## 3 Joint Angle and Delay Estimation (JADE)
### Overview/Summary of IS for ML DA JADE  
We start by approximating \(\mathbf{B}^{H}\mathbf{B}\) in (12) as follows:  
\[\mathbf{B}^{H}\mathbf{B}\approx P\ K\,\mathbf{I}_{Q}, \tag{14}\]  
where \(\mathbf{I}_{Q}\) denotes the \((Q\times Q)\) identity matrix. Then, we plug (14) into (12) to obtain:  
\[\mathcal{F}_{c}\big{(}\theta,\tau\big{)} \approx \frac{1}{PK}\ \mathbf{x}^{H}\mathbf{B}\mathbf{B}^{H}\mathbf{x} \approx\frac{1}{PK}||\mathbf{B}^{H}\mathbf{x}||^{2}, \tag{15}\] \[\approx \frac{1}{PK}\sum_{q=1}^{Q}\bigg{|}\sum_{k=1}^{K}[\tilde{\mathbf{A }}(\theta)\mathbf{D}_{k}(\tau)]^{H}\mathbf{x}(k)\bigg{|}^{2}.\]  
Now, injecting the expressions of \(\tilde{\mathbf{A}}(\theta)\) and \(\mathbf{D}_{k}(\tau)\) into (15), we obtain the approximate CLF:  
\[\mathcal{F}_{c}(\theta,\tau)\approx\frac{1}{PK}\sum_{q=1}^{Q}\psi(\theta_{q}, \tau_{q}), \tag{16}\]  
where \(\psi(\theta,\tau)\) is the so-called periodogram of the observation signal [11] given by:  
\[\psi(\theta,\tau)=\bigg{|}\sum_{p=1}^{P}\sum_{k=1}^{K}e^{-j\pi(p-1)\sin{( \theta)}}e^{-j2\pi\tilde{\mathbf{A}}\tilde{\mathbf{f}}\tau}x_{p}^{*}(k)\bigg{|} ^{2}. \tag{17}\]  
Relying on the observations made above, we summarize the IS process in [11] of generating the \(R\) realizations of the AoA-TD couples according to the following steps:  
**Step (1):** we start by evaluating the periodogram \(\psi(\theta_{i},\tau_{j})\) in (17) at all grid points \((\theta_{i},\tau_{j})\in\overline{\Gamma(\frac{-\pi}{2},\frac{\pi}{2},\delta _{\theta})}\times\Gamma(0,\tau_{\text{max}},\delta_{\tau})\) where \(\Lambda(v_{\text{min}},v_{\text{max}},\delta_{v})\) denotes the set of points obtained over the interval \([v_{\text{min}},v_{\text{max}}]\) with a uniform sampling step \(\delta_{v}\). **Step (2):**: we evaluate the so-called joint pseudo-pdfs [11] set of values over the above AoA-TD grid as follows:  
\[\Phi(\theta_{i},\tau_{j})\!\!=\!\!\frac{\exp\{\rho_{1}\psi(\theta_{i},\tau_{j})\} }{\sum_{i}\!\!\sum_{j}\!\exp\{\rho_{1}\psi(\theta_{i},\tau_{j})\}\delta_{\theta }\delta_{\tau}}, \tag{18}\]  
where \(\rho_{1}\) is a design parameter to be chosen properly later on. **Step (3):**: we evaluate the marginal pseudo-pdf of \(\tau_{j}\) as follows:  
\[\tilde{\phi}_{\tau}(\tau_{j})=\sum_{i}\tilde{\phi}_{\theta,\tau}(\theta_{i}, \tau_{j})\delta_{\theta}. \tag{19}\]  
Then, we can find the initial TD estimates that correspond to the \(Q\) maxima of (19) as:  
\[\big{[}\tilde{\tau}_{1}^{0},\tilde{\tau}_{2}^{0},\cdots,\tilde{\tau}_{Q}^{0} \big{]}=\underset{\tau}{\mathrm{argmax}}\,|_{Q}\Big{(}\tilde{\phi}_{\tau}( \tau)\Big{)}, \tag{20}\]  
where \(\underset{Q}{\mathrm{argmax}}\,|_{Q}(f)\) returns the \(Q\) maxima of the function \(f\). **Step (4):**: for \(q=1,2,\cdots,Q\), we compute the pseudo-CDF of \(\tau_{j}\) as follows:  
\[\mathcal{J}_{\tau_{q}}(\tau_{j})=\sum_{l\leq j}\tilde{\Phi}_{\tau_{q}}(\tau_{l })\delta_{\tau}\quad\forall\ \tau_{l}\ \in\lambda_{\tilde{\tau}_{q}^{0}}, \tag{21}\]  
where  
\[\lambda_{\tilde{\tau}_{q}^{0}}=\Gamma\Big{(}\tilde{\tau}_{q}^{0}-\Delta_{\tau },\tilde{\tau}_{q}^{0}+\Delta_{\tau},\delta_{\tau}\Big{)}. \tag{22}\]  
**Step (5):**: For \(q=1,2,\cdots,Q\), we generate \(R\) realizations \(\Big{\{}u_{q}^{(r)}\sim\mathcal{U}([0,1])\Big{\}}_{r=1}^{R}\).