# An Improved GreenOFDM Scheme for PAPR Reduction
## 3 From SLM-OFDM to GreenOFDM
The reason lies on the specific way in which the IFFTs are progressively computed and the possible available candidates. Indeed, in the former GreenOFDM method, the available candidates up to the \(v\)-th computed IFFT (with \(v\leq U\)) are \(C_{v}\propto v^{2}/4\) (Section 3.1 in [31]); while for the proposed method, the available candidates up to the \(v\)-th computed IFFT are \(C_{v}=v^{2}\). This means that every additional computed IFFT allows us to generate many more candidates for the proposed method and hence more _chances_ to find the symbol whose PAPR is below the imposed threshold, reducing the expected total number of operations (in [28], it was pointed-out that the IFFTs computations are the greatest contributors to complexity in terms of the total number of operations). Computer simulations were carried out to validate the improved complexity reduction capabilities of the proposed method. To be consistent with the comparisons, we have imposed the same set of parameters as in [28], which are \(10^{5}\) symbols iterations with \(N=64,128,256,512\) and \(1024\)\(QPSK\)-modulated subcarriers, oversampling factor \(L=4\) and \(U=1024\) IFFTs for the SLM-OFDM, \(U=64\) for the GreenOFDM and \(U=32\) for the proposed method. The same complexity reduction strategy applied to all the schemes as a function of the imposed threshold \(\gamma_{p_{i}}\) calculated with Equations (5) and (7) at \(p=10^{-3}\) was computed. The CCDF distribution of the total number of operations (multiplications, additions and comparisons), noted \(\mathcal{M}\), was computed for different imposed threshold \(\gamma_{p_{i}}\). The value of \(\mathcal{M}\) at CCDF(\(\mathcal{M}\))\(=10^{-3}\) was obtained for comparison purposes and the Computational Complexity Reduction Factor (CCRF) was calculated. Figure 8 depicts, on the left side, the CCRF\({}_{\mathcal{M}}\) between the original GreenOFDM and the SLM-OFDM as introduced in [28] i.e.,

\[\text{CCDF}_{\text{SLM}}(\gamma)\approx\left(1-\left(1-\exp\left\{-\gamma \right\}\right)^{2.8N}\right)^{C_{\text{SLM}}} \tag{4}\]7

On the right side we can find the CCRF\({}_{\mathcal{M}}\) between the GreenOFDMv2 and again the SLM-OFDM, i.e.,

\[\text{CCDF}_{\text{SLM}}(\gamma)\approx\left(1-\left(1-\exp\left\{-\gamma \right\}\right)^{2.8N}\right)^{C_{\text{SLM}}} \tag{4}\]8

As it can be seen, for the same imposed threshold, the CCRF for the GreenOFDMv2 is always lower than the GreenOFDM. Furthermore, almost a factor two of reduction is appreciated between both CCRFs, which is in line with the previously evoked factor 2 in terms of the reduced number of IFFTs. ## 6 Conclusions

In this paper, we present an efficient method to reduce the PAPR in OFDM. This method is a simplification of the recent GreenOFDM technique, which is itself an extension of the well-known SLM PAPR reduction technique. In all these techniques, several waveform candidates are generated from a certain number of IFFT to retain the one with the lowest PAPR as actual OFDM symbol. The PAPR performance of these methods is shown to be a decreasing function of the number of candidates. To increase the number of candidates for a given number of IFFT, the key idea of GreenOFDM was to combine two coupleable IFFT-based waveforms rather than only one in the SLM method. Moreover, the new version proposed in this paper allows us to use plainly the full set of IFFTs for each group of waveforms while maintaining their ability to be coupled, which was not the case for the original GreenOFDM version in [22] that split the IFFTs into two smaller separate subsets. Consequently, compared to the first version of the GreenOFDM method, the proposed method adds a simple yet efficient improvement to reduce the PAPR by increasing the number of available candidates by a factor of 4 without increasing the number of IFFTs. Alternatively, a same PAPR performance is obtained by using only half the number of IFFTs.