### 3.1 The FOC-ITMR algorithm

The new proposed algorithm named as FOC-ITMR, which is based on reconstructing two Toeplitz matrices **C**
_{1} and **C**
_{2} by using two parallel ULAs, is described in detail in this subsection. Firstly, we define **C**
_{1} and **C**
_{2} with the cumulant elements \( {c}_k^m \) and \( {\tilde{c}}_k^m \) arranging as follows

$$ \begin{array}{l}{c}_k^m= cum\left[{x}_m^m(t),{\left({x}_m^m(t)\right)}^{*},{\left({x}_m^m(t)\right)}^{*},{x}_k^m(t)\right]\\ {}\kern1.4em = cum\left[{\displaystyle \sum_{i=1}^Q{\gamma}_i{s}_1(t)}+{\displaystyle \sum_{i= Q+1}^P{s}_i(t)+{n}_{x,0}(t)},{\displaystyle \sum_{i=1}^Q{\gamma}_i^{*}{s}_1^{*}(t)}+{\displaystyle \sum_{i= Q+1}^P{s}_i^{*}(t)}+{n}_{x,0}^{*}(t),\right.\kern0.2em \\ {}\left.{\displaystyle \sum_{i=1}^Q{\gamma}_i^{*}{s}_1^{*}(t)}+{\displaystyle \sum_{i= Q+1}^P{s}_i^{*}(t)}+{n}_{x,0}^{*}(t),\kern0.3em {\displaystyle \sum_{i=1}^Q{\gamma}_i{s}_1(t){e}^{- j\frac{2\pi}{\lambda}{d}_x\left( k- m\right) \cos {\theta}_i}}+{\displaystyle \sum_{i= Q+1}^P{s}_i(t){e}^{- j\frac{2\pi}{\lambda}{d}_x\left( k- m\right) \cos {\theta}_i}}+{n}_{x, k}(t)\right]\\ {}\kern1.4em ={\left({\displaystyle \sum_{i=1}^Q{\gamma}_i^{*}}\right)}^2\left({\displaystyle \sum_{i=1}^Q{\gamma}_i}\right){\displaystyle \sum_{i=1}^Q{\gamma}_i{e}^{- j\frac{2\pi}{\lambda}{d}_x\left[\left( k-1\right)-\left( m-1\right)\right] \cos {\theta}_i}}\kern0.1em cum\left[{s}_1(t),{s}_1^{*}(t),{s}_1^{*}(t),{s}_1(t)\right]\\ {}+{\displaystyle \sum_{i= Q+1}^P{e}^{- j\frac{2\pi}{\lambda}{d}_x\left[\left( k-1\right)-\left( m-1\right)\right] \cos {\theta}_i}} cum\left[{s}_i(t),{s}_i^{*}(t),{s}_i^{*}(t),{s}_i(t)\right]\\ {}={\displaystyle \sum_{i=1}^P{d}_i{e}^{- j\frac{2\pi}{\lambda}{d}_x\left[\left( k-1\right)-\left( m-1\right)\right] \cos {\theta}_i}}\end{array} $$

(6)

$$ \begin{array}{l}{\tilde{c}}_k^m= cum\left[{x}_m^m(t),{\left({x}_m^m(t)\right)}^{*},{\left({x}_m^m(t)\right)}^{*},{y}_k^m(t)\right]\\ {}\kern1.5em = cum\left[{\displaystyle \sum_{i=1}^Q{\gamma}_i{s}_1(t)}+{\displaystyle \sum_{i= Q+1}^P{s}_i(t)}+{n}_{x,0}(t),{\displaystyle \sum_{i=1}^Q{\gamma}_i^{*}{s}_1^{*}(t)}+{\displaystyle \sum_{i= Q+1}^P{s}_i^{*}(t)}+{n}_{x,0}^{*}(t),\right.\kern0.1em \\ {}\left.\kern0.2em {\displaystyle \sum_{i=1}^Q{\gamma}_i^{*}{s}_1^{*}(t)}+{\displaystyle \sum_{i= Q+1}^P{s}_i^{*}(t)}+{n}_{x,0}^{*}(t),{\displaystyle \sum_{i=1}^Q{\gamma}_i{s}_1(t){e}^{- j\frac{2\pi}{\lambda}{d}_x\left( k- m\right) \cos {\theta}_i}{e}^{j\frac{2\pi}{\lambda}{d}_y \cos {\beta}_i}}+{\displaystyle \sum_{i= Q+1}^P{s}_i(t){e}^{- j\frac{2\pi}{\lambda}{d}_x\left( k- m\right) \cos {\theta}_i}{e}^{j\frac{2\pi}{\lambda}{d}_y \cos {\beta}_i}}+{n}_{y,0}(t)\right]\kern0.9em \\ {}\kern1.4em ={\left({\displaystyle \sum_{i=1}^Q{\gamma}_i^{*}}\right)}^2\left({\displaystyle \sum_{i=1}^Q{\gamma}_i}\right){\displaystyle \sum_{i=1}^Q{\gamma}_i{e}^{- j\frac{2\pi}{\lambda}{d}_x\left[\left( k-1\right)-\left( m-1\right)\right] \cos {\theta}_i}{e}^{j\frac{2\pi}{\lambda}{d}_y \cos {\beta}_i}} cum\left[{s}_1(t),{s}_1^{*}(t),{s}_1^{*}(t),\right.\\ {}\kern0.5em {s}_1\left.(t)\right]+{\displaystyle \sum_{i= Q+1}^P{e}^{- j\frac{2\pi}{\lambda}{d}_x\left[\left( k-1\right)-\left( m-1\right)\right] \cos {\theta}_i}{e}^{j\frac{2\pi}{\lambda}{d}_y \cos {\beta}_i}}\kern0.1em cum\left[{s}_i(t),{s}_i^{*}(t),{s}_i^{*}(t),{s}_i(t)\right]\kern0.1em \\ {}\kern1.4em ={\displaystyle \sum_{i=1}^P{d}_i{e}^{- j\frac{2\pi}{\lambda}{d}_x\left[\left( k-1\right)-\left( m-1\right)\right] \cos {\theta}_i}{e}^{j\frac{2\pi}{\lambda}{d}_y \cos {\beta}_i}}\end{array} $$

(7)

where

$$ {d}_i=\left\{\begin{array}{c}\hfill {\gamma}_i{\tilde{\rho}}_{4,{s}_1}\hfill \\ {}\hfill {\rho}_{4,{s}_i}\kern0.4em \hfill \end{array}\right.\kern2em \begin{array}{c}\hfill i=\kern0.4em 1,\cdots, Q\hfill \\ {}\hfill i= Q+1,\cdots, P\hfill \end{array} $$

(8)

with \( {\rho}_{4,{S}_i}= c u m\left[{S}_i(t),{S}_i^{*}(t),{S}_i^{*}(t),{S}_i(t)\right] \) and \( {\tilde{\rho}}_{4,{S}_1}={\left({\displaystyle {\sum}_{i=1}^Q{\gamma}_i^{*}}\right)}^2\left({\displaystyle {\sum}_{i=1}^Q{\gamma}_i}\right){\rho}_{4,{S}_1} \).

Then, the Toeplitz matrices C_{1} and C_{2} can be constructed, respectively,

$$ \begin{array}{l}\kern0.7em {\mathbf{C}}_1=\left[\begin{array}{cccc}\hfill {c}_1^1\hfill & \hfill {c}_1^2\hfill & \hfill \dots \hfill & \hfill {c}_1^N\hfill \\ {}\hfill {c}_2^1\hfill & \hfill {c}_2^2\hfill & \hfill \cdots \hfill & \hfill {c}_2^N\hfill \\ {}\hfill \vdots \hfill & \hfill \vdots \hfill & \hfill \cdots \hfill & \hfill \vdots \hfill \\ {}\hfill {c}_N^1\hfill & \hfill {c}_N^2\hfill & \hfill \cdots \hfill & \hfill {c}_N^N\hfill \end{array}\right]\kern0.5em \\ {}\kern2.2em =\left[\begin{array}{cccc}\hfill {\displaystyle \sum_{i=1}^P{d}_i{e}^{- j\frac{2\pi}{\lambda}{d}_x(0) \cos {\theta}_i}}\hfill & \hfill {\displaystyle \kern2em \sum_{i=1}^P{d}_i{e}^{- j\frac{2\pi}{\lambda}{d}_x\left(-1\right) \cos {\theta}_i}}\hfill & \hfill \kern1.5em \dots \hfill & \hfill {\displaystyle \kern2em \sum_{i=1}^P{d}_i{e}^{- j\frac{2\pi}{\lambda}{d}_x\left(1- N\right) \cos {\theta}_i}}\hfill \\ {}\hfill {\displaystyle \sum_{i=1}^P{d}_i{e}^{- j\frac{2\pi}{\lambda}{d}_x(1) \cos {\theta}_i}}\hfill & \hfill {\displaystyle \kern2em \sum_{i=1}^P{d}_i{e}^{- j\frac{2\pi}{\lambda}{d}_x(0) \cos {\theta}_i}}\hfill & \hfill \kern2em \cdots \hfill & \hfill {\displaystyle \kern2em \sum_{i=1}^P{d}_i{e}^{- j\frac{2\pi}{\lambda}{d}_x\left(2- N\right) \cos {\theta}_i}}\hfill \\ {}\hfill \vdots \hfill & \hfill \vdots \hfill & \hfill \cdots \hfill & \hfill \vdots \hfill \\ {}\hfill {\displaystyle \sum_{i=1}^P{d}_i{e}^{- j\frac{2\pi}{\lambda}{d}_x\left( N-1\right) \cos {\theta}_i}}\hfill & \hfill {\displaystyle \kern0.5em \sum_{i=1}^P{d}_i{e}^{- j\frac{2\pi}{\lambda}{d}_x\left( N-2\right) \cos {\theta}_i}}\hfill & \hfill \kern1em \cdots \hfill & \hfill {\displaystyle \kern2em \sum_{i=1}^P{d}_i{e}^{- j\frac{2\pi}{\lambda}{d}_x(0) \cos {\theta}_i}}\hfill \end{array}\right]\kern0.1em \\ {}\kern2.2em =\left[\begin{array}{cccc}\hfill 1\hfill & \hfill 1\hfill & \hfill \dots \hfill & \hfill 1\hfill \\ {}\hfill {e}^{- j\frac{2\pi}{\lambda}{d}_x(1) \cos {\theta}_1}\hfill & \hfill \kern3em {e}^{- j\frac{2\pi}{\lambda}{d}_x(1) \cos {\theta}_2}\hfill & \hfill \cdots \hfill & \hfill {e}^{- j\frac{2\pi}{\lambda}{d}_x(1) \cos {\theta}_P}\hfill \\ {}\hfill \vdots \hfill & \hfill \vdots \hfill & \hfill \cdots \hfill & \hfill \vdots \hfill \\ {}\hfill {e}^{- j\frac{2\pi}{\lambda}{d}_x\left( N-1\right) \cos {\theta}_1}\hfill & \hfill {e}^{- j\frac{2\pi}{\lambda}{d}_x\left( N-1\right) \cos {\theta}_2}\hfill & \hfill \cdots \hfill & \hfill {e}^{- j\frac{2\pi}{\lambda}{d}_x\left( N-1\right) \cos {\theta}_P}\hfill \end{array}\right]\kern0.5em \\ {}\kern3.2em \left[\begin{array}{cccc}\hfill {d}_1\hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill \\ {}\hfill \hfill & \hfill {d}_2\hfill & \hfill \hfill & \hfill \hfill \\ {}\hfill \hfill & \hfill \hfill & \hfill \cdots \hfill & \hfill \hfill \\ {}\hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill {d}_P\hfill \end{array}\right]\kern0.3em \left[\begin{array}{cccc}\hfill 1\hfill & \hfill \kern1em {e}^{j\frac{2\pi}{\lambda}{d}_x(1) \cos {\theta}_1}\hfill & \hfill \kern1em \dots \hfill & \hfill \kern1em {e}^{j\frac{2\pi}{\lambda}{d}_x\left( N-1\right) \cos {\theta}_1}\hfill \\ {}\hfill 1\hfill & \hfill \kern1em {e}^{j\frac{2\pi}{\lambda}{d}_x(1) \cos {\theta}_2}\hfill & \hfill \kern2em \cdots \hfill & \hfill \kern1em {e}^{j\frac{2\pi}{\lambda}{d}_x\left( N-1\right) \cos {\theta}_2}\hfill \\ {}\hfill \vdots \hfill & \hfill \vdots \hfill & \hfill \cdots \hfill & \hfill \vdots \hfill \\ {}\hfill 1\hfill & \hfill \kern1em {e}^{j\frac{2\pi}{\lambda}{d}_x(1) \cos {\theta}_P}\hfill & \hfill \kern1.9em \cdots \hfill & \hfill \kern1em {e}^{j\frac{2\pi}{\lambda}{d}_x\left( N-1\right) \cos {\theta}_P}\hfill \end{array}\right]\kern2.2em \\ {}=\mathbf{AD}{\mathbf{A}}^H\end{array} $$

(9)

$$ \begin{array}{l}{\mathbf{C}}_2=\left[\begin{array}{cccc}\hfill {\tilde{c}}_1^1\hfill & \hfill {\tilde{c}}_1^2\hfill & \hfill \dots \hfill & \hfill {\tilde{c}}_1^N\hfill \\ {}\hfill {\tilde{c}}_2^1\hfill & \hfill {\tilde{c}}_2^2\hfill & \hfill \cdots \hfill & \hfill {\tilde{c}}_2^N\hfill \\ {}\hfill \vdots \hfill & \hfill \vdots \hfill & \hfill \cdots \hfill & \hfill \vdots \hfill \\ {}\hfill {\tilde{c}}_N^1\hfill & \hfill {\tilde{c}}_N^2\hfill & \hfill \cdots \hfill & \hfill {\tilde{c}}_N^N\hfill \end{array}\right]\kern0.3em \\ {}\kern1.5em =\left[\begin{array}{cccc}\hfill {\displaystyle \sum_{i=1}^P{d}_i{e}^{- j\frac{2\pi}{\lambda}{d}_x(0) \cos {\theta}_i}{e}^{j\frac{2\pi}{\lambda}{d}_y \cos {\beta}_i}}\hfill & \hfill {\displaystyle \kern2em \sum_{i=1}^P{d}_i{e}^{- j\frac{2\pi}{\lambda}{d}_x\left(\hbox{-} 1\right) \cos {\theta}_i}{e}^{j\frac{2\pi}{\lambda}{d}_y \cos {\beta}_i}}\hfill & \hfill \kern2em \dots \hfill & \hfill {\displaystyle \kern2em \sum_{i=1}^P{d}_i{e}^{- j\frac{2\pi}{\lambda}{d}_x\left(1\hbox{-} N\right) \cos {\theta}_i}{e}^{j\frac{2\pi}{\lambda}{d}_y \cos {\beta}_i}}\hfill \\ {}\hfill {\displaystyle \sum_{i=1}^P{d}_i{e}^{- j\frac{2\pi}{\lambda}{d}_x(1) \cos {\theta}_i}{e}^{j\frac{2\pi}{\lambda}{d}_y \cos {\beta}_i}}\hfill & \hfill {\displaystyle \kern2em \sum_{i=1}^P{d}_i{e}^{- j\frac{2\pi}{\lambda}{d}_x(0) \cos {\theta}_i}{e}^{j\frac{2\pi}{\lambda}{d}_y \cos {\beta}_i}}\hfill & \hfill \kern2.3em \cdots \hfill & \hfill {\displaystyle \kern2em \sum_{i=1}^P{d}_i{e}^{- j\frac{2\pi}{\lambda}{d}_x\left(2- N\right) \cos {\theta}_i}{e}^{j\frac{2\pi}{\lambda}{d}_y \cos {\beta}_i}}\hfill \\ {}\hfill \vdots \hfill & \hfill \vdots \hfill & \hfill \cdots \hfill & \hfill \vdots \hfill \\ {}\hfill {\displaystyle \sum_{i=1}^P{d}_i{e}^{- j\frac{2\pi}{\lambda}{d}_x\left( N-1\right) \cos {\theta}_i}{e}^{j\frac{2\pi}{\lambda}{d}_y \cos {\beta}_i}}\hfill & \hfill {\displaystyle \kern1em \sum_{i=1}^P{d}_i{e}^{- j\frac{2\pi}{\lambda}{d}_x\left( N-2\right) \cos {\theta}_i}{e}^{j\frac{2\pi}{\lambda}{d}_y \cos {\beta}_i}}\hfill & \hfill \kern1em \cdots \hfill & \hfill {\displaystyle \kern1em \sum_{i=1}^P{d}_i{e}^{- j\frac{2\pi}{\lambda}{d}_x(0) \cos {\theta}_i}{e}^{j\frac{2\pi}{\lambda}{d}_y \cos {\beta}_i}}\hfill \end{array}\right]\\ {}\kern1.5em =\left[\begin{array}{cccc}\hfill 1\hfill & \hfill 1\hfill & \hfill \dots \hfill & \hfill 1\hfill \\ {}\hfill {e}^{- j\frac{2\pi}{\lambda}{d}_x(1) \cos {\theta}_1}\hfill & \hfill {e}^{- j\frac{2\pi}{\lambda}{d}_x(1) \cos {\theta}_2}\hfill & \hfill \cdots \hfill & \hfill {e}^{- j\frac{2\pi}{\lambda}{d}_x(1) \cos {\theta}_P}\hfill \\ {}\hfill \vdots \hfill & \hfill \vdots \hfill & \hfill \cdots \hfill & \hfill \vdots \hfill \\ {}\hfill {e}^{- j\frac{2\pi}{\lambda}{d}_x\left( N-1\right) \cos {\theta}_1}\hfill & \hfill {e}^{- j\frac{2\pi}{\lambda}{d}_x\left( N-1\right) \cos {\theta}_2}\hfill & \hfill \cdots \hfill & \hfill {e}^{- j\frac{2\pi}{\lambda}{d}_x\left( N-1\right) \cos {\theta}_P}\hfill \end{array}\right]\left[\begin{array}{cccc}\hfill {d}_1\hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill \\ {}\hfill \hfill & \hfill {d}_2\hfill & \hfill \hfill & \hfill \hfill \\ {}\hfill \hfill & \hfill \hfill & \hfill \cdots \hfill & \hfill \hfill \\ {}\hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill {d}_P\hfill \end{array}\right]\kern0.4em \\ {}\kern2.5em \left[\begin{array}{cccc}\hfill {e}^{j\frac{2\pi}{\lambda}{d}_y \cos {\beta}_1}\hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill \\ {}\hfill \hfill & \hfill {e}^{j\frac{2\pi}{\lambda}{d}_y \cos {\beta}_2}\hfill & \hfill \hfill & \hfill \hfill \\ {}\hfill \hfill & \hfill \hfill & \hfill \cdots \hfill & \hfill \hfill \\ {}\hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill {e}^{j\frac{2\pi}{\lambda}{d}_y \cos {\beta}_P}\hfill \end{array}\right]\left[\begin{array}{cccc}\hfill 1\hfill & \hfill {e}^{j\frac{2\pi}{\lambda}{d}_x(1) \cos {\theta}_1}\hfill & \hfill \dots \hfill & \hfill {e}^{j\frac{2\pi}{\lambda}{d}_x\left( N-1\right) \cos {\theta}_1}\hfill \\ {}\hfill 1\hfill & \hfill {e}^{j\frac{2\pi}{\lambda}{d}_x(1) \cos {\theta}_2}\hfill & \hfill \cdots \hfill & \hfill {e}^{j\frac{2\pi}{\lambda}{d}_x\left( N-1\right) \cos {\theta}_2}\hfill \\ {}\hfill \vdots \hfill & \hfill \vdots \hfill & \hfill \cdots \hfill & \hfill \vdots \hfill \\ {}\hfill 1\hfill & \hfill {e}^{j\frac{2\pi}{\lambda}{d}_x(1) \cos {\theta}_P}\hfill & \hfill \cdots \hfill & \hfill {e}^{j\frac{2\pi}{\lambda}{d}_x\left( N-1\right) \cos {\theta}_P}\hfill \end{array}\right]\\ {}\kern1.4em =\mathbf{ADV}{\mathbf{A}}^H\end{array} $$

(10)

where

$$ \begin{array}{l}\mathbf{A}=\left[\mathtt{a}\left({\theta}_1\right),\mathtt{a}\left({\theta}_2\right),\dots, \mathtt{a}\left({\theta}_P\right)\right]\\ {}\kern1em =\left[\begin{array}{cccc}\hfill 1\hfill & \hfill 1\hfill & \hfill \dots \hfill & \hfill 1\hfill \\ {}\hfill {e}^{- j\frac{2\pi}{\lambda}{d}_x(1) \cos {\theta}_1}\hfill & \hfill {e}^{- j\frac{2\pi}{\lambda}{d}_x(1) \cos {\theta}_2}\hfill & \hfill \cdots \hfill & \hfill {e}^{- j\frac{2\pi}{\lambda}{d}_x(1) \cos {\theta}_P}\hfill \\ {}\hfill \vdots \hfill & \hfill \vdots \hfill & \hfill \cdots \hfill & \hfill \vdots \hfill \\ {}\hfill {e}^{- j\frac{2\pi}{\lambda}{d}_x\left( N-1\right) \cos {\theta}_1}\hfill & \hfill {e}^{- j\frac{2\pi}{\lambda}{d}_x\left( N-1\right) \cos {\theta}_2}\hfill & \hfill \cdots \hfill & \hfill {e}^{- j\frac{2\pi}{\lambda}{d}_x\left( N-1\right) \cos {\theta}_P}\hfill \end{array}\right]\end{array} $$

(11)

$$ \mathbf{D}= diag\left({d}_1,{d}_2,\cdots, {d}_P\right) $$

(12)

$$ \begin{array}{l}\mathbf{V}= diag\left[ v\left({\beta}_1\right), v\left({\beta}_2\right),\cdots, v\left({\beta}_P\right)\right]\\ {}\kern1em =\kern0.4em \left[\begin{array}{cccc}\hfill {e}^{j\frac{2\pi}{\lambda}{d}_y \cos {\beta}_1}\hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill \\ {}\hfill \hfill & \hfill {e}^{j\frac{2\pi}{\lambda}{d}_y \cos {\beta}_2}\hfill & \hfill \hfill & \hfill \hfill \\ {}\hfill \hfill & \hfill \hfill & \hfill \cdots \hfill & \hfill \hfill \\ {}\hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill {e}^{j\frac{2\pi}{\lambda}{d}_y \cos {\beta}_P}\hfill \end{array}\right]\end{array} $$

(13)

Clearly, Eq. (11) means that the matrix **A** is a Vandermonde matrix as long as *θ*
_{
i
}, (*i* = 1, ⋯, *P*) comes from different angles. Therefore, **A** is a column full-rank matrix, namely, rank (**A**) = *P*, whose columns are linearly independent. From the expression of *d*
_{
i
} in (8), we can see that *d*
_{
i
} is a non-zero constant. Thus, from (12), it is easy to know that the matrix **D** is of rank P. Moreover, according to Eq. (13), we confirm that **V** satisfies the condition of full rank for different angles *β*
_{
i
}.

The eigenvalue decomposition of **C**
_{1} can be written as

$$ {\mathbf{C}}_1={\displaystyle \sum_{i=1}^P{\eta}_i{\mathbf{g}}_i{\mathbf{g}}_i^H} $$

(14)

where {*η*
_{1},…, *η*
_{
P
}} and {**g**
_{1},…, **g**
_{
P
}} are the non-zero eigenvalues and corresponding eigenvectors of the matrix **C**
_{1}. The pseudo-inverse of the matrix **C**
_{1} is

$$ {\mathbf{C}}_1^{\dagger }={\displaystyle \sum_{i=1}^P{\eta}_i^{-1}{\mathbf{g}}_i{\mathbf{g}}_i^H} $$

(15)

Due to the fact that **A** is a column full-rank matrix, from (9), we can attain

$$ \mathbf{D}{\mathbf{A}}^H={\left({\mathbf{A}}^H\mathbf{A}\right)}^{-1}{\mathbf{A}}^H{\mathbf{C}}_1 $$

(16)

Combining (10) with (16), the alternative expression of **C**
_{2} can be achieved as follows

$$ \begin{array}{l}{\mathbf{C}}_2=\mathbf{ADV}{\mathbf{A}}^H\\ {}\kern1.3em =\mathbf{AV}\mathbf{D}{\mathbf{A}}^H\\ {}\kern1.3em =\mathbf{AV}{\left({\mathbf{A}}^H\mathbf{A}\right)}^{-1}{\mathbf{A}}^H{\mathbf{C}}_1\end{array} $$

(17)

Right multiplying both sides of (17) by \( {\mathbf{C}}_1^{\dagger}\mathbf{A} \)

$$ {\mathbf{C}}_2{\mathbf{C}}_1^{\dagger}\mathbf{A}=\mathbf{A}\mathbf{V}{\left({\mathbf{A}}^H\mathbf{A}\right)}^{-1}{\mathbf{A}}^H{\mathbf{C}}_1{\mathbf{C}}_1^{\dagger}\mathbf{A} $$

(18)

Substituted (14) and (15) into (18)

$$ \begin{array}{l}{\mathbf{C}}_2{\mathbf{C}}_1^{\dagger}\mathbf{A}=\mathbf{AV}{\left({\mathbf{A}}^H\mathbf{A}\right)}^{-1}{\mathbf{A}}^H\kern0.2em \left({\displaystyle \sum_{i=1}^P{\eta}_i{\mathbf{g}}_i{\mathbf{g}}_i^H}\right)\left({\displaystyle \sum_{i=1}^P{\eta}_i^{-1}{\mathbf{g}}_i{\mathbf{g}}_i^H}\right)\mathbf{A}\\ {}\kern3.1em =\mathbf{AV}{\left({\mathbf{A}}^H\mathbf{A}\right)}^{-1}{\mathbf{A}}^H{\displaystyle \sum_{i=1}^P{\mathbf{g}}_i{\mathbf{g}}_i^H}\mathbf{A}\end{array} $$

(19)

Notice that \( {\displaystyle \sum_{i=1}^P{\mathbf{g}}_i{\mathbf{g}}_i^H} \) is an identity matrix, that is \( {\displaystyle \sum_{i=1}^P{\mathbf{g}}_i{\mathbf{g}}_i^H}=\mathbf{I} \). Thus, Eq. (19) can be further rewritten as

$$ \begin{array}{l}{\mathbf{C}}_2{\mathbf{C}}_1^{\dagger}\mathbf{A}=\mathbf{AV}{\left({\mathbf{A}}^H\mathbf{A}\right)}^{-1}\left({\mathbf{A}}^H\mathbf{A}\right)\\ {}\kern3.1em =\mathbf{AV}\end{array} $$

(20)

From (20), 2-D angle parameters, which are obtained by performing EVD on \( {\mathbf{C}}_2{\mathbf{C}}_1^{\dagger } \) denoted as Toeplitz-based generalized DOA matrix, lie in **A** and **V**. By performing EVD on \( {\mathbf{C}}_2{\mathbf{C}}_1^{\dagger } \)

$$ {\mathbf{C}}_2{\mathbf{C}}_1^{\dagger }={\displaystyle \sum_{i=1}^P{\xi}_i{\mathbf{u}}_i{\mathbf{u}}_i^H} $$

(21)

where *ξ*
_{
i
} and **u**
_{
i
} are the non-zero eigenvalues and the corresponding eigenvectors of the matrix \( {\mathbf{C}}_2{\mathbf{C}}_1^{\dagger } \). Thus, the *P* bigger signal subspaces can be obtained from (21). It can be verified that the subspaces spanned by **A** and the signal subspaces **U** = [**u**
_{1}, ⋯, **u**
_{
i
} ⋯, **u**
_{
P
}] are the same, where **u**
_{
i
} is the *i*th column of the matrix **U**.

Define **h**
_{
i
} = **u**
_{
i
}/**u**
_{
i
} (1), where **u**
_{
i
}(*j*), (*j* = 1, ⋯, *N*) denotes the *j*th element of vector **u**
_{
i
}, and then we can get

$$ {\kappa}_i=\frac{1}{N-1}{{\displaystyle \sum_{m=1}^{N-1} arg\left[\frac{{\mathbf{h}}_i\left( m+1\right)}{{\mathbf{h}}_i(m)}\right]}}_{.} $$

(22)

where **h**
_{
i
}(*j*), (*j* = 1, ⋯, *N*) represents the *j*th element of vector **h**
_{
i
}.

Therefore, by combining \( \boldsymbol{a}\left({\theta}_i\right)={\left[1,{e}^{- j\frac{2\pi}{\lambda}{d}_x \cos {\theta}_i},\dots, {e}^{- j\frac{2\pi}{\lambda}{d}_x\left( N-1\right) \cos {\theta}_i}\right]}^T \) and \( v\left({\beta}_i\right)={e}^{j\frac{2\pi}{\lambda}{d}_y \cos {\beta}_i} \), the estimated 2-D DOAs can be obtained

$$ {\theta}_i= \arccos \left(\frac{\lambda}{2\pi {d}_x}{\kappa}_i\right) $$

(23)

$$ {\beta}_i= \arccos {\left[\frac{\lambda}{2\pi {d}_y} arg\left({\xi}_i\right)\right]}_{.} $$

(24)

Till now, 2-D DOAs of incoming signals, namely, *θ*
_{
i
} and *β*
_{
i
} can be achieved automatically paired according to Eqs. (23) and (24) without additional computations for parameter pair-matching.

The proposed algorithm with finite sampling data can be implemented as follows

Step 1: Compute the cumulant elements \( {c}_k^m \) and \( {\tilde{c}}_k^m \) according to (6) and (7), respectively;

Step 2: Reconstruct the two Toeplitz matrices **C**
_{1} and **C**
_{2} by (9) and (10);

Step 3: Obtain the pseudo-inverse matrix \( {\mathbf{C}}_1^{\dagger } \) by performing the EVD of the matrix **C**
_{1};

Step 4: Perform EVD of \( {\mathbf{C}}_2{\mathbf{C}}_1^{\dagger } \) to obtain the non-zero eigenvalues and the corresponding eigenvector;

Step 5: Estimate the 2-D DOAs of incident coherent source signals via (22)–(24).

### 3.2 Location analysis

In this subsection, the advantage of the proposed algorithm is discussed. As for two *N* × *N* dimension Toeplitz matrices, the maximum number of signals that can be distinguished is *N*−1 by the proposed FOC-ITMR method. Assume that the number of each subarray in [26] is 2 *M* + 1, the FOC-TMR method can distinguish *M* signals. According to the parameters set in [25], the FOC-FSS method can tell the same number of signals as [26]. In other words, based on the same array configuration and the same number of sensors, the proposed algorithm has twice larger array aperture than the compared FOC-TMR algorithm in [26]. Therefore, the proposed algorithm can not only resolve more signals than the compared method in [26] but also achieve better estimation performance.