# High-Precision Mutual Coupling Coefficient Estimation for Adaptive Beamforming

Authors: Ziang Feng; Guoping Hu; Qichao Ge; Hao Zhou
Journal: Wireless Communications and Mobile Computing
 (2020)

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## Abstract

Here, a high-precision mutual coupling coefficient estimation method is proposed that is more suitable for adaptive beamforming than traditional algorithms. According to the relationship between the designed transition matrix and the signal, the proposed algorithm selects the transition matrix corresponding to the high-power signal. The high-precision estimation of the mutual coupling coefficient is obtained by using the selected transition matrix estimation, which yields relatively good estimation accuracy for the mutual coupling coefficient when the desired signal-to-noise ratio (SNR) is low and relatively robust adaptive beamforming with unknown mutual coupling. Simulation results demonstrate the validity of the proposed method.


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## Body

## 1. Introduction

Signal processing technology, such as direction of arrival (DOA) and robust adaptive beamforming (RAB), has been widely used in radar, sonar, communication, etc. [1–7]. As a kind of array error, mutual coupling seriously affects the performance of various signal processing algorithms [8–11]. To avoid the influence of mutual coupling, a middle subarray-based (MSB) approach is proposed in [12], and a maximum interelement spacing constraint (MISC) array is designed in [13]. In addition, many algorithms for calibrating mutual coupling have been proposed. Since the mutual coupling matrix (MCM) can be modeled as a banded symmetric Toeplitz matrix for a uniform linear array (ULA), a subspace-based method is proposed in [14], and a fourth-order cumulant- (FOC-) based method is proposed in [15]. In addition, an iterative autocalibration algorithm based on the eigendecomposition of the sampling covariance matrix for a uniform circular array (UCA) is proposed to calibrate unknown mutual coupling since the MCM has a complex symmetric circular Toeplitz structure in a UCA in [16]. Furthermore, a joint DOA estimation and mutual coupling self-calibration for ULA-based bistatic multiple-input-multiple-output (MIMO) radar is proposed in [17]. Based on [16], a parameter estimation method for direction-dependent mutual coupling is proposed in [18]. However, the MSB approach reduces the degree of freedom (DOF) of the array, and the complex structure of the MISC array increases the difficulty of signal processing, while the subspace-based method in [14] constructs a high-dimensional matrix, and the algorithms proposed in [16, 18] both need an iterative process. To reduce the computational complexity in estimating mutual coupling, two low-complexity algorithms for direction-dependent mutual coupling and direction-independent mutual coupling were proposed in [19, 20], respectively.To improve the robustness of adaptive beamforming in the presence of unknown mutual coupling, a middle subarray-plus-reconstruction-based (MSRB) method combining the MSB algorithm with interference-plus-noise covariance matrix (INCM) reconstruction [21] is proposed in [22]. However, the MSRB approach requires a large array aperture for high performance. Similarly, the desired signal steering vector with unknown mutual coupling is calibrated by using the specific structure of the MCM, and the new beamformer is obtained by combining a diagonal loading beamformer with the desired signal steering vector estimation [11]. To reduce the computational load, a subspace-plus-reconstruction-based (SRB) beamformer is then designed by incorporating the subspace-based mutual coupling coefficient estimation method [14], and INCM reconstruction is designed in [10]. Obviously, the algorithms proposed in [20] can be utilized to design beamformers to further reduce the computational load. However, the algorithm proposed in [20] cannot obtain a high accuracy for the mutual coupling coefficients when there is a high power difference between signals since the accuracy of the subspace-based method is positively correlated with the signal-to-noise ratio (SNR).To further improve the robustness of adaptive beamforming to unknown mutual coupling, a novel subspace-based algorithm is proposed to estimate mutual coupling coefficients and is utilized to design a novel adaptive beamformer. Different from the algorithms proposed in [20], we add the process of selecting several suitable transitional matrices and calculating their inverses. After estimating a group of transitional matrices, we select the transitional matrix corresponding to the maximum spectral peak to estimate the mutual coupling coefficient vector and the MCM. Then, the signal steering vector is calibrated with the estimated MCM. Finally, by combining the estimated MCM and the INCM reconstruction, we propose a novel adaptive beamforming algorithm. According to simulation, compared with several existing approaches, the proposed mutual coupling estimation method has a higher estimation accuracy, especially in the case of SNR differences between several signals. The designed beamformer is more robust than the existing beamformer to mutual coupling.


## 2. Signal Mode

We assume that a ULA ofN sensors is impinged by L narrowband uncorrelated signals, and the noise is additive white Gaussian noise. We assume that the signal and noise are statistically independent. When the direction-independent mutual coupling effect is considered, the received snapshot at the kth time instant can be expressed as
(1)xk=xsk+nk=C∑l=1Lslkaθl+nk,where xsk and nk stand for the N×1 vector of the signal and noise, respectively. θl is the lth signal DOA, and slk and aθl∈ℂN×1 are the corresponding complex envelope and steering vector, respectively. C∈ℂN×N denotes the MCM, which can be molded as a banded symmetric Toeplitz matrix for a ULA since the mutual coupling coefficients between the elements are inversely proportional to their distance. In general, we assume that the mutual coupling coefficient becomes zero when the spacing of two elements exceeds an interelement spacing of P; hence, the mutual coupling coefficient vector and the MCM can be defined as
(2)c=c0,c1,c2,⋯,cP−1T,C=ToeplitzcN,cN,where c0=1 and cN=cT,0T∈ℂN×1.Recall that in Equation (1), the covariance matrix of xk can be given by
(3)R=ExkxHk=CARsAHCH+σn2IN,where E· and ⋅H represent the expectation and conjugate transpose, respectively. A=a1,a2,⋯,aL∈ℂN×L denotes the manifold matrix, and Rs=diagσ12,σ22,⋯,σL2∈ℝL×L is the covariance matrix of signals, where σl2 is the power of the lth signal. σn2 is the power of the noise, and IN∈ℝN×N is an identity matrix.In practice, the sampling covariance matrix is usually used in lieu of the covariance matrix, and the sampling covariance matrixR^x can be expressed as
(4)R^x=1K∑k=1KxkxHk,where K stands for the number of snapshots. After eigendecomposing R^x, we can obtain
(5)R^x=UΛUH=UsΛsUsH+UnΛnUnH,where U is the eigenvector matrix and Λ denotes the corresponding eigenvalue matrix. Λs and Λn are diagonal matrices that contain L large eigenvalues and the remaining small eigenvalues, respectively. Us and Un are the corresponding eigenvector matrices. In general, Us is called the signal subspace, and Un is called the noise subspace. Additionally, Caθll=1L can span the signal subspace.For instance, when DOA estimation is performed using a subspace algorithm, the spatial spectral function shown below is usually used [13]:
(6)Pθ=1aHθUnUnHaθ.Obviously, if the MCM is unknown, the spectral peaks do not correspond to the true DOA of the signals sinceUn is orthogonal to Caθ, not aθ. That is, if C can be estimated, then the DOA of signals can be estimated.


## 3. Proposed Algorithm

### 3.1. Mutual Coupling Coefficient Estimation

According to the banded symmetric Toeplitz structure of MCMC, we can obtain [19]
(7)Caθ=Tθc,where
(8)Tθ=E1aθ,E2θaθ,⋯,EPθaθ,where
(9)Epij=1,ifCij=cp0,otherwise,p=0,1,⋯,P−1.Based on the orthogonality betweenCaθl and the noise subspace Un, we can obtain
(10)aHθlCHUnUnHCaθl=0.Recall that Equations (7) and (10) can be rewritten as
(11)cHQθlc=0,where Qθ is a transitional matrix and is defined as
(12)Qθ=THθUnUnHTθ,which is a P×P matrix, and its dimensions are smaller than those of the transitional matrix in [11, 13].WhenP≤N−L, Qθ is a nonsingular matrix for a general θ since the ranks of Tθ and Un are P and N−L, respectively. However, if θ is one DOA of the incident signals, Qθ is a singular matrix, and its determinant is zero [23]. When Qθ is a singular matrix, we can find that Equation (11) holds and that c is an eigenvector of the matrix Qθ corresponding to the eigenvalue zero, since c is a nonzero vector. That is to say, the degree of freedom of the proposed algorithm is N−P.Namely, the mutual coupling coefficient vectorc can be estimated by
(13)cl=vlminvlmin1,(14)c^=1L∑l=1Lcl,where vlmin denotes the eigenvector corresponding to the minimum eigenvalue of Qθl, and vlmin1 is the first entry of vlmin.From Equation (12), we can find that the matrix Qθl can be easily calculated by the known DOA of the incident signals, but the DOAs are unknown. Hence, we construct a new spectral function
(15)Pdetθ=1detQθ,where det⋅ stands for the determinant of a matrix. Finally, Qθl can be obtained through spectral peak searching since the first L peaks correspond to Qθll=1L.Since the noise subspace is estimated by the sampling covariance matrix, the correlation between the signal steering vector and the signal subspace improves with increasing SNR. Namely, as the SNR increases, the orthogonality of the signal and noise subspace becomes more obvious. That is, at a low SNR, the signal steering vector and the noise subspace may not be orthogonal, i.e.,(16)aHθrCHUnUnHCaθr≠0,where θr stands for the DOA corresponding to a signal with a low SNR.Further, we can obtain(17)cHQθrc≠0.Distinctly,c is not the eigenvector of the matrix Qθr. In other words, if there are two signals, one with a high SNR and the other with a low SNR, such as an interference signal and a desired signal simultaneously incident on an array, the estimation accuracy decreases when Equations (13) and (14) are used to estimate the mutual coupling coefficients. In this case, we can eigendecompose only Qmax corresponding to the maximum peak of Equation (15); hence, the estimation of c can be given by
(18)c^=vm−minvm−min1,where vm−min is the eigenvector corresponding to the minimum eigenvalue of the matrix Qmax, and vm−min1 is the first entry of vm−min.Obviously, when multiple incident SNRs are similar, the use of Equation (18) to estimate the mutual coupling coefficients results in errors. However, in most cases, the SNRs of incident signals are not the same. For example, in RAB, due to the simultaneous existence of interference signals and the desired signal, the estimation accuracy of mutual coupling coefficients is higher when using Equation (18) than when using Equations (13) and (14).


### 3.2. Adaptive Beamforming with Unknown Mutual Coupling

Based on Equation (18), when there are one desired signal and L interference signals, the received data can be calibrated as
(19)x^k=C^−1xk=C^−1Cs0kaθ0+∑l=1Lslkaθl+C^−1nk≈s0kaθ0+∑l=1Lslkaθl+C^−1nk.where C^≜Toeplitzc^N,c^N, c^N=c^T,0T∈ℂN×1, θ0 is the direction of the desired signal, and s0k is the corresponding signal complex envelope.Recall that in Equation (3), the covariance matrix can be rewritten as
(20)Rx^=Ex^kx^Hk=σ02aθ0aHθ0+∑l=1Lσl2aθlaHθl+σn2C^−1C^−H.Distinctly, after the received data are calibrated, the noise covariance matrix is no longerσn2IM but is σn2C^−1C^−H; namely, the white Gaussian noise becomes nonwhite Gaussian noise, which seriously affects the performance of the beamformer. Therefore, we compensate for Rx^ to obtain
(21)R~x^=R^x^−σ^n2C^−1C^−H+σ^n2IM,where σ^n2 is the noise power estimate, which can be expressed as
(22)σ^n2=1M−L−1∑m=L+2Mλ^m,where λ^m (m=1,2,⋯,M) in descending order are the eigenvalues of the sample covariance matrix R^x^. Similarly, R^x^ is given by
(23)R^x^=1K∑k=1Kx^kx^Hk.Further, the INCM can be reconstructed by [24]
(24)R~i+n=R~x^−σ^02aθ0aHθ0+γIM,where γ is a diagonal loading factor set to further reduce the impact of nonwhite Gaussian noise and σ^02 can be calculated as
(25)σ^02=1aHθ0R~x^‐1aθ0.Under the minimum variance distortionless response principle for RAB, the weight vector is usually expressed as [25]
(26)wopt=Ri+n−1aθ0aHθ0Ri+n−1aθ0,where Ri+n denotes the ideal INCM without mutual coupling.UsingR~i+n in lieu of Ri+n yields
(27)w^opt=R~i+n−1aθ0aHθ0R~i+n−1aθ0.However, Equation (11) cannot be used to estimate the MCM whenM−L−1<P, since Qθ is a singular matrix for any θ. Namely, the DOF of the proposed algorithm is limited for a fixed ULA.The main computational load of the designed beamformer is caused by estimating mutual coupling coefficients. The process of estimating mutual coupling coefficients consists of two parts: one is the construction of spectral function, and the other is the search of spectral peaks. Hence, the computational load of the proposed method is approximatelyOM3+S×OP3, where OM3 is due to the construction of spectral function, S×OP3 is caused by the search of spectral peaks, and S is the number of spectral peak searches.In other words, the proposed beamformer can be summarized as follows:Step 1. Estimate the covariance matrix Rx~ using Equation (4), and eigendecompose it using Equation (5).Step 2. Construct the transformation matrix and the transitional matrix using Equations (7) and (12), respectively.Step 3. Estimate the transitional matrix Qmax corresponding to the highest SNR signal by searching for the highest peak of Equation (15), and estimate the mutual coupling coefficient vector using Equation (18).Step 4. Reconstruct the INCM by calibrating the received data and compensating for calibration errors using Equation (24).Step 5. Calculate the weight using Equation (27).


## 3.1. Mutual Coupling Coefficient Estimation

According to the banded symmetric Toeplitz structure of MCMC, we can obtain [19]
(7)Caθ=Tθc,where
(8)Tθ=E1aθ,E2θaθ,⋯,EPθaθ,where
(9)Epij=1,ifCij=cp0,otherwise,p=0,1,⋯,P−1.Based on the orthogonality betweenCaθl and the noise subspace Un, we can obtain
(10)aHθlCHUnUnHCaθl=0.Recall that Equations (7) and (10) can be rewritten as
(11)cHQθlc=0,where Qθ is a transitional matrix and is defined as
(12)Qθ=THθUnUnHTθ,which is a P×P matrix, and its dimensions are smaller than those of the transitional matrix in [11, 13].WhenP≤N−L, Qθ is a nonsingular matrix for a general θ since the ranks of Tθ and Un are P and N−L, respectively. However, if θ is one DOA of the incident signals, Qθ is a singular matrix, and its determinant is zero [23]. When Qθ is a singular matrix, we can find that Equation (11) holds and that c is an eigenvector of the matrix Qθ corresponding to the eigenvalue zero, since c is a nonzero vector. That is to say, the degree of freedom of the proposed algorithm is N−P.Namely, the mutual coupling coefficient vectorc can be estimated by
(13)cl=vlminvlmin1,(14)c^=1L∑l=1Lcl,where vlmin denotes the eigenvector corresponding to the minimum eigenvalue of Qθl, and vlmin1 is the first entry of vlmin.From Equation (12), we can find that the matrix Qθl can be easily calculated by the known DOA of the incident signals, but the DOAs are unknown. Hence, we construct a new spectral function
(15)Pdetθ=1detQθ,where det⋅ stands for the determinant of a matrix. Finally, Qθl can be obtained through spectral peak searching since the first L peaks correspond to Qθll=1L.Since the noise subspace is estimated by the sampling covariance matrix, the correlation between the signal steering vector and the signal subspace improves with increasing SNR. Namely, as the SNR increases, the orthogonality of the signal and noise subspace becomes more obvious. That is, at a low SNR, the signal steering vector and the noise subspace may not be orthogonal, i.e.,(16)aHθrCHUnUnHCaθr≠0,where θr stands for the DOA corresponding to a signal with a low SNR.Further, we can obtain(17)cHQθrc≠0.Distinctly,c is not the eigenvector of the matrix Qθr. In other words, if there are two signals, one with a high SNR and the other with a low SNR, such as an interference signal and a desired signal simultaneously incident on an array, the estimation accuracy decreases when Equations (13) and (14) are used to estimate the mutual coupling coefficients. In this case, we can eigendecompose only Qmax corresponding to the maximum peak of Equation (15); hence, the estimation of c can be given by
(18)c^=vm−minvm−min1,where vm−min is the eigenvector corresponding to the minimum eigenvalue of the matrix Qmax, and vm−min1 is the first entry of vm−min.Obviously, when multiple incident SNRs are similar, the use of Equation (18) to estimate the mutual coupling coefficients results in errors. However, in most cases, the SNRs of incident signals are not the same. For example, in RAB, due to the simultaneous existence of interference signals and the desired signal, the estimation accuracy of mutual coupling coefficients is higher when using Equation (18) than when using Equations (13) and (14).


## 3.2. Adaptive Beamforming with Unknown Mutual Coupling

Based on Equation (18), when there are one desired signal and L interference signals, the received data can be calibrated as
(19)x^k=C^−1xk=C^−1Cs0kaθ0+∑l=1Lslkaθl+C^−1nk≈s0kaθ0+∑l=1Lslkaθl+C^−1nk.where C^≜Toeplitzc^N,c^N, c^N=c^T,0T∈ℂN×1, θ0 is the direction of the desired signal, and s0k is the corresponding signal complex envelope.Recall that in Equation (3), the covariance matrix can be rewritten as
(20)Rx^=Ex^kx^Hk=σ02aθ0aHθ0+∑l=1Lσl2aθlaHθl+σn2C^−1C^−H.Distinctly, after the received data are calibrated, the noise covariance matrix is no longerσn2IM but is σn2C^−1C^−H; namely, the white Gaussian noise becomes nonwhite Gaussian noise, which seriously affects the performance of the beamformer. Therefore, we compensate for Rx^ to obtain
(21)R~x^=R^x^−σ^n2C^−1C^−H+σ^n2IM,where σ^n2 is the noise power estimate, which can be expressed as
(22)σ^n2=1M−L−1∑m=L+2Mλ^m,where λ^m (m=1,2,⋯,M) in descending order are the eigenvalues of the sample covariance matrix R^x^. Similarly, R^x^ is given by
(23)R^x^=1K∑k=1Kx^kx^Hk.Further, the INCM can be reconstructed by [24]
(24)R~i+n=R~x^−σ^02aθ0aHθ0+γIM,where γ is a diagonal loading factor set to further reduce the impact of nonwhite Gaussian noise and σ^02 can be calculated as
(25)σ^02=1aHθ0R~x^‐1aθ0.Under the minimum variance distortionless response principle for RAB, the weight vector is usually expressed as [25]
(26)wopt=Ri+n−1aθ0aHθ0Ri+n−1aθ0,where Ri+n denotes the ideal INCM without mutual coupling.UsingR~i+n in lieu of Ri+n yields
(27)w^opt=R~i+n−1aθ0aHθ0R~i+n−1aθ0.However, Equation (11) cannot be used to estimate the MCM whenM−L−1<P, since Qθ is a singular matrix for any θ. Namely, the DOF of the proposed algorithm is limited for a fixed ULA.The main computational load of the designed beamformer is caused by estimating mutual coupling coefficients. The process of estimating mutual coupling coefficients consists of two parts: one is the construction of spectral function, and the other is the search of spectral peaks. Hence, the computational load of the proposed method is approximatelyOM3+S×OP3, where OM3 is due to the construction of spectral function, S×OP3 is caused by the search of spectral peaks, and S is the number of spectral peak searches.In other words, the proposed beamformer can be summarized as follows:Step 1. Estimate the covariance matrix Rx~ using Equation (4), and eigendecompose it using Equation (5).Step 2. Construct the transformation matrix and the transitional matrix using Equations (7) and (12), respectively.Step 3. Estimate the transitional matrix Qmax corresponding to the highest SNR signal by searching for the highest peak of Equation (15), and estimate the mutual coupling coefficient vector using Equation (18).Step 4. Reconstruct the INCM by calibrating the received data and compensating for calibration errors using Equation (24).Step 5. Calculate the weight using Equation (27).


## 4. Simulations

In this section, we implement several simulations to validate the effectiveness and superiority of the proposed algorithm. A ULA withN=12 elements spaced a half-wavelength apart is used. The additive noise in the elements is modeled as spatially and temporally independent complex Gaussian noise with zero mean and unit variance. We assume that the mutual coupling coefficient vector is
(28)c=1,0.90e−jπ/3,0.75ejπ/4,0.45e−jπ/10,0.15e−jπ/6T.According to the previous analysis, an iterative algorithm and two subspace-based methods without iteration are selected for comparison, and the three algorithms are called Liao’s method [14], Elbir’s method [18], and Wen’s method [20]. In each trial, the number of angular sectors is selected as 12, and ε=10−4 in Elbir’s method. In addition, the Cramér-Rao bound (CRB) of the real and imaginary parts of the mutual coupling coefficients is provided in simulations. Note that the calculation of CRB is performed as in [26], and the CRB is modified to correspond to the real and imaginary parts of the mutual coupling coefficients.Example 1.
The directions of the desired signal and two interferences are assumed to be 10°, -20°, and 40°, respectively, and the corresponding SNR and interference-to-noise ratio (INR) are set to 10 dB and 40 dB, respectively. The number of snapshots is 500, and 100 Monte Carlo runs are performed. In this experiment, the results of the estimation of mutual coupling coefficients are listed in Table1 (a single run). The proposed method successfully estimates all mutual coefficients with high accuracy. Figure 1 displays the root mean square error (RMSE) of the real and imaginary parts of all mutual coupling coefficients versus SNR and the number of snapshots, where the RMSE is calculated by
(29)RMSE=1MP−1∑m=1M∑p=1P−1w^mp−wp2,where M is the number of Monte Carlo runs, wp denotes the real or imaginary part of cp, and w^mp stands for the estimated value of wp in the mth trial.Table 1
Mutual coupling coefficient estimation (SNR=10dB).
Mutual coupling coefficientsReal part (α)Imaginary part (β)α^αβ^βc10.44880.4500-0.7766-0.7794c20.52880.53030.52840.5303c30.42640.4280-0.13920.1391c40.13030.1299-0.0742-0.0750Figure 1
(a) RMSE of the mutual coupling coefficients versus SNR,K=500, INR=40dB. (b) RMSE of the mutual coupling coefficients versus the number of snapshots, SNR=10dB, INR=40dB.
(a)(b)Figure1(a) displays the RMSE values of the estimation of the real and imaginary parts of all the mutual coupling coefficients versus the SNR. The RMSE of the real and imaginary parts of all the mutual coupling coefficients versus the number of snapshots is shown in Figure 1(b). As we can see, the mutual coupling coefficient estimation accuracy of the proposed algorithm is always close to the CRB; when the difference in the signal power is large, the performance of the proposed method is closer to the CRB, and the larger the difference is, the more obvious the advantage of the proposed algorithm. However, when the power of each signal is similar, the performance of the proposed method is worse than that of Liao’s method and Wen’s algorithm but is still better than that of Elbir’s algorithm. These results occur because when all the signal powers are similar, the mutual coupling coefficients estimated by all the transitional matrices are close to each other, and the mutual coupling coefficients obtained using more information have higher accuracies. Additionally, when the SNR of the desired signal is significantly higher than the INR of the interference signal, the proposed method still yields better performance.Example 2.
The performance of the proposed beamformer and several classical robust adaptive beamformers, such as LSMI, MSB, MSRB, and SRB, is investigated in this example. In addition, the simulation compares a simplified SRB (SSRB) beamformer obtained utilizing Wen’s method to replace the mutual coupling coefficient estimation algorithm in the SRB beamformer. The output signal-to-interference-plus-noise ratio (SINR) versus the SNR with the number of snapshots fixed at 500 and the output SINR versus the number of snapshots with the SNR fixed at 10 dB are analyzed. Figure2(a) displays the output SINR of different beamformers versus SNR, while Figure 2(b) shows the output SINR of those approaches versus the number of snapshots.Figure 2
(a) Output SINR versus SNR,INR=40dB, K=500. (b) Output SINR versus the number of snapshots, SNR=10dB, INR=40dB.
(a)(b)In Figure2, the output SINR of the proposed beamformer is always close to the optimal SINR. In addition, at a high SNR, its performance is close to that of the SRB beamformer and SSRB beamformer, which is obviously better than that of the other algorithms. However, at a low SNR, the performance of the designed beamformer is obviously higher than that of the SRB beamformer and the SSRB beamformer due to the high-precision estimation of the mutual coupling matrix. The performance of the MSRB beamformer is limited by the array aperture and is always lower than that of the proposed beamformer. In addition, the designed beamformer has a faster convergence rate than the SRB beamformer and SSRB beamformer and is always close to the optimal output SINR.Example 3.
The performance of the designed beamformer with the unknown direction error is analyzed in this example. It is assumed that there is a random direction error of the desired signal, and it is uniformly distributed in−2°,2° in each trial. Namely, the direction error of the desired signal changes from run to run but remains fixed in one trial. The results are shown in Figure 3.Figure 3
(a) Output SINR versus SNR with the unknown direction error,INR=40dB, K=500. (b) Output SINR versus the number of snapshots with the unknown direction error, SNR=10dB, INR=40dB.
(a)(b)Compared with Figure2, the performance of the proposed beamformer, the SRB beamformer, and the SSRB beamformer is basically unchanged in Figure 3 because these three beamformers can estimate the desired signal direction when estimating the mutual coupling coefficients. The performance of the MSRB beamformer is only slightly degraded, but that of the MSB beamformer is seriously degraded because INCM reconstruction is utilized in the MSRB beamformer but not in the latter. Note that the convergence rate of the proposed beamformer is also very fast compared to those of other beamformers.Example 4.
The performance of the designed beamformer with incoherent local scattering.
The influence of incoherent local scattering is considered in this experiment. Generally, incoherent local scattering always occurs and seriously affects the performance of the beamformer. Here, we assume that in the case of incoherent local scattering, the received desired signal can be expressed as(30)xsk=aθ0s0k+∑q=13sqkaθq,where θqq=1,2,3 denotes the direction of the qth local scattering signal and is subject to uniform distribution in θ0−2∘,θ0+2∘ and sqk and aθq are the corresponding signal waveform and the steering vector, respectively. The simulation results are shown in Figure 4.Figure 4
(a) Output SINR versus SNR with incoherent local scattering,INR=40dB, K=500. (b) Output SINR versus the number of snapshots with incoherent local scattering, SNR=10dB, INR=40dB.
(a)(b)Figure4 displays the simulation result of each tested beamformer with incoherent local scattering. Compared with Figure 2, since local scattering signals disturb the sampling covariance matrix, the performance of all tested beamformers deteriorates to varying degrees, especially at high SNRs. Since the incoherent scattering signal causes serious errors in the estimation of the mutual coupling coefficient at a high SNR, the performance of all algorithms is seriously degraded at a high SNR. However, at a low SNR, the influence of the incoherent scattered signal is small, and the proposed algorithm only utilizes the highest power signal to estimate mutual coupling coefficients; thus, the performance of the proposed beamformer is always close to the optimal output SINR. In addition, the convergence rate of the proposed beamformer is very fast compared to those of other beamformers.


## 5. Conclusion

In this paper, a modified subspace-based mutual coupling coefficient estimation algorithm for beamforming is proposed. The main contribution of this manuscript includes two parts: (1) to improve the mutual coupling coefficient estimation accuracy, we propose a strategy to improve the estimation accuracy by choosing appropriate transition matrices; (2) to improve the robustness of the beamformer to unknown mutual coupling, a beamformer is designed combining the calibrated steering vector and the interference-plus-noise covariance matrix reconstruction method. The proposed beamformer has superior performance than exiting algorithms especially when there is a big power gap between different interferences. Simulations demonstrate the superiority of the modified subspace-based mutual coupling coefficient estimation algorithm and the robustness of the designed beamformer to unknown mutual coupling.



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*Source: 6669547-2020-12-21.xml*