## 4 Cramer-Rao Bound Derivation and Analysis
\in \mathbb{C}^{2K\times 2K} \tag{25}\]  
where \(\mathbf{R}\) and \(\mathbf{P}\) are the array and signal covariance matrix, respectively:  
\[\mathbf{R}\triangleq\text{E}\Big{\{}\mathbf{x}(t)\mathbf{x}^{H}(t)\Big{\}}=\lim_{N\to \infty}\frac{1}{N}\sum_{t=1}^{N}\mathbf{x}(t)\mathbf{x}^{H}(t)=\mathbf{AP}^{H}+\sigma_{v}^{2 }\mathbf{I}\in\mathbb{C}^{M\times M} \tag{26}\]  
\[\mathbf{P}\triangleq\text{E}\Big{\{}\mathbf{s}(t)\mathbf{s}^{H}(t)\Big{\}}=\lim_{N\to \infty}\frac{1}{N}\sum_{t=1}^{N}\mathbf{s}(t)\mathbf{s}^{H}(t)\in\mathbb{C}^{K\times K} \tag{27}\]  
CRB of 2D DOA estimation for signals with known waveforms and unknown complex amplitudes is a generalization of a similar result in [40] as:  
\[\mathbf{CRB}_{d}(\mathbf{\omega})=\Big{[}\mathbf{\Omega}-\text{Re}\Big{\{}\big{[}\widetilde {\mathbf{U}}\quad\widetilde{\mathbf{V}}\big{]}^{H}\widetilde{\mathbf{W}}^{-1}\big{[} \widetilde{\mathbf{U}}\quad\widetilde{\mathbf{V}}\big{]}\Big{\}}\Big{]}^{-1}\in\mathbb{ C}^{2K\times 2K} \tag{28}\]  
where \(\mathbf{\Omega}\) is defined as:  
\[\mathbf{\Omega}\triangleq\frac{2N}{\sigma_{v}^{2}}\text{Re}\Big{\{}(\mathbf{D}^{H}\bm {D})\odot\mathbf{P}_{-}^{T}\Big{\}}\in\mathbb{C}^{2K\times 2K} \tag{29}\]  
\[\mathbf{P}_{-}\triangleq \begin{bmatrix}\mathbf{P}&\mathbf{P}\\ \mathbf{P}&\mathbf{P}\end{bmatrix} \in\mathbb{C}^{2K\times 2K} \tag{30}\]\(\widetilde{\mathbf{U}}\), \(\widetilde{\mathbf{V}}\) and \(\widetilde{\mathbf{W}}\) are defined as:  
\[ \begin{cases}\widetilde{\mathbf{u}}\triangleq\frac{2}{\sigma_{v}^{2}}\sum_{t=1}^{N} \mathbf{S}^{H}(t)A^{H}\mathbf{D}_{\theta}\mathbf{S}(t)=\frac{2N}{\sigma_{v}^{2}}(\mathbf{A}^{H} \mathbf{D}_{\theta})\odot\mathbf{P}^{T}\in\mathbb{C}^{K\times K}\\ \widetilde{\mathbf{V}}\triangleq\frac{2}{\sigma_{v}^{2}}\sum_{t=1}^{N}\mathbf{S}^{H}(t )\mathbf{A}^{H}\mathbf{D}_{\theta}\mathbf{S}(t)=\frac{2N}{\sigma_{v}^{2}}(\mathbf{A}^{H}\mathbf{D} _{\varphi})\odot\mathbf{P}^{T}\in\mathbb{C}^{K\times K}\\ \widetilde{\mathbf{W}}\triangleq\frac{2}{\sigma_{v}^{2}}\sum_{t=1}^{N}\mathbf{S}^{H}(t )\mathbf{A}^{H}\mathbf{A}\mathbf{S}(t)=\frac{2N}{\sigma_{v}^{2}}(\mathbf{A}^{H}\mathbf{A})\odot \mathbf{P}^{T}\in\mathbb{C}^{K\times K}\end{cases}  \tag{31}\]  
\[ \begin{cases}\mathbf{S}(t)\triangleq\text{diag}\{\mathbf{s}(t)\}=\text{diag}\{s_{1}(t),s_{2}(t),\cdots,s_{K}(t)\}\in\mathbb{C}^{K\times K}\\ \mathbf{Y}(t)\triangleq\text{diag}\{\mathbf{y}(t)\}=\text{diag}\{y_{1}(t),y_{2}(t), \cdots,y_{K}(t)\}\in\mathbb{C}^{K\times K}\end{cases}  \tag{32}\]  
The derivation details are shown in the Appendix A. Here, we will provide some insights into \(\mathbf{CRB}_{d}(\mathbf{\omega})\). If the source signals are uncorrelated with each other, i.e., \(\mathbf{P}=\text{diag}\{\sigma_{s1}^{2},\sigma_{s2}^{2},\cdots,\sigma_{sK}^{2}\}\) is a diagonal matrix, where \(\sigma_{sk}^{2}=\text{E}\big{\{}|s_{k}(t)|^{2}\big{\}}\) is the power of \(s_{k}(t)\), (28) for UCA will reduce to:  
\[ \begin{split}\mathbf{CRB}_{d}(\theta_{k},\varphi_{k})& =\frac{\sigma_{v}^{2}}{2N\sigma_{sk}^{2}}\text{Re}^{-1}\Bigg{\{} \begin{bmatrix}\mathbf{a}_{\theta_{k}}^{\prime}(\theta_{k},\varphi_{k})\mathbf{a}_{ \theta_{k}}^{\prime}(\theta_{k},\varphi_{k})&0\\ 0&\mathbf{a}_{\varphi_{k}}^{\prime}(\theta_{k},\varphi_{k})\mathbf{a}_{\varphi_{k}}^{ \prime}(\theta_{k},\varphi_{k})\end{bmatrix}\Bigg{\}}\\ &=\frac{\lambda^{2}\sigma_{v}^{2}}{4 MN\pi^{2}r^{2}\sigma_{sk}^{2}}\text{diag}^ {-1}\big{\{}\cos^{2}(\theta_{k}),\sin^{2}(\theta_{k})\big{\}},\quad k=1,2, \cdots,K\end{split}  \tag{33}\]  
as \(\mathbf{a}_{\theta_{k}}^{\prime\ H}(\theta_{k},\varphi_{k})\mathbf{a}_{\varphi_{k}}^{ \prime}(\theta_{k},\varphi_{k})=0\), \(\mathbf{a}_{\theta_{k}}^{\prime\ H}(\theta_{k},\varphi_{k})\mathbf{a}(\theta_{k}, \varphi_{k})=0\) and \(\mathbf{a}_{\varphi_{k}}^{\prime\ H}(\theta_{k},\varphi_{k})\mathbf{a}(\theta_{k}, \varphi_{k})=0\). That is to say, in such a case, the CRB of any one of multiple sources is the same as the CRB when it is the only source. Hence, the CRB of 2D DOA estimation of different sources are independent with each other and can be calculated separately.