Recursive Updating Algorithm for Robust Adaptive Beamforming in a Uniform Circular Array

Abstract

When the phase-mode transformation technique is used in a uniform circular array, its output performance is known to degrade owing to the approximations applied to the formulation. In this paper, we develop a robust recursive updating algorithm based on the worst-case performance optimization and phase-mode transformation in the uniform circular array, which provides efficient robustness not only against the signal steering vector mismatches, but also against the transformation errors. The proposed algorithm belongs to the class of the diagonal loading technique and the transformation matrix belongs to a certain ellipsoid set. The weight vector is updated by the Lagrange multiplier method, in which the parameters are derived simply. The proposed algorithm has a closed-form solution, in which we analyse the reasonable ranges of two key parameters. The convergence performance and the output SINR performance are also analysed. In additional, the implementation complexity costs of the proposed algorithm and MVDR algorithm are presented in this paper. Our robust algorithm has a low implementation complexity cost and achieves the mean output array SINR consistently close to the optimal one. Simulation results are presented to compare the performances of our algorithm with the conventional algorithms.

Appendix

1.1 A. The Ellipsoid Set Covering ${\varvec{U}}$

The first part of the array correlation matrix ${\varvec{U}}$ is given by

$$\begin{aligned} {\varvec{U}}_{1} & = \Delta {\varvec{G\hat{R}}}_{xx} {\varvec{Z}}^{H} \Delta {\varvec{Q}}^{H} \\ & = \left[ {\begin{array}{*{20}l} {z_{ - \nu } {\varvec{v}}_{ - \nu }^{H} {\hat{\varvec{R}}}_{xx} z_{ - \nu }^{ * } {\varvec{v}}_{ - \nu } } & \cdots & {z_{ - \nu } {\varvec{v}}_{ - \nu }^{H} {\hat{\varvec{R}}}_{xx} z_{\nu }^{ * } {\varvec{v}}_{\nu } } \\ \vdots & \ddots & \vdots \\ {z_{\nu } {\varvec{v}}_{\nu }^{H} {\hat{\varvec{R}}}_{xx} z_{ - \nu }^{ * } {\varvec{v}}_{ - \nu } } & \cdots & {z_{\nu } {\varvec{v}}_{\nu }^{H} {\hat{\varvec{R}}}_{xx} z_{\nu }^{ * } {\varvec{v}}_{\nu } } \\ \end{array} } \right] \\ \end{aligned}$$

(65)

where the diagonal error matrix $\Delta {\varvec{G}}$ and the matrix ${\varvec{Z}}$ are written in the following form for simplicity

$${\varvec{Z}} = \left[ {\begin{array}{*{20}c} {{\varvec{v}}_{ - \nu }^{H} } \\ {{\varvec{v}}_{ - \nu + 1}^{H} } \\ \vdots \\ {{\varvec{v}}_{\nu - 1}^{H} } \\ {{\varvec{v}}_{\nu }^{H} } \\ \end{array} } \right],\quad \Delta {\varvec{G}} = \left[ {\begin{array}{*{20}c} {z_{ - \nu } } & \cdots & {\varvec{0}} \\ \vdots & \ddots & \vdots \\ {\varvec{0}} & \cdots & {z_{\nu } } \\ \end{array} } \right]$$

(66)

where the each $z_{i}$ must satisfy the inequation $\left| {z_{i} } \right| \le \delta_{{i,\text{ }}} (i = - \nu , \ldots ,\nu )$.

The $i{\text{th}}$ column of (65) is equal to

$${\varvec{g}}_{1,i} = z_{ - \nu + i - 1}^{ * } \Delta {\varvec{G\hat{R}}}_{xx} {\varvec{v}}_{ - \nu + i - 1}$$

(67)

We consider the fact that the upper bound of $z_{i}$ is the following inequation

$$\vartheta (\Delta {\varvec{G}}\Delta {\varvec{G}}^{H} ) \le { \text{max} }\left\{ {\delta_{i}^{2} ,i = - \nu , \ldots ,\nu } \right\}$$

(68)

It is shown that the vector ${\varvec{g}}_{1,i}$ belongs to the following ellipsoid set

$$\begin{aligned} {\varvec{F}}_{1,i} ({\varvec{0}},{\varvec{T}}_{1,i} ) & = \left\{ {{\varvec{T}}_{1,i}^{1/2} {\varvec{\upupsilon}}|\text{ }\left\| {\varvec{\upupsilon}} \right\| \le 1} \right\} \hfill \\ {\varvec{T}}_{1,i}^{1/2} & = \delta_{ - \nu + i - 1} \varepsilon {\hat{\varvec{R}}}_{xx} \hfill \\ \end{aligned}$$

(69)

where the parameter $\varepsilon^{2}$ can be chosen as

$$\varepsilon^{2} = { \text{max} }\{ \delta_{i}^{2} ,\text{ }i = - \nu , \ldots ,\nu \}$$

(70)

The second part of the array correlation matrix is ${\varvec{U}}_{2} = \Delta {\varvec{G\hat{R}}}_{xx} {\varvec{G}}_{0}^{H}$, and we can write the $i{\text{th}}$ column of the matrix ${\varvec{U}}_{2}$ as

$${\varvec{g}}_{2,i} = \Delta {\varvec{G\hat{R}}}_{xx} {\varvec{v}}_{ - \nu + i - 1} {\varvec{Q}}_{ - \nu + i - 1}^{ * }$$

(71)

We can conclude that the vector ${\varvec{g}}_{2,i}$ belongs to the ellipsoid set [1]

$$\begin{aligned} {\varvec{F}}_{2,i} ({\varvec{0}},{\varvec{T}}_{2,i} ) & = \left\{ {{\varvec{T}}_{2,i}^{1/2} {\varvec{\upupsilon}}|\text{ }\left\| {\varvec{\upupsilon}} \right\| \le 1} \right\} \hfill \\ {\varvec{T}}_{2,i}^{1/2} & = \varepsilon \left| {{\varvec{Q}}_{ - \nu + i - 1} } \right|{\hat{\varvec{R}}}_{xx} \hfill \\ \end{aligned}$$

(72)

Similarly, we can conclude that the third part ${\varvec{U}}_{3} = {\varvec{G}}_{0}^{{}} {\hat{\varvec{R}}}_{xx} \Delta {\varvec{G}}^{H}$ belongs to the ellipsoid set

$$\begin{aligned} {\varvec{F}}_{3,i} ({\varvec{0}},{\varvec{T}}_{3,i} ) & = \left\{ {{\varvec{T}}_{3,i}^{1/2} {\varvec{\upupsilon}}|\text{ }\left\| {\varvec{\upupsilon}} \right\| \le 1} \right\} \hfill \\ {\varvec{T}}_{3,i}^{1/2} & = \delta_{ - \nu + i - 1} {\varvec{G}}_{0} {\hat{\varvec{R}}}_{xx} \hfill \\ \end{aligned}$$

(73)

1.2 B. The Detailed Proof of (24)

Proof

For an arbitrary ${\varvec{w}}$, we can prove that

$${\varvec{w}}^{H} \left( {\beta^{2} \Delta {\varvec{G}}\Delta {\varvec{G}}^{H} - \Delta {\varvec{Gaa}}^{H} \Delta {\varvec{G}}^{H} } \right){\varvec{w}} \ge 0,\text{ }\forall {\varvec{w}}$$

(74)

The inequality (74) is rewritten as

$${\varvec{w}}^{H} \Delta {\varvec{G}}\left( {\beta^{2} {\varvec{I}} - {\varvec{aa}}^{H} } \right)\Delta {\varvec{G}}^{H} {\varvec{w}} \ge 0,\text{ }\forall {\varvec{w}}$$

(75)

We can derive (75) as

$$\begin{aligned} & {\hat{\varvec{w}}}_{b}^{H} \left( {\beta^{2} {\varvec{I}} - {\varvec{aa}}^{H} } \right){\hat{\varvec{w}}}_{b} \ge 0,\text{ }\forall {\hat{\varvec{w}}}_{b} \hfill \\ & \Rightarrow \left\| {{\hat{\varvec{w}}}_{b}^{H} {\varvec{a}}} \right\| \le \beta \left\| {{\hat{\varvec{w}}}_{b} } \right\| \hfill \\ \end{aligned}$$

(76)

where the vector ${\hat{\varvec{w}}}_{b} = \Delta {\varvec{G}}^{H} {\varvec{w}}$.

Considering (76) for Euclidean norm, it is easy that we can conclude the inequality

$$\left\| {{\hat{\varvec{w}}}_{b}^{H} {\varvec{a}}} \right\| \le \left\| {{\hat{\varvec{w}}}_{b} } \right\|\left\| {\varvec{a}} \right\| = \beta \left\| {{\hat{\varvec{w}}}_{b} } \right\|$$

(77)

Therefore, the inequality (24) is true.

1.3 C. The Derivation of the Lagrange Multiplier $l$

The quadratic constraint of the optimization problem (28) is written as

$$h({\hat{\varvec{w}}}) = \text{ }1$$

(78)

where

$$h({\hat{\varvec{w}}}) = {\hat{\varvec{w}}}^{H} (k)\left( {\xi^{2} {\varvec{I}} - {\varvec{dd}}^{H} } \right){\hat{\varvec{w}}}(k) + {\varvec{d}}^{H} {\hat{\varvec{w}}}(k) + \text{ }{\hat{\varvec{w}}}^{H} (k){\varvec{d}}$$

(79)

Inserting (31) into (78), we can obtain the Lagrange multiplier $l$ as

$$l = \frac{1}{{2u^{2} \left[ {{\varvec{J}}^{H} (k)(\xi^{2} {\varvec{I}} - {\varvec{dd}}^{H} ){\varvec{J}}(k)} \right]}}\left( {\zeta (k) + \rho (k)} \right)$$

(80)

where

$$\begin{aligned} {\varvec{J}}(k) & = \left( {\xi^{2} {\varvec{I}} - {\varvec{dd}}^{H} } \right){\hat{\varvec{w}}}(k) + {\varvec{d}} \hfill \\ \zeta (k) & = u\left[ {{\varvec{q}}^{H} (k)\left( {\xi^{2} {\varvec{I}} - {\varvec{dd}}^{H} } \right){\varvec{J}}(k) + \varepsilon^{2} {\varvec{J}}^{H} (k){\varvec{Iq}}(k)} \right] \hfill \\ & \quad + u{\varvec{J}}^{H} (k){\varvec{d}} + {\varvec{d}}^{H} {\varvec{J}}(k) - u{\varvec{J}}^{H} (k){\varvec{dd}}^{H} {\varvec{q}}(k) \hfill \\ \rho^{\text{H}} (k)\rho (k) & = \zeta^{H} (k)\zeta (k) - 4u^{2} {\varvec{J}}^{H} (k)\left( {\xi^{2} {\varvec{I}} - {\varvec{dd}}^{H} } \right) \hfill \\ & \quad \cdot {\varvec{J}}(k)\left( {\phi (k) - 1} \right) \hfill \\ \phi (k) & = {\varvec{q}}^{H} (k)\left( {\xi^{2} {\varvec{I}} - {\varvec{dd}}^{H} } \right){\varvec{q}}(k) + {\varvec{q}}^{H} (k){\varvec{d}} + {\varvec{d}}^{H} {\varvec{q}}(k) \hfill \\ \end{aligned}$$

(81)

In (39), ${\varvec{q}}(k) = [{\varvec{I}} - u{\varvec{U}}]{\hat{\varvec{w}}}(k)$.

From the Eqs. (80) and (81), we note that the solving process of the parameter $l$ is very complicated. To decrease the computation cost, we can simply compute $l$ by the linear combination method. According to [26], we consider a linear combination with the form

$${\hat{\varvec{U}}}_{d} = \gamma \varvec{I} + \eta {\hat{\varvec{G}}}$$

(82)

where ${\hat{\varvec{U}}} = \left( {{\varvec{G}}_{0} + \Delta {\varvec{G}}} \right)\left( {\varvec{R}_{\varvec{N}} + r{\varvec{I}}} \right)\left( {{\varvec{G}}_{0} + \Delta {\varvec{G}}} \right)^{H}$, $\gamma > 0$, and $\eta > 0$.

Then, we can obtain the enhanced covariance matrix as

$${\varvec{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{U} }} = {\hat{\varvec{U}}} + \frac{\gamma }{\eta }\varvec{\rm I}$$

(83)

Contrasting the diagonal loading covariance matrix ${\hat{\varvec{U}}} + l\xi^{2} {\varvec{\rm I}}$ in (33) and ${\hat{\varvec{U}}} + \frac{\gamma }{\eta }{\varvec{\rm I}}$ in (83), we derive that the loading factor $l\xi^{2}$ can be replaced by ${\gamma \mathord{\left/ {\vphantom {\gamma \eta }} \right. \kern-0pt} \eta }$. In this way, we can obtain the parameter $l$ as

$$l = \frac{\gamma }{\eta } \cdot \frac{1}{{\xi^{2} }}$$

(84)

Contrasting (80) and (84), the proposed algorithm can decrease the complexity cost. Next, we must get the parameters $\gamma$ and $\eta$ by minimizing the mean squared error (MSE) of ${\hat{\varvec{U}}}_{d}$ as

$$f = { \text{min} }\,E\left\{ { \, \left\| {{\hat{\varvec{U}}}_{d} - {\varvec{U}}} \right\|^{2} } \right\}$$

(85)

The minimization problem is derived as [27]

$$\begin{aligned} f & = E\left\{ {\left\| {\gamma \varvec{\rm I} - \left( {1 - \eta } \right){\varvec{U}} + \eta \left( {{\hat{\varvec{U}}} - {\varvec{U}}} \right)} \right\|^{2} } \right\} \\ &= \left\| {\gamma \varvec{\rm I} - \left( {1 - \eta } \right){\varvec{U}}} \right\|^{2} + \eta^{2} E\left\{ {\left\| {{\hat{\varvec{U}}} - {\varvec{U}}} \right\|^{2} } \right\} \\ &= \gamma^{2} M - 2\gamma \left( {1 - \eta } \right)tr({\varvec{U}}) + \left( {1 - \eta } \right)^{2} \left\| {\varvec{U}} \right\|^{2} + \eta^{2} E\left\{ {\left\| {{\hat{\varvec{U}}} - {\varvec{U}}} \right\|^{2} } \right\} \\ \end{aligned}$$

(86)

By minimizing (86) with respect to $\gamma$, we can obtain the optimal $\gamma_{0}$ for fixed $\eta$ as

$$\gamma_{0} = \frac{{\left( {1 - \eta_{0} } \right) \, tr\left( {\varvec{U}} \right)}}{M}$$

(87)

Inserting $\gamma_{0}$ into (86) and replacing $\eta_{0}$ by $\eta$, we give another unconstrained minimization problem

$$\mathop { \text{min} }\limits_{\eta } \, \frac{{\left( {1 - \eta } \right)^{2} \left\{ {\left\| {\varvec{U}} \right\|^{2} M - tr^{2} \left( {\varvec{U}} \right)} \right\}}}{M} + \eta^{2} E\left\{ {\left\| {{\hat{\varvec{U}}} - {\varvec{U}}} \right\|^{2} } \right\}$$

(88)

Computing the gradient of (88) and equating it to zero yields the optimal solution for $\eta$ as

$$\eta_{0} = \frac{\alpha }{\alpha + \chi }$$

(89)

where $\alpha = \frac{{\left\| {\varvec{U}} \right\|^{2} M - tr^{2} \left( {\varvec{U}} \right)}}{M},\quad \chi = E\left\{ {\left\| {{\hat{\varvec{U}}} - {\varvec{U}}} \right\|^{2} } \right\}$

To estimate $\gamma_{0}$ and $\eta_{0}$ from the received available data, we need to obtain the estimation values for $\chi$ and $\alpha$. In practice, the exact array correlation matrix ${\varvec{U}}$ is unavailable. Therefore, ${\varvec{U}}$ is replaced by ${\hat{\varvec{U}}}$ to estimate $\alpha$ as

$$\hat{\alpha } = \frac{{\left\| {{\hat{\varvec{U}}}} \right\|^{2} M - tr^{2} \left( {{\hat{\varvec{U}}}} \right)}}{M}$$

(90)

Let $\hat{r}_{m}$ and $r_{m}$ denote the $m{\text{th}}$ columns of ${\hat{\varvec{U}}}$ and ${\varvec{U}}$, respectively. We have

$$E\left\{ {\left\| {{\hat{\varvec{U}}} - {\varvec{U}}} \right\|^{2} } \right\} = \sum\limits_{m = 1}^{M} {E\left\{ { \, \left\| {\hat{r}_{m} - r_{m} } \right\|^{2} } \right\}}$$

(91)

Alternatively, we can use the unbiased estimate $\left\| {{\hat{\varvec{U}}} - {\varvec{U}}} \right\|^{2}$ for $E\left\{ {\left\| {{\hat{\varvec{U}}} - {\varvec{U}}} \right\|^{2} } \right\}$.

Thereby, we can estimate $\chi$ as [26]

$$\hat{\chi } = \left\| {{\hat{\varvec{U}}} - {\varvec{U}}} \right\|^{2}$$

(92)

Applying (90) and (92), we can derive the estimation values for $\gamma_{0}$ and $\eta_{0}$, respectively

$$\hat{\gamma }_{0} = \frac{{\left( {1 - \hat{\eta }_{0} } \right) \, tr\left( {\hat{\varvec{U}}} \right)}}{M}$$

(93)

and

$$\hat{\eta }_{0} = \frac{{\hat{\alpha }}}{{\hat{\alpha } + \hat{\chi }}}$$

(94)

Finally, the diagonal loading factor $\lambda \xi^{2}$ is replaced by the estimation ratio ${{\hat{\gamma }_{0} } \mathord{\left/ {\vphantom {{\hat{\gamma }_{0} } {\hat{\eta }}}} \right. \kern-0pt} {\hat{\eta }}}_{0}$. Thus we can obtain the expression of the Lagrange multiplier $l$ as

$$\hat{l} = \frac{{\hat{\gamma }_{0} }}{{\hat{\eta }_{0} }} \cdot \frac{1}{{\xi^{2} }}.$$

(95)