# Single-Carrier Rotation-Interleaved Space-Time Code for Frequency-Selective Fading Channels +
## Appendix B Frequency Domain Iterative Receiver  
The detailed formulation of the frequency domain iterative receiver applicable to the proposed rotation-interleaved multi-stream space-time code is described as follows. The iterative receiver consists of the soft-cancellation, MMSE frequency domain equalization (MMSE-FDE) and the decoding. The MMSE-FDE aims to find the estimates of the input coded symbols transmitted from antenna \(i\in\{1,\cdots,M_{T}\}\). Recall that \(K\) denotes the size of transmitted block. Denote \(\mathbf{O}_{ij}\) as the channel matrix corresponding to the \((i,j)\)th transmit-receive antenna pair and \(\mathbf{O}_{ij}\in\mathcal{C}^{K\times K}\) is a circulant matrix with \([h_{ij}(0),h_{ij}(1),\cdots,h_{ij}(L-1),0,\cdots,0]\) as its first row. By performing the eigen-decomposition, \(\mathbf{O}_{ij}\) can be written as \(\mathbf{O}_{ij}=\mathbf{F}^{H}\mathbf{I}_{ij}\mathbf{F}\), where \(\mathbf{F}\) is the FFT-matrix and \(\mathbf{\Gamma}_{ij}\) is a diagonal matrix containing the channel coefficients in the frequency domain. Denote \(\mathbf{\Xi}_{i}=[\mathbf{\Gamma}_{i1}^{H},\cdots,\mathbf{\Gamma}_{iM_{R}}^{H}] ^{H}\) and \(\mathbf{\Omega}=[\mathbf{\Xi}_{1},\cdots,\mathbf{\Xi}_{M_{T}}]\). The MMSE-FDE will take into consideration the quality of the soft-cancellation, which is explained as follows. Let us denote \(\overline{s}_{i}^{i}\) as the soft estimation of \(s_{t}^{i}\), the transmitted symbol from the \(i\)th antenna at time \(t\), which is obtained from the decoding process in the last iteration, and the estimates are assumed to be \(\overline{s}_{t}^{i}\approx\mathbb{E}[s_{t}^{i}|\mathbf{Y}]\). Then, the covariance matrix of estimation error can be approximated by \(\mathbf{\Lambda}_{t}=\text{diag}(E_{s}-|\overline{s}_{t}^{1}|^{2},E_{s}-| \overline{s}_{t}^{2}|^{2},\cdots,E_{s}-|\overline{s}_{t}^{M_{T}}|^{2})\). To further simplify the computation of the MMSE filter coefficients, we calculate the average of the estimation error as \(\varrho=1/(KM_{T})\sum_{t=1}^{K}\sum_{t=1}^{M_{T}}|s_{t}^{i}|^{2}\) and therefore \(\mathbf{\Lambda}_{t}\approx\varrho\mathbf{I}_{M_{T}}\). Denote the frequency domain representation of the received signal vector at receive antenna \(j\) as \(\mathbf{\underline{r}}_{j}=\mathbf{F}(\mathbf{y}_{j})^{T}\) and \(\mathbf{\underline{r}}=[\mathbf{z}_{1}^{H},\cdots,\mathbf{z}_{M_{R}}^{H}]^{H}\). In addition, denote \(\overline{s}^{i}=[\overline{s}_{1}^{i},\cdots,\overline{s}_{K}^{i}]^{T}\) and its frequency domain representation as \(\overline{\mathbf{s}}^{i}=\mathbf{F}\overline{s}^{i}\). Then, denote \(\overline{\mathbf{\underline{r}}}=\mathbf{\Omega}[(\overline{\mathbf{s}}^{1}) ^{H},\cdots,(\overline{\mathbf{s}}^{M_{T}})^{H}]^{H}\). Define \(\mathbf{\overline{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{ \mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{ \mathbf{ \mathbf{ \mathbf{ \mathbf{ \mathbf{ \mathbf \mathbf{                                     \leftleft\|\mathbf{ \mathbf{ \mathbf{ \mathbf{      \mathbf{             \leftleft\{ \ \ \ \ \ \ \leftleft\{ \ \ \left\left\{\ \left\{\ \left\left\{\leftleft\{\leftleftleft\{\leftleftleftleft({ \left({ \leftleftleftleft(The extrinsic LLRs, which are subsequently fed to the decoder as intrinsic LLRs, can be derived by using the equivalent output form (1) and the bits-to-symbol mapping. If BPSK is used, the extrinsic LLR sequences of the code bits are simply given by  
\[\lambda_{e}[c_{t}^{i}]=\frac{4\text{Re}\{e^{-\beta_{t}^{i}}z_{t}^{i}\}}{\gamma_{ i}^{-1}-(E_{s}-\varrho)} \tag{10}\]  
where \(c_{t}^{i}\) and \(\theta_{t}^{i}\) are the transmitted code bit and the respective rotation applied at time \(t\) on \(i\)th antenna, respectively. Note that \(c_{t}^{i}\) and \(\theta_{t}^{i}\) belong to a particular stream and we drop the stream index for simplicity.