Novel Robust Adaptive Beamformer in the Presence of Gain-Phase Errors

In this paper, we consider the problem of robust adaptive beamforming (RAB) for a linear array in the presence of gain-phase errors. By setting a few calibrated auxiliary elements on one side of a linear array, we propose a novel gain error and phase error estimation algorithm to calibrate gain-phase errors. According to the proposed estimation algorithm, the unknown gain error and phase error are jointly estimated by calculating the eigenvector corresponding to the minimum eigenvalue of the available transitional matrices. Furthermore, we calibrate the received data and compensate for the nonwhite noise caused by the calibration process. Then, the interference-plus-noise covariance matrix (INCM) is reconstructed based on the calibrated steering vector. By incorporating the reconstructed INCM with the minimum variance distortionless response principle, a novel robust adaptive beamformer for a linear array with gain-phase errors is designed. The proposed beamformer can obviously improve RAB performance in a linear array with gain-phase errors. The robustness and superiority of the designed beamformer are demonstrated in simulations.

Not applicable.

This work was supported in part by the National Natural Science Foundation of China under Grants 61901511 and 61871471, in part by the National Science Foundation in Shaanxi Province of China under Grant 2019JM-322, in part by the Natural Science Basic Research Plan in Shaanxi Province of China under Grant 2020JQ-478, and in part by the Research Program of the National University of Defense Technology under Grant ZK19-10.

Authors and Affiliations

1. Graduate College, Air Force Engineering University, Xi’an, 710051, China

   Qichao Ge & Ziang Feng

2. Air and Missile Defense College, Air Force Engineering University, Xi’an, 710051, China

   Yongshun Zhang

3. College of Information and Communication, National University of Defense Technology, Xi’an, 710100, China

   Xiangyang Liu

Contributions

Q. G and Z. F contributed to data curation; Q. G helped with methodology; Y. Z contributed to project administration; and X. L helped with funding acquisition.

Corresponding author

Correspondence to Qichao Ge.

Conflicts of interest

The authors declare no conflicts of interest.

Code Availability

The program and code used in this manuscript are available. If anyone wants to use these programs, please contact the author directly. The author will send the program to anyone who requests it through email for the first time after receiving the request.

Publisher's Note

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Appendix A

The proof by contradiction approach is used to prove the feasibility of the proposed joint estimation algorithm.

It is assumed that \( \theta_{l} ,l = 1,2, \ldots ,L \) are the DOAs of the incident signals and that there is another angle \( \theta_{i} \ne \theta_{l} , \) \( l = 1,2, \ldots ,L \) that makes \( {\mathbf{Q}}\left( {\theta_{i} } \right) \) a singular matrix. Through the previous analysis, we can obtain

Substituting (36) into (61) yields

That is,

Therefore, \( {\tilde{\mathbf{a}}}\left( {\theta_{i} } \right) \) is orthogonal to the noise subspace, that is,

Since \( {\tilde{\mathbf{a}}}\left( {\theta_{l} } \right),l = 1,2, \ldots ,L \) can also span the signal subspace, we can obtain

Therefore, there is a set \( k_{l} ,\;\;l = 1,2, \ldots ,L,\;\;k_{1} k_{2} \ldots k_{L} \ne 0 \) such that

In an array without grating lobes, the steering vector corresponding to each angle (from 0 to \( \pi \)) is uncorrelated; namely, \( {\tilde{\mathbf{a}}}\left( {\theta_{l} } \right),\;l = 1,2, \ldots ,L \) and \( {\tilde{\mathbf{a}}}\left( {\theta_{i} } \right) \) are uncorrelated. Therefore, in the set \( k_{l} ,\;\;l = 1,2, \ldots ,L,\;\;k_{1} k_{2} \ldots k_{L} \ne 0 \), only one element is equal to 1, and the rest are all equal to 0. That is, \( \theta_{i} \) is one DOA of the incident signals, which is the opposite of the assumption. That is, all the angles that make \( {\mathbf{Q}}\left( \theta \right) \) a singular matrix are the DOA estimations of the signals.

Appendix B

Here, we define the parameter vector

where

where \( G = M - P + 1 \).

Therefore, the \( (m,n) \)th element of the Fisher information matrix (FIM) is given by Wang et al. [20]:

where

where \( {\mathbf{p}} = \text{diag} ({\mathbf{R}}_{s} ) = [\sigma_{0}^{2} ,\sigma_{1}^{2} , \ldots ,\sigma_{L - 1}^{2} ]^{T} \) and \( \text{vec} ( \cdot ) \), \( ( \cdot )^{ * } \), and \( \odot \) denote the vectorization of a matrix, the conjugate, and the Khatri–Rao product, respectively.

From (67), we obtain

where \( {\varvec{\upvarepsilon}} = [\varepsilon_{ 2} ,\varepsilon_{ 3} , \ldots ,\varepsilon_{G} ] \), \( {\boldsymbol{\varsigma }} = [\varsigma_{ 2} ,\varsigma_{ 3} , \ldots ,\varsigma_{G} ] \).

Based on (71), we can compute the derivatives in (72) and obtain

where \( {\tilde{\mathbf{A}}} = {\boldsymbol{\varPsi}{\bf A}} \).

Furthermore,

where \( {\mathbf{I}}_{G - 1} \in {\mathbb{R}}^{(G - 1) \times (G - 1)} \) is an identity matrix,

where \( {\mathbf{i}}_{G - 1} { = }[1,1, \ldots ,1] \in {\mathbb{R}}^{{1 \times \left( {G - 1} \right)}} \), \( {\mathbf{I}}_{L} \in {\mathbb{R}}^{L \times L} \) is an identity matrix and

Recall that from (69), the FIM can be expressed as

Since \( {\mathbf{R}}_{{\tilde{x}}} \) is a positive definite matrix, \( ({\mathbf{R}}_{{\tilde{x}}}^{T} \otimes {\mathbf{R}}_{{\tilde{x}}} )^{ - 1} \) is also a positive definite matrix. Therefore, we can define

Hence, (83) can be rewritten as

Finally, the CRB of each parameter is given by

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