## Appendix A Derivation of the CRB of 2D DOA with Known Waveforms and Unknown Complex Amplitudes  
Similar to [40], define the vector consisting of all real-valued unknown variables in (6) as:  
\[\mathbf{\alpha}\triangleq\left[\sigma_{v}^{2},\mathbf{\gamma}^{T},\mathbf{\gamma}^{T}, \mathbf{\theta}^{T},\mathbf{\varphi}^{T}\right]^{T}\in\mathbb{R}^{(1+4K)\times 1} \tag{11}\]  
where \(\sigma_{v}^{2}\), \(\mathbf{\gamma}\), \(\mathbf{\theta}\) and \(\mathbf{\varphi}\) are defined in Section 2. The corresponding Fisher information matrix can be expressed as:  
\[\mathbf{F}(\mathbf{\alpha})=\begin{bmatrix}\text{var}^{-1}(\sigma_{v}^{2})&&\mathbf{0}& \\ &\widetilde{W}&-\widetilde{W}&\widetilde{\mathbf{U}}&\widetilde{\mathbf{V}}\\ &\widetilde{W}&\widetilde{W}&\widetilde{\mathbf{U}}&\widetilde{\mathbf{V}}\\ \mathbf{0}&\widetilde{\mathbf{U}}^{T}&\widetilde{\mathbf{U}}^{T}&\\ &\widetilde{\mathbf{V}}^{T}&\widetilde{\mathbf{V}}^{T}&\mathbf{\Omega}\\ \end{bmatrix}\in\mathbb{C}^{(1+4K)\times(1+4K)} \tag{12}\]  
where \(\mathbf{\Omega}\) is defined in (29), and the rest is defined as follows:  
\[\text{var}(\sigma_{v}^{2})\triangleq\frac{\sigma_{v}^{4}}{MN} \tag{13}\]  
\[\begin{cases}\mathbf{U}\triangleq\frac{2}{\sigma_{v}^{2}}\sum_{t=1}^{N}\mathbf{Y}^{H} (t)\mathbf{A}^{H}\mathbf{D}_{\theta}\mathbf{S}(t)\in\mathbb{C}^{K\times K}\\ \mathbf{V}\triangleq\frac{2}{\sigma_{v}^{2}}\sum_{t=1}^{N}\mathbf{Y}^{H}(t)\mathbf{A}^{H} \mathbf{D}_{\varphi}\mathbf{S}(t)\in\mathbb{C}^{K\times K}\\ \mathbf{W}\triangleq\frac{2}{\sigma_{v}^{2}}\sum_{t=1}^{N}\mathbf{Y}^{H}(t)\mathbf{A}^{H} \mathbf{A}\mathbf{Y}(t)\in\mathbb{C}^{K\times K}\\ \end{cases} \tag{14}\]  
where \(M\) and \(N\) are the number of sensors and snapshots, \(\mathbf{Y}(t)\) and \(\mathbf{S}(t)\) are defined in (32), \(\mathbf{D}_{\theta}\) and \(\mathbf{D}_{\varphi}\) are defined in (23) and \(\mathbf{A}\) is defined in Section 2.  
Given the relationship between CRB and the Fisher information matrix [31], CRB for \(\mathbf{\omega}=[\mathbf{\theta}^{T},\mathbf{\varphi}^{T}]^{T}\) with known waveforms and unknown complex amplitudes is:  
\[\begin{split}\mathbf{CRB}_{d}(\mathbf{\omega})&=\begin{cases} \mathbf{\Omega}-\begin{bmatrix}\widetilde{\mathbf{U}}&\widetilde{\mathbf{U}}^{T}\\ \widetilde{\mathbf{V}}^{T}&\widetilde{\mathbf{V}}^{T}\end{bmatrix}\begin{bmatrix} \widetilde{\mathbf{W}}&-\widetilde{\mathbf{W}}\\ \widetilde{\mathbf{W}}&\widetilde{\mathbf{W}}\end{bmatrix}^{-1}\begin{bmatrix} \widetilde{\mathbf{U}}&\widetilde{\mathbf{V}}\\ \widetilde{\mathbf{U}}&\widetilde{\mathbf{V}}\end{bmatrix}\end{cases}^{-1}\\ &=\begin{cases}\mathbf{\Omega}-\text{Re}\Big{\{}[\mathbf{U}\quad\mathbf{V} ]^{H}\mathbf{W}^{-1}[\mathbf{U}\quad\mathbf{V}]\Big{\}}\Big{\}}^{-1}\end{cases}\end{split} \tag{10}\]  
As we can see from (11) and (10), CRB seems like a function of \(\{y_{k}(t)\}_{k=1}^{K}\) and \(\mathbf{\gamma}\), and varies along them. To reduce CRB to a fixed expression independent of them, substitute relationship \(\mathbf{U}=\mathbf{\Gamma}^{-H}\widetilde{\mathbf{U}}\), \(\mathbf{V}=\mathbf{\Gamma}^{-H}\widetilde{\mathbf{V}}\) and \(\mathbf{W}=\mathbf{\Gamma}^{-H}\widetilde{\mathbf{W}}\mathbf{\Gamma}^{-1}\) into (10), where \(\mathbf{\Gamma}\) is defined in Section 2, \(\widetilde{\mathbf{U}}\), \(\widetilde{\mathbf{V}}\) and \(\widetilde{\mathbf{W}}\) are defined in (31) and CRB can be further simplified to (28).