# An Improved GreenOFDM Scheme for PAPR Reduction

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## Abstract

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## 1. Introduction

- (i)
- (ii)
- in order to correctly estimate the PAPR of OFDM signals in the digital domain, the computational complexity of the digital modulator is increased due to oversampling [4].
- (iii)
- the high PAPR of these signals generally increases the Power Amplifier’s (PA’s) power dissipation as its linear region has to be extended to be able to accommodate signals with wide amplitude excursions. Not doing so results in non-linear distortions and out-of-band radiations that degrade the system performances [5].

## 2. OFDM and PAPR

## 3. From SLM-OFDM to GreenOFDM

Algorithm 1 The SeLected Mapping (SLM)-Orthogonal Frequency Division Multiplexing (OFDM) algorithm. |

Require:$\{{X}_{k}\}$, $\{{\varphi}_{u,k}\}$ with ${\varphi}_{u,k}\in \{\pm 1\}$ |

minPAPR $\leftarrow +\infty $ |

for $(u\leftarrow 0\mathbf{to}U-1)$ do |

$\{{X}_{u,k}\}\leftarrow \{{X}_{k}\xb7{\varphi}_{u,k}\}$; |

$\{{x}_{u}[n]\}\leftarrow \mathrm{IFFT}\{{X}_{u,k}\}$; |

if $(\mathrm{PAPR}\{{x}_{u}[n]\}<\mathrm{minPAPR})$ then |

minPAPR $\leftarrow \mathrm{PAPR}\{{x}_{u}[n]\}$; |

$\{{x}_{\tilde{u}}[n]\}\leftarrow \{{x}_{u}[n]\}$; |

end if |

end for |

SEND $\{{x}_{\tilde{u}}[n]\}$; |

Algorithm 2 The GreenOFDM algorithm. |

Require: $\{{x}_{{g}_{1}}[n]\}=\mathrm{IFFT}\{{X}_{k}\xb7{\phi}_{{g}_{1},k}\}$, $\{{x}_{{g}_{2}}[n]\}=\mathrm{IFFT}\{{X}_{k}\xb7i\xb7{\phi}_{{g}_{2},k}\}$ |

minPAPR $\leftarrow +\infty $; |

for $({g}_{1}\leftarrow 0\mathbf{to}\frac{U}{2}-1)$ do |

for $({g}_{2}\leftarrow \frac{U}{2}\mathbf{to}U-1)$ do |

$\{{x}_{{g}_{1},{g}_{2}}[n]\}\leftarrow \frac{\left\{{x}_{{g}_{1}}[n]+{x}_{{g}_{2}}[n]\right\}}{\sqrt{2}}$; |

if $(\mathrm{PAPR}\{{x}_{{g}_{1},{g}_{2}}[n]\}<\mathrm{minPAPR})$ then |

minPAPR $\leftarrow \mathrm{PAPR}\{{x}_{{g}_{1},{g}_{2}}[n]\}$; |

$\{{x}_{{\tilde{g}}_{1},{\tilde{g}}_{2}}[n]\}\leftarrow \{{x}_{{g}_{1},{g}_{2}}[n]\}$; |

end if |

end for |

end for |

SEND $\{{x}_{{\tilde{g}}_{1},{\tilde{g}}_{2}}[n]\}$; |

## 4. An Improved GreenOFDM Version

Algorithm 3 The GreenOFDMv2 algorithm. |

Require:$\{{x}_{u}[n]\}=\mathrm{IFFT}\{{X}_{k}\xb7{\varphi}_{u,k}\}$ |

minPAPR $\leftarrow +\infty $; |

for $({u}_{1}\leftarrow 0\mathbf{to}U-1)$ do |

for $({u}_{2}\leftarrow 0\mathbf{to}U-1)$ do |

if ${u}_{1}=={u}_{2}$ then |

$\{{x}_{{u}_{1},{u}_{2}}[n]\}\leftarrow \{{x}_{{u}_{1}}[n]\}$; |

else |

$\{{x}_{{u}_{1},{u}_{2}}[n]\}\leftarrow \frac{\left\{{x}_{{u}_{1}}[n]+i\times {x}_{{u}_{2}}[n]\right\}}{\sqrt{2}}$; |

end if |

if $(\mathrm{PAPR}\{{x}_{{u}_{1},{u}_{2}}[n]\}<\mathrm{minPAPR})$ then |

minPAPR $\leftarrow \mathrm{PAPR}\{{x}_{{u}_{1},{u}_{2}}[n]\}$; |

$\{{x}_{{\tilde{u}}_{1},{\tilde{u}}_{2}}[n]\}\leftarrow \{{x}_{{u}_{1},{u}_{2}}[n]\}$; |

end if |

end for |

end for |

SEND $\{{x}_{{\tilde{u}}_{1},{\tilde{u}}_{2}}[n]\}$; |

## 5. Simulation Results and Discussion

#### 5.1. PAPR Complementary Cumulative Distribution Function (CCDF)

#### 5.2. PAPR Threshold: Simulations and Approximated Formula

#### 5.3. Discussion on Additional Possible Complexity Reduction

Algorithm 4 Computational complexity reduction in the GreenOFDMv2 scheme. |

Require:$\{{X}_{k}\},\{{\varphi}_{u,k}\}\phantom{\rule{4.pt}{0ex}}\mathrm{with}\phantom{\rule{4.pt}{0ex}}{\varphi}_{u,k}\in \{\pm 1\},{\gamma}_{{p}_{i}}$ |

$\{{x}_{0:U-1}[n]\}\leftarrow \mathrm{zeros}(U,LN)$; |

for $({u}_{1}\leftarrow 0\mathbf{to}U-1)$ do |

$\{{x}_{{u}_{1}}[n]\}=\mathrm{IFFT}\{{X}_{k}.{\varphi}_{u,k}\}$ |

for $({u}_{2}\leftarrow {u}_{1}\mathbf{to}0\mathbf{step}-1)$ do |

if ${u}_{1}=={u}_{2}$ then |

$\{\mathrm{flag},\{{x}_{{u}_{1},{u}_{2}}[n]\}\}\leftarrow \mathrm{Hierarchical}-\mathrm{sample}\left\{{\gamma}_{{p}_{i}},\{{x}_{{u}_{1}}[n]\}\right\}$; |

else |

$\{\mathrm{flag},\{{x}_{{u}_{1},{u}_{2}}[n]\}\}\leftarrow \mathrm{Hierarchical}-\mathrm{sample}\left\{{\gamma}_{{p}_{i}},\left\{{x}_{{u}_{1}}[n]\right\},\left\{{x}_{{u}_{2}}[n]\right\}\right\}$; |

end if |

if $(\mathrm{flag})$ then |

$\{{x}_{{\tilde{u}}_{1},{\tilde{u}}_{2}}[n]\}\leftarrow \{{x}_{{u}_{1},{u}_{2}}[n]\}$; |

STOP(); |

end if |

end for |

end for |

SEND $\{{x}_{{\tilde{u}}_{1},{\tilde{u}}_{2}}[n]\}$; |

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

CCDF | Complementary Cumulative Distribution Function |

CCRF | Computational Complexity Reduction Factor |

IAPR | Instantaneous-to-Average Power Ratio |

IFFT | Inverse Fast Fourier Transform |

OFDM | Orthogonal Frequency Division Multiplexing |

PA | Power Amplifier |

PAPR | Peak-to-Average Power Ratio |

SLM | SeLected Mapping |

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**Figure 4.**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).

**Figure 5.**The value of $\gamma [\mathrm{dB}]$ to attain CCDF$(\gamma )={10}^{-3}$ for different values of U and N for OFDM, SLM-OFDM, GreenOFDM and GreenOFDMv2, obtained by simulation (marks) and approximated expressions (dashed lines).

**Figure 6.**The value of $\gamma [\mathrm{dB}]$ to attain CCDF$(\gamma )={10}^{-3}$ for different values of U and N for SLM-OFDM, GreenOFDM and GreenOFDMv2 obtained by simulation (marks). Approximated expressions in Equation (5) are also plotted in dashed lines for GreenOFDMv2.

**Figure 7.**The value of $\gamma [\mathrm{dB}]$ to attain CCDF$(\gamma )={10}^{-3}$ for different values of U and N for SLM-OFDM, GreenOFDM and GreenOFDMv2 obtained by simulation (marks). Approximated expressions in Equation (5) are also plotted in dashed lines. Left side corresponds to the results for 16-QAM mapping and right side corresponds to the results for 64-QAM. QAM—Quadrature Amplitude Modulation.

**Figure 8.**The Computational Complexity Reduction Factor (CCRF) between the GreenOFDM and SLM-OFDM as presented in [28] (

**left**-side) and the CCRF between the proposed method and the conventional SLM-OFDM (

**right**-side) with complexity reduction as a function of ${\gamma}_{{p}_{i}}[\mathrm{dB}]$ and N.

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**MDPI and ACS Style**

Gulfo Monsalve, J.L.; Ros, L.; Brossier, J.-M.; Mestdagh, D. An Improved GreenOFDM Scheme for PAPR Reduction. *Telecom* **2020**, *1*, 196-210.
https://doi.org/10.3390/telecom1030014

**AMA Style**

Gulfo Monsalve JL, Ros L, Brossier J-M, Mestdagh D. An Improved GreenOFDM Scheme for PAPR Reduction. *Telecom*. 2020; 1(3):196-210.
https://doi.org/10.3390/telecom1030014

**Chicago/Turabian Style**

Gulfo Monsalve, Jorge Luis, Laurent Ros, Jean-Marc Brossier, and Denis Mestdagh. 2020. "An Improved GreenOFDM Scheme for PAPR Reduction" *Telecom* 1, no. 3: 196-210.
https://doi.org/10.3390/telecom1030014