A Variable Diagonal Loading Beamformer with Joint Uncertainties of Steering Vector and Covariance Matrix

Abstract

In this paper, a novel robust adaptive beamforming is proposed in which both the uncertainties of steering vector and covariance matrix are taken into account. First we develop a min–max optimization problem which aims to find a steering vector with the maximum output power under the worst-case covariance mismatch. Then we relax this min–max optimization problem to a max–min optimization problem which can be solved by using the Karush–Kuhn–Tucker optimality conditions. It is also shown that the proposed technique can be interpreted in terms of variable diagonal loading where the optimal loading factors are related to both the correlations (between the eigenvectors and the signal of interest) and the eigenvalues of the data covariance matrix. The effectiveness of the proposed approach is supported by computer simulation results.

Appendix: Proof of (24)

Let us start with the equation

$$\begin{aligned} \underset{j=1}{\overset{N}{\sum }}\frac{z_j^{2}}{\left[ 1+{\uplambda }\left( \widehat{\gamma _j}+\beta _j\right) \right] ^{2}}=\epsilon _1. \end{aligned}$$

(31)

Differentiating with respect to $\beta _i$, we have

$$\begin{aligned} \underset{j=1}{\overset{N}{\sum }} \frac{-2z_j^2\left[ \left( \widehat{\gamma }_j+\beta _j\right) \frac{\partial {\uplambda }}{\partial \beta _i}+{\uplambda }\delta (i-j)\right] }{\left[ 1+{\uplambda }\left( \widehat{\gamma }_j+\beta _j\right) \right] ^3}=0 \end{aligned}$$

(32)

where $\delta (i)$ is the impulse function. Therefore,

$$\begin{aligned} \frac{\partial {\uplambda }}{\partial \beta _i}=- \frac{\frac{{\uplambda }z_i^2}{\left[ 1+{\uplambda }\left( {\widehat{\gamma}}_i+\beta _i\right) \right] ^3}}{\sum \nolimits _{j=1}^{N} \frac{z_j^2\left( {\widehat{\gamma}}_j+\beta _j\right) }{\left[ 1+{\uplambda }\left( {\widehat{\gamma}}_j+\beta _j\right) \right] ^3}} \end{aligned}$$

(33)

Using (22) and differentiating it with respect to $\beta _i$, we have

$$\begin{aligned} -\frac{\partial f}{\partial \beta _i}= & {} \underset{j=1}{\overset{N}{\sum }} \frac{z_j^2 \delta (i-j)}{\left( \widehat{\gamma }_j+\beta _j+\frac{1}{{\uplambda }}\right) ^2}\nonumber \\&+ \underset{j=1}{\overset{N}{\sum }} \frac{-2z_j^2\left( \widehat{\gamma }_j+\beta _j\right) \left[ \delta (i-j)-\frac{1}{{\uplambda }^2}\frac{\partial {\uplambda }}{\partial \beta _i}\right] }{\left( \widehat{\gamma }_j+\beta _j+\frac{1}{{\uplambda }}\right) ^3}\nonumber \\= & {} \frac{z_i^2\left[ {\uplambda }^2-{\uplambda }^3(\widehat{\gamma }_i+\beta _i)\right] }{\left[ 1+{\uplambda }(\widehat{\gamma }_j+\beta _j)\right] ^3} +2{\uplambda }\frac{\partial {\uplambda }}{\partial \beta _i}\underset{j=1}{\overset{N}{\sum }} \frac{z_j^2\left( \widehat{\gamma }_j+\beta _j\right) }{\left[ 1+{\uplambda }(\widehat{\gamma }_j+\beta _j)\right] ^3} \end{aligned}$$

(34)

Substituting $\frac{\partial {\uplambda }}{\partial \beta _i}$ from (33), we have

$$\begin{aligned} -\frac{\partial f}{\partial \beta _i}= & {} \frac{z_i^2\left[ {\uplambda }^2-{\uplambda }^3\left( \widehat{\gamma }_i+\beta _i\right) \right] }{\left[ 1+{\uplambda }\left( \widehat{\gamma }_i+\beta _i\right) \right] ^3} +\frac{-2{\uplambda }^2 z_i^2}{\left[ 1+{\uplambda }\left( \widehat{\gamma }_i+\beta _i\right) \right] ^3}\nonumber \\= & {} -\frac{{\uplambda }^2 z_i^2}{\left[ 1+{\uplambda }\left( \widehat{\gamma }_i+\beta _i\right) \right] ^2} \end{aligned}$$

(35)

which is equivalent to (24).