# Adaptive beamforming via steering vector estimation

In the era of 5G, the problem of spectrum congestion has become serious, especially with the popularity of smart devices. Hence, it is challengeable for adaptive beamforming to achieve effective interference suppression in complex propagation environments. In addition, the large number of antennas adopted in massive MIMO systems are also eager to face the array mismatch problem, which degrade the performance of adaptive beamformers.

Both interference multipath and array mismatch will degrade the accuracy of interference-plus-noise covariance matrix reconstruction. Due to the fact that the adaptive beamformer is a function of interference-plus-noise covariance matrix and desired signal steering vector, we consider improving the performance of adaptive beamformer by estimating the steering vector of the desired signal.
In detail, maximizing the Capon beamformer output power

$\hat{p}(\boldsymbol{a}) = \frac{1}{\boldsymbol{a}^{\mathrm{H}} \hat{\boldsymbol{R}}^{-1} \boldsymbol{a}} \nonumber$

leads to the following convex optimization problem

$\begin{array}{rcl}&\min\limits_{\boldsymbol{e}_{\bot}}& (\bar{\boldsymbol{a}} + \boldsymbol{e}_{\bot})^{\mathrm{H}} \hat{\boldsymbol{R}}^{-1} (\bar{\boldsymbol{a}} + \boldsymbol{e}_{\bot}) \nonumber \\&\mbox{subject to}& \bar{\boldsymbol{a}}^{\mathrm{H}} \boldsymbol{e}_{\bot} = 0 \nonumber \\ && (\bar{\boldsymbol{a}} + \boldsymbol{e}_{\bot})^{\mathrm{H}} \hat{\boldsymbol{R}} (\bar{\boldsymbol{a}} + \boldsymbol{e}_{\bot}) \le \bar{\boldsymbol{a}}^{\mathrm{H}} \hat{\boldsymbol{R}} \bar{\boldsymbol{a}}\end{array}$

where $\hat{\boldsymbol{R}}$ is the sample covariance matrix of the array received data, $\boldsymbol{a}$ and $\bar{\boldsymbol{a}}$ respectively denote the actual steering vector and the presumed steering vector, $\boldsymbol{e}_{\bot}$ is the orthogonal component of the mismatch vector $\boldsymbol{e} = \boldsymbol{a} - \bar{\boldsymbol{a}}$. Here, this inequality constraint is used to prevent the corrected steering vector from converging to any interferer steering vector. The numerical solution of the above optimization formulation is available.

Substituting the estimated steering vector $\tilde{\boldsymbol{a}} = \bar{\boldsymbol{a}} + \boldsymbol{e}_{\bot}$ into the minimum variance distortionless response (MVDR) beamformer, the signal steering vector estimation-based MVDR beamformer is given as

$\boldsymbol{w} = \frac{\hat{\boldsymbol{R}}^{-1} \tilde{\boldsymbol{a}}} {\tilde{\boldsymbol{a}}^{\mathrm{H}}\hat{\boldsymbol{R}}^{-1} \tilde{\boldsymbol{a}}} \nonumber$

is obtained.

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