# Single-Carrier Rotation-Interleaved Space-Time Code for Frequency-Selective Fading Channels +
## 2 System Model  
In what follows, a MIMO channel with \(M_{T}\) transmit antennas and \(M_{R}\) receive antennas is considered. The space-time coded system model and its performance analysis for the frequency-selective channel are briefly presented here. The input data stream is encoded into blocks of space-time codeword of dimension \(M_{T}\) by \(K\), where \(K\) is the number of symbols over time. To facilitate block-based single-carrier transmission, the coded sequence (block of \(K\) symbols) to be sent from each transmit antenna is inserted with a cyclic prefix of length greater or equal to the maximum channel memory order \(L-1\), which is known by the transmitter. At the receiver, the discrete-time received signal over the \(j\)-th receive antenna can be expressed as  
\[y_{t}^{j}=\sum_{l=0}^{L-1}\sum_{i=1}^{M_{T}}\sqrt{E_{s}}h_{ij}(l)c_{(t-l)\,mod \,K}^{i}+n_{t^{\prime}}^{j},\,t=0,\cdots,K-1 \tag{1}\]  
where \(c_{t}^{i}\) is the coded symbol transmitted at the \(t\)-th time slot over \(i\)-th transmit antenna. We denote \(h_{ij}(l)\) as the channel fading coefficient corresponding to the \(l\)-th path between the \(i\)-th and \(j\)-th antenna. We assume that the transmitter has no knowledge about the channel while the receiver has perfect channel information. The channel fading coefficients \(h_{ij}(l)\), \(i=1,\cdots,M_{T},j=1,\cdots,M_{R},l=0,\cdots,L-1\) are assumed to be i.i.d. zero-mean complex Gaussian random variables with the same variance \(\sigma_{ij}^{2}(0)=\cdots=\sigma_{ij}^{2}(L-1)\) and \(\sum_{t=0}^{L-1}\sigma_{ij}^{2}(l)=1,\,\,\forall i,j\), where \(\sigma_{ij}^{2}(l)\triangleq E\{|h_{ij}(l)|^{2}\}\). The noise samples, \(n_{t}^{j}\), are complex white Gaussian with variance \(N_{0}\). The received signals can now be analyzed as below. First, Equation (1) can be put into a row-vector form as  
\[\mathbf{y}_{j}=\sum_{l=0}^{L-1}\sqrt{E_{s}}\mathbf{h}_{j}(l)\underbrace{ \mathbf{C}(0)\mathbf{\Pi}^{l}}_{\mathbf{C}(l)}+\mathbf{n}_{j} \tag{2}\]  
\[\text{where }\mathbf{y}_{j}=[y_{0}^{j},\cdots,y_{K-1}^{j}],\,\mathbf{n}_{j}=[n_{0 }^{j},\cdots,n_{K-1}^{j}],\,\mathbf{h}_{j}(l)=[h_{1j}(l),h_{2j}(l),\cdots,h_{ M_{Tj}}(l)],\]  
\[\mathbf{\Pi}=\left[ \begin{array}{ccccc}0&1&0&\cdots&0\\ 0&0&1&\ddots&0\\ \vdots&\ddots&\ddots&\ddots&\vdots\\ 1&0&\cdots&\cdots&0\end{array} \right] \tag{3}\]  
and  
\[\mathbf{C}(0)=\left[ \begin{array}{ccccc}c_{0}^{1}&\cdots&c_{K-1}^{1}\\ c_{0}^{2}&\cdots&c_{K-1}^{2}\\ \vdots&\vdots&\vdots\\ c_{0}^{M_{T}}&\cdots&c_{K-1}^{M_{T}}\end{array} \right]. \tag{4}\]  
In (2), we have used the fact that \(\mathbf{C}(l)=\mathbf{C}(0)\mathbf{\Pi}^{l}\), that is, \(\mathbf{C}(l)\) is the result of cyclically shifting the columns in \(\mathbf{C}(0)\) by \(l\) positions to the right. Note that \(\mathbf{C}(0)\) is the traditional definition of a space-time codeword in the flat fading case. Thus, each additional multipath would introduce a delayed version of this space-time codeword into the received signals. For this reason, \(\mathbf{C}(l)\) can also be interpreted as coming from a set of virtual transmit antennas [18]. Stacking all these delayed versions of the same codeword gives  
\[\mathbf{C}=\left[ \begin{array}{c}\mathbf{C}(0)\\ \vdots\\ \mathbf{C}(L-1)\end{array} \right]. \tag{5}\]  
Let \(\mathbf{h}_{j}=[\mathbf{h}_{j}(0)\mathbf{h}_{j}(1)\cdots\mathbf{h}_{j}(L-1)]\), then Equation (2) can be expressed as  
\[\mathbf{y}_{j}=\sqrt{E_{s}}\mathbf{h}_{j}\mathbf{C}+\mathbf{n}_{j}. \tag{6}\]  
Note that Equation (6) is in the exact form as in the flat fading case where a single space-time codeword is considered. By stacking up all received vectors due to different receive antennas, we obtain  
\[\mathbf{Y}=\left[ \begin{array}{c}\mathbf{y}_{1}\\ \vdots\\ \mathbf{y}_{M_{R}}\end{array} \right]=\sqrt{E_{s}}\mathbf{H}\mathbf{C}+\mathbf{ N} \tag{7}\]  
where \(\mathbf{H}=[\mathbf{h}_{1}^{T},\cdots,\mathbf{h}_{M_{R}}^{T}]^{T}\) and \(\mathbf{N}=[\mathbf{n}_{1}^{T},\cdots,\mathbf{n}_{M_{R}}^{T}]^{T}\).