## Appendix I --Detailed Derivation of Matrices in (101)  
Note that matrices \(\mathbf{\Omega}_{\mathbf{p}}^{\mathrm{H}}\mathbf{\Omega}_{\mathbf{p}}\), \(\mathbf{\Omega}_{\mathrm{Re}\{\widetilde{\mathbf{\rho}}\}}^{\mathrm{H}}\mathbf{\Omega}_{ \mathrm{Re}\{\widetilde{\mathbf{\rho}}\}}\), \(\mathbf{\Omega}_{\mathrm{Re}\{\widetilde{\mathbf{\sigma}}\}}^{\mathrm{H}}\mathbf{\Omega}_ {\mathrm{Re}\{\widetilde{\mathbf{\sigma}}\}}\), \(\mathbf{\Omega}_{\mathbf{p}}^{\mathrm{H}}\mathbf{\Omega}_{\mathrm{Re}\{\widetilde{\mathbf{ \sigma}}\}}\), and \(\mathbf{\Omega}_{\mathrm{Re}\{\widetilde{\mathbf{\rho}}\}}^{\mathrm{H}}\mathbf{\Omega}_{ \mathrm{Re}\{\widetilde{\mathbf{\sigma}}\}}\) are given in (100) to (102). Therefore, to calculate the matrices in (101), we only need to derive the expressions for matrices \(\mathbf{\Omega}_{\mathrm{Re}\{\widetilde{\mathbf{\sigma}}\}}^{\mathrm{H}}\mathbf{\Omega}_ {\mathrm{Re}\{\widetilde{\mathbf{\sigma}}\}}\), \(\mathbf{\Omega}_{\mathbf{p}}^{\mathrm{H}}\mathbf{\Omega}_{\mathrm{Re}\{\widetilde{\mathbf{ \sigma}}\}}\), \(\mathbf{\Omega}_{\mathrm{Re}\{\widetilde{\mathbf{\sigma}}\}}^{\mathrm{H}}\mathbf{\Omega}_ {\mathrm{Re}\{\widetilde{\mathbf{\sigma}}\}}\), and \(\mathbf{\Omega}_{\mathrm{Re}\{\widetilde{\mathbf{\sigma}}\}}^{\mathrm{H}}\mathbf{\Omega}_ {\mathrm{Re}\{\widetilde{\mathbf{\sigma}}\}}\). It follows from (99) that  
\[\mathbf{\Omega}_{\mathrm{Re}\{\widetilde{\mathbf{\sigma}}\}}^{\mathrm{H}}\mathbf{\Omega}_ {\mathrm{Re}\{\widetilde{\mathbf{\sigma}}\}}=\text{blkdiag}[\ \ |\beta_{1}|^{2}\cdot|\vec{| \vec{|s}|}|_{2}^{2}\cdot\mathbf{I}_{M}\ ^{\ \ },\ |\beta_{2}|^{2}\cdot||\vec{|s}| |_{2}^{2}\cdot\mathbf{I}_{M}^{\ \ },\ \cdots\ ^{\ },\ |\beta_{N}|^{2}\cdot||\vec{|s}| |_{2}^{2}\cdot\mathbf{I}_{M}^{\ }]\,, \tag{104}\]  
\[\begin{array}{c}\mathbf{\Omega}_{\mathrm{Re}\{\widetilde{\mathbf{\sigma}}\}}^{ \mathrm{H}}\mathbf{\Omega}_{\mathrm{Re}\{\widetilde{\mathbf{\sigma}}\}}=\ \ \big{[}\begin{smallmatrix}\frac{k}{\kappa}|\beta_{1}|^{2}\cdot|\vec{|s}|_{2}^{2} \cdot\mathrm{exp}\{-\mathrm{i}\omega_{k}\gamma_{1}(\mathbf{p})\}\cdot\big{(}\frac{ \partial^{2}\vec{|s}|_{2}^{2}}{\partial\vec{|s}|^{2}}\big{)}^{\mathrm{H}}\ ^{\ },\ \frac{k}{\kappa}|\beta_{2}|^{2}\cdot|\vec{|s}|_{2}^{2}\cdot\mathrm{exp}\{- \mathrm{i}\omega_{k}\gamma_{2}(\mathbf{p})\}\cdot\big{(}\frac{\partial^{2}\vec{|s}| _{2}^{2}}{\partial\vec{|s}|^{2}}\big{)}^{\mathrm{H}}\ ^{\ },\ \cdots\\ \cdots\ \ \sum\limits_{k=1}^{k}|\beta_{N}|^{2}\cdot|\vec{|s}|_{2}^{2}\cdot\mathrm{ exp}\{-\mathrm{i}\omega_{k}\gamma_{k}(\mathbf{p})\}\cdot\big{(}\frac{ \partial^{2}\vec{|[PERSON]; [PERSON]; [PERSON]; [PERSON] A low-complexity ESPRIT-based DOA estimation method for co-prime linear arrays. _Sensors_**2016**, _16_, 1367. [CrossRef] [PubMed]
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