# Signal Detection for GSTFIM Systems with Carrier Frequency Offset +
## 2 GSTFIM Systems with CFO  
In this paper, we consider a GSTFIM system with \(N_{t}\) transmitter antennas, \(N_{r}\) receiver antennas, \(T\) symbol durations, \(N\) subcarriers, and an \(M\)-PSK/QAM constellation, as shown in Figure 1. \(N\) subcarriers are divided into \(K\) blocks each with \(N_{b}\) subcarriers, and only \(N_{a}\) out of \(N_{b}\) subcarriers are activated. The information bits \(b_{k}=b_{1}+b_{2}\), where \(k=\{1,2,\ldots,K\}\), are divided into two parts. The first \(b_{1}\) bits are used to generate \(N_{a}\) GSTSK symbols as  
\[\mathbf{S}_{i}^{k}=\sum_{l=1}^{L}s_{l}\mathbf{A}_{l}^{k}\in\mathbb{C}^{N_{l} \times T}, \tag{1}\]  
where \(i=1,\ldots,N_{a}\) and \(L\) is the number of activated dispersion matrices. The second \(b_{2}\) bits are employed to select \(N_{a}\) activated indices of the antennas out of \(N_{b}\), denoted by \(\mathbf{I}^{k}=\{I_{1},I_{2},\ldots,I_{N_{a}}\}\).  
Then, we map the GSTSK symbols \([\mathbf{S}_{1}^{k},\mathbf{S}_{2}^{k},\ldots,\mathbf{S}_{N_{a}}^{k}]^{T}\) to the block matrix as  
\[\mathbf{X}^{k}=[\mathbf{X}_{1}^{k},\mathbf{X}_{2}^{k},\ldots,\mathbf{X}_{N_{b} }^{k}]^{T}. \tag{2}\]  
Specifically, \(N_{a}\) elements of \(\mathbf{X}^{k}\) are expressed as \([\mathbf{S}_{1}^{k},\mathbf{S}_{2}^{k},\ldots,\mathbf{S}_{N_{a}}^{k}]^{T}\), whose indices are denoted by \(\mathbf{I}^{k}\), and whose other elements are zeros.  
Then, the transmit vector \(\mathbf{x}=[\mathbf{x}^{1},\mathbf{x}^{2},\ldots,\mathbf{x}^{K}]\in\mathbb{C} ^{NN_{l}\times T}\) in the time domain can be obtained by the IFFT as  
\[\mathbf{x}=IFFT([\mathbf{X}^{1},\mathbf{X}^{2},\ldots,\mathbf{X}^{K}]). \tag{3}\]  
After cyclic prefix (CP) insertion, passing through a wireless channel and CP removal, the time-domain receiver symbols are expressed as \(\mathbf{y}=[\mathbf{y}^{1},\mathbf{y}^{2},\ldots,\mathbf{y}^{K}]\in\mathbb{C} ^{NN_{l}\times T}\). Then frequency-domain symbols can be obtained by FFT transformation as  
\[[\mathbf{Y}^{1},\mathbf{Y}^{2},\ldots,\mathbf{Y}^{K}]=FFT([\mathbf{y}^{1}, \mathbf{y}^{2},\ldots,\mathbf{y}^{K}]). \tag{4}\]  
We use \(e_{m}\) and \(f_{n}\) to denote the CFOs between each pair of transmitter and receiver antennas, respectively, where \(m=\{1,2,\ldots,N_{t}\}\) and \(n=\{1,2,\ldots,N_{r}\}\). \(e_{m}\) and \(f_{n}\) are assumed to be different and remain the same during the symbol duration. Thus, the frequency-domain model with the CFO in the \(l\)-th subcarrier can be expressed as  
\[\mathbf{Y}_{l}=\sum_{p=1}^{N}{(\mathbf{H}_{p}\odot\mathbf{S}_{p-l})\mathbf{ X}_{p}}+\mathbf{Z}_{l}=(\mathbf{H}_{l}\odot\mathbf{S}_{1})\mathbf{X}_{l}+ \sum_{p=1,p,\
eq l}^{N}{(\mathbf{H}_{p}\odot\mathbf{S}_{p-l+1})\mathbf{X}_{p} }+\mathbf{Z}_{l}, \tag{5}\]  
Figure 1: Transmitter structure of the GSTFIM system.  
where \(l=n_{b}+N_{b}(k\cdot 1)\in\{1,2,\ldots,N\}\). \(\mathbf{X}_{l}\) and \(\mathbf{X}_{p}\) can be expressed as \(\mathbf{X}_{N_{b}}^{k}\) and \(\mathbf{X}_{n^{\prime}b^{\prime}}^{k^{\prime}}\), respectively, where  
\[k^{\prime}=\left\lfloor\frac{p}{N_{b}}\right\rfloor+1,n^{\prime}_{b}=\text{ mod}\,(p,N_{b}), \tag{6}\]  
\(\mathbf{H}_{l}\in\mathbb{C}^{N_{r}\times N_{l}}\) represents the channel frequency domain impulse response and \(\mathbf{Z}_{l}\in\mathbb{C}^{n_{r}\times T}\) denotes the noise vector, whose elements obey the Gaussian distribution \(CN(0,\sigma^{2})\). \(\mathbf{S}_{p}\in\mathbb{C}^{N_{r}\times N_{t}}\) denotes the CFO matrix, which is expressed as  
\[\mathbf{S}_{p}=\left[\begin{array}{cccc}S_{p}^{(1,1)}&S_{p}^{(1,2)}&\cdots &S_{p}^{(1,N_{t})}\\ S_{p}^{(2,1)}&S_{p}^{(2,2)}&\ddots&S_{p}^{(2,N_{t})}\\ \vdots&\ddots&\ddots&\vdots\\ S_{p}^{(N_{t},1)}&S_{p}^{(N_{t},2)}&\cdots&S_{p}^{(N_{t},N_{t})}\end{array} \right], \tag{7}\]  
where \(p=\{1,2,\ldots,N\}\) and \(S_{p}^{(n,m)}\) denotes the CFO coefficient [18], which is expressed by  
\[S_{p}^{(n,m)}=\frac{\sin\pi(p+e_{m}+f_{n})}{N\sin\frac{\pi}{N}(p+e_{m}+f_{n})} e^{j\pi(1-\frac{1}{N})(p+e_{m}+f_{n})}, \tag{8}\]  
where \(n\in\{1,2,\ldots,N_{r}\}\) and \(m\in\{1,2,\ldots,N_{t}\}\).