# An Improved GreenOFDM Scheme for PAPR Reduction
## 3 From SLM-OFDM to GreenOFDM
Left side corresponds to the results for 16-QAM mapping and right side corresponds to the results for 64-QAM. QAM—Quadrature Amplitude Modulation. ### Discussion on Additional Possible Complexity Reduction

As seen before, the proposed method requires only the square-root of the number of IFFTs of the original SLM method for a same PAPR reduction performance (i.e., a same number of candidates), making it a feasible technique from a complexity point-of-view. In addition to the reduction in the number of required IFFTs, the strategy proposed in [28,31] to reduce the implementation complexity of GreenOFDM can also be exploited for GreenOFDMv2 to further reduce the computational complexity. The probabilistic strategy exploits the knowledge of the CCDF of the PAPR. Indeed, to attain a certain performance in terms of PAPR (at a CCDF of PAPR \(p\), with a value of \(\gamma_{p}\) deterministically predicted through Equation (5)), there is no need to find the lowest PAPR symbol candidate among all the candidates but the first that satisfies a PAPR below the threshold \(\gamma\) (denoted as \(\gamma_{p_{0}}\), \(p_{0}\) being typically chosen to \(10^{-3}\)). In addition, the idea of finding the first candidate with acceptable PAPR is complemented with the fact that calculating the PAPR requires to compute the instant power for each sample (see Equation (2)). Indeed, if the Instantaneous-to-Average Power Ratio (IAPR) of a sample of a certain candidate exceeds the imposed _threshold_, there is no need to continue with that candidate and it can be discarded to continue with another one. These facts are taken into account and implemented through the _IFFTs-on-demand_ and _hierarchical sampling_ methods (please refer to [31] Sections 3.1 and 3.2 for the details of each method) in order to drastically reduce the computational complexity. Furthermore, in [31] it was shown that relaxing the PAPR reduction capability to a slightly higher threshold \(\gamma_{p_{i}}\) with

\[\text{CCDF}_{\text{SLM}}(\gamma)\approx\left(1-\left(1-\exp\left\{-\gamma \right\}\right)^{2.8N}\right)^{C_{\text{SLM}}} \tag{4}\]6

attains considerably higher computational reductions with a low impact in the PAPR reduction which provides a good performance-complexity trade-off. The same methods are implementable in the GreenOFDMv2 as described in Algorithm 4. In this algorithm, the IFFTs are computed step-by-step as the candidates are progressively calculated in the 'Hierarchical-sample' method that returns a flag in addition of the candidate itself. This flag being at '1' means that the corresponding candidate PAPR is below the imposed threshold \(\gamma_{p_{i}}\) and can be chosen for transmission. Otherwise the candidate has at least an IAPR exceeding \(\gamma_{i}\) and is hence discarded. ```
\(\{X_{k}\},\{\phi_{u,k}\}\) with \(\phi_{u,k}\in\{\pm 1\},\gamma_{p_{i}}\) \(\{x_{0:UL-1}[n]\}\leftarrow\text{zeros}(U,LN)\); for\((u_{1}\gets 0\)to\(U-1)\)do \(\{x_{u_{1}}[n]\}=\text{IFFT}\{X_{k}.\phi_{u,k}\}\) for\((u_{2}\gets u_{1}\)to\(0\)step\(-1)\)do if\(u_{1}==u_{2}\)then \(\{\text{flag},\{x_{u_{1},u_{2}}[n]\}\}\leftarrow\text{Hierarchical-sample} \left\{\gamma_{p_{i}},\{x_{u_{1}}[n]\}\right\}\); else \(\{\text{flag},\{x_{u_{1},u_{2}}[n]\}\}\leftarrow\text{Hierarchical-sample} \left\{\gamma_{p_{i}},\{x_{u_{1}}[n]\},\{x_{u_{2}}[n]\}\right\}\); endif if\((\text{flag})\)then \(\{x_{u_{1},u_{2}}[n]\}\leftarrow\{x_{u_{1},u_{2}}[n]\}\); STOP(); endif endfor endfor \(\text{SEND}\{x_{\hat{a}_{1},\hat{a}_{2}}[n]\}\);
```

**Algorithm 4** Computational complexity reduction in the GreenOFDMv2 scheme. All these strategies have allowed for a relevant computational reduction of the former GreenOFDM in [28], especially in terms of the number of IFFTs around fifty percent (with \(N=64\) subcarriers, \(U=64\) IFFTs, and a back-off of 0.3 dB for the PAPR threshold). The reduction of the computational complexity of GreenOFDMv2 is expected to be better than those in the former version of GreenOFDM.