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Steering vector optimization using subspace-based constraints for robust adaptive beamforming

Abstract

To address the issue of steering vector mismatch, a robust adaptive beamforming design via steering vector optimization is proposed in this paper. Different from conventional studies, this paper resolves the exact desired signal (DS) steering vector through formulating an array output power maximization problem subjected to noise subspace (NS) based and interference subspace (IS) based constraints. Under the condition that the NS is ready to be attained while the IS is hard to be got, two efficient interference-plus-noise covariance matrix (INCM) reconstruction means, i.e. direct DS matrix removal from sample covariance matrix and indirect DS blocking from training data and matrix transition, are derived to estimate the IS with high accuracy. Herein, after resolving DS steering vector, the weight vectors are thereby extracted with orthogonal projection (OP) criterion. Numerical simulations verify that the devised methods can outperform the existing ones and obtain almost optimal performance across a wide range of DS power.

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Correspondence to Pan Zhang.

Appendices

Appendix 1

Derivation Of (18).

The eigen-decomposition of the DSNCM \({\tilde{\mathbf{R}}}_{SN}\) in (17) can be given as:

$$ {\tilde{\mathbf{R}}}_{SN} { = }\sum\limits_{i = 1}^{M} {\mu_{i} {\mathbf{p}}_{i} {\mathbf{p}}_{i}^{{\text{H}}} } $$
(30)

where \(\mu_{i} ,i = 1,2, \cdots ,M\) denote the eigen-values in descend order, and \({\mathbf{p}}_{i} ,i = 1,2, \cdots ,M\) denote the corresponding eigen-vectors. Therefore, the inverse of the DSNCM \({\tilde{\mathbf{R}}}_{SN}\) can be represented as:

$$ {\tilde{\mathbf{R}}}_{SN}^{ - 1} { = }\sum\limits_{i = 1}^{M} {\frac{{{\mathbf{p}}_{i} {\mathbf{p}}_{i}^{{\text{H}}} }}{{\mu_{i} }}} $$
(31)

As we can notice, if \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\sigma }_{i}^{{2}} \gg \tilde{\sigma }_{n}^{2} ,i = - l, - l + 1, \cdots ,l\) and \(2l + 1 < M\) are fulfilled, \({\tilde{\mathbf{R}}}_{SN}^{ - 1}\) can be approximated as:

$$ {\tilde{\mathbf{R}}}_{SN}^{ - 1} { = }\sum\limits_{i = 1}^{2l + 1} {\frac{{{\mathbf{p}}_{i} {\mathbf{p}}_{i}^{{\text{H}}} }}{{\tilde{\sigma }_{i}^{2} + \tilde{\sigma }_{n}^{2} }}} + \sum\limits_{2l + 2}^{M} {\frac{{{\mathbf{p}}_{i} {\mathbf{p}}_{i}^{{\text{H}}} }}{{\tilde{\sigma }_{n}^{2} }}} \cong \sum\limits_{2l + 2}^{M} {\frac{{{\mathbf{p}}_{i} {\mathbf{p}}_{i}^{{\text{H}}} }}{{\tilde{\sigma }_{n}^{2} }}} $$
(32)

In consideration of that \(\sum\nolimits_{i = 1}^{M} {{\mathbf{p}}_{i} {\mathbf{p}}_{i}^{{\text{H}}} } = {\mathbf{I}}\) is also held, \({\tilde{\mathbf{R}}}_{SN}^{ - 1}\) can be further altered as:

$$ {\tilde{\mathbf{R}}}_{SN}^{ - 1} \cong \frac{1}{{\tilde{\sigma }_{n}^{2} }}\left( {{\mathbf{I}} - \sum\limits_{i = 1}^{2l + 1} {{\mathbf{p}}_{i} {\mathbf{p}}_{i}^{{\text{H}}} } } \right) $$
(33)

The subspace spanned by \([{\mathbf{p}}_{1} ,{\mathbf{p}}_{2} , \cdots ,{\mathbf{p}}_{2l + 1} ]\) is equivalent to that spanned by \([{\mathbf{a}}(\tilde{\theta }_{0} - l\Delta ),{\mathbf{a}}(\tilde{\theta }_{0} - l\Delta + \Delta ), \cdots ,{\mathbf{a}}(\tilde{\theta }_{0} + l\Delta )]\), so \({\tilde{\mathbf{R}}}_{SN}^{ - 1}\) owns the blocking feature as:

$$ {\tilde{\mathbf{R}}}_{SN}^{ - 1} {\mathbf{a}}(\tilde{\theta }_{0} + i\Delta ) \cong {\mathbf{0}},i = - l, - l + 1, \cdots ,l $$

Appendix 2

Derivation of (24).

Alternatively, the revised quasi INCM \({\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{R} }}_{IN,2}\) in (23) can also be denoted as:

$$ {\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{R} }}_{IN,2} = \sum\limits_{i = 1}^{J} {\sigma_{i}^{2} {\tilde{\mathbf{R}}}_{SN}^{ - 1} {\mathbf{a}}(\theta_{i} ){\mathbf{a}}^{{\text{H}}} (\theta_{i} )({\tilde{\mathbf{R}}}_{SN}^{ - 1} )^{{\text{H}}} } + \tilde{\sigma }_{n}^{2} {\mathbf{I}} $$

Clearly, it is simple to draw the following conclusion using the equivalence between (23) and (35):

$$ \sum\limits_{i = 1}^{J} {\sigma_{i}^{2} {\tilde{\mathbf{R}}}_{SN}^{ - 1} {\mathbf{a}}(\theta_{i} ){\mathbf{a}}^{{\text{H}}} (\theta_{i} )({\tilde{\mathbf{R}}}_{SN}^{ - 1} )^{{\text{H}}} } = \sum\limits_{i = 1}^{J} {{\mathbf{v}}_{i} (\lambda_{i} - \tilde{\sigma }_{n}^{2} ){\mathbf{v}}_{i}^{{\text{H}}} } $$
(36)

For the sake of recovering \(\sum\nolimits_{i = 1}^{J} {\sigma_{i}^{2} {\mathbf{a}}(\theta_{i} ){\mathbf{a}}^{{\text{H}}} (\theta_{i} )}\), we need to pre-multiply and post-multiply both sides of (36) by \({\tilde{\mathbf{R}}}_{SN}\) and \({\tilde{\mathbf{R}}}_{SN}^{{\text{H}}}\), that is:

$$ \sum\limits_{i = 1}^{J} {\sigma_{i}^{2} {\mathbf{a}}(\theta_{i} ){\mathbf{a}}^{{\text{H}}} (\theta_{i} )} = \sum\limits_{i = 1}^{J} {{\tilde{\mathbf{R}}}_{SN} {\mathbf{v}}_{i} (\lambda_{i} - \tilde{\sigma }_{n}^{2} ){\mathbf{v}}_{i}^{{\text{H}}} {\tilde{\mathbf{R}}}_{SN}^{{\text{H}}} } $$
(37)

Then, we add the obtained noise component \(\tilde{\sigma }_{n}^{2} {\mathbf{I}}\) to (37), which leads to the reconstructed INCM as:

$$ {\hat{\mathbf{R}}}_{IN} = \sum\limits_{i = 1}^{J} {{\tilde{\mathbf{R}}}_{SN} {\mathbf{v}}_{i} (\lambda_{i} - \tilde{\sigma }_{n}^{2} ){\mathbf{v}}_{i}^{{\text{H}}} {\tilde{\mathbf{R}}}_{SN}^{{\text{H}}} } + \tilde{\sigma }_{n}^{2} {\mathbf{I}} $$
(38)

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Zhang, P. Steering vector optimization using subspace-based constraints for robust adaptive beamforming. Multidim Syst Sign Process 32, 1083–1102 (2021). https://doi.org/10.1007/s11045-021-00775-y

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Keywords

  • Adaptive beamforming
  • Steering vector optimization
  • Subspace-based constraints
  • INCM reconstruction
  • OP criterion