# Signal Detection for GSTFIM Systems with Carrier Frequency Offset +
## 3 Proposed MMSE Detectors for GSTFIM Systems  
To mitigate the ICI caused by the CFO and the effect of the channel, Ref. [24] proposed an iterative algorithm to process the receiver symbol \(\mathbf{Y}_{l}\) based on the MMSE criterion as in [19; 25]. In this section, we generalize the MMSE-based detector in [24] in two different ways based upon either a single carrier or block carriers, which are termed \"SC-MMSE\" and \"BC-MMSE\", respectively. First, the MMSE-based detector in [24] is given below. **Step 1:** Initialize the a priori probability of the transmitter symbol \(\mathbf{X}_{l}\) as \(P(\mathbf{X}_{l}=\beta_{i})=\frac{1}{2^{8}}\), where \(b^{3}=\frac{b_{1}}{N_{a}}\) and \(\beta_{i}\in\mathbf{\
abla}\), \(\mathbf{\
abla}\) denotes the set of all GSTSK symbols and the \(N_{t}\times T\) dimension all-zero matrix. Then, initialize the number of iterations as \(\Sigma=1\). **Step 2:** Compute the mean and variance of \(\mathbf{X}_{l}\) based on the initial a priori probability \(P(\mathbf{X}_{l}=\beta_{i})\) as follows  
\[ \begin{split}\text{E}(\mathbf{X}_{l})&=\sum_{\beta_ {i}\in\mathbf{\
abla}}\beta_{i}P(\mathbf{X}_{l}=\beta_{i}),\\ \text{D}(\mathbf{X}_{l})&=\sum_{\beta_{i}\in\mathbf{ \
abla}}\beta_{i}\beta_{i}^{,H}P(\mathbf{X}_{l}=\beta_{i})-\text{E}(\mathbf{ X}_{l})\text{E}(\mathbf{X}_{l})^{H}.\end{split}  \tag{9}\]  
**Step 3:** Eliminate the ICI caused by the CFO based on the model (6) as follows  
\[\mathbf{W}_{l}=\mathbf{Y}_{l}-\sum_{p=1,p\
eq l}^{N}\,(\mathbf{H}_{p}\odot \mathbf{S}_{p-l+1})\text{E}(\mathbf{X}_{p}), \tag{10}\]  
where \(\mathbf{Y}_{l}\) and \(\mathbf{W}_{l}\) can be approximated to  
\[\mathbf{Y}_{l}=(\mathbf{H}_{l}\odot\mathbf{S}_{1})\mathbf{X}_{l}+\sum_{p=l- \frac{\mathbb{C}-1}{2},p\
eq l}^{l+\frac{\mathbb{C}-1}{2}}(\mathbf{H}_{p}\odot \mathbf{S}_{p-l+1})\mathbf{X}_{p}+\mathbf{Z}_{l}, \tag{11}\]  
\[\mathbf{W}_{l}=\mathbf{Y}_{l}-\sum_{p=l-\frac{\mathbb{C}-1}{2},p\
eq l}^{l+ \frac{\mathbb{C}-1}{2}}(\mathbf{H}_{p}\odot\mathbf{S}_{p-l+1})\text{E}( \mathbf{X}_{p})\,, \tag{12}\]and \(G(G\leq N)\) is the number of adjacent subcarriers that affects the \(l\)-th subcarrier, while the ICI of the other carriers is approximated to zero. **Step 4:** Apply the MMSE criterion to the model (5) in the subcarrier to obtain  
\[\mathbf{\tilde{X}}_{l}{=}\mathbf{C}_{l}[\mathbf{Y}_{l}-\sum_{p=l-\frac{G-1}{2},p\
eq l}^{l+\frac{G-1}{2}-1}\left(\mathbf{H}_{p}\odot\mathbf{S}_{p-l+1} \right)\mathrm{E}(\mathbf{X}_{p})]=\mathbf{C}_{l}\mathbf{W}_{l}, \tag{13}\]  
where  
\[ \begin{array}{l}\mathbf{C}_{l}=\frac{L}{Q}(\mathbf{H}_{l}\odot\mathbf{S}_{1 })]^{H}\{\frac{L}{Q}(\mathbf{H}_{l}\odot\mathbf{S}_{1})(\mathbf{H}_{l}\odot \mathbf{S}_{1})^{H}+\\ \sum_{p=l-\frac{G-1}{2},p\
eq l}^{l+\frac{G-1}{2}}(\mathbf{H}_{p}\odot \mathbf{S}_{p-l+1})\mathrm{D}(\mathbf{X}_{p})(\mathbf{H}_{p}\odot\mathbf{S}_ {p-l+1})^{H}\ \ +\sigma^{2}\mathbf{I}_{N_{l}\times N_{r}}\}^{-1}.\end{array}  \tag{14}\]  
**Step 5:** Calculate the posterior probability of the transmitter symbols \(\mathbf{X}_{l}\) as  
\[P(\mathbf{\tilde{X}}_{l}|\mathbf{X}_{l}=\beta_{i})=\psi_{l}e^{-\frac{\| \mathbf{X}_{l}-\Delta_{l}\|^{2}}{\omega_{l}^{2}}}, \tag{15}\]  
where  
\[\Delta_{l}=\mathrm{E}(\mathbf{\tilde{X}}_{l}|\mathbf{X}_{l}{=}\beta_{i})= \mathbf{C}_{l}(\mathbf{H}_{l}\odot\mathbf{S}_{1})\beta_{i}, \tag{16}\]  
\[\mathrm{D}_{l}=\mathrm{D}(\mathbf{\tilde{X}}_{l}|\mathbf{X}_{l}{=}\beta_{i})= \frac{L}{Q}\mathbf{C}_{l}(\mathbf{H}_{l}\odot\mathbf{S}_{1})-\frac{L}{Q} \mathbf{C}_{l}(\mathbf{H}_{l}\odot\mathbf{S}_{1})(\mathbf{H}_{l}\odot \mathbf{S}_{1})^{H}\mathbf{C}_{l}^{H}, \tag{17}\]  
\[\omega_{l}^{2}=\frac{trace[D_{l}]}{N_{l}T}. \tag{18}\]  
\(\psi_{l}\) denotes the normalization coefficient. **Step 6:** Let \(\Sigma=\Sigma+1\) and \(P(\mathbf{X}_{l}=\beta_{i})=P(\mathbf{\tilde{X}}_{l}|\mathbf{X}_{l}=\beta_{i})\), and then go back to Step 2. If \(\Sigma>\Sigma_{\max}\), the detector proceeds to the next step. **Step 7:** Reconstruct \(\mathbf{\tilde{X}}^{k}\) by the probability \(P(\mathbf{X}_{l}=\mathbf{f}_{i})\).