# Signal Detection for GSTFIM Systems with Carrier Frequency Offset +
## 4 Proposed Detectors Based on the MP Criterion  
In this section, the MP-based [27; 28; 29; 30] detectors as shown in Figure 2 are proposed to achieve a better performance. According to the CFO model in (11), the receiver symbol \(\mathbf{Y}_{l}\) can be regarded as the sum of \(G\) adjacent symbols \(\mathbf{X}_{p}\) of the \(l\)-th subcarrier, where \(p=(l-(G-1)/2,l+(G-1)/2)\). In other words, for the \(k\)-th subcarrier, its transmitter symbol \(\mathbf{X}_{k}\) has come out \(G\) times. Thus, these \(G\) CFO models (11) can be considered as a diversity model of \(\mathbf{X}_{k}\). Assume that index \(h\), which represents one of these \(G\) adjacent subcarriers around subcarrier \(k\), satisfies the condition of \(|h-k|\leqslant(G-1)/2\). Thus these diversity models of \(\mathbf{X}_{k}\) can be described by (28). \[\mathbf{Y}_{h}-(\mathbf{H}_{k}\odot\mathbf{S}_{k-h+1})\mathbf{X}_{k}=\sum_{m= h-(G-1)/2,m\
eq k}^{h+(G-1)/2}\left(\mathbf{H}_{m}\odot\mathbf{S}_{m-h+1} \right)\mathbf{X}_{m}+\mathbf{Z}_{h}. \tag{28}\]  
\[\mathbf{u}_{h}^{k}=E\left(\sum_{m=h-(G-1)/2,m\
eq k}^{h+(G-1)/2}(\mathbf{H}_{ m}\odot\mathbf{S}_{m-h+1})\mathbf{X}_{m}+\mathbf{Z}_{h}\right)=\sum_{m=h-(G-1)/2,m \
eq k}^{h+(G-1)/2}(\mathbf{H}_{m}\odot\mathbf{S}_{m-h+1})E(\mathbf{X}_{m}) \tag{29}\]  
\[\mathbf{v}_{h}^{k}=D\left(\sum_{m=h-(G-1)/2,m\
eq k}^{h+(G-1)/2}(\mathbf{H}_{ m}\odot\mathbf{S}_{m-h+1})\mathbf{X}_{m}+\mathbf{Z}_{h}\right)=\sum_{m=h-(G-1)/2,m \
eq k}^{h+(G-1)/2}(\mathbf{H}_{m}\odot\mathbf{S}_{m-h+1})D(\mathbf{X}_{m})( \mathbf{H}_{m}\odot\mathbf{S}_{m-h+1})^{H}+\sigma^{2}\mathbf{I}_{N_{t}} \tag{30}\]Then, the mean \(\mathbf{u}_{h}^{k}\) and variance \(\mathbf{v}_{h}^{k}\) of the interference term on the right-hand side of (28) are calculated in (29) and (30), respectively, where the mean \(E(\mathbf{X}_{m})\) and variance \(D(\mathbf{X}_{m})\) of \(\mathbf{X}_{m}\) are calculated using the a priori probability \(P(\mathbf{X}_{m}=\mathbf{fi}_{i})\) of \(\mathbf{X}_{m}\) and the calculation is shown in (9). According the to the MP criterion, the posterior probability of \(\mathbf{X}_{k}\) can be calculated by the above mean and variance as  
\[P_{h}(\mathbf{X}_{k}=\mathbf{fi}_{i})\propto\exp\left(\frac{||\mathbf{Y}_{h}- (\mathbf{H}_{k}\odot\mathbf{S}_{k-h+1})\mathbf{X}_{k}-\mathbf{u}_{h}^{k}||_{F }^{2})}{trace(\mathbf{v}_{h}^{k})/N_{r}}\right), \tag{31}\]  
where \(trace(\cdot)\) denotes the sum of the diagonal elements of a matrix. The overall posterior probability of \(\mathbf{X}_{k}\) can be obtained by combining the above \(G\) posterior probabilities in (28) as follows  
\[P(\mathbf{X}_{k}=\mathbf{fi}_{i})=C_{k}\prod_{|h-k|\leq(G-1)/2}P_{h}(\mathbf{ X}_{k}=\mathbf{fi}_{i}) \tag{32}\]  
where \(C_{k}\) is a normalization coefficient. Then the probability \(P(\mathbf{X}_{k}=\mathbf{fi}_{i})\) is used in (10) for the next iteration. In general, the iterative process can be handled as Steps 1-7 using (9) and (25)-(29). After several iterations, similar to Step 7 of the MMSE-based detector in Section 3, the estimated GSTFIM symbol \(\mathbf{\hat{X}}^{k}\) is reconstructed by the probability \(P(\mathbf{X}_{k}=\mathbf{fi}_{i})\). Here, the  
Figure 2: Block diagram of the MP-criterion-based detector. two MP detectors based on the single carrier and block carriers are dubbed \"SC-MP\" and \"BC-MP\", respectively. The proposed SC-MP and BC-MP detectors are different from the detector of [30] in the following two aspects. (a) The detector of [30] is conceived for generalized space-and-frequency index modulation (GSFIM) systems, and is derived from spatial- and frequency-domain-based matrix operations. The proposed detectors are conceived for GSTFIM systems, and are derived on spatial- and time-domain-based matrix operation. Specifically, the mean and variance of the GSTFIM transmitter symbols in the \(l\)-th subcarrier are calculated based on matrix \(\mathbf{X}_{l}\in\mathbb{C}^{N_{l}\times T}\) in (9) rather than matrix \(\mathbf{X}_{g}\in\mathbb{C}^{N_{l}\times N_{f}}\).