## 2 Signal Architecture and Problem Description  
We consider a relaying broadcast scheme with source \((\mathcal{S})\), relay \((\mathcal{R})\), and destination \((\mathcal{D})\) terminals, as described in Figure 1. The half-duplex mechanism is used by \(\mathcal{R}\), such that it sends and receives data at distinct times. An STBC option \(\alpha\) is selected by \(\mathcal{S}\) from a collection of alternative choices \(\Psi\). A packet of transmitted information is divided into portions, each of which is made up of \(z_{\alpha}\) symbols chosen at random from a signal constellation. The \(b\)th portion \(\mathrm{d}^{(b)}=\left[d_{0}^{(b)},d_{1}^{(b)},\cdots,d_{z_{\alpha}-1}^{(b)}\right]\) is converted by the encoder of STBC \(\alpha\) into a sent matrix \(\mathbf{M}_{\alpha}^{(b)}\), which is then transmitted from \(n_{\alpha}\) antennas at \(u_{\alpha}\) time intervals. To illustrate, we assume that \(\Psi=\{c_{0},c_{1},c_{2},c_{3}\}\) where \(c_{i}\) is the \(i\)th STBC option, and its transmitted matrix is described as  
\[\mathbf{M}_{\alpha=c_{0}}^{(b)}\Big{(}\mathrm{d}^{(b)}=\left[d_{0}^{(b)},d_{1 }^{(b)}\right]\Big{)}= \begin{array}{c}\text{Time}\rightarrow\\ \text{Space}\downarrow\end{array} \left[ \begin{array}{c}d_{0}^{(b)}\\ d_{1}^{(b)}\end{array} \right]\,\] (1a) \[\mathbf{M}_{\alpha=c_{1}}^{(b)}\Big{(}\mathrm{d}^{(b)}=\left[d_{0}^{(b)},d_{1 }^{(b)}\right]\Big{)}= \begin{array}{c}\text{Time}\rightarrow\\ \text{Space}\downarrow\end{array} \left[ \begin{array}{cc}d_{0}^{(b)}&-d_{1 }^{(b)*}\\ d_{1}^{(b)}&d_{0}^{(b)*}\end{array} \right]\,\] (1b) \[\mathbf{M}_{\alpha=c_{2}}^{(b)}\Big{(}\mathrm{d}^{(b)}=\left[d_{0}^{(b)},d_{1 }^{(b)},d_{2}^{(b)}\right]\Big{)}= \begin{array}{c}\text{Time}\rightarrow\\ \text{Space}\downarrow\end{array} \left[ \begin{array}{cc}d_{0}^{(b)}&-d_{1 }^{(b)*}&d_{2}^{(b)*}&0\\ d_{1}^{(b)}&d_{0}^{(b)}&0&-d_{2}^{(b)*}\\ d_{2}^{(b)}&0&-d_{0}^{(b)*}&d_{1}^{(b)*}\end{array} \right]\,\] (1c) \[\mathbf{M}_{\alpha=c_{3}}^{(b)}\Big{(}\mathrm{d}^{(b)}\!=\!\left[d_{0}^{(b)},d_ {1}^{(b)},d_{2}^{(b)},d_{3}^{(b)}\right]\Big{)}\!=\] \[ \begin{array}{c}\text{Time}\rightarrow\\ \text{Space}\downarrow\end{array} \left[\begin{array}{cccccccc}d_{0}^{(b)}&-d_ {1}^{(b)}&-d_{3}^{(b)}&d_{0}^{(b)*}&-d_{0}^{(b)*}&-d_{0}^{(b)*}&-d_{0}^{(b)*} \\ d_{1}^{(b)}&d_{0}^{(b)}&d_{3}^{(b)}&-d_{2}^{(b)*}&d_{1}^{(b)*}&d_{0}^{(b)*}&d_ {3}^{(b)*}&-d_{2}^{(b)*}\\ d_{2}^{(b)}&-d_{0}^{(b)}&d_{0}^{(b)}&d_{1}^{(b)*}&d_{2}^{(b)*}&d_{3}^{(b)*}&d _{0}^{(b)*}&d_{1}^{(b)*}\\ d_{3}^{(b)}&d_{2}^{(b)}&-d_{1}^{(b)}&d_{0}^{(b)}&d_{3}^{(b)*}&d_{2}^{(b)The collected \(n\)th element at the relay, \(y_{\mathcal{R}}(n)\), and the destination, \(y_{\mathcal{D}}(n)\), during the first stage is mathematically represented, respectively, as follows:  
\[y_{\mathcal{R}}(n)=\sum_{q=0}^{n_{n}-1}h_{\mathcal{SR}}^{(q)}\sum_{m=0}^{u_{n}-1 }\sum_{b=0}^{\infty}\overline{w}_{q}^{(a)}(n)\delta(n-m-bu_{a})+g_{\mathcal{R}}( n), \tag{2}\]  
\[y_{\mathcal{D}}^{(1)}(n)=\sum_{q=0}^{n_{n}-1}h_{\mathcal{SD}}^{(q,1)}\sum_{m=0} ^{u_{n}-1}\sum_{b=0}^{\infty}\overline{w}_{q}^{(a)}(n)\delta(n-m-bu_{a})+g_{ \mathcal{D}}^{(1)}(n), \tag{3}\]  
where \(h_{\mathcal{SR}}^{(q)}\) and \(h_{\mathcal{SD}}^{(q,t)}\) are the link coefficients between the source antenna \(q\) and relay, and between the source antenna \(q\) and destination at the time interval \(t\), respectively, \(\overline{w}_{q}^{(a)}(n)\) is the \(n\)th element of vector \(\mathbf{W}_{q}^{(a)},g_{\mathcal{R}}(n)\) and \(g_{\mathcal{D}}^{(1)}(n)\) are the noise contributions at the relay and destination nodes, respectively, and \(\delta(\cdot)\) is the Kronecker delta function. Since the \(\mathcal{R}\) terminal only receives data in one time slot, the channel coefficient between the source and relay omits the superscript value \(t\). The \(\mathcal{D}\) node, on the other hand, collects data in two time slots.