## 3 Direction-of-Arrival Estimators with Known Waveforms
### Maximum Likelihood Estimator with Known Waveforms  
We regard the source signals as deterministic process due to known waveforms, hence, likelihood function of \(\{\mathbf{x}(t)\}_{t=1}^{N}\) is:  
\[\begin{split} P(\mathbf{\theta},\mathbf{\varphi},\mathbf{\gamma},\sigma_{v}^{2 })&=\prod_{t=1}^{N}\!\!\frac{1}{(\pi\sigma_{v}^{2})^{M}}\exp\! \left\{-\frac{1}{\sigma_{v}^{2}}||\mathbf{x}(t)-\mathbf{B}\mathbf{y}(t)||^{2}\right\}\\ &=\frac{1}{(\pi\sigma_{v}^{2})^{MN}}\exp\!\left\{-\frac{1}{\sigma _{v}^{2}}\sum_{t=1}^{N}||\mathbf{x}(t)-\mathbf{B}\mathbf{y}(t)||^{2}\right\}\end{split} \tag{6}\]  
Since the noise in (1) is a zero-mean white Gaussian random process, it is obvious that maximizing the likelihood function with respect to \(\mathbf{\theta},\mathbf{\varphi}\) and \(\mathbf{\gamma}\) is equivalent to minimizing the following cost function:  
\[L(\mathbf{\theta},\mathbf{\varphi},\mathbf{\gamma})=\sum_{t=1}^{N}||\mathbf{x}(t)-\mathbf{B}\mathbf{y} (t)||^{2} \tag{7}\]  
In this case, the LS approach is the same as the ML method [31], so that the LS or ML estimate of \(\mathbf{B}\) is given as follows:  
\[\begin{split}\hat{\mathbf{B}}&=\left(\sum_{t=1}^{N}\!\bm {x}(t)\mathbf{y}^{H}(t)\right)\!\left(\sum_{t=1}^{N}\!\mathbf{y}(t)\mathbf{y}^{H}(t)\right) ^{-1}\\ &=\hat{\mathbf{R}}_{xy}\hat{\mathbf{R}}_{yy}^{-1}\end{split} \tag{8}\]  
where \(\hat{\mathbf{R}}_{xy}=1/N\sum_{t=1}^{N}\mathbf{x}(t)\mathbf{y}^{H}(t)\) and \(\hat{\mathbf{R}}_{yy}=1/N\sum_{t=1}^{N}\mathbf{y}(t)\mathbf{y}^{H}(t)\) are estimates of auto-correlation and cross-correlation. As \(\mathbf{R}_{yy}\) is nonsingular, so does \(\hat{\mathbf{R}}_{yy}\) when \(N\) is large enough. It is easy to find that \(\hat{\mathbf{B}}\) is a asymptotic consistent estimate of \(\mathbf{B}\):  
\[\begin{split}\lim_{N\to\infty}\hat{\mathbf{B}}&=\lim_{N \to\infty}\hat{\mathbf{R}}_{xy}\hat{\mathbf{R}}_{yy}^{-1}\\ &=\lim_{N\to\infty}\left[\frac{1}{N}\!\sum_{t=1}^{N}\!\left(\mathbf{B }\mathbf{y}(t)+\mathbf{n}(t)\right)\mathbf{y}^{H}(t)\right]\hat{\mathbf{R}}_{yy}^{-1}\\ &=\lim_{N\to\infty}\left(\mathbf{B}\hat{\mathbf{R}}_{yy}+\hat{\mathbf{R}}_{ny }\right)\hat{\mathbf{R}}_{yy}^{-1}\\ &=\mathbf{B}\end{split} \tag{9}\]  
where \(\lim_{N\to\infty}\hat{\mathbf{R}}_{ny}=\mathbf{R}_{ny}=\mathbf{0}\) since the noise vectors and the known waveforms are uncorrelated.  
Once we obtain \(\hat{\mathbf{B}}\), based on above analysis and the structure of \(\mathbf{B}\), i.e., \(\mathbf{b}_{k}=\gamma_{k}\mathbf{a}(\theta_{k},\,\varphi_{k})\), where \(\mathbf{b}_{k}\) is the \(k\)th column of \(\mathbf{B}\), the problem of ML estimation of \(\mathbf{\theta}\) and \(\mathbf{\varphi}\) is decoupled into \(K\) independent maximization problems as follows:  
\[\big{(}\hat{\theta}_{k},\,\hat{\varphi}_{k}\big{)}=\operatorname*{arg\,max}_{ \theta,\varphi}|\mathbf{a}^{H}(\theta,\varphi)\hat{\mathbf{b}}_{k}|^{2},\quad k=1,2, \cdots,K \tag{10}\]where \(\hat{b}_{k}\) is the \(k\)th column of \(\hat{B}\). Though the maximization in (10) can be performed by a 2D spectrum peak search, the computational cost is unattractive. To avoid the high complexity of 2D search, we propose a quite simple and efficient interferometer-based estimator, which reduces the search problem to a closed-form solution.