## 3 Proposed SLM PAPR Scheme
### Proposed Low-Complexity Conversion Vectors  
As with the proposed SLM PAPR scheme in [10], the IFFT operation is replaced by the conversion vectors \(T_{i}\) for \(i~{}=~{}1,\cdots,U\) and the candidate OFDM signals are generated by performing circular convolution of the IFFT of the original OFDM symbol block with the conversion vectors. The basic architecture of the SLM PAPR reduction scheme using conversion vectors in [10] is shown in Figure 3. It was indicated that the computational complexity at the transmitter can be greatly reduced, and each conversion vector requires only \(3N\) complex additions. Figure 1: Equivalent structure of SLM in generating candidate signals in the time domain. Figure 2: General form of selected mapping (SLM) scheme in the time domain. From the previous discussion, we know that when the conversion vectors are properly designed in the time domain, conversion vectors can be an equivalent channel and the effect of the conversion matrix can be removed by the channel equalization operation. Based on the conversion vector structure, in order to implement the low-complexity SLM scheme without SI, the proposed conversion vector \(\mathbf{p}^{(u)}\) can be designed according to the following rules:  
1. The support \(L_{u}\) of \(\mathbf{p}^{(u)}\) is chosen as a small integer to restrict the length of the equivalent channel;
2. There are few nonzero elements whose number is denoted as \(K\) in \(\mathbf{p}^{(u)}\) to reduce the complexity in generating the candidate signals with Equation (11);
3. The real and imaginary parts of each nonzero element in \(\mathbf{p}^{(u)}\) are chosen in the set \(\{0,\pm 1\}\) to avoid the complex multiplications in generating the candidate signals (ignoring the constant factor to maintain the power constraint) [10]. Generally speaking, there are many choices for the exact values of \(L_{u}\) and \(K\). The support \(L_{u}\) in constraint (1) and the number \(K\) may vary depending on the application. Therefore, the proposed scheme can achieve a flexible compromise between computational complexity and BER performance. It can be used for any system with the parameters chosen according to the application. For example, to have comparable complexity with the available works, the values of \(K\) can be selected as \(K=3\) or \(K=4\) and the values of \(L_{u}\) can be selected as smaller than eight samples. Given the parameters for \(K\) and \(L_{u}\), we can properly design the vectors \(\mathbf{p}^{(u)}\) to generate the candidate signals with low-complexity without the transmission of SI. To achieve good BER performance, we can design the vectors \(\mathbf{p}^{(u)}\) with the following two steps. In the first step, we construct a set \(\Re~{}=~{}\{\mathbf{r}^{(1)},\mathbf{r}^{(2)}\ldots,\mathbf{r}^{(V)}\}\) with \(V\) (\(V\) can be much larger than \(U\)) vectors according to the above three rules for the given values of \(K\) and \(L_{u}\) by varying the values and positions of the nonzero elements in \(\mathbf{r}^{(i)}\), \(1\leq i\leq V\). One example of generating \(\Re\) used in this paper is shown in Table 1, where \(\mathbf{T}~{}=~{}\{T_{0},T_{1},\ldots,T_{K-1}\}\) (\(T_{K}~{}=~{}L_{u}-1\)) is the set of the indices of the nonzero elements and \(K\) is the number of nonzero elements in \(\mathbf{r}^{(i)}\), \(1\leq i\leq V\), and \(\mathbf{a}~{}=~{}\left\{a^{(0)},a^{(1)},\ldots,a^{(K-1)}\right\}\) is the set of the values of the nonzero elements. Clearly, given the parameters of \(K\) and \(L_{u}\), a large number of vectors \(\mathbf{r}^{(i)}\) can be generated. In the second step, we take the FFT of the \(V\) vectors in \(\Re\) to generate \(V\) phase rotation vectors. Then, we choose \(U\)-1 phase rotation vectors from them such that the smallest amplitude