Modified Robust Capon Beamforming with Approximate Orthogonal Projection onto the Signal-Plus-Interference Subspace

Abstract

In this paper, a new robust adaptive beamforming technique based on a modification of the robust Capon beamforming approach is introduced. It is shown that this technique leads to an approximate eigenspace projection-based array steering vector estimation. The steering vector is estimated as an approximation for the orthogonal projection of the presumed steering vector of the desired signal onto the signal-plus-interference subspace. Also, it is demonstrated that the optimal diagonal loading factor corresponds to the minimum of the estimated beamformer output power. Furthermore, the formulation of the proposed method provides the possibility of estimating the direction-of-arrival of the desired signal. This estimation is then used to update the presumed steering vector. An important virtue of the proposed method is its better or comparable performance with less computational complexity than the recent existing methods.

Appendix

1.1 Appendix A

The following inverse of a partitioned matrix [17] can be used to evaluate the inverse on the left-hand side of (10)

$$\begin{aligned} \left( \begin{array}{cc} \mathbf A \ &{} \mathbf C \\ \mathbf D \ &{} \mathbf B \end{array}\right) ^{-1} = \left( \begin{array}{c} 0 \\ \mathbf I \end{array}\right) \mathbf B ^{-1} \left( {\begin{array}{cc} 0\&\mathbf I \end{array}}\right) + \left( \begin{array}{c} \mathbf I \\ -\mathbf B ^{-1}{} \mathbf D \end{array}\right) (\mathbf A -\mathbf C {} \mathbf B ^{-1}{} \mathbf D )^{-1} \left( {\begin{array}{cc} \mathbf I&-\mathbf CB ^{-1} \end{array}}\right) \end{aligned}$$

(29)

where $\mathbf A \in C^{m\times m}$ , $\mathbf B \in C^{n\times n}$ , $\mathbf C \in C^{m\times n}$, $\mathbf D \in C^{n\times m}$. We need to find the inverse $(\mathbf A ^H_s\mathbf A _s)^{-1}$ as follows:

$$\begin{aligned} (\mathbf A ^H_s\mathbf A _s)^{-1}= \left( \begin{array}{cc} \Arrowvert \mathbf a _o\Arrowvert ^2\ &{} \mathbf{a }_o^H\mathbf A _i\\ \mathbf A _i^H\mathbf a _o\ &{} \mathbf A _i^H\mathbf A _i \end{array}\right) ^{-1} \end{aligned}$$

(30)

with the appropriate associations

$$\begin{aligned} (\mathbf A ^H_s\mathbf A _s)^{-1}&= \left( \begin{array}{c} 0 \\ \mathbf I \end{array}\right) (\mathbf A ^H_i\mathbf A _i)^{-1} \left( {\begin{array}{cc} 0\&\mathbf I \end{array}}\right) + \dfrac{1}{\Arrowvert \mathbf a _o\Arrowvert ^2-\mathbf a _o^H\mathbf A _i(\mathbf A ^H_i\mathbf A _i)^{-1}{} \mathbf A _i^H\mathbf a _o}\ \nonumber \\ {}&\quad .\left( \begin{array}{c} 1 \\ -(\mathbf A ^H_i\mathbf A _i)^{-1}{} \mathbf A _i^H\mathbf a _o \end{array}\right) \left( {\begin{array}{cc} 1\&-\mathbf a _o^H\mathbf A _i(\mathbf A ^H_i\mathbf A _i)^{-1} \end{array}}\right) \end{aligned}$$

(31)

Multiplying the left-hand side of (31) by $\mathbf A _s$ and the right-hand side by $\mathbf A _s^H$ and rearranging (31) will be expressed as (10).

1.2 Appendix B

The matrix $(\mathbf I +\lambda \mathbf R )$ can be written as follows:

$$\begin{aligned} \mathbf I +\lambda \mathbf R =(1+\lambda \sigma ^2_n)\mathbf I +\lambda \sigma ^2_s\mathbf a _o\mathbf a _o^H+\lambda \mathbf A _i\varvec{\varSigma }_i\mathbf A ^H_i=\mathbf R _{n\lambda }+\lambda \sigma ^2_s\mathbf a _o\mathbf a _o^H \end{aligned}$$

(32)

where $\mathbf R _{n\lambda }=(1+\lambda \sigma ^2_n)\mathbf I +\lambda \mathbf A _i\varvec{\varSigma }_i\mathbf A ^H_i$. Using the well-known matrix inversion lemma, the inverse becomes

$$\begin{aligned} (\mathbf I +\lambda \mathbf R )^{-1}=(\mathbf R _{n\lambda }+\lambda \sigma ^2_s\mathbf a _o\mathbf a _o^H)^{-1}=\mathbf R _{n\lambda }^{-1}-\mathbf R _{n\lambda }^{-1}\left( \dfrac{\lambda \sigma ^2_s\mathbf a _o\mathbf a _o^H}{1+\lambda \sigma ^2_s\mathbf a _o^H\mathbf R _{n\lambda }^{-1}{} \mathbf a _o}\right) \mathbf R _{n\lambda }^{-1} \end{aligned}$$

(33)

Again by using the same lemma, the inverse of $\mathbf R _{n\lambda }$ can be written as follows:

$$\begin{aligned} \mathbf R _{n\lambda }^{-1}=\dfrac{1}{1+\lambda \sigma ^2_n}[\mathbf I -\mathbf A _i((\lambda _d+\sigma ^2_n)\varvec{\varSigma }_i^{-1}+\mathbf A _i^H\mathbf A _i)^{-1}{} \mathbf A _i^H]=\dfrac{1}{1+\lambda \sigma ^2_n}(\mathbf I -\mathbf P ) \end{aligned}$$

(34)

where $\mathbf P$ is the second term within the square brackets and $\lambda _d=1/\lambda$. By substituting (34) into (33) , multiplying both sides by $\bar{\mathbf{a }}$ and simplifying gives

$$\begin{aligned} (\mathbf{I }+\lambda \mathbf{R })^{-1}{\bar{\mathbf{a }}}&=\dfrac{1}{1+\lambda \sigma ^{2}_{n}}\left[ (\mathbf{I }-\mathbf{P }){\bar{\mathbf{a }}}-\dfrac{\mathbf{a }_{o}^{H}(\mathbf{I }-\mathbf{P }){\bar{\mathbf{a }}}}{\mu (\lambda )+\mathbf{a }_{o}^{H}(\mathbf{I }-\mathbf{P })\mathbf{a }_{o}}(\mathbf{I }-\mathbf{P })\mathbf{a }_{o}\right] \nonumber \\ (1+\lambda \sigma ^2_n)(\mathbf{I }+\lambda \mathbf{R })^{-1}{\bar{\mathbf{a }}}&=(\mathbf{I }-\mathbf{P })[{\bar{\mathbf{a }}}-\eta (\lambda )\mathbf{a }_{o}] \nonumber \\&={\bar{\mathbf{a }}}-[\eta (\lambda )\mathbf{a }_{o}+\mathbf{P }({\bar{\mathbf{a }}}-\eta (\lambda )\mathbf{a }_{o})] \end{aligned}$$

(35)

where $\mu (\lambda )=(1+\lambda \sigma ^2_n)/(\lambda \sigma ^2_s)$ and $\eta (\lambda )=\mathbf a _o^H(\mathbf I -\mathbf P )\bar{\mathbf{a }}/ (\mu (\lambda )+\mathbf a _o^H(\mathbf I -\mathbf P )\mathbf a _o)$. From (35), we obtain

$$\begin{aligned} \hat{\mathbf{a }}_o(\lambda )&=\eta (\lambda )\mathbf a _o+\mathbf P (\bar{\mathbf{a }}-\eta (\lambda )\mathbf a _o) \nonumber \\&=\bar{\mathbf{a }}-(1+\lambda \sigma ^2_n)(\mathbf I +\lambda \mathbf R )^{-1}\bar{\mathbf{a }} \end{aligned}$$

(36)

Note that the noise power $\sigma ^2_n$ can be estimated from the eigenvalue decomposition of the covariance matrix. Here, the minimum of the eigenvalues is taken as an estimate of the noise power.