The Cominlab TEPN (Toward Energy Proportional Network) project gather four research teams to contribute on energy efficiency in Wireless Networks. The project is organized in four work packages, and the work is done around six thesis. The following table gives the thesis title and implied persons.

Table 1: presentation of the thesis

Title | PhD Student | Advisors |

WLAN density and energy efficiency | Dareen Shehadeh | Nicolas Montavont, Alberto Blanc |

Delay Tolerant Users: The Solution to End-to-End Energy Efficiency | Samantha Gamboa (parially funded) | Alexander Pelov, Nicolas Montavont |

Spectral Efficiency - Energy Efficiency Trade-off in Interference Limited Wireless Networks | Ahmad-Mahbubul Alam | Jean-Yves Baudais, Philippe Mary, Xavier Lagrange |

Decision Making for Green Cognitive Networks | Navikkumar Modi | Christophe Moy, Philippe Mary |

Jointed PAPR Optimization and Pre-Distrortion technics | Ali Cheaito | Matthieu Crussière, Yves Louet, Jean-François Helard |

Study of a New Multi-Carrier Waveform | Marwa Chafii | Jacques Palicot, Rémi Gribonval |

Cross-Layer Design Analysis for Energy Efficient Wireless Networks | Laudin Molina | Alberto Blanc, Nicolas Montavont |

## Context and objectives

Next to 5% of the global electricity consumption is due to networks in general, and yet the expected data transported in wireless networks is expected to grow by 60% / year. Facing the increasing energy consumption to operate wireless networks, TEPN aims at minimizing wireless network energy consumption. The primarly goal is to make the energy consumption of a wireless network a linear function of the load. This objective is captured in Figure 1.

Figure 1: TEPN objectives: make more proportional the energy consumption of a wireless network to the load. On the left (this Figure is taken from the project EARTH), we see current base station energy consumption: even without active users (no output power), a base station is consuming a large amount of energy (around 650W). This energy consumption increases with the load. On the right, it is the TEPN objective : tend to 0 consumption when there is no load, and have the energy consumption more dependant on the load

In TEPN, we adopt the simplified equation from the EARTH project to represent the energy consumption at a Base Station. This equation is presented as follows:

P_{in }= N_{TRX} (P_{0} + D_{p} P_{out}) (Eq 1)

In Equation (1), P_{in} is the base station energy consumption, N_{TRX }is the number of transmission chains of the base station, P_{0} is the energy consumption that is independant from any tranmission, only when the Base Station is turned ON, D_{p} is the slope of the energy consumption, which is a function of the power, and P_{out} is the power radiated at the antenna for the transmission.

In order to make wireless network more energy efficient, we deploy several efforts at different levels. From a global perspective, we study WLAN and cellular networks, and evaluate the Base Station switch ON and OFF methods. Current deployment are generally dense, because they have been designed to serve at the peak usage when there is a maximum number of clients. However, this peak can only be observed at a certain time of a day, otherwise the load is highly variable and can be actually very low. Thesis 1, 2 and 3 (as presented in Table 1) tackle this issue by studying cell switching algorithms to turn on and off base station when they are not needed to serve users (typically when the load is low). We also approach these algorithm from a system point of view, by studying theoritically the trade-off between spectral efficiency and energy efficiency for an entire wireless network.

Thesis 4 is focusing on machine learning technics in general to provide algorithm for Green communication. One outcome of this thesis is to study how to allocate bandwith / frequency to base station. Through a machine learning process, base stations could learn about their environment, and take a decision on which frequency to use for communication in a dynamic manner.

Thesis 5 and 6 are working on lower layers and closer to the hardware. They are dealing with amplificators energy consumption, which represent 80% of current Base Station energy consumption. They tackle PAPR (Peak to Average Power Ratio) reduction technics, including a study of a new wave form that would require less energy by nature, because of its form, and pre-distrortion technics that can be applied on the signal, also with the effect of energy reduction.

The preliminary results are presented next.

## WLAN density study

- PhD Student: D. Shehadeh
- Partner involved: IRISA OCIF

WiFi is probably the most popular short-range wireless access technology with an exponential increase of Access Points (AP) deployments for the past few years [1]. They are being deployed in companies, schools, public and private areas; resulting in a high APs density and overlapping coverage areas. Usually all APs are switched on all the time, even when the number of users served by the wireless access network is very low, resulting in a high and partially wasteful energy consumption. Even though a single AP consumes only few watts, the total energy consumed could reach hundreds of watts in large deployments. For example, according to [2] the ADSL boxes (which are usually APs too) in France consume 5TWh per year. Tsui et al. [3] detected over 103000 APs in the city of Taipei through war-walking, while Achtzehn et al. [4] estimated the density of APs deployed in different cities in Germany to vary between 488 AP per km2 in industrial areas, and 6103 AP per km2 in urban residential areas. Faced by AP proliferation and with the goal of reducing the energy consumption, recent research efforts have proposed algorithms to dynamically switch on and off a set of APs depending on the traffic load.

In continuity with these efforts, we study the Wifi AP deployment in an urban environment and evaluate the AP redundancy in providing network coverage. We provide a methodology and a set of tools to measure this density. Using the Android application Wi2Me to performs network scanning, we gather real data about APs through war-walking in the centre of Rennes - France. After processing these data with various filters, we apply two different selection algorithms to select a minimal set of APs that are enough to provide the same coverage, assuming that the non-selected APs could be switched off.

### Description of the experiment

Figure 2(a) : First path : a loop Figure 2(b) : second path: zizag in narrow streets

We walked for more than 20 hours in the center of Rennes (France) (see Figure 2) carrying smartphones running the Wi2Me application [14] (“war-walking”). This application scans periodically all the WiFi channels and logs all the available APs, generating a set of Wi2Me Traces, containing the time, the GPS location and other information about the discovered AP. We then process one or more of these traces to produce a Coverage Matrix, as presented in Figure 3. The Coverage Matrix is a representation of the AP availability: each AP is presented in a column, while we have each scanning event in rows. A black box (or a binary one) means that an AP has been seen during the given scanning event. We can see that AP coverage is usually not continuous: we observe alternative white and black areas for some APs. Some of these gaps are due to link error during the scanning process, while others are simply obstacles that avoid a mobile terminal to correctly receive the AP signal at certain locations in a street.

Figure 3: The coverage matrix, which reprense the AP availabilities along a path

In these experiments, we discovered around 8000 APs, and 17 APs per scanning, which reveal a really dense deployment.

### Proposed algorithms

Our goal is to evaluate how many AP could be enough to cover the studied areas. Clearly, there is no need for so many APs to cover the path: a mobile terminal can be currently served by 17 different APs at any location on the map. We proposed two algorithms to compute what we call the minimal AP sets.

**Algorithm 1: The greedy**

- Calculate the total coverage value for the APs, i.e., the sum of all the ones in the corresponding column of the Input Matrix.
- Select the AP with the largest total coverage value and add it to the Minimum APs Set.
- Delete the lines where there is a one in the column of the selected AP from the Input Matrix.
- Delete the column of the selected AP from the Input Matrix.
- Go to Step 1 and repeat until all the lines are deleted.

**Algorithm 2: The continuous Greedy**

- Starting from the beginning of the path, i.e., the first line of the Input Matrix, identify the available APs (the columns where there are ones in this line).
- Calculate for each of these APs the continu- ous coverage (sum of successive ones starting from the current position until the first zero).
- Choose the AP with the longest continuous coverage and add it to the Minimum APs Set.
- Go to the last line of the continuous coverage of the selected AP.
- Repeat steps 2 to 4 until the end of the matrix i.e end of the path.

### Main results

After applying the algorithm, we found that around** 6.5%** of the detected APs are sufficient to provide full coverage of the path. This percentage varies from 4.25% to 10.91% depending on the dataset, the phone, and the AP selection algorithm. In other words, we can maintain the coverage of a path while switching off at least 89% of the existing APs. This means that the potential of energy saving obtained by switching off the non selected APs is about 89% too. We estimated the energy consumed by the selected APs using the results of [17], which reported that the average consumption of a single AP is about 6 W. Results show that the selected APs consume around 540 W in average for the Loop path and 1215 W for the Zigzag path. This reflects a significant reduction in energy consumption when compared to the initial estimated consumption of 7200 W and 19 323 W for the two paths respectively.

Results also show that the minimal AP set offers a good overlapping. The number of APs seen at any position of the path varies between 1 and 6, with a mean of approximately 2, and an overlapping of two or more APs along more than 60% of the path. Compared to the initial situation where we had an average of 17 APs present at each position of the path, this represents a large reduction in overlapping. The overlapping in the minimal set is useful to insure smoother handovers.

Figure 4: Coverage of the minimal AP sets

Finally Figure 4 shows the contribution of each AP from the minimal AP set. By contribution, we mean the coverage that each of these APs provide to the user. We can see that there is a large portion of the set that does not provide a large coverage: with the greedy algorithm, 75% of the APs are contributing for less than 2% of the total coverage of the path, and we observed that using only the 60% of the *best* AP still provides 98% of path coverage.

## Systemic approach: spectral efficiency VS energy efficiency

- PhD Student: A. Alam
- Partner involved: IETR SCN, IRISA REOP

The increasing demand for high spectral efficiency (SE) to increase the user experience on innovative medias leads to design energy-consuming wireless netwoks. On the other hand, the energy efficiency (EE), defined as the number of bits that can be successfully sent over the energy consumed, has emerged as a new constraint for wireless networks design. It is well known that high SE and high EE can not be achieved in the same time when the static power consumption is zero. The SE-EE tradeoff is hence a relevant way to characterize some functionning points in the network. Several works have dealt with energy-efficient communications, e.g. [1] and references therein, starting from Shannon theory. Moreover, Rodoplu *et al.* have provided a work of notable importance considering the EE of large wireless networks and deriving scaling laws on EE w.r.t. the size of the network [2]. The authors have extended the work of Gupta and Kumar on the capacity scaling laws of ad-hoc networks to the EE metric. On the other hand several authors have studied the EE in cellular networks based on stochastic geometry tools, e.g. [3], [4]. However, these works do not consider the static power consumption of a base station implying differences in the EE-SE tradeoff.

In this work, we first attempt to charaterize the EE-SE tradeoff in interference-limited wireless networks considering regular hexagonal network structure in a first time.

Figure 5: EE-SE tradeoff (log-log scale)

In a point-to-point communication, the upper-bound of the EE-SE tradeoff defines limit of the achievable EE-SE tradeoffs, as the capacity region defines the limit of the achievable rate. Without static power nor interference, this region is the noise limited region in Figure 5. Static power and interference reduce this region of achievable tradeoff depending on the consumption and intererence models.

### System model

We consider an hexagonal planning network and we investigate the EE-SE tradeoff for one user in a cell for several frequency reuse factor K. The power consumption model is as described by (Eq 1) with N_{TRX}=1.

Assuming that interfering base stations form a continuum of interferer from the considered cell point of view, we adapt the fluid model introduce by Kelif *et al*. [5]. They proposed an expresssion of the interference to signal ratio (ISR) called f-parameter in their work, but which are accurate for a certain path loss exponent and a dense reuse factor (K=1). We can show that the SE-EE tradeoff in path loss condition can be expressed as:

(Eq 2)

with N_{TRX}, P_{0} and D_{P} given in (Eq 1), W the system bandwidth, c the path-loss coefficient, r the transmitter-receiver distance, the path-loss exponent, K the reuse factor and N_{0} the noise spectral density.

### Main results

Fig. 6 summarizes the EE-SE tradeoff for three reuse factors, i.e. K=1, 3 and 4 while Fig. 7 draws the evolution of the optimal point in the EE-SE tradeoff, i.e. maximal EE, according to the frequency reuse factor.

Fig. 6: SE-EE tradeoff labeled on several resuse factor K Fig. 7: Evolution of the EE-SE optimal for w.r.t. the reuse factor

We oberve in particular that for interference and circuit limited systems, i.e. for non-zero static power consumption, the EE=f(SE) behaves linearly and achieves an optimal point before decreases sharply. We remark also that an agressive reuse factor, as it is done today for 4G system, is non-optimal for the EE-SE tradeoff. The maximal EE point is obtained for K=3. The next step is to integrate the Shadowing into the study.

### References

[1] E. V. Belmega, S. Lasaulce and M. Debbah "A Survey on Energy-Efficient Communications", *In proc. IEEE PIMRC 2010*.

[2] V. Rodoplu and T. H. Meng "Bits-per-Joule Capacity of Energy-Limited Wireless Networks", *IEEE Transactions on Wireless Communications*, vol. 6, no. 3, 2007.

[3] W. Nie, X. Wang, F.-C. Zheng and W. Zhang, "Energy-Efficient Base Station Cooperation in Downlink Heterogeneous Cellular Networks" *In proc. Globecom 2014*.

[4], J.-M. Gorce, D. Tsilimantos, P. Ferrand, H. V. Poor, "Energy-Capacity Tradeoff Bounds in a Downlink Typical Cell", *In proc. IEEE PIMRC 2014*.

[5] J.-M. Kelif, M. Coupechoux and P. Godlewski "A Fluid Model for Performance Analysis in Cellular Networks", *EURASIP Journal on Wireless Communications and Networking*, volume 2010.

## Machine Learning for Green Cognitive Radio

- PhD Student: N. Modi
- Partner involved: IETR SCN, IETR SCEE

Nowadays, energy efficiency is critical concern for the design of new mobile system. Limited battery capacity and heat dissipation capabilities initiate a need of an energy efficient design of mobile system. Cognitive radio can be seen as a possible solution to these types of problem, because of self-adaptability feature of cognitive radio. The aim of the study is to merge some techniques deriving from reinforcement learning [1,2,3] and statistical decision making [4] in order to optimize the energy consumption in a cognitive unit. The reinforcement techniques, which are a recursive approach of trials and rewards, offer a promising solution for on-line learning and decision making. We use a model based on multi-armed bandit (MAB) algorithm coming from the machine learning domain in order to find vacant frequency bands for transmission. We address here the Opportunistic Spectrum Access (OSA) scenario, which is one of cognitive radio approach to mitigate the spectrum scarcity issue. In OSA context, secondary users (SU) use spectrum not used by primary users (PU) which have a license for that particular band, when PU let it unoccupied. The deal is that SU should never interfere with PU. OSA should not degrade primary network operation. Upper confidence bound (UCB) algorithms have been proposed a decade ago in order to solve the MAB asymptotically. We investigate the work of [1,2,3] on the classic MAB to the Markov MAB cases. We have shown that OSA scenario can be modeled as a MAB problem.

In this work, we consider Gilbert-Elliot channel model where goal of the reinforcement learning policy is to suggest the channel which has more probability to be vacant, from set of K channels to sense. Figure 8 shows the Gilbert-Elliot channels for the cognitive radio. Each channel has two states such as free or occupied. The goal of secondary users is the take benefit of one vacant channel at each time step. Therefore, SU must interfere as little as possible with PUs, in other words, it must sense at each iteration if there is a PU in the channel he wants to use before transmitting.

Figure 8: Channel Occupancy (Gilbert-Elliot channel model)

**Multi-Arm Bandit Problem**

Choosing between K channels in OSA is equivalent to choosing between K gambling machines of armed bandits! This is why OSA can be modeled as a multi-armed bandit problem. At the beginning of the game, we have no idea about which machine will make you win the more, in other words, we have no idea of which channel is more vacant. So the idea is to play all the machines and receive reward in order to learn the statistics of the all machines. The final goal is to maximize the obtained reward in the long run. At each time we have to decide which machine is the best for the next time step. Moreover, we also have to find an optimal machine which offers reward more often. In OSA scenario, each player is considered as a secondary user, each arm is a frequency band. The observed reward from playing an arm is referred as a state of channel free or occupied. Finally the action of a user is defined as sensing of a channel.

**Markov Multi-Arm Bandit Problem**

Markov MAB problem is more suitable for modeling OSA scenario. In this formulation state of the channel is modeled as a Markov chain with two states, i.e. free and occupied. Reward is assumed to be a function of the observed state of a channel at a certain time. The possible advantage of Markov MAB framework over the classical MAB is that here rewards are not assumed to be binary Bernoulli distributed, thus it allows to model reward as a function of channel condition. In our work, we consider both formulation of Markov MAB problem such as Rested MAB and Restless MAB.

**Rested MAB**: In rested formulation, only sensed channel changes the state and offers reward. All other passive arms remain frozen unless played. This problem is independent of the time elapsed between two consecutive actions of each arm [5]. In our first investigation, we applied rested bandit for the OSA scenario.

**Restless MAB**: Other case is the restless Markov MAB formulation where at each action step, all arms may change states and offer rewards as per the current state of those arms. Here, passive arms also change states therefore next observed states of passive arms are dependent on the time elapsed between two consecutive actions on those arm. In restless formulation, optimal policy is no longer staying with the one arm. For an optimal policy, it is required to switch among arms to obtain optimal reward based on the past observations [6].

**Main Results**

Figure 9 shows the cumulative regret achieved by playing an UCB1 policy for the OSA problem modeled as Markov MAB framework. Regret is the expected reward loss after n sensing due to the fact that the policy does not always sense the optimal channel. From the Figure 9, we can see that UCB1 policy achieves logarithmic order regret for the Markov MAB framework.

Figure 9: UCB1: cumulative regret for the rested Markov MAB

Whereas Figure 10 shows the optimal channel selection percentage and successful transmission percentage for the UCB1 policy. Optimal channel selection percentage is defined as a number of times given policy played an optimal channel from total number of time steps. We also introduce another performance analysis measure defined as a successful transmission percentage (STP) which is defined as a number of times vacant slot is detected from total time steps.

Figure 10: UCB1: Optimal arm selection and STP for the rested Markov MAB

A patent is currently pending on an improvement of UCB that we have realized but it cannot be detailed here not to divulgate the invention.

** References**

[1] Charles Clancy, Joe Hecker, and Erich Stuntebeck, “Applications of Machine Learning to Cognitive Radio Networks.” IEEE Wireless Communications Magazine, vol. 14, no.4, pp. 47-52, 2007.

[2] Jouini W, Moy C, Palicot J, "Decision making for cognitive radio equipment: analysis of the first 10 years of exploration", EURASIP Journal on Wireless Communications and Networking 2012.

[3] W. Jouini, "Contribution to Learning and Decision Making under Uncertainty for Cognitive Radio". PhD dissertation, SUPELEC, 2012.

[4] Bourbia S, Le Guennec D, Grati K, Gazel A, "Statistical Decision Making Method for Cognitive Radio", ICT 2012, 19th International Conference on Telecommunications, Jounieh, Lebanon, 23-25 April 2012

[5] Cem Tekin and Mingyan Liu. "Online algorithms for the multi-armed bandit problem with markovian rewards". In: Communication, Control, and Computing (Allerton), 2010 48th Annual Allerton Conference on. IEEE. 2010

[6] Haoyang Liu, Keqin Liu, and Qing Zhao. "Learning in a Changing World: Restless Multiarmed Bandit With Unknown Dynamics". In: IEEE Transactions on Information Theory 59.3 (2013).

## PAPR reduction

- PhD Students: A. Cheaito, M. Chafii
- Partner involved: IETR SCEE, IRISA METISS, IETR SCN

## 1. Jointed PAPR Optimization and Pre-Distrortion technics

Current communication systems are requesting high connectivity, reliable transmission in mobility and increasing spectral efficiency. LTE, WiMAX, WiFi, DVB and other communication systems today use Orthogonal Frequency Division Multiplexing (OFDM) which is considered as one of the key technologies able to fulfill all these demands. However, OFDM has one major drawback which is the large fluctuations of the signal magnitude commonly characterized by the so-called Peak-to-Average Power Ratio (PAPR). Indeed, signals with a high PAPR value may experience strong distortions in and out of the band during the passage through nonlinear components such as the Power Amplifier (PA). In the literature, two main approaches are usually advocated to solve the problems of the PAPR and the non-linearity of the PA. The first one is to carry out PAPR reduction methods consisting in reducing the dynamics of the signal by means of dedicated signal processing. Among the large variety of PAPR reduction algorithms, the most popular are clipping, coding, Selected Mapping (SLM) and Tone Reservation (TR). The second approach which can be used in a complementary way with the former is to make use of linearization techniques that tries to compensate for the non-linearity of the PA. On this aspect, different solutions are possible such as Digital Pre- Distortion (DPD) , Linear Amplification with Non-linear Component (LINC) method and Feedback. Hence, an efficient implementation of a communication system in which the PAPR of the signals plays an important role should embed a PAPR reduction technique followed by a linearization process. Such implementation can for example be found in high or medium power transmits stations using OFDM at the physical layer, e.g. DVB-T2 towers or LTE nodes. Today however, all these treatments remain static and do not take into account transmission conditions. A smart solution for future implementations would be to control the PAPR reduction and linearization stages in a flexible way according to some predefined parameters so that they become adaptive and self-configurable. These parameters are metrics widely used to measure the performance of the linearization and the PAPR reduction. Noise Power Ratio (NPR), Adjacent Channel Power Ratio (ACPR) and Error Vector Magnitude (EVM) are examples of these parameters. In particular, EVM is a common figure of merit for assessing the quality of digital modulated telecommunication signals. Assuming a transmitter implementation with adaptive PAPR reduction and linearization processes, one can imagine controlling the DPD and the PAPR reduction stages to meet a various EVM target values related to different qualities of service. In that perspective, we are interested in the analytical derivation of the EVM of an OFDM signal after nonlinear amplification when DPD is used or not. Although some upper-bounds of the second order moment of the predistortion error and of the EVM can be found in the literature, to the best of our knowledge, no analytical expression exists in the literature. Our main contribution then consists in providing such a result as a function of the transition factor of the power amplifier characteristics and of the accuracy factor of the DPD function.

### **System description **

Fig.11 represents the block diagram of a simplified transmission chain equipped with a DPD stage receding the power amplifier. We denote x_{1}(t) the multicarrier signal generated by the system, x_{2}(t) the pre-distorted signal and z(t) the amplified output signal. As our contribution focuses on the impact of the predistortion in the computation of the EVM expression, no particular PAPR reduction process is considered here. The model of power amplifier used is the Rapp model given by (1), and predistortion function corresponding to Rapp model is given by (2).where a and b are the predistortion and PA knee factor respectively and A is the amplitude of the saturation output voltage of the amplifier.

However, The EVM of the amplified signal Z_{k} is expressed as follows:

Where

represents the second order moment of the error magnitude and ** E{|X_{k}|^{2}} **is the average signal power. Note that

**is the expectation function.**

*E{.}*

Figure 11: Transmitter block diagram

### **Main results**

After the theoretical calculation, we have these two EVM expressions without and with DPD given by (5) and (6) respectively:

Eventually, we obtain EVM expressions in the form of a series expansion involving Gamma functions and depending on the knee factor b, the predistortion knee factor b when the DPD is activated and the saturation power A^{2} of the PA, as well as on the signal PAPR and the average power P_{r}.

Figure 12: Approximated and exact EVM with nonlinear amplification and without predistortion

Figure 13: Approximated and exact EVM with nonlinear amplification and with predistortion

Fig. 12 depicts the theoretical (exact and approximated) and simulated EVM as a function of the IBO, without predistortion. In practice, note that the IBO commonly applied to the input signal is always more than 10dB to mitigate distortions. We consider knee factors b of 1:5; 2 and 2:5, and an OFDM signal with PAPR = 10dB. The approximated EVM results are given for O = 4 and O = 6. From the curves, we can observe that our proposed equation of the EVM matches exactly the exact and the simulated EVM for only O = 4 and O = 6. Fig. 13 shows the theoretical (exact and approximated) and simulated EVM when predistortion is used, with various couples of values for a and b, and with O = 6 and O = 12. Once again, our theoretical analyzes match perfectly the exact and simulated one proving the consistency of the proposed approximated EVM expressions.

This very recent work is a first step in the analytical study of a fully adaptive transmitter model, being able to be flexibly controlled according to the transmission conditions such as the input power, the amplifier and predistortion characteristics and the PAPR. This study is a part of the global optimization approach of the transmitter efficiency and linearity, which could be very useful for example to the current deployment of the DVB-T2 transmitters.

## 2. Study of a New Multi-Carrier Waveform

Another way to reduce the fluctuations of the OFDM envelope is to act on the modulation scheme to construct a signal with low variations. The idea is to construct a new waveform that satisfies the constraint of a low PAPR. This idea is different from the other PAPR reduction techniques that are acting on the signal before and/or after the modulation while keeping the same modulation scheme.

To start our investigation, we consider a generalized system that we name Generalized Waveforms for Multi-Carrier (GWMC) system, which is a modulation system based on any modulation scheme. We model the waveforms with a family of functions {g_{m}} _{m}_{ ∈ [0, M-1]}, the index m stands for the carrier index and M is the number of carrier. The objective is to find the optimal modulation functions that gives, at the output of the modulation, a multi-carrier signal with low PAPR.

In our investigation, we follow the following steps:

- We consider the following two constraints:

The first constraint is that the temporal support of every function g_{m} has to be greater or equal than the GWMC symbol period. The second constraint is satisfied all the time in practice, it states that every function g_{m} should have a temporal decay.Under the previous constraints in Eq.(1) end Eq.(2), we derive a general expression of the PAPR (Eq.(3)), and we show that the PAPR depends indeed on the family modulation functions [1]. This conclusion is consistent with our objective, and proves that we can change the PAPR by changing the waveform. The general expression of the PAPR allows us to calculate the theoretical expression of the PAPR for any GWMC system. For more applications and discussions about this new general expression, the reader can refer to [2].

- Based on the previous PAPR expression, we model the PAPR reduction problem as a constrained optimization problem as expressed in Eq.(4). We optimize on the family of modulation functions {g
_{m}}_{m}_{ ∈ [0, M-1]}, the PAPR performance under two constraints on the waveforms [3].

- In [4], we give a solution of the optimization problem. We conclude that, for all the modulation schemes that satisfy the previous constraints, the OFDM has an optimal PAPR performance. At the best of our knowledge, this is the first work that gives an analytical proof of the optimality of the OFDM in PAPR performance. In addition to that, the study identifies a large family of mutli-carrier systems which are optimum in terms of the PAPR performance, and thus have the same PAPR performance than the OFDM. It is true that we were hoping to find a new waveform better than OFDM as a solution of the optimization problem, but the result that we find is still important in the sense that he limited the optimality of the OFDM to the previous constraints and clarify why in the literature [5] [6], other MCM systems have the same or worse PAPR performance than OFDM. In addition to that, this result gives an idea about getting a better PAPR than OFDM, which is actually to release the first constraint.
- The OFDM is optimal in terms of PAPR performance, when the previous two constraints hold. But if the family {g
_{m}}_{m}_{ ∈ [0, M-1]}, does not satisfy the first condition, it is possible to find a better PAPR performance than OFDM. That means that if there exists at least an index m_{0}such that the waveform g_{m0}has a temporal support less than the symbol period, and that means that its amplitude vanishes at least in time interval, then we can get a better PAPR performance than OFDM. The example in the literature is the Wavelet OFDM using the Inverse Discrete Wavelet Transform based on Haar wavelet for modulation, the PAPR is reduced by 2db [7]. The perspective of the work is to study the impact of releasing the first constraint on the system, and then acting on the temporal support of the function to construct a new waveform with low PAPR.

**Publications**

[1] M. Chafii, J. Palicot, and R. Gribonval, “Closed-form approximations of the Peak-to-Average Power Ratio for Multi-Carrier Modulation systems”, Eusipco, Lisbon, Portugal, 2014.

[2] M. Chafii, J. Palicot, and R. Gribonval, “ A PAPR Upper Bound of Generalized Waveforms for Multi-Carrier Modulation systems, ISCCSP, 6^{th} International Symposium on Communications, Control and Signal Processing, Athens, Greece 2014.

[3] M. Chafii, J. Palicot, and R. Gribonval, “Closed-form approximations of the Peak-to-Average Power Ratio for Multi-Carrier Modulation systems and their applications”, EURASIP Journal on Advances in Signal Processing, July 2014.

[4] M. Chafii, J. Palicot, and R. Gribonval, “Is OFDM an optimum Multi-Carrier Modulation system in terms of PAPR performance?” to be submitted.

**References**

[5] A. Skrzypczak, P. Siohan, and J. P. Javaudin,”Peak-to-Average Power Ratio Issues for Pulse-Shaped Multicarrier Modulations,” in Advances on processing for multiple carrier schemes: OFDM and OFDMA, Faouzi Bader and Nizar Zorba, pp. 43-90. Nova Science Publishers, Inc., 2011.

[6] A. Kliks, “New Transmission and Reception Techniques of the Generalized Multicarrier Signals, Ph.D. thesis, Poznan University of Technology, 2011.

[7] J. Zakaria, M. Salleh, “Wavelet-based OFDM Analysis: BER Performance and PAPR Profile for Various Wavelets”, IEEE Symposium on Industrial Electronics and Applications, September 2012, Bandung, Indonesia.