Robust Adaptive Beamforming via Virtual Interpolation-Based Atomic Norm Minimization

Abstract

The interpolation concept can fill the holes in the difference coarray of nonuniform linear arrays (NLAs), thus ensuring effective enhancement of the degrees of freedom and expansion of the array aperture. However, compared to interpolation-based direction of arrival (DOA) estimation techniques, the application of interpolation to beamforming approaches has not been universally investigated. In this study, two robust adaptive beamforming algorithms are proposed. First, the observed data is interpolated and the signal of interest (SOI) is subtracted. Second, the vector completion problem of the received data associated with the derived virtual array is formulated, which can be implemented with the utilization of atomic norm minimization. Unlike previous algorithms, the proposed method obtains the interference plus noise covariance matrix (INCM) and the DOA of the SOI by solving the vector completion problem rather than matrix completion. Thereafter, this problem is solved iteratively in two convex steps, whose closed-form expressions for the alternating direction method of multipliers are derived to enhance computational efficiency. The first proposed algorithm calculates the weight vector using rows corresponding to the physical sensor locations of the INCM. The second one fills the gaps of NLAs using the virtual filling technique to form a consecutive array and directly applies the resulting INCM to compute the weight vector. Numerical simulations demonstrate the superior overall performance of the two proposed methods under nonideal scenarios.

Appendices

Appendix A

First, we substitute ${\mathbf{y}} - p_{0} {\mathbf{b}}_{0}$ into the left side of (18), it gains

$$ \begin{gathered} \tau ({\mathbf{y}}_{q} ) = \tau ({\mathbf{y}} - p_{0} {\mathbf{b}}_{0} ) = \tau ({\mathbf{y}}) - p_{0} \tau ({\mathbf{b}}_{0} ) = \sum\limits_{q = 1}^{Q} {p_{q} \tau ({\mathbf{b}}_{q} )} \hfill \\ = p_{1} \left[ {\begin{array}{*{20}l} 1 \hfill & {\quad e^{{ - j\pi \sin \theta_{1} }} } \hfill & {\quad \cdots } \hfill & {\quad e^{{ - j\pi (L - 1)\sin \theta_{1} }} } \hfill \\ {e^{{j\pi \sin \theta_{1} }} } \hfill & {\quad 1} \hfill & \quad \hfill & \quad \hfill \\ \vdots \hfill & \quad \hfill & {\quad \ddots } \hfill & {\quad \vdots } \hfill \\ {e^{{j\pi (L - 1)\sin \theta_{1} }} } \hfill & \quad \hfill & {\quad \cdots } \hfill & {\quad 1} \hfill \\ \end{array} } \right] + \cdots \hfill\\ + p_{Q} \left[ {\begin{array}{*{20}l} 1 \hfill & {\quad e^{{ - j\pi \sin \theta_{Q} }} } \hfill & {\quad \cdots } \hfill & {\quad e^{{ - j\pi (L - 1)\sin \theta_{Q} }} } \hfill \\ {e^{{j\pi \sin \theta_{Q} }} } \hfill & {\quad 1} \hfill & \quad \hfill & \quad \hfill \\ \vdots \hfill & \quad \hfill & {\quad \ddots } \hfill & {\quad \vdots } \hfill \\ {e^{{j\pi (L - 1)\sin \theta_{Q} }} } \hfill & \quad \hfill & {\quad \cdots } \hfill & {\quad 1} \hfill \\ \end{array} } \right] \hfill \\ \end{gathered} $$

(40)

Analyzing the $q$-th term in (40), we get

$$\begin{aligned}& p_{q} \left[ {\begin{array}{*{20}c} 1 & {e^{{ - j\pi \sin \theta_{q} }} } & \cdots & {e^{{ - j\pi (L - 1)\sin \theta_{q} }} } \\ {e^{{j\pi \sin \theta_{q} }} } & 1 & {} & {} \\ \vdots & {} & \ddots & \vdots \\ {e^{{j\pi (L - 1)\sin \theta_{q} }} } & {} & \cdots & 1 \\ \end{array} } \right]\\ & \quad = p_{q} \left[ {\begin{array}{*{20}c} 1 \\ {e^{{j\pi \sin \theta_{q} }} } \\ \vdots \\ {e^{{j\pi (L - 1)\sin \theta_{q} }} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} 1 & { \, e^{{ - j\pi \sin \theta_{q} }} \, } & { \cdots \, } & {e^{{ - j\pi (L - 1)\sin \theta_{q} }} } \\ \end{array} } \right] = p_{q} {\mathbf{b}}_{q} {\mathbf{b}}_{q}^{\text{H}}\end{aligned} $$

(41)

Therefore, the following formula is derived:

$$ \tau ({\mathbf{y}}_{q} ) = p_{1} {\mathbf{b}}_{1} {\mathbf{b}}_{1}^{\text{H}} + p_{2} {\mathbf{b}}_{2} {\mathbf{b}}_{2}^{\text{H}} + \cdots + p_{Q} {\mathbf{b}}_{Q} {\mathbf{b}}_{Q}^{\text{H}} = \sum\limits_{q = 1}^{Q} {p_{q} {\mathbf{b}}_{q} {\mathbf{b}}_{q}^{\text{H}} } $$

(42)

This completes the proof.

Appendix B

For ${\mathbf{y}}^{i + 1}$, let

$$ \begin{aligned} \nabla_{{\mathbf{y}}} \Gamma \left( {{\mathbf{y}},p_{0}^{i} ,{\mathbf{u}}^{i} ,t^{i} ,{\mathbf{Z}}^{i} ,{\mathbf{F}}^{i} } \right) &= \nabla_{{\mathbf{y}}} \left\| {{\mathbf{y}} \oplus {\mathbf{g}} - {\mathbf{r}}_{I} } \right\|_{2}^{2} \\ & \quad + \nabla_{{\mathbf{y}}} \left\langle {{\mathbf{F}}^{i} ,{\mathbf{Z}}^{i} - \left[ {\begin{array}{*{20}l} {\tau ({\mathbf{u}}^{i} )} \hfill & {\quad {\mathbf{y}} - p_{0}^{i} {\hat{\mathbf{b}}}_{0} } \hfill \\ {\left( {{\mathbf{y}} - p_{0}^{i} {\hat{\mathbf{b}}}_{0} } \right)^{\text{H}} } \hfill & {\quad t^{i} } \hfill \\ \end{array} } \right]} \right\rangle \hfill \\ &\quad + \nabla_{{\mathbf{y}}} \rho \left\| {{\mathbf{Z}}^{i} - \left[ {\begin{array}{*{20}l} {\tau ({\mathbf{u}}^{i} )} \hfill & {\quad {\mathbf{y}} - p_{0}^{i} {\hat{\mathbf{b}}}_{0} } \hfill \\ {\left( {{\mathbf{y}} - p_{0}^{i} {\hat{\mathbf{b}}}_{0} } \right)^{\text{H}} } \hfill & {\quad t^{i} } \hfill \\ \end{array} } \right]} \right\|_{\text{F}}^{2} \hfill \\ &\quad = {\mathbf{y}} \oplus {\mathbf{g}} - {\mathbf{r}}_{I} - {\mathbf{f}}_{12}^{i} + 2\rho {\mathbf{y}} - 2\rho {\mathbf{z}}_{12}^{i} - 2\rho p_{0}^{i} {\hat{\mathbf{b}}}_{0} = 0 \hfill \\ \end{aligned} $$

(43)

where ${\mathbf{Z}}^{i} = \left[ {\begin{array}{*{20}l} {{\mathbf{Z}}_{11}^{i} } \hfill & {\quad {\mathbf{z}}_{12}^{i} } \hfill \\ {{\mathbf{z}}_{21}^{i} } \hfill & {\quad z_{22}^{i} } \hfill \\ \end{array} } \right]$, ${\mathbf{z}}_{12}^{i} = {\mathbf{z}}_{21}^{{i,{\text{H}} }}$, ${\mathbf{F}}^{i} = \left[ {\begin{array}{*{20}c} {{\mathbf{F}}_{11}^{i} } & {\quad {\mathbf{f}}_{12}^{i} } \\ {{\mathbf{f}}_{21}^{i} } & {\quad f_{22}^{i} } \\ \end{array} } \right]$, and ${\mathbf{f}}_{12}^{i} = {\mathbf{f}}_{21}^{{i,{\text{H}} }}$. Subsequently, the update form for ${\mathbf{y}}^{i + 1}$ is calculated as

$$ {\mathbf{y}}^{i + 1} = \frac{1}{{2\rho + {\mathbf{g}}}} \oplus ({\mathbf{r}}_{I} + {\mathbf{f}}_{12}^{i} + 2\rho {\mathbf{z}}_{12}^{i} + 2\rho p_{0}^{i} {\hat{\mathbf{b}}}_{0} ) $$

(44)

In the case of $p_{0}^{i + 1}$, we make

$$ \begin{aligned} &\nabla _{{p_{0} }} \Gamma \left( {{\mathbf{y}}^{{i + 1}} ,p_{0} ,{\mathbf{u}}^{i} ,t^{i} ,{\mathbf{Z}}^{i} ,{\mathbf{F}}^{i} } \right) \\ &\quad = \nabla _{{p_{0} }} \left\langle {{\mathbf{F}}^{i} ,{\mathbf{Z}}^{i} - \left[ {\begin{array}{*{20}c} {\tau ({\mathbf{u}}^{i} )} & {{\mathbf{y}}^{{i + 1}} - p_{0} {\boldsymbol{\hat{b}}}_{0} } \\ {({\mathbf{y}}^{{i + 1}} - p_{0} {\boldsymbol{\hat{b}}}_{0} )^{\text{H} } } & {t^{i} } \\ \end{array} } \right]} \right\rangle \\ & \qquad + \nabla _{{p_{0} }} \rho \left\| {{\mathbf{Z}}^{i} - \left[ {\begin{array}{*{20}c} {\tau ({\mathbf{u}}^{i} )} & {{\mathbf{y}}^{{i + 1}} - p_{0} {\boldsymbol{\hat{b}}}_{0} } \\ {({\boldsymbol{y}}^{{i + 1}} - p_{0} {\boldsymbol{\hat{b}}}_{0} )^{\text{H} } } & {t^{i} } \\ \end{array} } \right]} \right\|_{\text{F} }^{2} \\ & \quad = {\boldsymbol{\hat{b}}}_{0}^{\text{H} } {\mathbf{f}}_{{12}}^{i} + 2\rho {\boldsymbol{\hat{b}}}_{0}^{\text{H} } ({\mathbf{z}}_{{12}}^{i} - {\mathbf{y}}^{{i + 1}} + p_{0} {\boldsymbol{\hat{b}}}_{0} ) = 0 \\ \end{aligned} $$

(45)

The closed form for $p_{0}^{i + 1}$ is yielded:

$$ p_{0}^{i + 1} = \tfrac{1}{{\left\| {{\hat{\mathbf{b}}}_{0} } \right\|_{2}^{2} }}{\hat{\mathbf{b}}}_{0}^{\text{H}} \left( {{\mathbf{y}}^{i + 1} - {\mathbf{z}}_{12}^{i} - \tfrac{1}{2\rho }{\mathbf{f}}_{12}^{i} } \right) $$

(46)

For ${\mathbf{u}}^{i + 1}$, let

$$ \begin{aligned} & \nabla _{{\mathbf{u}}} \Gamma \left( {{\mathbf{y}}^{{i + 1}} ,p_{0}^{{i + 1}} ,{\mathbf{u}},t^{i} ,{\mathbf{Z}}^{i} ,{\mathbf{F}}^{i} } \right) = \nabla _{{\mathbf{u}}} \frac{\mu }{2}\left\{ {\frac{1}{L}\text{Tr} \left( {\tau ({\mathbf{u}})} \right) + t^{i} } \right\} \\ & \qquad + \nabla _{{\mathbf{u}}} \left\langle {{\mathbf{F}}^{i} ,{\mathbf{Z}}^{i} - \left[ {\begin{array}{*{20}l} {\tau ({\mathbf{u}})} \hfill & {\quad {\mathbf{y}}^{{i + 1}} - p_{0}^{{i + 1}} {\boldsymbol{\hat{b}}}_{0} } \hfill \\ {({\mathbf{y}}^{{i + 1}} - p_{0}^{{i + 1}} {\boldsymbol{\hat{b}}}_{0} )^{\text{H} } } \hfill & {\quad t^{i} } \hfill \\ \end{array} } \right]} \right\rangle + \nabla _{{\mathbf{u}}} \rho \left\| {{\mathbf{Z}}^{i} - \left[ {\begin{array}{*{20}l} {\tau ({\mathbf{u}})} \hfill & {\quad {\mathbf{y}}^{{i + 1}} - p_{0}^{{i + 1}} {\boldsymbol{\hat{b}}}_{0} } \hfill \\ {({\mathbf{y}}^{{i + 1}} - p_{0}^{{i + 1}} {\boldsymbol{\hat{b}}}_{0} )^{\text{H} } } \hfill & {\quad t^{i} } \hfill \\ \end{array} } \right]} \right\|_{\text{F} }^{2} = 0 \\ \end{aligned} $$

(47)

The l-th entry of ${\mathbf{u}}^{i + 1}$ is derived as

$$ {\mathbf{u}}_{(l)}^{i + 1} = \frac{1}{2\rho (L - l + 1)}\left\{ {{\text{Tr}} \left( {{\mathbf{F}}_{11}^{i} ,l} \right) + 2\rho {\text{Tr}} \left( {{\mathbf{Z}}_{11}^{i} ,l} \right) - \mu {\mathbf{e}}_{1} - 2\rho {\mathbf{e}}_{1} {\text{Tr}} \left( {{\mathbf{Z}}_{11}^{i} ,l} \right)} \right\},l = 1,2, \ldots ,L $$

(48)

Here, ${\text{Tr}} \left( {{\mathbf{X}},l} \right) = \sum\nolimits_{\chi } {X_{l + \chi - 1, \, \chi } } ,\chi = 1,2, \ldots ,L - l + 1$ and ${\mathbf{e}}_{1}$ stands for a zero vector, except that its first entry is 1.

In the case of $t^{i + 1}$, we make

$$ \begin{gathered} \nabla_{t} \Gamma \left( {{\mathbf{y}}^{i + 1} ,p_{0}^{i + 1} ,{\mathbf{u}}^{i + 1} ,t,{\mathbf{Z}}^{i} ,{\mathbf{F}}^{i} } \right) = \nabla_{t} \frac{\mu }{2}\left\{ {\frac{1}{L}{\text{Tr}} \left( {\tau ({\mathbf{u}}^{i + 1} )} \right) + t} \right\} \hfill \\ + \nabla_{t} \left\langle {{\mathbf{F}}^{i} ,{\mathbf{Z}}^{i} - \left[ {\begin{array}{*{20}l} {\tau ({\mathbf{u}}^{i + 1} )} \hfill & {\quad {\mathbf{y}}^{i + 1} - p_{0}^{i + 1} {\hat{\mathbf{b}}}_{0} } \hfill \\ {({\mathbf{y}}^{i + 1} - p_{0}^{i + 1} {\hat{\mathbf{b}}}_{0} )^{\text{H}} } \hfill & {\quad t} \hfill \\ \end{array} } \right]} \right\rangle \\ \qquad + \nabla_{t} \rho \left\| {{\mathbf{Z}}^{i} - \left[ {\begin{array}{*{20}l} {\tau ({\mathbf{u}}^{i + 1} )} \hfill & {\quad {\mathbf{y}}^{i + 1} - p_{0}^{i + 1} {\hat{\mathbf{b}}}_{0} } \hfill \\ {({\mathbf{y}}^{i + 1} - p_{0}^{i + 1} {\hat{\mathbf{b}}}_{0} )^{\text{H}} } \hfill & {\quad t} \hfill \\ \end{array} } \right]} \right\|_{\text{F}}^{2} \hfill \\ = \frac{\mu }{2} - f_{22}^{i} + 2\rho (t - z_{22}^{i} ) = 0 \hfill \\ \end{gathered} $$

(49)

Therefore, we update $t^{i + 1}$ as

$$ t^{i + 1} = z_{22}^{i} + \frac{1}{2\rho }f_{22}^{i} - \frac{\mu }{4\rho } $$

(50)

For ${\mathbf{Z}}^{i + 1}$, it can be formulated as

$$ \begin{aligned} {\mathbf{Z}}^{i + 1} &= \mathop {\arg \min }\limits_{{{\mathbf{Z}} \ge 0}} \Gamma \left( {{\mathbf{y}}^{i + 1} ,p_{0}^{i + 1} ,{\mathbf{u}}^{i + 1} ,t^{i + 1} ,{\mathbf{Z}},{\mathbf{F}}^{i} } \right) \\ &= \mathop {\arg \min }\limits_{{{\mathbf{Z}} \ge 0}} \left\langle {{\mathbf{F}}^{i} ,{\mathbf{Z}} - \left[ {\begin{array}{*{20}l} {\tau ({\mathbf{u}}^{i + 1} )} \hfill & {\quad {\mathbf{y}}^{i + 1} - p_{0}^{i + 1} {\hat{\mathbf{b}}}_{0} } \hfill \\ {\left( {{\mathbf{y}}^{i + 1} - p_{0}^{i + 1} {\hat{\mathbf{b}}}_{0} } \right)^{\text{H}} } \hfill & {\quad t^{i + 1} } \hfill \\ \end{array} } \right]} \right\rangle \\ & \qquad + \rho \left\| {{\mathbf{Z}} - \left[ {\begin{array}{*{20}l} {\tau ({\mathbf{u}}^{i + 1} )} \hfill & {\quad {\mathbf{y}}^{i + 1} - p_{0}^{i + 1} {\hat{\mathbf{b}}}_{0} } \hfill \\ {\left( {{\mathbf{y}}^{i + 1} - p_{0}^{i + 1} {\hat{\mathbf{b}}}_{0} } \right)^{\text{H}} } \hfill & {\quad t^{i + 1} } \hfill \\ \end{array} } \right]} \right\|_{\text{F}}^{2} \hfill \\ & = \mathop {\arg \min }\limits_{{{\mathbf{Z}} \ge 0}} \left\| {{\mathbf{Z}} - \left[ {\begin{array}{*{20}l} {\tau ({\mathbf{u}}^{i + 1} )} \hfill & {\quad {\mathbf{y}}^{i + 1} - p_{0}^{i + 1} {\hat{\mathbf{b}}}_{0} } \hfill \\ {\left( {{\mathbf{y}}^{i + 1} - p_{0}^{i + 1} {\hat{\mathbf{b}}}_{0} } \right)^{\text{H}} } \hfill & {\quad t^{i + 1} } \hfill \\ \end{array} } \right] + \frac{1}{2\rho }{\mathbf{F}}^{i} } \right\|_{\text{F}}^{2} \hfill \\ \end{aligned} $$

(51)

Based on the eigenvalue decomposition, we obtain

$$ \left[ {\begin{array}{*{20}l} {\tau ({\mathbf{u}}^{i + 1} )} \hfill & {\quad {\mathbf{y}}^{i + 1} - p_{0}^{i + 1} {\hat{\mathbf{b}}}_{0} } \hfill \\ {({\mathbf{y}}^{i + 1} - p_{0}^{i + 1} {\hat{\mathbf{b}}}_{0} )^{\text{H}} } \hfill & {\quad t^{i + 1} } \hfill \\ \end{array} } \right] - \frac{1}{2\rho }{\mathbf{F}}^{i} = {\mathbf{D}} {\varvec{\Lambda}} {\mathbf{D}}^{\text{H}} $$

(52)

Then, ${\mathbf{Z}}^{i + 1} = {\mathbf{D}}{\varvec{\Lambda}}^{\!+}{\mathbf{D}}^{\text{H}}$ is obtained, where ${{\varvec{\Lambda}}}^{ + }$ is gained by substituting the negative entries in ${{\varvec{\Lambda}}}$ with 0.

In the case of ${\mathbf{F}}^{i + 1}$, we have

$$ \begin{gathered} \nabla_{{\mathbf{F}}} \Gamma \left( {{\mathbf{y}}^{i + 1} ,p_{0}^{i + 1} ,{\mathbf{u}}^{i + 1} ,t^{i + 1} ,{\mathbf{Z}}^{i + 1} ,{\mathbf{F}}} \right) \hfill \\ = \nabla_{{\mathbf{F}}} \left\langle {{\mathbf{F}},{\mathbf{Z}}^{i + 1} - \left[ {\begin{array}{*{20}l} {\tau ({\mathbf{u}}^{i + 1} )} \hfill & {{\mathbf{y}}^{i + 1} - p_{0}^{i + 1} {\hat{\mathbf{b}}}_{0} } \hfill \\ {\left( {{\mathbf{y}}^{i + 1} - p_{0}^{i + 1} {\hat{\mathbf{b}}}_{0} } \right)^{\text{H}} } \hfill & {t^{i + 1} } \hfill \\ \end{array} } \right]} \right\rangle \\ = {\mathbf{Z}}^{i + 1} - \left[ {\begin{array}{*{20}l} {\tau \left( {{\mathbf{u}}^{i + 1} } \right)} \hfill & {\quad {\mathbf{y}}^{i + 1} - p_{0}^{i + 1} {\hat{\mathbf{b}}}_{0} } \hfill \\ {\left( {{\mathbf{y}}^{i + 1} - p_{0}^{i + 1} {\hat{\mathbf{b}}}_{0} } \right)^{\text{H}} } \hfill & {t^{i + 1} } \hfill \\ \end{array} } \right] \hfill \\ \end{gathered} $$

(53)

Thereafter, ${\mathbf{F}}^{i + 1}$ is updated as

$$ {\mathbf{F}}^{i + 1} = {\mathbf{F}}^{i} + \rho \left\{ {{\mathbf{Z}}^{i + 1} - \left[ {\begin{array}{*{20}l} {\tau ({\mathbf{u}}^{i + 1} )} \hfill & {{\mathbf{y}}^{i + 1} - p_{0}^{i + 1} {\hat{\mathbf{b}}}_{0} } \hfill \\ {\left( {{\mathbf{y}}^{i + 1} - p_{0}^{i + 1} {\hat{\mathbf{b}}}_{0} } \right)^{\text{H}} } \hfill & {t^{i + 1} } \hfill \\ \end{array} } \right]} \right\} $$

(54)

ADMM is completed once the maximum number of iterations, that is, $i_{\max }$, is achieved or the stopping criterion is met.

Appendix C

The deviation between ${\hat{\mathbf{y}}}$ and ${\mathbf{r}}_{I}$, excluding their interpolated elements, can be expressed as

$$ \begin{gathered} \left\| {{\hat{\mathbf{y}}} \oplus {\mathbf{g}} - {\mathbf{r}}_{I} } \right\|_{2}^{2} = \left\| {{\hat{\mathbf{y}}} \oplus {\mathbf{g}} - {\mathbf{y}}^{{\mathbf{*}}} \oplus {\mathbf{g}} + {\mathbf{y}}^{{\mathbf{*}}} \oplus {\mathbf{g}} - {\mathbf{r}}_{I} } \right\|_{2}^{2} \hfill \\ = \left\| {{\hat{\mathbf{y}}} \oplus {\mathbf{g}} - {\mathbf{y}}^{{\mathbf{*}}} \oplus {\mathbf{g}}} \right\|_{2}^{2} + \left\| {{\mathbf{y}}^{{\mathbf{*}}} \oplus {\mathbf{g}} - {\mathbf{r}}_{I} } \right\|_{2}^{2} + 2\left\langle {{\hat{\mathbf{y}}} \oplus {\mathbf{g}} - {\mathbf{y}}^{{\mathbf{*}}} \oplus {\mathbf{g}},{\mathbf{y}}^{{\mathbf{*}}} \oplus {\mathbf{g}} - {\mathbf{r}}_{I} } \right\rangle \hfill \\ \end{gathered} $$

(55)

According to (55), we get

$$ \left\| {{\hat{\mathbf{y}}} \oplus {\mathbf{g}} - {\mathbf{y}}^{{\mathbf{*}}} \oplus {\mathbf{g}}} \right\|_{2}^{2} = \left\| {{\hat{\mathbf{y}}} \oplus {\mathbf{g}} - {\mathbf{r}}_{I} } \right\|_{2}^{2} - \left\| {{\mathbf{y}}^{{\mathbf{*}}} \oplus {\mathbf{g}} - {\mathbf{r}}_{I} } \right\|_{2}^{2} - 2\left\langle {{\hat{\mathbf{y}}} \oplus {\mathbf{g}} - {\mathbf{y}}^{{\mathbf{*}}} \oplus {\mathbf{g}},{\mathbf{y}}^{{\mathbf{*}}} \oplus {\mathbf{g}} - {\mathbf{r}}_{I} } \right\rangle $$

(56)

Introducing the Cauchy–Schwarz inequality into the last term of (56) yields

$$ \left\langle {{\hat{\mathbf{y}}} \oplus {\mathbf{g}} - {\mathbf{y}}^{{\mathbf{*}}} \oplus {\mathbf{g}},{\mathbf{y}}^{{\mathbf{*}}} \oplus {\mathbf{g}} - {\mathbf{r}}_{I} } \right\rangle \le \left\| {{\hat{\mathbf{y}}} \oplus {\mathbf{g}} - {\mathbf{y}}^{{\mathbf{*}}} \oplus {\mathbf{g}}} \right\|_{2} \left\| {{\mathbf{y}}^{{\mathbf{*}}} \oplus {\mathbf{g}} - {\mathbf{r}}_{I} } \right\|_{2} $$

(57)

Based on (23), (57), and $\left\| {{\mathbf{y}}^{{\mathbf{*}}} \oplus {\mathbf{g}} - {\mathbf{r}}_{I} } \right\|_{2} \le \left\| {\tau ({\mathbf{y}}^{{\mathbf{*}}} ) \oplus {\mathbf{G}} - {\mathbf{R}}_{I} } \right\|_{\text{F}}$, (56) can be transformed into [35]

$$ \left\| {{\hat{\mathbf{y}}} \oplus {\mathbf{g}} - {\mathbf{y}}^{{\mathbf{*}}} \oplus {\mathbf{g}}} \right\|_{2}^{2} \le \tfrac{\mu }{2}\left\{ {\tfrac{1}{L}{\text{Tr}} \left( {\tau ({\mathbf{u}}^{{\mathbf{*}}} - {\hat{\mathbf{u}}})} \right) + t^{{\mathbf{*}}} - \hat{t}} \right\} + 2\left\| {{\hat{\mathbf{y}}} \oplus {\mathbf{g}} - {\mathbf{y}}^{{\mathbf{*}}} \oplus {\mathbf{g}}} \right\|_{2} \left\| {\tau ({\mathbf{y}}^{{\mathbf{*}}} ) \oplus {\mathbf{G}} - {\mathbf{R}}_{I} } \right\|_{\text{F}} $$

(58)

where ${\mathbf{G}} = \tau ({\mathbf{g}})$, and ${\mathbf{R}}_{I} = \tau ({\mathbf{r}}_{I} )$ stands for the covariance matrix of the derived virtual array [39]. Subsequently, the off-the-shelf conclusion in Lemma 1 [24] is exploited, described as

Lemma 1

If ${\mathbf{x}}(k),k = 1,2, \cdots ,K$ is a Gaussian random vector following the distribution ${\mathbf{x}}(k)\sim CN(0,{\mathbf{R}}_{x} )$,

$$ {\text{P}} \left\{ {\left\| {{\mathbf{R}}_{x} - {\overline{\mathbf{R}}}_{x} } \right\|_{\text{F}} \ge \tfrac{1}{\sqrt K }{\text{Tr}} ({\mathbf{R}}_{x} )} \right\} \le 2{\text{e}}^{ - 2c\sqrt K } $$

(59)

Here, ${\text{P}} \left\{ \cdot \right\}$ represents the probability. Without loss of generality, generalizing (59) to the virtual domain yields [35, 39]

$$ \left\| {\tau ({\mathbf{y}}^{{\mathbf{*}}} ) \oplus {\mathbf{G}} - {\mathbf{R}}_{I} } \right\|_{\text{F}} \le \tfrac{1}{\sqrt K }{\text{Tr}} \left( {\tau ({\mathbf{y}}^{{\mathbf{*}}} ) \oplus {\mathbf{G}}} \right) \le \tfrac{1}{\sqrt K }{\text{Tr}} \left( {\tau ({\mathbf{y}}_{q}^{{\mathbf{*}}} + p_{0}^{{\mathbf{*}}} {\mathbf{b}}_{0}^{{\mathbf{*}}} )} \right) = \tfrac{1}{\sqrt K }\left\{ {{\text{Tr}} \left( {\tau ({\mathbf{u}}^{{\mathbf{*}}} )} \right) + Lp_{0}^{{\mathbf{*}}} } \right\} = \mu_{1} $$

(60)

with a probability of, at least, $1 - 2{\text{e}}^{ - 2c\sqrt K }$. In (60), we consider ${\mathbf{u}}^{{\mathbf{*}}}$ to be a precise estimate of ${\mathbf{y}}_{q}^{{\mathbf{*}}}$. Then, defining $\mu_{2} = \tfrac{\mu }{2}\left\{ {\tfrac{1}{L}{\text{Tr}} \left( {\tau ({\mathbf{u}}^{{\mathbf{*}}} - {\hat{\mathbf{u}}})} \right) + t^{{\mathbf{*}}} - \hat{t}} \right\}$ and denoting $\left\| {{\hat{\mathbf{y}}} \oplus {\mathbf{g}} - {\mathbf{y}}^{{\mathbf{*}}} \oplus {\mathbf{g}}} \right\|_{2}$ by $y_{g}$, (58) is expressed in the following general form:

$$ y_{g}^{2} - 2\mu_{1} y_{g} - \mu_{2} = \left( {y_{g} - \mu_{1} - \sqrt {\mu_{1}^{2} + \mu_{2} } } \right)\left( {y_{g} - \mu_{1} + \sqrt {\mu_{1}^{2} + \mu_{2} } } \right) \le 0 $$

(61)

Thus, we get $y_{g} - \mu_{1} - \sqrt {\mu_{1}^{2} + \mu_{2} } \le 0$, which means

$$ \left\| {{\hat{\mathbf{y}}} \oplus {\mathbf{g}} - {\mathbf{y}}^{{\mathbf{*}}} \oplus {\mathbf{g}}} \right\|_{2} \le \mu_{1} + \sqrt {\mu_{1}^{2} + \mu_{2} } $$

(62)

This completes the proof.