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 C1 and C2 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 ith column of the matrix U.
Define h
i
= u
i
/u
i
(1), where u
i
(j), (j = 1, ⋯, N) denotes the jth 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 jth 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.