# An Improved GreenOFDM Scheme for PAPR Reduction
## 3 From SLM-OFDM to GreenOFDM
```
0:\(\{x_{g_{1}}[n]\}=\text{IFFT}\{X_{k}\cdot\varphi_{g_{1,k}}\}\), \(\{x_{g_{2}}[n]\}=\text{IFFT}\{X_{k}\cdot i\cdot\varphi_{g_{2,k}}\}\) \(\min\)PAPR \(\leftarrow\)\(+\infty\); for\((g_{1}\gets 0\)to \(\frac{U}{2}-1)\)do for\((g_{2}\leftarrow\frac{U}{2}\)to \(U-1)\)do \(\{x_{g_{1},g_{2}}[n]\}\leftarrow\frac{\{x_{g_{1}}[n]+x_{g_{2}}[n]\}}{\sqrt{2}}\); if\((\text{PAPR}\{x_{g_{1},g_{2}}[n]\}<\min\)PAPR) then \(\min\)PAPR \(\leftarrow\) PAPR\(\{x_{g_{1},g_{2}}[n]\}\); \(\{x_{g_{1},g_{2}}[n]\}\leftarrow\{x_{g_{1},g_{2}}[n]\}\); endif endfor \(\text{SEND}\{x_{g_{1},g_{2}}[n]\}\);
```

**Algorithm 2** The GreenOFDM algorithm. ## 4 An Improved GreenOFDM Version

The goal of the improved version is to increase the total number of candidates without increasing the number of IFFTs. As aforementioned, the key idea of the GreenOFDM technique is based on the use of two different groups (respectively generated from \(\{\pm 1\}\) or \(\{\pm i\}\) orthogonal sequences) of IFFTs. In the original version of the GreenOFDM technique, only the first half of the IFFTs were used for the first group (\(\{x_{g_{1}}[n]\}=\text{IFFT}\{X_{k}\cdot\varphi_{g_{1},k}\}\) with \(g_{1}<\frac{U}{2}\)), the second half being reserved for the second group (\(\{x_{g_{2}}[n]\}=\text{IFFT}\{X_{k}\cdot\varphi_{g_{2},k}\}\), with \(g_{2}\geq\frac{U}{2}\)). In the proposed improved version, referred from now on as GreenOFDMv2, the full set of IFFTs will be used both for the first group, and then for the second group after a slight modification exploiting the linearity of the IFFT and ensuring orthogonality between the two groups. The IFFTs of the initial sequence \(\{X_{k}\}\) weighted by pure real binary sequences \(\{\phi_{u,k}\}\) with values in \(\{\pm 1\}\) compose directly the first group:

\[\{x_{u}[n]\}=\text{IFFT}\{X_{k}\cdot\phi_{u,k}\},\ \ \text{with}\ \ 0\leq u\leq U-1\]

Figure 2: Original GreenOFDM block diagram [22, 31]. The complex waveforms of the second group correspond conceptually to the IFFTs of the initial sequence \(\{X_{k}\}\) weighted by pure imaginary binary sequences \(i\times\{\phi_{u,k}\}\) with values in \(\{\pm i\}\), but are generated in a less complex way. Indeed, due to the linearity of the IFFT, the complex waveforms of the second group \(\{\tilde{x}_{u}[n]\}\) are equal to the complex waveforms of the first group \(\{x_{u}[n]\}\), only modified by a factor of \(i\):

\[\{\tilde{x}_{u}[n]\} = \text{IFFT}\{X_{k}\cdot i.\phi_{u,k}\}\quad\text{ with }\;0\leq u\leq U-1\] \[= i\times\text{IFFT}\{X_{k}\cdot\phi_{u,k}\}\] \[= i\times\left\{x_{u}[n]\right\},\]

So, the second group can be obtained from the IFFT outputs of the first group \(\{x_{u}[n]\}\) by only shifting the complex values (\(\Re+i\cdot\Im\)) into (\(-\Im+i\cdot\Re\)), eliminating the need to compute the IFFTs again. For a given number of IFFT, the size of each group is then doubled compared to the original GreenOFDM method. The improved GreenOFDM method is described in Algorithm 3. It can be observed that the first and second groups are redefined. The first group contains the \(U\) IFFTs (the equivalent to the SLM-OFDM generated candidates), and the second group is trivially generated by exploiting the linearity of the IFFT. ```
\(\{x_{u}[n]\}=\text{IFFT}\{X_{k}\cdot\phi_{u,k}\}\) \(\text{minPAPR}\leftarrow+\infty\); for\((u_{1}\gets 0\)to\(U-1)\)do for\((u_{2}\gets 0\)to\(U-1)\)do if\(u_{1}==u_{2}\)then \(\{x_{u_{1},u_{2}}[n]\}\leftarrow\{x_{u_{1}}[n]\}\); else \(\{x_{u_{1},u_{2}}[n]\}\leftarrow\frac{\{x_{u_{1}}[n]+i\times x_{u_{2}}[n]\}}{ \sqrt{2}}\); endif if\((\text{PAPR}\{x_{u_{1},u_{2}}[n]\}<\text{minPAPR})\)then \(\text{minPAPR}\leftarrow\text{PAPR}\{x_{u_{1},u_{2}}[n]\}\); \(\{x_{\tilde{u}_{1},\tilde{u}_{2}}[n]\}\leftarrow\{x_{u_{1},u_{2}}[n]\}\); endif endfor endfor \(\text{SEND}\{x_{\tilde{u}_{1},\tilde{u}_{2}}[n]\}\);
```

**Algorithm 3** The GreenOFDMv2 algorithm.