# PAPR of linearly precoded massive MIMO signals

Marco Guzzo, Inbar Fijalkow

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Marco Guzzo, Inbar Fijalkow. PAPR of linearly precoded massive MIMO signals. IEEE Conference on Standards for Communications and Networking (CSCN 2025), IEEE, Sep 2025, Bologna, Italy. hal-05250568

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Submitted on 11 Sep 2025

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# PAPR of linearly precoded massive MIMO signals

Marco GUZZO, Inbar FIJALKOW

ETIS, UMR 8051

CY Cergy Paris University, ENSEA, CNRS

Cergy, France

{marco.guzzocholet, inbar.fijalkow}@ensea.fr

**Abstract**—We discuss the non dependence of the peak to average power ratio (PAPR) with respect to the number of antennas at the base station (BS) in the case of perfect channel state information (CSI) for downlink multi-user massive MIMO systems despite what was supposed and intuited in the literature.

**Index Terms**—Multi-user massive MIMO, linear precoder, PAPR, energy efficiency

## I. INTRODUCTION

MASSIVE MIMO involves the use of many antennas at the base station of a wireless network. In a multi-user configuration, the base station serves a number of users relatively small regarding to the number of antennas. As a result, robustness and data rate can be increased [1]. The deployment of wireless massive MIMO systems makes it possible to use inexpensive, low-power single-antenna user equipment, with the base station being the only device that requires more costly technology [2].

The power consumption relies on the signal power range and peak power at the input of the power amplifier (PA). The common metric to evaluate the signal dynamic range around its average value is the peak to average power ratio (PAPR) that we will investigate next.

## II. MULTI-USER MASSIVE MIMO MODEL

We assume a flat-fading downlink scenario in which an *M*-antennas BS is sending zero-mean uncorrelated symbols $s_k$ with power $\sigma_s^2$ (i.e. $\mathbb{E}[ss^\dagger] = \sigma_s^2 I_K$), to $k \in \{1, \dots, K\}$ single-antenna users over the $K \times M$ channel $\mathbf{H}$. We assume perfect CSI. The channel matrix is given by $\mathbf{H} = \text{Diag}(\zeta_1, \zeta_2, \dots, \zeta_K)\hat{\mathbf{H}} = \mathbf{\Sigma}\hat{\mathbf{H}}$ where the elements of $\hat{\mathbf{H}}$ are circularly symmetric complex Gaussian random variables. The parameters $\zeta_1, \zeta_2, \dots, \zeta_K$ are the individual channel gains encompassing for path loss and shadowing.

We introduce the parameter $\gamma = \frac{M}{K}$, the inverse antenna load. The massive MIMO configuration compels $\gamma > 1$. The linear precoder $\mathbf{P}$ takes advantage of the transmit diversity by weighting linearly the information stream: $\mathbf{x} = \mathbf{P}\mathbf{s}$. The Zero Forcing precoder $\mathbf{P} = \mathbf{H}^\dagger(\mathbf{H}\mathbf{H}^\dagger)^{-1}$ has been chosen because it can be used for LTE signals and it removes all the interferences between users. The chain equation can be summarized by $\mathbf{r} = \mathbf{H}\mathbf{P}\mathbf{s}$.

This work was supported by the ANR under the France 2030 program, grant "NF-PERSEUS : ANR-22-PEFT-0004".

## III. PAPR OF THE MULTI-USER MASSIVE MIMO

The analysis of the sent signal amplitude variations is done using the PAPR defined as the ratio between a signal's peak power and average power,

$$ \mathrm{PAPR}_{\mathrm{x}}=\frac{p_{\max , \mathrm{x}}}{p_{\mathrm{avg}, \mathrm{x}}}=\frac{\max _{m \in\{1, \ldots, M\}}\left|\mathbf{x}_{m}\right|^{2}}{\mathbb{E}\left[\left|\mathbf{x}_{m}\right|^{2}\right]} \quad(1) $$

The simulated curves are drawn from 100.000 Monte-Carlo runs with $K \in \{10, 20\}$, $\sigma_s^2 = 10$ and $\forall k, \zeta_k^2 = 1$. Fig. 1 displays the PAPR with respect to $\gamma$.

<figure><img src="image_2.png" /><figcaption>Fig. 1. PAPR (in dB) for different precoded signals vs. *M*</figcaption></figure>

Despite common belief [1], [3], [4], the power dispersion around the mean power of the precoded signal does not increase linearly with the number of antennas *M* but is nearly constant above a certain threshold (typically $\gamma \ge 2$). Thanks to [4], we know how the average power decreases with $\gamma$ so that we suggest that the peak power follows the same tendency with respect to $\gamma$ for the Zero Forcing precoder.

## REFERENCES

[1] E. G. Larsson, O. Edfors, F. Tufvesson, and T. L. Marzetta, "Massive MIMO for next generation wireless systems," IEEE Commun. Mag., vol. 52, no. 2, pp. 186-195, Feb. 2014

[2] R. Zayani, H. Shaiek, and D. Roviras, "PAPR-Aware Massive MIMO-OFDM Downlink", IEEE Access, Vol. 7, pp. 25474-25484, 2019.

[3] H. Q. Ngo, E. G. Larsson and T. L. Marzetta, "Energy and Spectral Efficiency of Very Large Multiuser MIMO Systems," in IEEE Transactions on Communications, vol. 61, no. 4, pp. 1436-1449, April 2013

[4] Amodh Kant Saxena, Inbar Fijalkow, and A. Lee Swindlehurst, "Analysis of One-Bit Quantized Precoding for the Multiuser Massive MIMO Downlink", IEEE Transactions on Signal Processing, Vol. 65, N. 17, September, 2017