## 3 The Proposed Algorithm
### Noncircular Signal-Based Extended Matrix Construction  
Based on the new received data matrix \(\tilde{\textbf{X}}\), a reduced dimensional transformation is introduced in the following. For the \(p\)-th target, the transmit-receive steering vector \(\tilde{\textbf{a}}(\theta_{p})\) can be expressed as:  
\[\tilde{\textbf{a}}(\theta_{p})=(1,\dots,z_{p}^{\tilde{N}-1};z_{p},\dots,z_{p}^{ \tilde{N}};\dots;z_{p}^{\tilde{M}-1},\dots,z_{p}^{\tilde{M}+\tilde{N}-2})^{ \mathbb{T}} \tag{13}\]  
where \(p=1,2,\dots,P\). Many repeated terms are contained in \(\tilde{\textbf{a}}(\theta_{p})\), and the non-repeated elements can make up another vector \(\textbf{b}(\theta_{p})=[1,\exp(j\pi\text{sin}\theta_{p}),\cdots,\exp(j\pi( \tilde{M}+\tilde{N}-2)\text{sin}\theta_{p})]^{\mathbb{T}}\in\mathbb{C}^{( \tilde{M}+\tilde{N}-1)\times 1}\). The relationship between \(\tilde{\textbf{a}}(\theta_{p})\) and \(\textbf{b}(\theta_{p})\) can be derived as \(\tilde{\textbf{a}}(\theta_{p})=\textbf{G}\textbf{b}(\theta_{p})\), in which **G** is represented as:  
\[\textbf{G}=[\textbf{L}_{0}^{\mathbb{T}},\textbf{L}_{1}^{\mathbb{T}},\cdots, \textbf{L}_{\tilde{M}-1}^{\mathbb{T}}]^{\mathbb{T}} \tag{14}\]  
where \(\textbf{G}\in\mathbb{C}^{\tilde{M}\tilde{N}\times(\tilde{M}+\tilde{N}-1)}\), and \(\textbf{L}_{m}=[\textbf{0}_{\tilde{N}\times m},\textbf{I}_{\tilde{N}},\textbf {\tilde{0}}_{\tilde{N}\times(\tilde{M}-m-1)}]\in\mathbb{C}^{\tilde{N}\times( \tilde{M}+\tilde{N}-1)}\) for \(m=0,1,...,\tilde{M}-1\). Thus, based on **G**, a reduced dimensional transformation matrix can be constructed as \(\textbf{R}=(\textbf{G}^{\mathbb{H}}\textbf{G})^{(-\frac{1}{2})}\textbf{G}^{ \mathbb{H}}\), then \(\tilde{\textbf{X}}\) in Equation (12) turns into:  
\[\begin{split}\textbf{Y}&=(\textbf{G}^{\mathbb{H}} \textbf{G})^{(-\frac{1}{2})}\textbf{G}^{\mathbb{H}}\textbf{G}\textbf{B} \textbf{D}\textbf{\Lambda}_{p}\textbf{S}_{r}+\textbf{R}\tilde{\textbf{N}}\\ &=\textbf{F}\textbf{B}\textbf{D}\textbf{\Lambda}_{p}\textbf{S}_ {r}+\tilde{\textbf{N}}_{d}\end{split} \tag{15}\]  
where \(\textbf{Y}\) and \(\tilde{\textbf{N}}_{d}=\textbf{R}\tilde{\textbf{N}}\) are the new reduced dimensional data matrix and the new noise matrix, respectively. \(\textbf{F}=(\textbf{G}^{\mathbb{H}}\textbf{G})^{(\frac{1}{2})}\in\mathbb{C}^ {(\tilde{M}+\tilde{N}-1)\times(\tilde{M}+\tilde{N}-1)}\) can be directly calculated as:  
\[\textbf{F}=\text{diag}[1,\sqrt{2},...,\underbrace{\text{min}(\sqrt{\tilde{M} },\sqrt{\tilde{N}}),...,\text{min}(\sqrt{\tilde{M}},\sqrt{\tilde{N}})}_{| \tilde{M}-\tilde{N}|+1},...,\sqrt{2},1] \tag{16}\]  
By the transformation, the dimension of the data matrix is reduced from \(\tilde{M}\tilde{N}\times J\) to \((\tilde{M}+\tilde{N}-1)\times J\). In addition, as \(\textbf{R}\textbf{R}^{\mathbb{H}}=\textbf{I}_{\tilde{M}+\tilde{N}-1}\), the transformation does not bring in the additional spatial colored noises. In order to enlarge the array aperture by exploiting the noncircular signals, the new received data matrix \(\textbf{Y}\in\mathbb{C}^{(\tilde{M}+\tilde{N}-1)\times J}\) can be extended as \(\textbf{Z}=[\textbf{Y}^{\mathbb{T}},\textbf{Y}^{\mathbb{H}}]^{\mathbb{T}} \in\mathbb{C}^{2(\tilde{M}+\tilde{N}-1)\times J}\). Since **F**, **D** and \(\textbf{S}_{r}\) are real-valued matrices, **Z** is expressed as:  
\[\textbf{Z}=\left[\begin{array}{c}\textbf{F}\textbf{B}\textbf{D}\textbf{ \Lambda}_{p}\\ \textbf{F}\textbf{B}^{*}\textbf{D}\textbf{\Lambda}_{p}^{*}\end{array}\right] \textbf{S}_{r}+\left[\begin{array}{c}\tilde{\textbf{N}}_{d}\\ \textbf{N}_{d}^{*}\end{array}\right] \tag{17}\]