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PAPR Distribution for Single Carrier M-QAM Modulations

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Abstract

Single carrier modulations such as quadrature amplitude modulation (QAM) are recently becoming an attractive and complementary alternative to multiple carrier modulations. High order QAM provides spectral efficiency advantage at the price of larger dynamic range. This characteristic leads to enlarge the peak-to-average power ratio (PAPR) and so, to reduce the energy efficiency. This study provides an analysis of the PAPR distribution for QAM based systems. We focus on the distribution of the PAPR since it is the main evaluation means of PAPR reduction techniques. We present an analytic expression of the probability density function of the PAPR for limited length frames. The analysis shows that the expression of the PAPR usually found in the literature is valid only for long frames and is the asymptotic limit of the formula we propose. According to the simulation results, the distribution we suggest accurately describes the PAPR for long frames and is a good approximation for short frames.

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Authors and Affiliations

  1. Université Bretagne Loire, Université de Nantes, UMR CNRS 6164 - Institute of Electronics and Telecommunications of Rennes (IETR), Polytech Nantes, Nantes, France

    Kouakou Kouassi, Guillaume Andrieux & Jean-François Diouris

Authors
  1. Kouakou Kouassi
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  2. Guillaume Andrieux
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  3. Jean-François Diouris
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Correspondence to Guillaume Andrieux.

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Appendix: Proof of the Eq. (29)

Appendix: Proof of the Eq. (29)

The limit (28) can be written

$$\begin{aligned} \wp _{\infty } = \lim \limits _{N \rightarrow \infty }\frac{N\left( \sqrt{M}-1\right) ^2}{2 {\sigma }^2}\left( {\mu } \sqrt{ 1+\frac{8 {\sigma }^2}{N {\mu }^2}}- {\mu }\right) \end{aligned}$$
(31)

Using the Taylor expansion

$$\begin{aligned} \sqrt{1+x}=1+\frac{x}{2}+o(x^2) \end{aligned}$$
(32)

we get:

$$\begin{aligned} \wp _{\infty } = \frac{2\left( \sqrt{M}-1\right) ^2}{\mu } \end{aligned}$$
(33)

using the expression of \(\mu\) from (13) we get

$$\begin{aligned} \wp _{\infty } = 3\cdot \frac{ \sqrt{M}-1 }{\sqrt{M}+1}. \end{aligned}$$
(34)

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Kouassi, K., Andrieux, G. & Diouris, JF. PAPR Distribution for Single Carrier M-QAM Modulations. Wireless Pers Commun 104, 727–738 (2019). https://doi.org/10.1007/s11277-018-6046-1

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  • DOI: https://doi.org/10.1007/s11277-018-6046-1

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