# Enhancing PAPR and Throughput for DFT-s-OFDM System Using FTN and IOTA Filtering
## 3 Proposed IOTA-Based FTN-DFT-s-OFDM Waveform Design
### Receiver Design  
Even with an IOTA filter, the existence of ISI still requires a more complicated receiver to recover the original transmission signals. In the sequel, we design the receiving algorithm for the proposed waveform based on frequency-domain equalization (FDE). The receiver design should lay the basis for the conventional DFT-S-OFDM waveform while considering the specific characteristics of FTN and IOTA filters \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Specifications** \\ \hline \(\tau_{0}\) & \(1/\sqrt{2}\) \\ \(v_{0}\) & \(1/\sqrt{2}\) \\ \(t\) & \([-2\sqrt{2},2\sqrt{2}]\) \\ \(K\) & \(15\) \\ \(\mathcal{Q}\) & \(8\) \\ \(\mathbf{h}_{\text{EGF}}(t)\) & \(=2^{1}\text{e}^{-\pi t^{2}}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of IOTA filter [29; 30]. Figure 2: OOB performance of IOTA filter. [PERSON] [31] proposes a frequency domain equalization (FDE) receiver structure based on minimum mean square error (MMSE), which can achieve low demodulation complexity, especially for long channel FTN schemes. Specifically, in the FDE scheme, a short cyclic prefix (CP) is added to each transport block and a finite-tap cyclic matrix is adopted to approximate the ISI generated by FTN signaling. Thus, an efficient FFT method and a low-complexity MMSE detection algorithm can be used in the receiver. FDE can achieve near-optimal BER performance without increasing demodulation complexity and power consumption. In the AWGN channel, the output signal after a matched filter given (5) can be expressed as  
\[\hat{g}(t)=y(t)*g(t)=\sqrt{E_{s}}\sum_{n}x(n)\hat{g}(t-n\tau T)+\eta(t), \tag{9}\]  
where \(\hat{g}(t)=\int g(r)g^{*}(\tau-t)d\tau\), \(\eta(t)=\int n(\tau)g^{*}(\tau-t)d\tau\) and \(E_{s}\) represents the average power of transmitting signal. Assuming that the time synchronization is perfect between the transmitter and receiver, the sampled value of the \(k\)-th signal at the receiver can be written as  
\[\hat{g}_{k} =\hat{g}(k\tau T) \tag{10}\] \[=\sqrt{E_{s}}\sum_{n}x(n)\hat{g}(k\tau T-n\tau T)+\eta(k\tau T)\] \[=\sqrt{E_{s}}x(n)\hat{g}(0)+\sqrt{E_{s}}\sum_{n\
eq k}x(n)\hat{g} (k\tau T-n\tau T)+\eta(k\tau T),\]  
where the first part is the symbol at present, the second part is ISI and \(\eta(k\tau T)\) is the zero-mean random Gaussian variable. Figure 3 is the transceiver structure of MMSE-FDE scheme. A CP with length \(2v\) is first added after \(N\) modulated symbols to realize the symbol transmission based on the block. Then, the first and last \(v\) receiving signal samples are removed from the entire \(N+2v\) samples, and we obtain a received signal block with length \(N\), which can be expressed as  
\[\hat{\textbf{y}}=[\hat{g}_{1},\ldots,\hat{g}_{N}]^{T}\in\textbf{C }^{N} \tag{11}\] \[=\textbf{G}\textbf{x}+\textbf{\eta},\]  
where \(\textbf{x}=[x_{1},x_{2},\ldots,x_{n}]^{T}\) is the transmitting signal and \(\textbf{\eta}=[\eta_{1},\eta_{2},\ldots,\eta_{N}]^{T}\) is the corresponding channel noise component. The \(k-th\) row of tap coefficient matrix \(\textbf{G}\in\textbf{R}^{N\times N}\) is  
\[\textbf{G}=\left[ \begin{array}{ccccccccc}g(-v\tau T)&\cdots&g(v\tau T)&0& \cdots&\cdots&\cdots&\cdots&0\\ 0&g(-v\tau T)&\cdots&g(v\tau T)&0&\cdots&\cdots&\cdots&0\\ 0&0&g(-v\tau T)&\cdots&g(v\tau T)&0&\cdots&\cdots&0\\ \vdots&\ddots&\ddots&\ddots&\ddots&\ddots&\ddots&\ddots&\vdots\\ 0&0&0&\cdots&0&g(-v\tau T)&\cdots&g(v\tau T)&0\\ g((2v-1)\tau T)&\cdots&g(v\tau T)&0&\cdots&0&g(-v\tau T)&\cdots&g((2v-2)\tau T) \\ g((2v-2)\tau T)&\cdots&g(v\tau T)&0&\cdots&0&g(-v\tau T)&\cdots&g((2v-3)\tau T) \\ \vdots&\ddots&\ddots&\ddots&\ddots&\ddots&\ddots&\ddots&\vdots\\ g(0)&\cdots&g(v\tau T)&0&\cdots&0&g(-v\tau T)&\cdots&g(-\tau T)\end{array}  \right].