## 3 Proposed Receiver
### Non-Linear Detection on DFTS-OFDM System  
The main contribution in this paper is the efficient integration of the DFTS precoding in the non-linear data detection process in the receiver. This has to be emphasized: one of the problems when using DFTS in the receiver is the difficulty of applying the non-linear detection algorithms while maintaining low computational complexity, since the equivalent channel matrix \(\mathbf{G}_{D}\mathbf{F}_{D}\) does not conserve a band structure. One solution to this issue is to find an operator that, upon being applied to the received signal, re-establishes the equivalent channel matrix's band structure. In this sense, the inverse Fourier transform \(\mathbf{F}_{D}^{H}\) is used to complete the Cramer-Loeve operator in the \(\mathbf{G}_{D}\) channel matrix. In mathematical terms, the vector received in the data position is obtained as:  
\[\mathbf{F}_{D}^{H}\mathbf{u}_{D} =\mathbf{F}_{D}^{H}\mathbf{G}_{D}\mathbf{F}_{D}\boldsymbol{\chi} +\mathbf{F}_{D}^{H}\mathbf{G}_{D,p}\mathbf{s}_{p}+\mathbf{F}_{D}^{H}\mathbf{z }_{D}, \tag{25}\] \[\mathbf{v} =\mathbf{K}\boldsymbol{\chi}+\mathbf{F}_{D}^{H}\mathbf{G}_{D,p} \mathbf{s}_{p}+\mathbf{z}_{D}, \tag{26}\]  
where \(\mathbf{v}=\mathbf{F}_{D}^{H}\mathbf{u}_{D}\), \(\boldsymbol{\chi}=\mathbf{F}_{D}^{H}\mathbf{s}_{D}\) and \(\mathbf{K}=\mathbf{F}_{D}^{H}\mathbf{G}_{D}\mathbf{F}_{D}\) is the equivalent channel matrix after applying the inverse Fourier transform to the received symbols in the data position with linear precoding. In order to simplify the notation, the noise vector \(\mathbf{z}_{D}=\mathbf{F}_{D}^{H}\mathbf{z}_{D}\) maintains the same nomenclature since the orthonormal transforms does not affect its statistics. Notice that the correlation characteristics and quasi-band structure of matrix \(\mathbf{G}_{D}\) imply that matrix \(\mathbf{K}\) also possess a quasiband structure after the transformation, as shown in Figure 3. This structure in particular allows detection algorithms to diminish their computational complexity, while performing close to the ML detector in terms of BER.  
Maximum Likelihood Detection Criteria  
The data detection using the maximum likelihood criterion can be determined from the Equation (26) by finding the vector \(\mathbf{\chi}\) that minimizes the following metrics:  
\[\mathbf{\hat{\chi}}=\operatorname*{argmin}_{\mathbf{\chi}\in\Omega^{N_{D}}}\|\mathbf{v}-\bm {K}\mathbf{\chi}\|^{2}, \tag{27}\]  
where \(\Omega\) is the constellation used for data modulation and \(\Omega^{N_{D}}\) is the set containing all possible combinations of the symbols transmitted through the \(N_{D}\) data subcarriers. This data detection method is highly computationally complex due to the exhaustive calculation of all of the Euclidean distances that are needed to estimate the vector \(\mathbf{\hat{\chi}}\). In order to reduce the complexity required by the detection, it can be applied a truncation to the matrix \(\mathbf{K}\), keeping \(B\) non-zero diagonals. These are determined using the following equation:  
\[B=2(\lambda+\lambda_{c})+1, \tag{28}\]  
where the \(B\) diagonals are distributed in 3 bands, one band is formed by the main diagonal and \(2\lambda\) adjacent diagonals, the second band is formed by the \(\lambda_{c}\) diagonals located in the upper right corner, and the third band is formed by \(\lambda_{c}\) diagonals located in the lower-left corner. The rest of the diagonals are truncated to zero. The structure of the truncated \(\mathbf{K}\) matrix is shown in Figure 3.