## 2 System Model  
We consider a SIMO OFDM system characterized by a single transmitting and \(P\) receiving antenna elements and \(K\) sub-carriers. At each time period, this system transmits over these sub-carriers \(K\) symbols \(\mathbf{s}=[s_{1},s_{2},\ldots,s_{K}]^{T}\), all belonging to an \(M\)-ary constellation alphabet \(C^{M}\), and are assumed to be known at pilot transmission periods (i.e., replaced a priori by 1) in the DA-type estimation scheme we are adopting here. The transmit data then goes through a multi-path channel consisting of different paths, whose number \(Q\) is assumed to be known. The resulting multi-path SIMO channel is characterized by \(Q\) different AoAs \(\mathbf{\theta}=[\theta_{1},\theta_{2},\cdots,\theta_{Q}]\in\mathcal{U}\big{(}[ \frac{-\pi}{2},\frac{\pi}{2}]\big{)}^{Q}\); \(Q\) different TDs \(\mathbf{\tau}=[\tau_{1},\tau_{2},\cdots,\tau_{Q}]\in\mathcal{U}\big{(}[0,\tau_{ \max}]\big{)}^{Q}\), where \(\tau_{\max}\) can be chosen to be as large as desired; \(\mathcal{U}\big{(}[v_{\min},v_{\max}]\big{)}\) denotes a uniform distribution over the interval \([v_{\min},v_{\max}]\); and \(Q\) complex gains \(\mathbf{\gamma}=[\gamma_{1},\gamma_{2},\cdots,\gamma_{Q}]^{T}\). All these three parameter vectors are assumed to be unknown. At the receiver side, the observation signal, \(x_{p}(k)\), over the \(p\)th antenna and the \(k\)th sub-carrier, is given by:  
\[x_{p}(k){=}h_{p}(k)s_{k}{+}n_{p}(k), \tag{1}\]  
where \(n_{p}(k)\) is an additive white Gaussian noise (AWGN) with zero mean and variance \(\sigma^{2}\), and \(h_{p}(k)\) is the channel frequency response (CFR), defined as follows:  
\[h_{p}(k)=\sum_{q=1}^{Q}\gamma_{q}e^{-j2\pi k\underline{\alpha}_{f}\tau_{q}}e^{ -j2\pi rp\sin\big{(}\theta_{q}\big{)}}, \tag{2}\]  
where \(\Delta_{f}\) is the sub-carrier spacing. By stacking, the scalar signal observation in (1) received at each \(k\)th sub-carrier into a single observation vector \(\mathbf{x}(k)=[x_{1}(k),x_{2}(k),\cdots,x_{p}(k)]^{T}\) is modeled as follows:  
\[\mathbf{x}(k){=}\,\mathbf{h}(k)s_{k}+\mathbf{n}(k), \tag{3}\]  
where \(\mathbf{n}(k)=[n_{1}(k),n_{2}(k),\ldots,n_{P}(k)]^{T}\) is an i.i.d. and spatially uncorrelated zero-mean Gaussian noise vector and \(\mathbf{h}(k)\) is the \(P\times 1\) CFR vector over all antennas, defined as:  
\[\mathbf{h}(k) = \sum_{q=1}^{Q}\mathbf{a}(\theta_{q})\gamma_{q}e^{-j2\pi k \underline{\alpha}_{f}\tau_{q}} \tag{4}\] \[= \bar{\mathbf{A}}(\mathbf{\theta})\mathbf{D}_{k}(\mathbf{\tau})\mathbf{\gamma}, \tag{5}\]  
where \(\bar{\mathbf{A}}(\mathbf{\theta})\triangleq[\mathbf{a}(\theta_{1}),\mathbf{a}( \theta_{2}),\ldots,\mathbf{a}(\theta_{Q})]\) is a \(P\times Q\) steering matrix, \(\mathbf{a}(\theta)\) is the \(P\times 1\) steering vector at AoA \(\theta\) defined for simplicity and without loss of generality here over a uniform linear array (ULA) as:  
\[\mathbf{a}(\theta)\triangleq\big{[}1,\;e^{-j\pi\sin(\theta)},\cdots,\;e^{-j(P -1)\pi\sin(\theta)}\big{]}^{T},\]  
and \(\mathbf{D}_{k}(\mathbf{\tau})\triangleq\text{diag}(e^{-j2\pi\underline{\alpha}_{ f}\tau_{1}},e^{-j2\pi\underline{\alpha}_{f}\tau_{2}},\cdots,e^{-j2\pi\underline{ \alpha}_{f}\tau_{Q}})\) is the \(Q\times Q\) TDs matrix, in which \(\bar{k}\) is some index translation of \(k=1,\cdots,K\) (e.g., \(\bar{k}=k-1\), \(\bar{k}=k-K/2-1\) if \(K\) is even, \(\bar{k}=k-(K-1)/2\) if \(K\) is odd, etc. ), with no impact at all on what follows.