# An Improved GreenOFDM Scheme for PAPR Reduction
## 3 From SLM-OFDM to GreenOFDM
In the particular case \(u_{1}=u_{2}\), there is no need to perform the additions \(\{x_{u_{1}}[n]+i\cdot x_{u_{2}}[n]\}\) because it does not change the PAPR of the initial waveform \(\{x_{u_{1}}[n]\}\). In the proposed method, these extra operations are avoided in that particular case by choosing the candidate directly from the IFFT output, i.e., we choose the candidate to be equal to \(\{x_{u_{1}}[n]\}\). The number of candidates that are generated in the proposed method, here after referred to as the _GreenOFDMv2_, is equal to the number of times the inner loop is run. It corresponds to \(C_{\text{Greenv2}}=U^{2}\). This means that we have 4 times more candidate waveforms as compared to the initial version of the GreenOFDM by computing the same number of IFFTs, hence a better PAPR reduction is expected. Figure 3 depicts the proposed method block diagram. ## 5 Simulation Results and Discussion

Computer simulations were carried out to evaluate the CCDF of PAPR of the GreenOFDMv2 method proposed in this work and to compare it with the previous methods (SLM-OFDM and the former GreenOFDM) and to the conventional OFDM. Also, the PAPR threshold, for a given CCDF, and its evolution were investigated to know and predict the required number of IFFTs to stay under a given PAPR threshold for the different techniques and different number of subcarriers. ### PAPR Complementary Cumulative Distribution Function (CCDF)

Results for \(10^{5}\) symbols with \(N=64\)\(QPSK\)-modulated subcarriers, with the oversampling factor \(L=4\) and \(U=16\) IFFTs for the SLM-OFDM, GreenOFDM and the GreenOFDMv2, are depicted in Figure 4 where the dashed lines correspond to the approximated expressions of the CCDF in Equations (3) and (4) and the marks correspond to the simulation results. Figure 3: The proposed _GreenOFDMv2_ method block diagram. The simulation results are well approximated by the CCDFs expressions for OFDM, SLM-OFDM (with \(C_{\text{SLM}}=U\)), GreenOFDM (by replacing \(C_{\text{SLM}}\) by \(C_{\text{Green}}=U^{2}/4\)) and GreenOFDMv2 (by replacing \(C_{\text{SLM}}\) by \(C_{\text{Green}2}=U^{2}\)). Moreover, it can be seen that the best performance for a fixed probability (\(\text{CCDF}(\gamma)\)) is obtained for the GreenOFDMv2 method as compared to SLM-OFDM and the previously proposed GreenOFDM, with a gain of 1.2 and 0.5 dB, respectively, for a \(\text{CCDF}(\gamma)=10^{-3}\) (the gain versus conventional OFDM is about 5 dB). This is because of the increase in the number of candidates (\(U^{2}\) for GreenOFDMv2 rather than \(U\) for SLM-OFDM or \(\frac{U^{2}}{4}\) for GreenOFDM). ### PAPR Threshold: Simulations and Approximated Formula

In this section we present a different way to analyse the PAPR reduction capabilities in SLM-based schemes. For a fixed probability \(\text{CCDF}(\gamma)=p\), it is possible to calculate the value of \(\gamma\) as a function of the number of subcarriers \(N\) and the number of candidates that depends on \(U\). To do so, Equation (4) is reformulated as follows:

\[\text{CCDF}_{\text{SLM}}(\gamma)\approx\left(1-\left(1-\exp\left\{-\gamma \right\}\right)^{2.8N}\right)^{C_{\text{SLM}}} \tag{4}\]0

where \(C\) is a generic notation for the number of candidates as a function of the number of IFFTs \(U\), i.e., \(C=C_{\text{SLM}}=U\) for SLM-OFDM, \(C=C_{\text{Green}}=\frac{U^{2}}{4}\) for GreenOFDM and \(C=C_{\text{Green}2}=U^{2}\) for the proposed method. The same way, for conventional OFDM, Equation (3) is reformulated as:

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

\[\text{CCDF}_{\text{SLM}}(\gamma)\approx\left(1-\left(1-\exp\left\{-\gamma \right\}\right)^{2.8N}\right)^{C_{\text{SLM}}} \tag{4}\]2 The Complementary Cumulative Distribution Function (CCDF) of Peak-to-Average-Power-Ratio (PAPR) for conventional OFDM, SLM-OFDM, GreenOFDM and GreenOFDMv2 with the same parameter \(U=16\) and for \(N=64\) subcarriers per symbol, obtained by simulation (marks) and approximated expressions (dashed lines).