to as the full-wave method [8]. Unfortunately, only the MoM can be used for the computation of (4) and other popular numerical techniques are not useful in the accurate calibration of the array. The multiple antenna induced emf method (MAIEM) was introduced recently to circumvent this limitation [5], [9]. Using this novel method, the coupling matrix is estimated using: −1 C˜ = −ZL (ZL + ZAN T ) ZAN T I T
(5)
where ZAN T is a diagonal matrix containing the input impedances of each of the elements, and I T is an N × K matrix whose rows contain the current distributions for the individual excitations of the elements with a unit-voltage source while the other elements are terminated into their respective load impedances. This method is independent of the numerical technique since the current distributions and input impedances can be numerically computed using techniques other than the MoM. B. Experimental Approach The coupling matrix can also be estimated by measurements taken for a single plane wave impinging from M different DOAs. The voltages measured at the array elements are combined into the matrix V : (6) V = V1 V2 · · · VM and the associated incident fields are combined into the matrix E: E = E1 E2 · · · EM (7)
The coupling matrix estimate is a least-square solution of V = ˜ CE: −1 C˜ = V E H EE H (8) where (·)H denotes the Hermitian transpose. Although this approach has the advantage of not relying on the accuracy of a numerical model of the array, it is also often employed using numerical techniques where the port voltages are computed in the receiving mode [6]. In fact, prior to [4], it was the only way to numerically estimate a non-square coupling matrix. Furthermore, it was the only alternative to the full-wave method prior to the MAIEM [5]. III. C ALIBRATION AND S TRUCTURE S CATTERING It is generally assumed that C˜ in (8) is a square matrix, i.e., matrices V and E have equal dimensions. However, as pointed out in [3] and [6], parasitic wires near the array elements can ˜ To do so, the number be accounted for in the calculation of C. of rows of E has to be increased to include the incident fields along the parasitic wires. Here, the same logic is applied to the array of eight dipoles in front of a rectangular conducting plate shown in Fig. 1. Contrary to [2] where a square C˜ was assumed for a similar arrangement, columns are now added to the matrix to account for structure scattering. The first eight columns are those associated with the eight dipoles. Then, a number of columns are added. Each of these additional columns accounts for a portion of the conducting plate. For all signal arriving with an elevation angle of θ = 90◦ and a conducting plate located at x = 0, the vectors in (7) are then given by: ⎤ ⎡ 2π ej λ (x1 cos φi +y1 sin φi ) ⎥ ⎢ .. ⎥ ⎢ . ⎥ ⎢ ⎥ ⎢ j 2π (x cos φ +y sin φ ) 8 i 8 i ⎥ ⎢ e λ Ei = ⎢ (9) ⎥ 2π j y sin φ i ⎥ ⎢ e λ 9 ⎥ ⎢ .. ⎥ ⎢ ⎦ ⎣ . 2π ej λ yK sin φi where λ and φi are the wavelength and azimuth DOA of the calibration signal, (x1 , y1 ) to (x8 , y8 ) are the locations of the eight vertical dipoles in the xy-plane, and y9 to yK are the center locations of each portion of the conducting plate. It is therefore implicitly assumed that the incident fields will be constant over a given portion of the conducting plate. This assumption is valid provided that the horizontal dimensions of the individual portions are electrically small. Hence the calibration accuracy will be dependent upon the number of such additional portions. As stated in [1], a necessary condition for obtaining a unique solution to (8) is that the number of calibration points, M , be at least equal to the number of array elements, N . Here, we reformulate this condition by ensuring that the number of calibration points, M , be at least equal to the number of rows of (9), K.
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consistent with the results presented in [2] where it was concluded that the coupling matrix concept was inaccurate in predicting radiation patterns. 10 5 0
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Fig. 1. 8-dipole array in front of a conducting plate. The first eight columns of the coupling matrix are associated with the dipoles while the other columns are associated with the different portions of the plate.
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IV. R ESULTS
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Fig. 2. Radiation pattern of the first element in an array of eight dipoles in front of a conducting plate predicted using coupling matrices of various dimensions through calibration.
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To demonstrate this new calibration approach, the array configuration of Fig. 1 was modeled using the software package FEKO [10]. The dipoles were modeled with thin wires having a length of 6.12 cm, spaced 4.9 cm apart, and centrally terminated into 50 Ω load impedances. The conducting plate had a width of 39.16 cm and a height of 24 cm, and was located 3 cm behind the dipoles. The frequency of all signals was set to 2.45 GHz. These dimensions and this frequency of operation were selected based on those used in [2]. Calibration was performed by simulating the numerical model illuminated by a single plane wave signal arriving from M = 19 different DOAs equally spaced between −90◦ ≤ φ ≤ 90◦ . The port voltages across the eight load impedances were then substituted into (6). Four different coupling matrices were estimated by varying the number K of rows of (7) and the appropriate substitution of phase shifts into the incident field vectors described in (9). The matrices obtained by (8) were calculated for K = 8, 10, 12 and 19. Therefore the matrices have dimensions of 8 × 8, 8 × 10, 8 × 12, and 8 × 19. Using these matrices, predictions were made on the radiation patterns of the array elements. This was done by calculating the port voltages using the following equation:
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Fig. 3. Radiation pattern of the fourth element in an array of eight dipoles in front of a conducting plate predicted using coupling matrices of various dimensions through calibration.
(10)
where E(φ) is the incident electric field vector whose entries are similar to those of (9) except that they are computed for all azimuth angles φ for which the radiation pattern is calculated. In Fig. 2, the amplitudes and phases of the port voltage of the first dipole, i.e., the first entry of v(φ), using the four different C˜ matrices are compared against the actual values directly computed in the receiving mode for −90◦ ≤ φ ≤ 90◦ by increments of 1◦ . Note that the amplitudes are normalized to those that would be received by a hypothetical perfectly matched isotropic antenna. The results confirm that increasing the dimension K of C˜ improves the accuracy of the radiation pattern prediction. The results for the fourth element, shown in Fig. 3, also corroborate this important finding. The curves showing the predictions using a square matrix (8 × 8) are
The importance of accurate predictions of radiation patterns can be better appreciated by investigating the performance for typical applications of receiving antenna arrays. In Fig. 4, it is shown that the resolution and accuracy of DOA estimation using the multiple signal classification (MUSIC) algorithm [11] is improved significantly with the addition of columns ˜ The results for three signal simultaneously arriving into C. from −45◦ , 60◦ and 85◦ are compared against those obtained by the numerical approach, covered in Section II-A, using either the full-wave method [4] or the MAIEM [5], [9]. These two methods are known to provide accurate radiation pattern predictions, and therefore yield very sharp peaks in the directions of the incoming signals. Although increasing the number of columns improves the performance as compared to
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the case where no calibration is performed, the improvement reaches a maximum for K = M . As mentioned previously, the number of columns of C˜ cannot exceed the number of calibration points, or (8) will not yield a unique solution. As a result, the performance for a calibration performed with dimensions of 8 × 19 is the best that can be achieved for 19 calibration points. To obtain better performance, it is necessary to increase the number of calibration points and the number of columns of C˜ accordingly.
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Fig. 5. Radiation pattern for an array of eight dipoles in front of a conducting plate using coupling matrices of various dimensions with a desired signal arriving from −45◦ and two undesired signals arriving from 60◦ and 85◦ .
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number of columns cannot exceed the number of calibration points. These findings may be valuable for practical antenna arrays that are often located close to near-field scatterers or mounting structures that can potentially affect the performance of the array.
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Fig. 4. MUSIC DOA spectrum for an array of eight dipoles in front of a conducting plate using coupling matrices of various dimensions with three incident signals arriving from −45◦ , 60◦ and 85◦ .
Another typical application of receiving arrays is interference nulling. Assuming that is known that a desired signal is arriving from −45◦ and two interfering signals are arriving from 60◦ and 85◦ , the signal to interference and noise ratio (SINR) can be maximized by forming a main beam towards the desired signal and two nulls towards the interfering signals. The resulting radiation patterns are shown in Fig. 5. The consequence of inaccurate calibration is barely noticeable by looking at the main beam. On the other hand, the nulls are seen to be highly dependent on the calibration. Using a square coupling matrix slightly reduces interference as compared to the uncalibrated case. However, it only forms a single null between the two interfering signals. Increasing the number of columns of the coupling matrix deepens the nulls significantly. Again, a larger number of calibration points would be required to reduce the interference even more and approach the performance of the full-wave or MAIEM methods. V. C ONCLUSIONS It was demonstrated that the coupling matrix concept can be used for accurately predicting the radiation patterns of receiving antenna array elements despite the presence of strong structure scattering. Even calibration performed by measurements can be accurate provided that the number of columns of the coupling matrix estimate is increased to account for incident signals reaching the structure causing the scattering. A factor limiting the accuracy of the resulting non-square coupling matrix is the number of calibration points, since the
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