# 2D-Unitary ESPRIT Based Multi-Target Joint Range and Velocity Estimation Algorithm for FMCW Radar
## 3 2D-Unitary ESPRIT Based Joint Range and Velocity Estimation Algorithm of Multiple Targets
### 2D-Unitary ESPRIT Algorithm for Joint Range and Velocity Estimation  
The 2D-Unitary ESPRIT algorithm is an extension of the 1D-Unitary ESPRIT algorithm [33] and is commonly used for 2D DOA estimation. The steering vector of the angular frequency \(u\) of the matrix \(\mathbf{Z}\) is written in conjugate centrosymmetric form when \(N_{s}\) is even:  
\[\mathbf{a}_{N_{s}}(u)=[e^{-j\frac{N_{u}}{2}u},\cdots,\;e^{-ju},e^{ju}\;,\; \cdots,\;e^{j\frac{N_{u}}{2}u}]^{\mathrm{T}} \tag{21}\]  
Similarly, the steering vectors \(\mathbf{a}_{M_{s}}(w)\) of the angular frequency \(w\) is obtained. Thus, the 2D-frequency vector of the signal matrix \(\mathbf{Z}\) is written as follows:  
\[\mathbf{A}(w,u)=\mathbf{a}_{M_{s}}(w)\mathbf{a}_{N_{s}}^{\mathrm{T}}(u) \tag{22}\]  
The real-valued realizations are obtained by using a bijective mapping between centro-Hermitian matrices and real matrices, where a complex matrix \(Q\in\mathbb{C}^{p\times q}\) is called centro-Hermitian if \(\Pi_{p}Q^{*}=Q\)[34]. The unitary matrix:  
\[\mathbf{Q}_{2K}=\frac{1}{\sqrt{2}} \begin{bmatrix}\mathbf{I}_{K}&j\mathbf{I}_{ K}\\ \prod_{K}&-j\prod_{K}\\ \end{bmatrix} ,\;K=\frac{N_{s}}{2} \tag{23}\]  
where \(\mathbf{I}_{K}\) is the unit matrix and \(\prod_{N}\) is the exchange matrix with ones on its antidiagonal and zeros elsewhere. Therefore, the complex matrix \(\mathbf{A}(w,u)\) is converted into a real matrix \(\mathbf{D}(w,\;u)\) as follows:  
\[ \begin{split}\mathbf{D}(w,\;u)&=\mathbf{Q}_{M_{s}}^{ \mathrm{H}}\mathbf{A}(w,\;u)\mathbf{Q}_{N_{s}}^{*}=\mathbf{Q}_{M_{s}}^{ \mathrm{H}}\mathbf{a}_{M_{s}}(w)\mathbf{a}_{N_{s}}^{\mathrm{T}}(u)\mathbf{Q}_ {N_{s}}^{*}\\ &=\mathbf{d}_{M_{s}}(w)\mathbf{d}_{N_{s}}^{\mathrm{T}}(u)\end{split}  \tag{24}\]  
where \(\mathbf{d}_{N_{s}}(u)=\mathbf{Q}_{N_{s}}^{\mathrm{H}}\mathbf{a}_{N_{s}}(u)\) and \(\mathbf{d}_{M_{s}}(w)=\mathbf{Q}_{M_{s}}^{\mathrm{H}}\mathbf{a}_{M_{s}}(w)\) are real-valued frequency vector. \(\mathbf{d}_{N_{s}}(u)\) satisfies rotation invariance as follows:  
\[e^{j\mu}\mathbf{Q}_{N_{s}-1}^{\mathrm{H}}\mathbf{J}_{1}\mathbf{Q}_{N_{s}} \mathbf{d}_{N_{s}}(u)=\mathbf{Q}_{N_{s}-1}^{\mathrm{H}}\mathbf{J}_{2}\mathbf{ Q}_{N_{s}}\mathbf{d}_{N_{s}}(u) \tag{25}\]where \(\mathbf{J}_{1}\) and \(\mathbf{J}_{2}\) are selection matrices of \((N_{s}-1)\times N_{s}\) written as \(\mathbf{J}_{1}= \begin{bmatrix}\mathbf{\Pi}_{N_{s}-1}&\mathbf{0}\end{bmatrix} \) and \(\mathbf{J}_{2}= \begin{bmatrix}\mathbf{0}&\mathbf{\Pi}_{N_{s}-1}\end{bmatrix} \). It is easily proved that \(\mathbf{Q}_{N_{s}-1}^{\mathrm{H}}\mathbf{J}_{2}\mathbf{Q}_{N_{s}}=\left( \mathbf{Q}_{N_{s}-1}^{\mathrm{H}}\mathbf{J}_{1}\mathbf{Q}_{N_{s}}\right)^{*}\). Thus, \(\mathbf{K}_{1}\) and \(\mathbf{K}_{2}\) denote the real and imaginary parts of \(\mathbf{Q}_{N_{s}-1}^{\mathrm{H}}\mathbf{J}_{2}\mathbf{Q}_{N_{s}}\) as follows:  
\[\mathbf{K}_{1}=\mathrm{Re}\{\mathbf{Q}_{N_{s}-1}^{\mathrm{H}}\mathbf{J}_{2} \mathbf{Q}_{N_{s}}\} \tag{26}\]  
\[\mathbf{K}_{2}=\mathrm{Im}\{\mathbf{Q}_{N_{s}-1}^{\mathrm{H}}\mathbf{J}_{2} \mathbf{Q}_{N_{s}}\} \tag{27}\]  
substituting (26) and (27) into (25), (25) can be rewritten as:  
\[e^{\frac{i}{2}}(\mathbf{K}_{1}-\mathbf{J}\mathbf{K}_{2})\mathbf{d}_{N_{s}}(u) =e^{-\frac{i}{2}}(\mathbf{K}_{1}+\mathbf{j}\mathbf{K}_{2})\mathbf{d}_{N_{s}}(u) \tag{28}\]  
Thus, the relation is satisfied by \(\mathbf{d}_{N_{s}}(u)\) as follows:  
\[\tan(\frac{u}{2})\mathbf{K}_{1}\mathbf{d}_{N_{s}}(u)=\mathbf{K}_{2}\mathbf{d}_ {N_{s}}(u) \tag{29}\]  
Similarly, the above relations are satisfied by \(\mathbf{d}_{M_{s}}(w)\) through replacing \(u\) with \(w\) and \(N_{s}\) with \(M_{s}\).