# Signal Detection for GSTFIM Systems with Carrier Frequency Offset +
## 3 Proposed MMSE Detectors for GSTFIM Systems
### BC-MMSE Detector  
The BC-MMSE detector is detailed as follows. For the \(k\)-th block, the GSTFIM symbol \(\mathbf{X}^{k}\) has \(2^{b_{k}}\) possible values \(\mathbf{X}=\{\mathrm{E}_{1},\mathrm{E}_{2},\ldots,\mathrm{E}_{2^{b_{k}}}\}\). For a specific symbol \(\mathrm{E}_{n}(n=1,\ldots,2^{b_{k}})\), the index set of the GSTSK symbol in \(\mathrm{E}_{n}\) is denoted by \(\mathrm{A}=\{a_{1},a_{2},\ldots,a_{N_{a}}\}\), and the index set of the all-zero matrix is \(\mathbf{C}=\{c_{1},c_{2},\ldots,c_{(N_{b}-N_{a})}\}\). As mentioned in Section 2, the mapping between the \(n_{b}\)-th subcarrier of the \(k\)-th block and the \(l\)-th of all the \(N\) subcarriers is \(l=n_{b}+N_{b}(k-1)\). Thus, the index set \(\mathrm{A}\) of the GSTSK symbol in all the \(N\) subcarriers is \(\mathrm{A}+N_{b}(k-1)=\{g_{1},\ g_{2},\ \ldots,g_{N_{a}}\}\), and the index set \(\mathrm{E}\) of the zero matrix in all the \(N\) subcarriers is \(\mathbf{C}+N_{b}(k-1)=\{h_{1},\ h_{2},\ \ldots,\ h_{(N_{b}-N_{a})}\}\). According to the probability vector \(P(\mathbf{X}_{l}=\mathbf{f}_{i})\), for the GSTSK symbols whose indices are \(\{g_{1},\ g_{2},\ \ldots,\ g_{N_{a}}\}\), their probability vector is expressed as  
\[\mathbf{P}_{SC-1}=\{p_{i_{1}}^{g_{1}},p_{i_{2}}^{g_{2}},\ldots,p_{i_{N_{a}}}^{ N_{a}}\}, \tag{24}\]  
where \(i_{1}\in\{1,2,\ldots,2^{b_{3}}\}\) indicates the index of the GSTSK symbol corresponding to index \(g_{1}\) in all the \(2^{b_{3}}\) possible GSTSK symbols, while \(p_{i_{1}}^{g_{1}}\) denotes the \(i_{1}\)-th value of the probability vector \(P(\mathbf{X}_{g_{1}}=\mathbf{f}_{i})\). However, for the all-zero matrices whose indices are \(\{h_{1},\ h_{2},\ \ldots,\ h_{(N_{b}-N_{a})}\}\), their probability vector is expressed as  
\[\mathbf{P}_{SC-2}=\{p_{(2^{b_{3}}+1)}^{h_{1}},p_{(2^{b_{3}}+1)}^{h_{2}},\ldots,p_{(2^{b_{3}}+1)}^{h_{(N_{b}-N_{a})}}\} \tag{25}\]  
where \(p_{(2^{b_{3}}+1)}^{h_{1}}\) is the \((2^{b_{3}}+1)\)-th element of the probability vector \(P(\mathbf{X}_{h_{1}}=\mathbf{f}_{i})\). Thus, the probability corresponding to the \(n\)-th symbol in the \(k\)-th block is  
\[\Theta_{n}^{k}=sum(\mathbf{P}_{SC-1}+\mathbf{P}_{SC-2}). \tag{26}\]  
The probability \(\mathbf{\Theta}^{k}\) of \(\mathbf{X}^{k}\) is obtained by traversing \(n\) from \(1\) to \(2^{b_{k}}\):  
\[\mathbf{\Theta}^{k}=\{\Theta_{1}^{k},\Theta_{2}^{k},\ldots,\Theta_{2^{b_{k}}}^{ k}\}. \tag{27}\]We denote the index of the max probability in \(\mathbf{\Theta}^{k}\) as \(o\), thus \(\mathbf{X}^{k}\) is estimated as \(\mathbf{\hat{X}}^{k}=\mathrm{E}_{0}\). In general, we propose two detectors SC-MMSE and BC-MMSE to reconstruct the GSTFIM symbol. The difference is that the SC-MMSE detector is realized by comparing the probability of the single carrier, while its BC-MMSE counterpart compares the probabilities of block carriers. The SC-MMSE detector has a lower complexity, but its performance is inferior due to the deletion of information. The BC-MMSE detector takes into account the structure of the GSTFIM symbol, and thus it offers better performance. It is worth noting that the proposed SC-MMSE and BC-MMSE detectors are different from those in [19] in the following two aspects. (a) The detector in [18] is conceived for SM-OFDM systems, which are derived on the vector-by-vector-based operation. By contrast, the proposed detectors target GSTFIM systems, and are derived from spatial- and time-domain-based matrix operations. Specifically, the mean and variance of all GSTFIM transmitter symbols are calculated based on matrix \(\mathbf{X}_{l}\in\mathbb{C}^{N_{t}\times T}\) in (10) rather than on vector \(\mathbf{x}_{l}\in\mathbb{C}^{N_{t}\times 1}\). In addition, the set of \(\mathbf{X}_{l}\) includes an all-zero matrix, which does not exist in SM-OFDM systems. (b) In the detector of [18], the posterior probability \(P(\mathbf{\hat{x}}_{k}|\mathbf{x}_{k}=\alpha_{i})\) of the estimated symbol \(\mathbf{\hat{x}}_{k}\) on the \(k\)-th subcarrier obtained by the MMSE criterion is utilized to reconstruct the detected SM symbol by comparing the posterior probability.