## 3 Direction-of-Arrival Estimators with Known Waveforms
### Interferometer-Based Estimator with Known Waveforms  
The interferometer-based estimator with known waveforms herein comes after the cost function in (10). In other words, it is equivalent to the ML estimators in performance to some degree. Firstly, we can infer from (10) that vector \(\hat{b}_{k}\) and \(\mathbf{a}(\theta_{k},\varphi_{k})\) are colinear so that they only differ in complex amplitude \(\gamma_{k}\) and share same phase-difference between different elements. Therefore, the phase-differences between any two of sensors for the \(k\)th source can be estimated by \(\hat{b}_{k}\). Finally, the estimated phase-differences are used to form a LS problem to obtain 2D DOA estimation in closed form. Note that our method is an extension of [26], as we bring in known waveforms and make use of baselines more comprehensively; therefore, our method will achieve much better performance in 2D DOA estimation. Take the estimation of \((\theta_{k},\varphi_{k})\) as an example to demonstrate our method; the others are the same. Based on above analysis, we start with calculating the phase-difference (free of noise) between the \(p\)th and the \(q\)th sensor (baseline \((p,q)\)) for the \(k\)th source as:  
\[ \begin{array}{ll}\eta_{k}^{p,q}&=\arg\{a_{p}(\theta_{k},\varphi_{k})a_{q}^{ *}(\theta_{k},\varphi_{k})\}=\arg\{\mathbf{b}_{k}(p)\mathbf{b}_{k}^{*}(q)\}\\ &=(2\pi r/\lambda)\sin\theta_{k}(\cos(\varphi_{k}-\beta_{p})-\cos(\varphi_{k}- \beta_{q}))\\ &=\frac{2\pi r}{\lambda}[\cos(\beta_{p})-\cos(\beta_{q})\quad\sin(\beta_{p})- \sin(\beta_{q})]\begin{bmatrix}\sin(\theta_{k})\cos(\varphi_{k})\\ \sin(\theta_{k})\sin(\varphi_{k})\end{bmatrix}\end{array}  \tag{11}\]  
where \(\mathbf{b}_{k}(p)\) denotes the \(p\)th element of \(\mathbf{b}_{k}\). To make sure that there is no phase ambiguity in \(\eta_{k}^{p,q}\), i.e., \(\eta_{k}^{p,q}\in\left[-\pi,\pi\right)\), we assume that \(r/\lambda\leq 1/4\), since \(|\cos(\varphi_{k}-\beta_{p})-\cos(\varphi_{k}-\beta_{q})|\leq 2\). Otherwise, a great many techniques are available for solving the problem of phase ambiguity [32; 33; 34; 35]. We refer the interested reader to these works for details and will not consider the ambiguity problem in this paper for simplicity. Obviously, \((\theta_{k},\varphi_{k})\) can be solved by phase-difference equations constructed from (11) using the LS method. Assume that the sensor interval of every baseline \((q-p)\) is fixed at \(=\Delta\) or \(\Delta-M\). To achieve the best performance, we shall make use of baselines as much as possible. Thus, we can exploit baseline\((1,1+\Delta)\), \((2,2+\Delta),\cdots,(M-\Delta,M)\) and \((M-\Delta+1,1)\), \((M-\Delta+2,2),\cdots,(M,\Delta)\) to form the equations, define:  
\[\mathbf{\eta}_{k}(\Delta)\triangleq[\eta_{k}^{1,1+\Delta},\eta_{k}^{2,2+\Delta}, \cdots,\eta_{k}^{M-\Delta,M},\eta_{k}^{M-\Delta+1,1},\eta_{k}^{M-\Delta+2,2}, \cdots,\eta_{k}^{M,\Delta}]^{T}\in\mathbb{R}^{M\times 1} \tag{12}\]  
\[\mathbf{C}(\Delta)\triangleq \begin{bmatrix}\cos(\beta_{1})-\cos(\beta_{1+\Delta}) \quad\sin(\beta_{1})-\sin(\beta_{1+\Delta})\\ \cos(\beta_{2})-\cos(\beta_{2+\Delta})\quad\sin(\beta_{2})-\sin(\beta_{2+ \Delta})\\ \vdots\quad\quad\quad\quad\quad\vdots\\ \cos(\beta_{M-\Delta})-\cos(\beta_{M})\quad\sin(\beta_{M-\Delta})-\sin(\beta_ {M})\\ \cos(\beta_{M-\Delta+1})-\cos(\beta_{1})\quad\sin(\beta_{M-\Delta+1})-\sin( \beta_{1})\\ \cos(\beta_{M-\Delta+2})-\cos(\beta_{2})\quad\sin(\beta_{M-\Delta+2})-\sin( \beta_{2})\\ \vdots\quad\quad\quad\quad\quad\vdots\\ \cos(\beta_{M})-\cos(\beta_{\Delta})\quad\sin(\beta_{M})-\sin(\beta_{\Delta}) \end{bmatrix} \in\mathbb{R}^{M\times 2} \tag{13}\]  
\[\mathbf{g}_{k}\triangleq\frac{2\pi r}{\lambda} \begin{bmatrix}\sin(\theta_{k})\cos( \varphi_{k})\\ \sin(\theta_{k})\sin(\varphi_{k})\end{bmatrix}