# An Improved GreenOFDM Scheme for PAPR Reduction
## 3 From SLM-OFDM to GreenOFDM  
In conventional SLM-OFDM, \(U\) different symbol waveforms are generated from the same data set \(\{X_{k}\}\). To do so, the input data set \(\{X_{k}\}\) is repeated \(U\) times and each copy is multiplied element-wise by pseudo-random sequences where the coefficients have unitary norm \(\phi_{u,k}\), \(0\leq u\leq U-1\); i.e., \(X_{u,k}=X_{k}\cdot\phi_{u,k}\) with \(|\phi_{u,k}|=1\). It is noteworthy to mention that in the following, \(U\) represents the total number of computed IFFTs as well as the total number of pseudo-random sequences. The time-domain symbol waveforms \(x_{u}[n]\) are computed by \(LN-\)IFFTs with inputs \(\left\{X_{u,k}\right\}\). The waveform that exhibits the lowest PAPR among the \(C_{\text{SLM}}=U\) candidates is selected for transmission. The SLM-OFDM algorithm is summarized in Algorithm 1 and its block diagram depicted in Figure 1.  
The function \(\text{CCDF}_{\text{SLM}}(\text{PAPR})\) decreases when the number of candidates \(C_{\text{SLM}}=U\) increases. However, higher values of \(C_{\text{SLM}}\) lead to larger values of the computational burden because of the required number of IFFTs. In order to mitigate this limitation of SLM-OFDM, the key idea of GreenOFDM is to generate the candidates by adding (with a normalisation factor \(\sqrt{2}\)) two coupleable IFFT-based waveforms. This allows for the generation of more candidates \(C_{\text{Green}}\) corresponding to all the possible combinations, whilst maintaining the same PAPR reduction behaviour of SLM-OFDM with respect to the number of candidates (i.e., Equation (4) with \(C_{\text{Green}}\) instead of \(C_{\text{SLM}}\)). The CCDF of PAPR of conventional SLM-OFDM is well approximated by [13]:  
\[\text{CCDF}_{\text{SLM}}(\gamma)\approx\left(1-\left(1-\exp\left\{-\gamma \right\}\right)^{2.8N}\right)^{C_{\text{SLM}}} \tag{4}\]```
0:\(\{X_{k}\}\), \(\{\phi_{u,k}\}\) with \(\phi_{u,k}\in\{\pm 1\}\) \(\min\)PAPR \(\leftarrow\)\(+\infty\); for\((u\gets 0\)to\(U-1)\)do \(\{X_{u,k}\}\leftarrow\{X_{k}\cdot\phi_{u,k}\}\); \(\{x_{u}[n]\}\leftarrow\) IFFT\(\{X_{u,k}\}\); if\((\)PAPR\(\{x_{u}[n]\}<\) minPAPR\()\)then \(\min\)PAPR \(\leftarrow\) PAPR\(\{x_{u}[n]\}\); \(\{x_{\tilde{u}}[n]\}\leftarrow\{x_{u}[n]\}\); endif endfor \(\mathrm{SEND}\{x_{\tilde{u}}[n]\}\);
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**Algorithm 1** The SeLected Mapping (SLM)-Orthogonal Frequency Division Multiplexing (OFDM) algorithm. The GreenOFDM technique generates \(C_{\mathrm{Green}}=U^{2}/4\) OFDM symbols candidates by first splitting the \(U\) set of copied and randomized data \(\{X_{u,k}\}\) into two groups: the first one represented by the index \(g_{1}\), with \(0\leq g_{1}\leq\frac{U}{2}-1\) and the second one represented by the index \(g_{2}\), with \(\frac{U}{2}\leq g_{2}\leq U-1\). The first group waveforms \(\{x_{g_{1}}[n]\}\) is obtained, like in SLM-OFDM, through IFFTs over \(\{X_{g_{1},k}=X_{k}\cdot\varphi_{g_{1},k}\}\) (with \(\{\varphi_{g_{1},k}\}=\{\phi_{u,k}\}\), \(u<\frac{U}{2}\) and \(\phi_{u,k}\in\pm 1\)). Similarly, the second group waveforms \(\{x_{g_{2}}[n]\}\) are obtained through IFFTs over \(\{X_{g_{2},k}=X_{k}\cdot\varphi_{g_{2},k}\}\) (with \(\{\varphi_{g_{2},k}\}=\{i\times\phi_{u,k}\}\), \(u\geq\frac{U}{2}\) and \(\phi_{u,k}\in\pm 1\)). As it can be seen, the difference between the two groups of waveforms is the fact that the sequences \(\{\varphi_{g_{2},k}\}\) use pure imaginary unitary values: \(\{\pm i\}\) whereas the sequences \(\{\varphi_{g_{1},k}\}\) use pure real unitary values: \(\{\pm 1\}\). In this way, individual elements of the two sequences are orthogonal in order not to cancel out some subcarriers after adding the two IFFT outputs. The last part of the GreenOFDM method, which depicts the way the \(\frac{U^{2}}{4}\) waveform candidates are obtained, is described in Algorithm 2. The equivalent block diagram of the GreenOFDM is depicted in Figure 2.

Figure 1: SLM-OFDM block diagram.