Source Localization Based on Hybrid Coarray for 1-D Mirrored Interferometric Aperture Synthesis

Mirrored coprime array design using sum‐difference coarray optimisation

“…Thus, the least squares solution of the above matrix A Equation () can be denoted as follows: [24–28] $$${\hat{\boldsymbol{y}}}_{\mathrm{SD}\,}^{LS}={\boldsymbol{T}}^{{\dagger}}\boldsymbol{y}={\left({\boldsymbol{T}}^{H}\boldsymbol{T}\right)}^{-1}{\boldsymbol{T}}^{H}\boldsymbol{y}$$$ However, the transform matrix T of MA under arbitrary geometry is always rank‐deficient, that is, Equation () is ill‐posed [28]. In this case, the solution either diverges or, if it exists, it is just a meaningless, poor‐quality approximation to the true value [29].…”

“…Each sensor receives both direct and reflected signals from the mirror simultaneously. Then, the array output vector x ðtÞ ∈ C P�1 is expressed as [27].…”

“…Thus, the least squares solution of the above matrix A Equation ( 8) can be denoted as follows: [24][25][26][27][28]…”

“…Each sensor receives both direct and reflected signals from the mirror simultaneously. Then, the array output vector $$\boldsymbol{x}(t)\in {\mathbb{C}}^{P\times 1}$$ is expressed as [27]. $$$\begin{array}{l}\begin{array}{rl}\hfill \boldsymbol{x}(t)& ={\boldsymbol{A}}_{M}s(t)+\boldsymbol{n}(t)\hfill \\ \hfill & =\left(\boldsymbol{A}-{\boldsymbol{A}}^{\ast }\right)\boldsymbol{s}(t)+\boldsymbol{n}(t)\hfill \\ \hfill & =\sum\limits _{k=1}^{K}\left[{\boldsymbol{a}}_{S}\left({\theta }_{k}\right)-{\boldsymbol{a}}_{S}^{\ast }\left({\theta }_{k}\right)\right]\boldsymbol{s}(t)+\boldsymbol{n}(t)\hfill \end{array}\hfill \end{array}$$$ where $$\boldsymbol{s}(t)={\left[{s}_{1}(t),\text{\ldots },{s}_{K}(t)\right]}^{T}\in {\mathbb{C}}^{K\times 1}$$ is the signal vector and $$\boldsymbol{n}(t)\in {\mathbb{C}}^{P\times 1}$$…”

“…The DCA and SCA for different mirror positions are shown in Figure 2, and h ≤ 2 is considered according to Equation (27). It is noted that both Equations ( 24) and ( 27) hold for MCA when h = 1 and h = 1.5.…”