Mabrouka El Guedri
email: mabrouka.elguedri@supelec.fr

Julien Bect

Christian Lajaunie
email: christian.lajaunie@ensmp.fr

Gilles Fleury
email: gilles.fleury@supelec.fr

Rédouane Seraoui

Thomas Garcia

Alexandre Girard

RJMCMC POINT PROCESS SAMPLER FOR SINGLE SENSOR SOURCE SEPARATION: AN APPLICATION TO ELECTRIC LOAD MONITORING

This paper presents an original method to separate the residential electric load into its major components. The method is explained in the particular case of space-heating, which is the most consuming electric end-use in France 1 . This is a source separation problem from a single mixture. The components to be retrieved are square signals characterized by a periodic regulation and a slowly timevarying duty cycles. A point process is used to model the electric load as a configuration of possibly overlapping square signals, given the priors on magnitude, duty cycle variations and the regulation periodicity. This stochastic process is simulated using a Reversible Jump Markov Chain Monte Carlo procedure. A simulated annealing scheme is used to achieve the posterior density maximization. First results on real data provided by Electricité de France are quite encouraging.

INTRODUCTION 1.Background on household electric load monitoring

The electric power industry and consumers recently face many challenges such as energy saving, market deregulation, power quality and greenhouse gas emissions reducing. Accurate and reliable information about the nature and the state of the electric systems will undoubtedly be helpful to meet these challenges. Actually, a good knowledge of the electric load and the targeted appliances help consumers understanding their bills and better control their consumption. It also provides utilities with detailed usage profiles of their customers, which is an efficient means to help in levelling peaks load and in planning future capacity.

A non-intrusive and economical solution may rely on information extracted from electric consumption measured at a centralized easily accessible part of a distribution network, namely the electricity meter.

1 70% of the whole households electric consumption in France (ADEME, 2005).

Non-intrusive electric load monitoring has been subject to several approaches over the last twenty years. General overviews can be found in [START_REF] Hart | Residential energy monitoring and computerized surveillance via utility power flows[END_REF][START_REF] Laughman | Power Signature Analysis[END_REF][START_REF] Leeb | A conjoint pattern recognition approach to nonintrusive load monitoring[END_REF]. The available solutions require measurements of the active and the reactive power, which carry out the finger-prints of the electric appliances. They are mostly made up of three steps. Event detection determines the appliances operating schedule. Load identification uses steady state powers and transient patterns, if available, to recognize the elementary components.

Energy estimation provides a breakdown of the daily energy into the major end-uses. As the structure of the electric load is very complex because of the diversity of electrical loads and of the consumers' habits, classical methods requires measurements of three voltages and currents with a specific device plugged in the electricity meter. We propose a novel approach using only the active power.

The measured power is a linear additive mixture of an unknown number of elementary signals. Our aim is to provide the most likely decomposition of the daily active power without any intrusion. We propose to incorporate knowledge on the primitives to be extracted as a priori knowledge. Our approach allows going further.

Problem statement

In this paper, we focus on the space-heating load decomposition. The observed signal is the active power .)ݐ(ݕ It is sampled at the sampling rate T e =2s. This observation is a sum of an unknown number K of periodic square waves ݕ (1 ≤ ݇ ≤ .)ܭ Each component is described by its period ܶ and its magnitude ܣ .

The observation is modelled as follows (equation1):

,ݐ∀ )ݐ(ݕ = ∑ ݕ )ݐ( + )ݐ(ܾ ୀଵ (1)
where b is an additive Gaussian noise. Each k th convector is defined on a compact support ሾݐ , ݐ ଵ ሿ. A convector period ܶ is almost constant and is defined as follows (equation 2)

ܶ = ܶ + ߝ (2)
where ܶ 00 ∈ ሼ40, 80ሽ is the theoretic periodicity and ߝ is an additive noise, modelled with a zero mean Gaussian law.

An example of a convector signal is given in Figure 1.

Notice that the width of the observed squares l varies over time. Time-variations of these parameters might be analyzed through the variations of the duty-cycle defined as follows ߩ = ݈ ܶ ൗ (Figure 1, where n represents the index of the squares).

The repartition of the duty-cycle variations (absolute values) obtained for a given convector is illustrated in Figure 2. The variation between two successive squares is constant. We propose to use a first autoregressive model to this parameter. Two samples of space-heating load are presented in Figure 3, Figure 4 and Figure 5. In the first case, two components are operating simultaneously. Even if the number of components is very small, the global load cannot be easily decomposed, especially because of the convector saturation ߩ = 1 (Figure 3, double arrow).

Another source of complexity of this problem is the interaction between the elementary components. The start up of a square and the shutdown of another one might occur at the same time. Real data show that this synchronism between convectors is realistic and occurs frequently (Figure 4, Figure 5). Moreover, magnitudes and periodicities of different components might have the same value, which makes the source separation problem quite difficult. We aim at extracting a plausible configuration of periodic square waves given the global consumption and priors. The electric load might be seen as a realization of a marked point process [START_REF] Van Lieshout | Point Processes and their applications[END_REF] of squares defined by a density function to be designed given priors on the duty cycle variations, the periodicity and the magnitude of a convector. The stochastic process is sampled using a Reversible Jump Markov Chain Monte Carlo (RJ-MCMC) sampler. A simulated annealing algorithm [START_REF] Kirkpatrick | Optimizing by simulated annealing[END_REF] achieves the posterior density maximization: an estimation of the model parameters in a Bayesian framework is performed this way. Some definitions and notations are presented is section 2.

In section 3, the estimation problem is presented in a Bayesian framework. The optimization algorithm and some details on the proposition kernels introduced in the MCMC sampler are described in section 4. Finally, first results on real data and future works are given in section 5.

POINT PROCESS FRAMEWORK FOR ELECTRIC LOAD MONITORING

Point processes models have been introduced for object extraction from remotely sensed images [START_REF] Baddeley | Stochastic geometry models in high-level vision[END_REF]. Marked point process models provide a natural setup for the inclusion of prior knowledge on the components of a given observation: parsimony, patterns structure and/or morphology, etc. These models were widely used in image processing [START_REF] Baddeley | Stochastic geometry models in high-level vision[END_REF][START_REF] Rue | Bayesian object identification[END_REF] to extract complex objects in a scene. They have recently been used in roads [START_REF] Stoica | A Gibbs point process for road extraction from remotely sensed images[END_REF], buildings [START_REF] Ortner | Building outline extraction from Digital Elevation Models using marked point processes[END_REF] or tree crowns [START_REF] Perrin | Tree crown extraction using marked point processes[END_REF] extraction. We refer to [START_REF] Baddeley | Stochastic geometry models in high-level vision[END_REF][START_REF] Descombes | Marked Point Process inImage Analysis: from Context to Geometry[END_REF] for mathematic definitions and more details about point processes.

In this work, we are interested in the case study of spaceheating electric load. The components to be extracted are the convectors, which might be considered as a set of squares. Notice that the convectors may overlap. Point processes models are adapted in the case of onedimensional signals where the objects interact with each other and may overlap. More precisely, the electric load decomposition problem requires to introduce two stochastic processes: a point process where the points are the convectors and a marked point process where the points are the squares and the marks are the square parameters. The proposed model is explained in the following subsections.

The state space

The observed signal P is defined on ܶ = ሾ0, ܶ ௫ ሿ. ܧ = ሼ ⊂ ,ܥ 1 ≤ || < +∞ሽ. Each object x of E is a finite set of convectors. The electric load components estimation is a large combinatorial problem. It may be solved by minimizing an energy on the state space of convectors. Efficient algorithm helping the state-space exploration are required. In the next sub-section, two point processes are used to model the squares and the electric load of convectors.

Point processes for electric load modelling

The electric load model is obtained through the definition of a stochastic process that describes the convectors and another one that describes the squares. Let ߤ be the probability distribution of a Poisson point process and ߥ be the intensity measure of this process defined on C. Consider a point process defined on C, i.e. an E-valued random variable. Each realization of X is a set of convectors. Given the reference probability distribution, a realization of X could be obtained in two steps. Given this distribution, one defines a Poisson process of probability distribution ߤ on S. Let us consider a marked point process where the points belong to ܶ = ሾ0, ܶ ௫ ሿ and the marks are stochastic parameters that belong ܯ = ሾ݈ , ݈ ௫ ሿ × ሾܣ , ܣ ௫ ሿ. A realization of this process is a finite set of squares (points in S), in other words a point in C. The observation )ݐ(ݕ (the space-heating active power) is a realization of a point process X defined by its density ݂(. ) with respect to the Poisson process probability distribution ߤ. One might then build a Markov chain which converges in law to X. The process density satisfies the following equation

݂() = ଵ exp (-ܷ()), (4
) where Z is a normalizing constant and ܷ() stands for the energy of the configuration .

APPLICATION TO ELECTRIC SPACE-HEATING DECOMPOSITION

Our aim is to estimate the configuration that maximizes the posterior density )ݕ|(݂ = ݂() given the active power Y. In a Bayesian framework, we need to build both an a priori density and a likelihood term.

Posterior density

As mentioned above, in a Bayesian framework [START_REF] Droesbeke | Méthodes bayésiennes en statistiques[END_REF], the marked process density is given as follows ݂() ∝ ݂ () ݂ ௗ௧ ,)|ݕ( (5) where the a priori density ݂ () includes priors on the parameters to be estimated, and the likelihood ݂ ௗ௧ )|ݕ( of a configuration given the observed signal. The energy term is split into a likelihood term and a prior (regularization) term as follows:

ܷ() = ܷ () + ܷ ௗ௧ () The relationship between energy terms and density functions is given by equations 6.

൜ ݂ () ∝ exp (-ܷ ))ݔ( ݂ ௗ௧ (ܲ |) ∝ exp (-ܷ ௗ௧ ()) (6)
Prior energy The regularization energy is defined by the following equation:

ܷ () = ∑ (ܷ (ܿ) - ∈ ߶ (ܿ)), (7)
where

ܷ (ܿ) = ߙ ߰ (ܿ) + ߙ ் ߰ ் (ܿ) + ߙ ఘ ߰ ఘ (ܿ) is a
linear mixture of three regularization terms respectively on the magnitude A, the periodicity T 0 and the duty cycle ρ.

Given real data analysis results, these parameters are modelled with Gaussian laws. Let ܶ ത and ܣ ҧ be respectively the period and the magnitude means of the convector c and ߪ ் , ߪ and ߪ ఘ the standard deviations. The functions ψ ୮ are defined as following:

ە ۖ ۔ ۖ ۓ ߰ (ܿ) = ∑ ((௨)ିҧ ) మ ଶఙ ಲ మ ௨∈ ߰ ் (ܿ) = ∑ (்(௨)ି் ത) మ ଶఙ మ ௨∈ ߰ ఘ (ܿ) = ∑ (ఘ(ାଵ)/ఘ()ିଵ) మ ଶఙ ഐ మ
The function ߶ is introduced to maximize the number of squares per convector and penalize great number of onesquare convectors. The chosen function is defined as below:

߶(ܿ) = ൜ ߜ if |ܿ| = 1 -ߚ|ܿ| ଶ otherwise
where ߜ ܽ݊݀ ߚ are pre-defined parameters.

Likelihood energy

This term measures the probability of data given a configuration of objects. Given the chosen model for spaceheating load, we propose to define this energy by the sum of each likelihood objects:

ܷ ௗ௧ () = ∑ ∑ ܷ )ݑ( ௨∈ ∈ , (8)
The energy term U l is designed as following:

The energy of a plausible square is negative. Otherwise, the energy of the proposed square is a positive convex function.

The gradient (on and off events) of a plausible square corresponds to a gradient of the observation in a pre-defined neighbourhood. The proposed likelihood energy definition requires a preprocessing of the electric load. In fact, the events (start up and trip) of the whole electric load have to be matched.

RJMCMC sampler and optimization algorithm

The space-heating load is modelled by a point process X defined by its density ݂() with respect to the Poisson process of intensity ߥ. A Markov chain (X m ) that converges to the distribution X Ρ is built using a RJ-MCMC samplers [START_REF] Green | Reversible Jump Monte Carlo Markov Chain computation and Bayesian model determination[END_REF].

In this work, some jumping kernels (birth and death of a square) and non-jumping kernels (dilatation of the magnitude or of the width of a given square, translation) are used to manage the chain transitions have been defined. The maximum a posterior, given by equation 9, ெ = ݔܽ݉݃ݎܽ ݂()) (9) is obtained using a simulated annealing algorithm [START_REF] Kirkpatrick | Optimizing by simulated annealing[END_REF].

RESULTS ON REAL DATA

The proposed model has been tested on real data gathered at two customers houses. In each case, the active powers of the elementary components have been also measured to evaluate the algorithms. Tests were carried on synthetic mixtures to evaluate the proposed model. Figure 7 presents results obtained for two convectors having nearly different magnitude and the same periodicity of regulation. Their duty-cycles are also very close. This signal is decomposed into two components (Figure 4blue) where the periodicity is well estimated. The squares widths are quite well estimated. Other tests have been carried out during this study taking into account convector saturation and several objects interactions. The obtained results, which are quite satisfying, and more details on the convergence of the MCMC sampler will be given in future works. We notice that some verifications have been carried out to ensure the convergence of the implemented sampler.

CONCLUSIONS AND FUTURE WORKS

The electric load decomposition given the active power sampled at one second is a single source separation problem. The load components are some square waves characterized by some priors. In this work, a novel method to estimate these components using the marked point processes framework is proposed. These models were widely used in image processing and in model selection. In our application, such processes provide a unique model to any electric appliance avoiding a constrained approach based on a specific model per class of electric appliances. In this paper, the proposed model in the case of spaceheating load is described and the posterior density is defined. The RJMCMC sampler embedded with a simulated annealing scheme is implemented using some transition kernels. First results on real data are satisfying and show that the model is suitable to deal with objects interactions (events in our application). Some improvements could be proposed, such the hyperparameters of the prior energy estimation. In future works, more details on the implemented sampler and on the optimization algorithm will be detailed. The proposed model will be extended to other electric appliances.

Figure 1 -Figure 2 -Figure 3 -Figure 4 -

 1234 Figure 1-Example of a convector electric load (T 00 =40s, A=1150 W)

Figure 5 -Figure 6 -

 56 Figure 5-Electric load of three convectors operating simultaneously

  It is the sum of the convectors active powers. Each convector might be considered as a set of squares. Each square u of a given convector c is entirely defined by three parameters (t, l, A) which are respectively the beginning time, the square width and the square magnitude. Theses parameters belong to ܯ = ሾ݈ , ݈ ௫ ሿ × ሾܣ , ܣ ௫ ሿ. The parameters min l and max l stand respectively for the minimal and the maximal widths of any square. The magnitudes min A and max A are respectively the minimal and the maximal nominal powers of any convector. These parameters are chosen based on information extracted from real data. The model of squares is illustrated in Figure 6. Let ܵ = ܶ × ܯ be the state space of squares. Notice that the state space of squares is a subset of ℝ ଷ . A configuration c of objects in S is an unordered list of objects in S, which can be written as = ሼݑ ଵ , ݑ ଶ , … , ݑ ሽ, where ݊ ≥ 1 and ∀ ݅, ݑ ∈ ܵ. Let ܥ be the set of all n-squares configurations , i.e. ܥ = ൛ | = ሼݑ ଵ , ݑ ଶ , … , ݑ ሽൟ. (3) The set of all the configurations is ܥ = ڂ ܥ ஶ ୀଵ . The state space of possible solutions is defined as follows:

  Firstly, one draws the convectors number n from a Poisson distribution  = e ିఔ() ఔ() ୬! . Secondly, n independent convectors are generated with respect to the distribution ఔ ఔ() .

Figure 7 -

 7 Figure 7-Decomposition of two convectors operating simultaneously ( s T T 40 02 01 = = ): the measured signals (blue), the estimates (red and cyan).


Nicolas Picard
email: nicolaspicard<nicolas.picard@unilim.fr

Does tacrolimus, in comparison with sirolimus, increase mycophenolic acid exposure in kidney transplant recipients?

Keywords: Animals, Clinical Trials as Topic, methods, standards, Drug Interactions, Humans, Immunosuppressive Agents, pharmacokinetics, Kidney Transplantation, methods, Mycophenolic Acid, analogs & derivatives, pharmacokinetics, Sirolimus, pharmacology, Tacrolimus, pharmacology

Does tacrolimus, in comparison with sirolimus, increase mycophenolic acid exposure in kidney transplant recipients?

Nicolas Picard

Table 1

Comparison of mycophenolic acid pharmacokinetics associated with sirolimus and tacrolimus comedication in renal transplant recipients at 3 to 6 months post transplant.

Dose normalized parameter

  pharmacokinetic parameters are shown as mean values SD. AUC, area under the curve; C , peak plasma concentration; C , trough plasma concentration; MPA, mycophenolic acid. Details on ± max min patient characteristics given in ( ).

Author Keywords TACROLIMUS

To the Editor

Based on their study, "Increased mycophenolic acid exposure in stable kidney transplant recipients on tacrolimus as compared with , Braun et al. concluded that mycophenolic acid (MPA) exposure is greater in those on sirolimus: implication for pharmacokinetics " tacrolimus treated patients than in those treated with sirolimus during maintenance immunosuppression after kidney transplant and suggested that it may reflect an intestinal interaction of tacrolimus and MPA ( ). 1

Knowledge of whether MPA exposure is differentially modified by tacrolimus and sirolimus is important because of the current clinical trend of substituting calcineurin inhibitors for mTOR inhibitors and of the growing evidence that MPA exposure should be controlled to optimize therapy.

However, given the interindividual pharmacokinetic variability of MPA, we feel that the number of patients studied by Braun et al. (n= 9 on sirolimus 14 on tacrolimus) was too low to rigorously demonstrate a pharmacokinetic interaction. We recently conducted a vs pharmacogenetic/pharmacokinetic study of MPA in a larger group of kidney transplant recipients and found no difference of MPA and MPA-glucuronide dose-normalized exposures between patients on sirolimus (n 42) and tacrolimus (n 28), with similar renal function ( )

(

). Table 1 Braun et al. suggested that MPA was better absorbed in the gastrointestinal tract when associated with tacrolimus than with sirolimus and argued that recent clinical findings from Mino et al. ( ) and experimental data involving an ex-vivo swine intestinal perfusion model ( 3) supported their hypothesis. In fact, in the perfusion model tacrolimus resulted in higher amounts of MPA-glucuronide with no reported 4 change in MPA, which rather suggests that tacrolimus decreases MPA oral bioavailability. Conversely, a well-designed drug interaction study in rats showed that MPA concentrations during monotherapy are not different from concentrations detected during tacrolimus coadministration, which is not in favour of an interaction ( ). On the other hand, the study from Milo et al. compared MPA 5 pharmacokinetics between kidney transplant recipients and patients with lupus nephritis receiving no calcineurin inhibitors but high doses of glucocorticoids, which makes any extrapolation to other situations very hazardous. The alternative hypothesis of unknown ' '

protein-protein interaction raised by Braun et al. seems very speculative.

The conclusions drawn by Braun et al. about the interaction between mycophenolate and tacrolimus appear thus to be conflicting with experimental findings as well as with observations in a larger population. However, the definitive proof can only be brought by a randomized cross-over clinical pharmacokinetics study in homogenous groups of transplant recipients.


Leonardo S Cardoso

Raul De Lacerda

Pierre Jallon

Mérouane Debbah

SDR4all: a Tool for Making Flexible Radio a Reality

Keywords: SDR4all, flexible radio, SDR, OFDM 1 Motivation

In this contribution, we describe the potential of SDR4all (Software Defined Radio for all) to solve the bottlenecks and reduce the innovation cycle related to the design and implementation of flexible radio algorithms. SDR4all is a programmable software tool with radio cards for wireless researchers, students and engineers. It enables to implement in software any wireless scheme between two laptops. Using an flexible orthogonal frequency division multiplexing (OFDM) based implementation example, we describe the tool and show the performance of the transmission on a real wireless channel at 2.4 Ghz ISM band.

One of the problems that make the conception to development cycle so slow in research and development (R&D), particularly in the telecommunications field, is the fact that there is a big gap between the theory and the practice. Telecommunication theorists usually ignore critical aspects of the implementation of radios that are usually overlooked in telecommunication courses. Bridging the gap between theory and practice can reduce the time-tomarket of ideas.

Matlab R has been proactive in this sense, offering tools and means to shorten the development cycle and allowing engineers to test their conceptions right inside their interface. However, in the area of telecommunications, not many options fill this need. While solutions do exist, they are generally hard to use and/or expensive. Existing projects and platforms (such as [1, 2, 3, 4]) aim at a building a configurable radio platform for the implementation and test of new technologies. In spite of their impressive capabilities, they are generally restricted to the use by professionals due to complicated programming languages (C, Assembly, VHDL, etc...), expensive hardware and testing equipment, lack of general knowhow in hardware devel-opment and radio frequency (RF) circuitry and to the fact that porting high level code to existing RF platforms is very time consuming. The need by telecommunication researchers/engineers/students for an easy to use tool for simple algorithm testing are left uncovered. Briefly, their needs are:

• Stress-test ideas with realistic constraints;

• Use real "models" of channels;

• Provide a proof of concept of an algorithm;

• Become acquainted with the issues related to radio systems;

• Create a practical view of the problems targeted;

• Verify the validity of the initial problem assumptions;

• Analyze the feasibility of the algorithms.

In this contribution, we propose SDR4all [5] as a solution to fill this gap. The SDR4all is based on the concept of software defined radio (SDR) [START_REF] Mitola | Cognitive radio: making software radios more personal[END_REF][START_REF] Mitola | Cognitive radio: An integrated agent architecture for software defined radio[END_REF][START_REF] Haykin | Cognitive radio: brain-empowered wireless communications[END_REF] to enable the implementation of algorithms that will be tested with real transmissions using actual hardware transceivers. The idea of this testbed is to enable students/researcher to test their ideas and algorithms on real transmissions while still keeping the simplicity of a high level programming language environment.

In the remainder of this work we will present the evaluation tool in detail in section 2 . We will then present an example transmission/reception chain and show its characteristics and features in section 3 as a means to showcase the capabilities of SDR4all. We then discuss about its metrics and results in section 4. We conclude this contribution in section 5

SDR Evaluation Tool

SDR4all is composed of a USB plug and play hardware part, and a software part. The hardware part is in charge of the RF and sampling processing while the radio transceiver's physical layer (PHY) is software driven, running in a computer.

In order to give a practical example, the hardware of the testbed was based on the USRP version 1 cards [9]. However, SDR4all can be used on other types of radio cards (such as the ones being developed by SDR4all [5]). These cards are divided into two parts: a mother-board and one or two daughter-boards. The mother-board is responsible for the RF control, communication over the USB link, analog-to-digital/digital-to-analog conversions and sampling. The daughter-board is responsible for the RF circuitry, including filters, amplifiers and oscillators. Note that the USRP version 1 accepts a wide range of daughterboards, available to allow testing in multiple bands [9].

As of now, SDR4all supports the RF circuitry provided by the RFX2400 daughter-board. The 2.4 GHz ISM band was chosen since it shares the same characteristics with the widely popular 802.11(b/g), bluetooth and WiMAX systems. Along with the RFX2400 daughterboard, dual VERT2450 antennas (TX and RX, each) are adopted. These are standard isotropic antennas made for the 2.4 and 5 GHz ISM band. The main parameters for the mother-board and RF circuitry are provided in As previously stated, the PHY layer is implemented in software, part of a Matlab R toolbox developed specifically to this end. In order for the toolbox to correctly communicate with the cards, a driver was built. At the moment, the driver enables non real-time communication between only a card and a computer, but multi-card, real-

USRP USRP USB link USB link driver driver Matlab R Matlab R OFDM TX OFDM RX Laptop Laptop
Figure 1: SDR4all overall layout time operation is currently under development. It is able to transmit a given vector of baseband I/Q symbols and to listen for a certain amount of time and provide a vector of the received baseband I/Q samples. The driver also allows to configure the bandwidth and the center frequency for both transmission and reception. The toolbox implements basic communication blocks, such as bit operations, modulation, packet formatting and so on. Full communication chains are also implemented. The overall schematics of SDR4all, for an OFDM chain, can be seen in figure 1.

OFDM Radio Chain

The demonstration is based on the OFDM transmission chain scheme currently available in the toolbox. The OFDM transmission chain is able to transmit variable size frames, comprised of two parts as seen in figure 2: a preamble and a payload. A data bearing mode and a channel sounder mode can be configured and will be further detailed. The preamble of size n pre symbols is transmitted as binary phase shift keying (BPSK) ones (1s) in time. It is used to assure correct detection of the frame as a whole and also to correctly detect phase offset variations due to imperfections in the phase lock loop (PLL), responsible for generating the carrier frequency at the chosen central frequency f c . Consider a received vector of complex samples s of size N. where | • | is the norm operator, s k is a complex baseband received sample at instant k ∈ {1, . . . , N}. For the choice of the window size M, we have picked the frame size. The frame is then decided to start at index

k ⋆ = argmax n∈{1,...,N-M} (e n ).
Although, maybe not the best frame detection technique, the energy detector is simple, computationally fast and provides rather robust results.

Once the frame is detected, the phase offset estimation takes part. To estimate the phase offset, we make use of the fact that the preamble is composed of same-phase symbols. Phase offset is estimated by

φ = 1 n pre k ⋆ +n pre -1 ∑ m=k ⋆ (∠[s m+1 ] -∠[s m ]) ,
where ∠(.) is the phase of a given complex value. Note that each subsequent phase offset calculation is averaged out to compensate for phase noise. The authors are aware that other techniques better adapted to the same task exist, but this one was chosen due to its simplicity. After the preamble, a silence gap of the same size of the preamble is inserted in order to allow for the estimation of the noise power. The phase offset of the whole frame is later corrected by

s ⋆ m = s m e i φ(m-k ⋆ ) , ∀ m ∈ {k ⋆ , . . . , k ⋆ + M}.
Still in the structure of the frame, a silent space is inserted between the preamble and the payload. This silence space is used for noise energy estimation for signal-tonoise ratio calculations.

As stated before, two OFDM modes can be used, each of which will configure the payload part of the frame differently. In the following we provide details on both modes.

Data Bearing Mode

A data bearing mode can be used to transmit useful data. The frame structure for the data bearing mode is shown in figure 3. It is capable of carrying packets of different sizes. The payload is composed of pilot and data blocks sandwiched into the discrete Fourier transform (DFT)modulated part of the frame. The fast Fourier transform (FFT) size of the packet is of n FFT , of which n zpad carriers are reserved for zero-pad purposes, leaving a total of n c = n FFTn zpad carriers for useful transmission. The payload part of s is reorganized into an OFDM symbol matrix R of size n c × M/n FFT given by rearranging s into a matrix of n FFT × M/n FFT and then discarding the middle n zpad rows. The R matrix is finally divided into a series of pilot R P n c ×n pilot and data R D n c ×n data sub-matrices.

The reference pilot block P is a randomly-generated symbols sequence, shared by both the transmitter and receiver. The pilots can be either modulated with a quadrature phase shift keying (QPSK) or be complex circularly symmetric. They are used to estimate the channel for subsequent equalization of the data symbols. The channel is estimated by

ĥi = 1 n pilot n pilot ∑ j=1 R P i, j P i, j , ∀ i ∈ {1, . . . , n c }, (1)
where the averaging is performed to smooth out the noise imperfections.

The data symbols are modulated with a 16-quadrature amplitude modulation (16-QAM) and transmitted. The sequence "pilot-data" is repeated until all the data to be transmitted is inside the payload. At reception the data is equalized by

Di, j = R D i, j ĥi , ∀ [i ∈ {1, . . . , n c }, j ∈ {1, . . . , n data }] .
Each data block is equalized by the channel estimation from its preceding pilot block. The parameters for the data bearing mode are summarized in table 2

Channel Sounder Mode

The OFDM chain also has a channel sounder mode, which is used to provide channel analysis. The frame structure for the channel sounder mode is shown in figure 4. In this mode, the payload is composed of pilots only. Similar to the data bearing mode, the FFT size of the packet is of n FFT , of which n zpad carriers are reserved for zero-pad purposes, leaving a total os N carriers of useful symbols for channel estimation. A total of n pilot OFDM symbols are transmitted in each pilot block. Again, these pilots symbols can be either modulated with QPSK or be complex circularly symmetric, known by both the transmitter and receiver. Channel estimation is carried out as in (1).

The parameters for the data bearing mode are summarized in table 3

Reception Trigger

Due to its non real-time constraint, the receiver can not decode the transmitted stream on the fly to detect the signal. That would require much more processing power and memory than Matlab R can cope with at the base-band rate. Thus, a triggering system was implemented to initiate the reception, to minimize the number of samples processed so that the detection of the packet can take place. To that end, the TCP/IP network was adopted as a triggering bearer. In the absence of a wired network, the wireless network can be used. In the case of a frequency superposition, the transmission/reception takes places one full second after the trigger has been sent to guarantee that the wireless network will not interfere with the toolbox packet transmission.

Metrics and Outputs

One of the best advantages of SDR4all relies on the fact that it processes actual base-band symbols. This allows for the creation of well tailored kinds of results and metrics, useful for either demonstrating the inner workings of the physical layer or for assessing the performance of an algorithm. In particular, this flexibility can be very useful for teaching and research scenarios.

As an example, we showcase the results provided by the OFDM chain. In addition to bit error analysis, symbol error analysis and signal to noise ratio (SNR), the OFDM chain also outputs the constellation of the data (shown in figure 5), estimated channels (in figure 6) and can also present the output of the transmitted data, in our case an image, as shown in figure 7. With access to the transmission/reception chain code, the user can also implement new metrics that suits his needs.

Conclusion

In this work, we have presented a new and easy to use SDR-based tool called SDR4all, able to empower telecommunications teachers and researchers towards the common goal of creation of state-of-the-art algorithms and techniques. Then we have shown how SDR4all is composed, providing details on its hardware and software structure. We have also provided an example of the capabilities of SDR4all, based on a OFDM-based transmission chain, highlighting its structure, configurability and available metrics and results. SDR4all is already being employed in universities and research labs and has proven to live up to its demands.

Figure 2 :

 2 Figure 2: General OFDM transmission chain frame configuration.

Figure 3 :

 3 Figure 3: OFDM frame structure for data bearing mode

Figure 4 :

 4 Figure 4: OFDM frame structure for channel sounder mode

Figure 7 :

 7 Figure 5: Constellation of the received symbols

table 1

 1

	[9].
	parameter	value
	operating band	ISM 2.4 ∼ 2.49 GHz
	base-band filtering	20 MHz
	channels	1 to 13 (802.11)
	total TX power	20 mW
	signal bandwidth	up to 16 MHz

Table 1 :

 1 Parameters for the hardware part.

Table 2 :

 2 . All parameters therein are selectable. Parameters for the data bearing mode.

	parameter	value
	f c	flexible inside the ISM band
	base band rate	1 MHz
	n pre	1000 syms
	n FFT	64
	n zpad	16 (8 at each end)
	N	48 carriers
	n pilots	30
	n data	30
	pilot structure	QPSK or circularly symmetric
	data modulation	16-QAM

Table 3 :

 3 . Again, all parameters therein are selectable. Parameters for the channel sounder mode.

	parameter	value
	f c	flexible inside the ISM band
	base band rate	1 MHz
	n pre	1000 syms
	n FFT	64
	n zpad	16 (8 at each end)
	N	48 carriers
	n pilots	30
	pilot structure QPSK or circularly symmetric

Acknowledgements

The authors would like to thank Digiteo for its support.


Gaoning He
email: gaoning.he@gmail.com

Mérouane Debbah
email: merouane.debbah@supelec.fr

Eitan Altman
email: altman@sophia.inria.fr

Game-Theoretic Techniques for Intelligent Wireless Networks

To cope with the fast development of decentralized wireless communications, game theory has been considered as a necessary and powerful mathematical tool to study the competition and cooperation between future intelligent wireless devices. This paper surveys the state-of-the-art of gametheoretical tools applied to wireless communications, focusing mainly on the analysis of wireless resource allocation problems.

INTRODUCTION

It has been nearly one and half century since the invention of wireless transmission using electromagnetic spectrum. From the early use of spark-gap telegraphy (1888, H. Hertz) to the first clear transmission of human speech in air (1919) and to the invention of frequency modulation (FM) until the early form of mobile phone cellular networks (1947, Bell Labs), wireless technologies and applications have been greatly changed. However, interference is still a Pandora's box in the design and analysis of today's multi-user wireless systems.

In wireless communications, multiple access transmission turns out to be a very common scenario, because (unlike the wire-line communications) radio spectrum is a shared common resource in the sense that every user has the possibility to transmit over it. Interference appears to be a problem (it can even block the transmission process) when several transmitters simultaneously access the same physical link. In this case, receivers obtain a mixed version of all the transmitted signals, and in general it is difficult and expensive to let the receiver distinguish and capture its desired information.

The control and reduction of multiuser interference is a fundamental problem in wireless communications. Many media access technologies together with efficient resource allocation algorithms have been introduced under this setting. All these technologies can be briefly summarized from the viewpoints of degrees of freedom:

• Frequency

Frequency Division Multiple Access (FDMA) is a channel access method which gives each user an individual allocation of one or several frequency sub-bands (or sub-channels).

FM broadcasting can be considered as a simple example of FDMA.

• Time

Time Division Multiple Access (TDMA) is a channel access method which allows several users to share the same frequency channel by dividing the signal into different time slots.

TDMA technology is widely used in the digital 2G cellular systems, e.g., IS-54 (1990) and GSM (1991).

• Code

Code Division Multiple Access (CDMA) is a channel access method employing spread-spectrum technology and a special coding scheme (where each transmitter is assigned a code) to allow multiple users to be multiplexed over the same physical link.

CDMA technology leads GSM in migration to 3G, e.g., IS-95B (1999) as 2.5G technology and CDMA2000 (2000) as 3G technology.

• Space

Space Division Multiple Access (SDMA) is a multiple input and multiple output (MIMO) based wireless communication technology. In traditional mobile cellular network systems, in order to provide radio coverage, the base station radiates the signal in all directions, since it has no information on the position of the mobile devices within the cell. SDMA is an advanced channel access method which enables access to a communication channel by identifying the user location and establishing a one-to-one mapping between the network frequency division and the identified spatial location. SDMA technology is considered as a key feature in 4G cellular systems.

With the increasing demand of wireless voice and data services, frequency spectrum has become a scarce resource for which everyone fights for. In fact, the spectral efficiency (efficiency measurement of using frequency spectrum) can be improved by radio resource management techniques such as efficient fixed or dynamic channel allocation, power control, link adaptation and diversity schemes.

Over the past two decades, resource allocation based on information theory [START_REF] Cover | Elements of Information Theory[END_REF] and optimization theory [START_REF] Boyd | Convex Optimization[END_REF][START_REF] Chiang | Geometric Programming for Communication Systems[END_REF] has played a central role. For example, waterfilling algorithm [START_REF] Cover | Elements of Information Theory[END_REF] is one of the most famous pioneering works that consider how to maximize the channel capacity. However, in order to achieve the information-theoretic limits (e.g., capacity region), it usually requires a central computing resource (a scheduler with comprehensive knowledge of the network information) to globally schedule and allocate the system resources. This process is centralized, it involves feedback and overhead communication whose load scales linearly with the number of transmitters and receivers in the network.

As the number of cells in the network increases, interference becomes the bottleneck. In the current cellular wireless arena, engineers frequently stumble on the scalability problem. As networks become more and more dense, classical methods based on interference avoidance/cancellation and techniques based on frequency and space reuse or power control are not be able to cope with interference due the increasing number of mobile terminals. The optimization based centralized network infrastructure begins to expose its weakness in many aspects, e.g., slow reconfiguration against varying environment, difficulty in centralizing network information, high cost of network operating expenses, increased computational complexity, etc.

It turns out that due to the different nature of the mobile devices as well as the high mobility of the network, future wireless networks are going towards more self-organizing resource allocations schemes in which mobile devices intelligently allocate resource in a decentralized manner [START_REF] Debbah | Mobile Flexible Networks: The challenges ahead[END_REF]. In recent years, there has been great research interest in selforganizing wireless networks. Tools of game theory [START_REF] Fudenberg | Game Theory[END_REF], as borrowed from economics1 , has emerged as an adequate framework for the analysis of wireless network resource allocation by modeling devices as selfish players.

TOOLS OF GAME THEORY

Game theory is a branch of applied mathematics that is used in the social sciences, primarily in economics (in order to model competition between companies), as well as computer science, biology, politics and many other areas. Game theory is a description of strategic interaction, which attempts to mathematically capture behavior in strategic situations, in which an individual's success in making choices depends on the choices of others. It enhances the understanding of conflict by devising theories, mathematical models and abstractions that serve to explain the nature and results of conflict.

Background

In game theory, there are some basic assumptions which are often utilized to facilitate the construction of tractable models for real situations. First, it is assumed that each individual (player ) in the game has a definite ordering of preferences over all outcomes of a given situation. These preferences take the form of a utility function (or payoff ). Second, the participants in games are sometimes considered to be rational. This means they always act in a way that maximizes their payoffs, they will always be capable of thinking through all possible outcomes and choosing that course of action which will result in the best possible outcome.

Thus, one way in which we can informally formulate a game is as a situation where there exist K decision-making entities or players (whose choices or actions) influence the outcomes of that situation. A player "attaches" a utility to each one of these outcomes and is motivated to act in a manner that maximizes his utility, given the choices of his opponents. This characterization of an K-player game is known as the strategic form of a game. Strategic-form game is an appropriate model to interpret the interaction between decision-makers. The model captures interaction between the players by allowing each player to be affected by the actions of all players (not only the player's own action). This paper will mainly focus on the applications of strategic-form games in wireless resource allocation problems.

A motivational example

What is a game problem? What can game theory tools bring to us? To better understand the concepts and principles behind game theory, we provide a simple example in the context of wireless communications, as follows:

Imagine a two-user multiple access channel (MAC), Fig. 1, in which two mobile devices (transmitters) x1 and x2 compete to transmit their signals towards a single base station (receiver) in a common wireless channel. There are two choices for both users: either to transmit with low power (denote by "Low") or transmit with high power (denote by "High"). They must decide simultaneously (without communication) which power level to choose. Typically, this problem can be modelled as a static game with three elements:

• A player set K = {x1, x2}.
• An action set A k = {Low, High}, k ∈ K, which is the same for both users.

• A payoff function set, which is described by the following matrix2 : Low High Low (win, win) (lose much, win much) High (win much, lose much) (lose, lose)

where, in each entry (a, b), the values a and b represent the payoff of player x1 and x2, respectively. Intuitively, from this payoff matrix, we have the following observations:

• If both mobiles choose to transmit with high power, they will surfer from the increased interference caused by the other one, which results in a "lose-lose" situation.

• If mobile x1 chooses to transmit with low power and mobile x2 chooses to transmit with high power, compared to the "lose-lose" case, mobile x1 will get a worse performance (denote by "lose much") and mobile x2 will benefit from the reduced interference and enjoy a better performance (denote by "win much").

• If both mobiles choose to transmit with low power, it will result in a "win-win" situation.

Obviously, to find the solution of this problem is beyond the capability of optimization theory, since user x1's best strategy depends on the strategy chosen by user x2, which user x1 does not know, and reciprocally for user x2.

Readers may guess that both mobiles must strictly prefer to transmit with low power. However, this "win-win" situation is not the solution of this game, i.e., it is not a natural outcome of rational players. It might be quite surprising that the only solution, the pure Nash equilibrium3 , of this game is the "lose-lose" situation.

Methodology to study game problems

In such a non-cooperative strategic-form game, what actions should the players choose? The famous Nash equilibrium represents a common solution concept that is mutual optimal, in the sense that no player has any motivation to deviate from it. In general, to analyze Nash equilibrium, there are three main questions that should be considered, which are "existence", "uniqueness" and "selection". Here, we briefly discuss this methodology in general situations [START_REF] Lasaulce | Methodologies for analyzing equilibria in wireless games[END_REF][START_REF] Altman | Operating Point Selection in Multiple Access Rate Regions[END_REF],

1. Existence -Does an equilibrium exist? 2. Uniqueness -Does there exist a unique equilibrium or multiple ones?

3. Selection -How to select a favouring equilibrium from the equilibrium set?

"Existence" is the very first question that naturally comes into our mind, since it is known that, in general, an equilibrium point does not necessarily exist. Mathematically speaking, proving the existence of an equilibrium is equivalent to proving the existence of a solution to a fixed-point problem [START_REF] Nash | Equilibrium points in n-points games[END_REF]. Since the existence of the fixed-point hints that there is some strategy set which is a best response to itself, therefore no player could do any better by deviating, and so it is an equilibrium. Fortunately, there are many scenarios based on usual channel models and performance metrics (e.g. Shannon transmission rate and rate regions have desirable convexity properties [START_REF] Boyd | Convex Optimization[END_REF] that are in favor of the existence of an equilibrium [START_REF] Rosen | Existence and Uniqueness of Equilibrium Points for Concave N-Person Games[END_REF]) where existing theorems are sufficient. "Uniqueness" is the second fundamental problem that we need to address when the existence is ensured. Ideally, we would prefer there to be a unique equilibrium because it is the simplest solution form for general game-theoretical problems, and it is important not only for predicting the state of the network but also crucial for convergence issues. Unfortunately, there are not so many general results on the topic of equilibrium uniqueness. One could find some useful results in the concave N-person games [START_REF] Rosen | Existence and Uniqueness of Equilibrium Points for Concave N-Person Games[END_REF], where it is shown that there exists exactly a unique equilibrium if the payoff functions satisfy the condition of diagonally strictly concave which can be easily verified.

However, there are many important scenarios where the equilibrium is not unique, e.g., routing games [START_REF] Roughgarden | Bounding the Inefficiency of Equilibria in Nonatomic Congestion Games[END_REF], coordination games [START_REF] Cooper | Coordination Games[END_REF], non-cooperative games with correlated constraints together with the concept of "generalized Nash equilibrium" [START_REF] Altman | Constrained Markov Games: Nash Equilibria[END_REF], etc. Natural questions that arise concern the selection of an appropriate equilibrium, i.e.,

• What can be done when one has to deal with a game having multiple equilibria?

• Are there some equilibria "dominating" others?

• Are there some equilibria more "fair" than others?

• What is a "good" selection rule to follow?

As a matter of fact, "equilibrium selection" is a mature theory in itself [START_REF] Harsanyi | A General Theory of Equilibrium Selection in Games[END_REF]. But here, instead of paying attention to the general theory, we are more interested in the applications of concave games, where Rosen has already introduced the notion of "normalized equilibria" [START_REF] Rosen | Existence and Uniqueness of Equilibrium Points for Concave N-Person Games[END_REF] that shows a very neat way to tackle this selection problem. Obviously, the selection rule is strongly related to the fairness criteria uses, e.g., max-min fairness [START_REF] Floyd | Connections with Multiple Congested Gateways in Packet-Switched Networks Part 1: One-way Traffic[END_REF], proportional fairness [START_REF] Kelly | Charging and Rate Control for Elastic Traffic[END_REF][START_REF] Kelly | Rate Control for Communication Networks: Shadow Prices, Proportional Fairness and Stability[END_REF], Jain's fairness [START_REF] Chiu | Analysis of the Increase and Decrease Algorithms for Congestion Avoidance in Computer Networks[END_REF], global optimization and normalized equilibrium, etc. Specifically, the authors of [START_REF] Altman | Operating Point Selection in Multiple Access Rate Regions[END_REF] have shown that the max-min fairness, proportional fairness and normalized equilibrium achieve the same rate allocation in the context of multiple access channels with multi-user detection. Up to now, equilibrium selection is still an open topic in many communication network models and applications, especially for the case of non-convex rate region, e.g., achievable rate region of collision channels and interference channels.

GAMES THEORY IN WIRELESS COM-MUNICATIONS

In recent years, there has been some great interest in selforganizing wireless networks in which mobile devices allocate resource in an intelligent way. There are three main ingredients that make game theory to be a fruitful theory for future wireless analysis and design:

1. Future generation wireless networks will most likely consist of intelligent radio devices, capable to sense the environment and effectively adjust their transmission parameters according to the current local channel conditions and QoS specifications.

2. Wireless channel is considered as a shared common resource (frequency, time, power, space, etc.) which implies competition and cooperation between wireless devices. In some sense, wireless resource allocation will not be simply considered as an optimal way to "share a cake", but a complex resource conflict between intelligent decision makers.

3. Wireless networks are highly structured which favors the success of game theoretic analysis.

The decision makers in the game are rational users or networks operators who control their communication devices. Indeed, game theory is an appropriate mathematical tool to model the interaction and to solve the resource conflicts existing in the wireless networks. Here, we illustrate several basic game theory tools that have already been used in the literature.

Wireless game with complete information

We call it a game with complete information, if the game structure (which includes play set, action set and payoff functions) is available to all players. For example, "chess" is a game with complete information, because the chess players, chess moves and each player's objective are known to all players.

Many efforts have been made in this direction, since it is considered as an initial step. Game theory has been widely applied to study the resource allocation and power control problems in various types of networks, such as fading MAC [START_REF] Lai | The Water-Filling Game in Fading Multiple-Access Channels[END_REF], orthogonal frequency division multiplexing (OFDM) [START_REF] He | Game-Theoretic Deployment Design of Small-Cell OFDM Networks[END_REF], multiple input and multiple output (MIMO) channels [START_REF] Palomar | Uniform Power Allocation in MIMO Channels: A Game Theoretic Approach[END_REF][START_REF] Lasaulce | Methodologies for analyzing equilibria in wireless games[END_REF][START_REF] Belmega | Power allocation games for MIMO multiple access channels with coordination[END_REF], and interference channels [START_REF] Altman | Game Theoretic approach for routing in dense ad-hoc networks[END_REF], etc.

The weakness of this game-theoretic approach is that the game models used in these previous works assume that the information/knowledge about other devices is available to all devices. However, this assumption is hardly met in practice. In practical wireless communication systems, devices can have local information but can barely access to global information on the network status.

Wireless game with incomplete information

In game theory, a strategic game with incomplete information is called a "Bayesian game". In contrast to games where devices have complete information about other devices, it is of great interest to investigate scenarios where devices have "incomplete information" about their components. The notion of "incomplete information" means that some players do not completely know the structure of the game. For example, "poker" is a game with incomplete information, because players are uncertain about the card states of other players, therefore, the exact object functions of other players are unknown.

Before talking about games with incomplete information, we need to clarify two other similar notions: "imperfect information" and "imperfect (or partial) channel state information (CSI)". The former "Imperfect information" is a common notion used in the sequential games [START_REF] Fudenberg | Game Theory[END_REF], in contrast to games of perfect information, a game is said to have imperfect information if not all players know all moves that have taken place (or the history of the game is not available to all players). The latter "imperfect/partial CSI" is a specific term widely used in the wireless communication society, it means that the channel information is not perfectly estimated/observed at the transmitter/receiver side.

It is important to note that the assumption of incomplete information is closely related to the two notions mentioned above. For example: we can consider the following communication scenario with imperfect CSI as a game of incomplete information: a device has perfect CSI about its own channel, but it has imperfect CSI about any other device's channel. Note that this is a common situation that usually happens in a wireless network, since it may be too "expensive" for every device to keep track of the time-variant channels of all other devices.

Unfortunately, in wireless communications, there are not many references using Bayesian games to model and solve the resource allocation problem. In [START_REF] Jean | Bayesian game-theoretic modeling of transmit power determination in a self-organizing CDMA wireless network[END_REF], a static Bayesian game-theoretic model is applied to the problem of transmit power determination in the uplink of a self-organizing CDMA wireless network. In another important work [START_REF] Adlakha | Competition in Wireless Systems via Bayesian Interference Games[END_REF], Bayesian game model is applied to study the two-user multichannel (or multi-carrier) interference channels, where each wireless device selects a power profile over the entire available bandwidth to maximize its data rate. The authors show that there exists a unique Bayesian equilibrium under the assumption of single-user detector (each user treats interference as noise and no interference cancellation techniques are used) and finite (typically selected) action set. In [START_REF] He | A Bayesian Game-Theoretic Approach for Distributed Resource Allocation in Fading Multiple Access Channels[END_REF], for multiuser fading MAC, under the assumption of discrete (finite) channel states, the authors show that there exists exactly one Bayesian equilibrium in the game. The efficiency of Bayesian equilibrium is studied in their numerical results.

Wireless game via Nash bargaining

Nash bargaining is a natural framework that allows us to define and design a fair assignment of wireless resource, e.g., transmission rate, between players which will play the role of bargainers. It is characterized by a set of axioms that are appealing in defining fairness or by a maximization of log-concave function on the set of feasible sum-rates. Nash bargaining solution (NBS) is an interesting solution concept since it can be seen as a natural extension of the proportional fairness criterion which is probably the most popular fairness. In such a game, players are faced with the problem to negotiate for a fair point in the convex set of feasible rates. If no agreement can be achieved by the players, the disagreement utilities (sum-rate) is obtained.

In fact, NBS is applied widely in network resource allocation. For example, in [START_REF] Han | Fair Multiuser Channel Allocation for OFDMA Networks Using Nash Bargaining Solutions and Coalitions[END_REF], a criterion based on NBS is applied to orthogonal frequency division multiple-access (OFDMA) networks. This method generalizes the proportional fairness and increases the efficiency of the system. In [START_REF] Shum | On the Fairness of Rate Allocation in Gaussian Multiple Access Channel and Broadcast Channel[END_REF], the ideas of max-min fairness, proportional fairness and NBS have been applied to the Gaussian multiple access channel (MAC) and the Gaussian broadcast channel (BC). Algorithms have been designed to locate the fair point in the capacity region. In [START_REF] Leshem | Cooperative game theory and the Gaussian interference channel[END_REF], cooperative game is used to provide preferred points on the boundary of the achievable rate region for a simple 2×2 interference channel. From their simulation results, the cooperative solution NBS is shown to significantly outperform the competitive NE. In [START_REF] He | Deployment Analysis of cooperative OFDM base stations[END_REF], NBS is studied as a tool to achieves point(s) on the Pareto frontier of the game theoretical rate region under asymmetric conditions for OFDM hot-spot networks. A stochastic algorithm is proposed.

CONCLUSIONS

In this paper, we have discussed the use of game theory in future wireless network. From a simple motivational example, we showed what is a game problem and we discussed the methodology to analyze the solution of a game problem. We have summarized several game theory tools that can be applied to study the wireless resource allocation problems.

Figure 1 :

 1 Figure 1: Two-user MAC

In the famous literature The Wealth of Nations, Adam Smith (A Scottish moral philosopher, pioneer of political economy, father of modern economics) expounded how rational self-interest and competition can lead to economic prosperity and well-being through macroeconomic adjustments.

Note that here we do not give specific values for the payoffs, but show it in a "win-lose" terminology.

A Nash equilibrium, named after John Nash, is a set of strategies, one for each player, such that no player has incentive to unilaterally change her action.


Gaoning He
email: gaoning.he@gmail.com

Sharon Betz
email: sharonbetz@gmail.com

Mérouane Debbah
email: merouane.debbah@supelec.fr

Game-Theoretic Deployment Design of Small-Cell OFDM Networks

We present a non-cooperative game-theoretic approach for the distributed resource allocation problem in the context of multiple transmitters communicating with multiple receivers through parallel independent fading channels, which is closely related with small-cell multi-user orthogonal frequency division multiplexing (OFDM) networks, e.g., Wi-Fi hotspots. We assume that all the transmitters are rational, selfish, and each one carries the objective of maximizing its own transmit rate, subject to its power constraint. In such a game-theoretic study, the central question is whether a Nash equilibrium (NE) exists, and if so, whether the network operates efficiently at the NE. We show, for independent fading channels, there almost surely exists a unique NE. Finally we present the behavior of average network performance at the NE through numerical results, and we compare the optimal centralized approach with our decentralized approach.

INTRODUCTION

Consider a scenario that multiple transmitters simultaneously sending information to their receivers through several independent channels or resources. In this paper we will use the example of small-cell OFDM networks [START_REF] Debbah | Mobile Flexible Networks: the route towards autonomic communications[END_REF]. In such a wireless network, we assume that multiple access points (APs) serve a small area (e.g., airports, restaurants, military bases, hotels, hospitals, libraries, supermarkets, etc.), simultaneously communicating to several mobile terminals (MTs) using OFDM over a number of dedicated sub-channels. An N -carrier OFDM system [START_REF] Bingham | Multicarrier Modulation for Data Transmission: An Idea Whose Time Has Come[END_REF] using a cyclic prefix or zeropadding [START_REF] Muquet | Novel Receiver and Decoding Schemes for Wireless OFDM Systems with Cyclic Prefix or Zero-padding[END_REF][START_REF] Muquet | Cyclic-Prefixing or Zero-Padding for Wireless Multicarrier Transmissions?[END_REF] to prevent inter-block interference is equivalent in the frequency domain to N flat fading parallel transmission sub-channels. In this scenario, each AP faces a problem of how to distribute the total available power among these N downlink sub-channels (subcarriers or clusters of subcarriers), i.e., should it allocate its total power to a single sub-channel, spread the power over all the sub-channels, or choose some subset of sub-channels on which to transmit? Here, we shall emphasize that a similar power alloca-tion problem can be considered for the uplink transmission where MTs are the transmitters who decide their transmit power strategies.

When this resource allocation problem is considered centrally to maximize the total achievable rate (corresponding to Shannon capacity [START_REF] Shannon | A mathematical theory of communication[END_REF] when single user detector is applied) for the small-cell OFDM network, it is an often studied optimization problem. Given other users' strategies, the problem of how to maximize a single user's sum-rate over all the sub-channels is a convex optimization problem [START_REF] Boyd | Convex Optimization[END_REF], whose solution is "waterfilling" [START_REF] Cover | Elements of Information Theory[END_REF][START_REF] Goldsmith | Capacity of Fading Channels with Channel Side Information[END_REF][START_REF] Tse | Fundamentals of Wireless Communication[END_REF]. The multi-user version of this problem is a non-convex optimization problem that may have multiple local optimal points [START_REF] Altman | Closed Form Solutions for Symmetric Water Filling Games[END_REF][START_REF] Song | Dynamic Spectrum Management for Next-generation DSL Systems[END_REF][START_REF] Papandriopoulos | Low-Complexity Distributed Algorithms for Spectrum Balancing in Multi-User DSL Networks[END_REF]. To solve the multi-user problem in a centralized way, it requires a scheduler to allocate the resources, which is similar to the multi-user waterfilling problem [START_REF] Yu | Competition and cooperation in multi-user communication environements[END_REF] in multiple access channel (MAC). This approach requires a central computing resource with perfect knowledge of the channel state information (CSIT), involving feedback and overhead communication whose load scales linearly with the number of transmitters and receivers in the network. To reduce the feedback load, selective multi-user diversity algorithms have been introduced in [START_REF] Gesbert | How Much Feedback is Multi-User Diversity Really Worth?[END_REF]. In contrast to centralized networks, we study in this paper distributed communication solutions that require no centralized control, reducing the overhead as well as the need for a knowledgeable powerful scheduler to allocate network resources. Now if each AP independently allocates the total power to maximize its own total achievable rate, considering all other users' transmissions as noise, this problem can be studied as a distributed non-cooperative game [START_REF] John | Non-cooperative Games[END_REF] where the selfish players are the APs who "play" the game by choosing their transmit power levels across sub-channels. Note that a selfish player may not act as a good neighbor to each other [START_REF] Popescu | Water Filling May Not Good Neighbors Make[END_REF], and in fact he does not care about any other player's performance neither the whole network performance at all. Nash equilibria [START_REF] John | Equilibrium Points in N-person Games[END_REF], which we will formally define in Section 3 but which are, informally, the best mutual strategies a rational player can play assuming that other players are also rational (using their own best strategies given their information), are important to study in such a distributed non-cooperative game because they represent the natural outcomes in competitive games. It is worth to mention that two special cases of this game have been studied in [START_REF] Yu | Distributed Multiuser Power Control for Digital Subscriber Lines[END_REF] and [START_REF] Altman | Closed Form Solutions for Symmetric Water Filling Games[END_REF]. In [START_REF] Yu | Distributed Multiuser Power Control for Digital Subscriber Lines[END_REF], the existence and uniqueness of NE were established for the two-player case. In [START_REF] Altman | Closed Form Solutions for Symmetric Water Filling Games[END_REF], the so called symmetric waterfilling game was studied (they assume, for a set of subcarriers and receivers, the channel gains from all transmitters are the same). They show that there exist an infinite number (a continuum) of NE in the game. However, up to now it is still not clear how to characterize the equilibrium set in the case of general independent fading channels with multiple players. The goal of this paper is therefore to address this fundamental problem.

The paper is organized in the following form: the problem formulation is introduced in Section 2. In Section 3, we study the existence of NE and we characterize the NE set. We shown in Section 4 the optimal centralized approach. Finally, some numerical results are provided in Section 5 followed by conclusions in Section 6.

SYSTEM MODEL

Communication Model

Consider a parallel Gaussian channels (in particular OFDM system) with M APs simultaneously sending information to N MTs over N sub-channels. Assume that each subchannel is pre-assigned to a different MT by a scheduler, therefore, each MT detects the signals only on the assigned sub-channel. We also assume that the channels have block fading so that the channel fading coefficients are constant during the period of each transmission block. Within a given transmission block, let G ∈ R M ×N ++ (positive real M × N matrix) be the channel gain matrix whose (m, n) entry is gm,n, the magnitude-squared of the fading channel gain on the downlink channel from AP m to MT n. We assume that G is a random M × N matrix with i.i.d. continuous entries (meaning that each entry gm,n is independent, identically and continuously distributed) due to independent fading channels. Assuming that signals from other APs are treated as noise, the signal to interference plus noise ratio (SINR) of the signal from AP m to MT n is γm,n = gm,npm,n σ 2 + j =m gj,npj,n

where pm,n ≥ 0 represents the power transmitted by the m th AP on the n th sub-channel. For the sake of simplicity, we assume that the variance of white Gaussian noise σ 2 is the same for each sub-channel n. The maximum achievable sum-rate for AP m is given by [START_REF] Cover | Elements of Information Theory[END_REF]

Rm = N n=1 log (1 + γm,n) , ∀m (2)
Each AP m has the power constraint

N n=1 pm,n ≤ Pm, ∀m (3)
for Pm > 0, ∀m.

Game Model

Here, we consider the previously mentioned model as a non-cooperative strategic game. In this game, the goal of each AP (player) m is to choose its own power vector pm = [pm,1 . . . pm,N ] T (subject to its total power constraint (3)) to maximize the sum-rate Rm. Let the long power vector p = p T 1 , . . . , p T M T denote the outcome of the game in terms of transmission power levels of all the M players on N subchannels. We can completely describe this non-cooperative OFDM game as

G M, {Pm} m∈M , {um} m∈M (4)
where the elements of the game are

• The player set: M = {1, . . . , M };

• The strategy set: {P1, . . . , PM }, where the strategy of player m is

Pm = pm : pm,n ≥ 0, ∀n, N n=1 pm,n ≤ Pm ; (5)
• The utility or payoff function set: {u1, . . . , uM }, with um (pm, p-m) = Rm expressed in [START_REF] Bingham | Multicarrier Modulation for Data Transmission: An Idea Whose Time Has Come[END_REF], where p-m denotes the power vector of length (M -1)N consisting of elements of p other than the m th element, i.e.,

p-m = [p T 1 , . . . , p T m-1 , p T m+1 , . . . , p T M ] T .

NASH EQUILIBRIUM

In such a non-cooperative setting, each player m acts selfishly, aiming to maximize its own payoff, given other players' strategies and regardless of the impact of its strategy may have on other players and thus on the overall performance. The process of such selfish behaviors usually results in Nash equilibrium, which represents a common solution concept for non-cooperative game theoretic problems [START_REF] John | Equilibrium Points in N-person Games[END_REF].

Definition 3.1. A power vector p ⋆ is said to be a Nash equilibrium if for every m ∈ M, um (p ⋆ m , p ⋆ -m ) ≥ um (pm, p ⋆ -m ) (6)
for all pm ∈ Pm.

From above, it is clear that a NE simply represents a particular "steady" state for a system, in the sense that, once reached, no player has any motivation to unilaterally deviate from it. In many cases, Nash equilibria represent the result of learning and evolution of all players in a system. Therefore, it becomes fundamentally important to predict and characterize such point(s) from the system design perspective of wireless networks. In the rest context of this paper, we will focus on charactering such point(s) in the power strategy space. The following questions will be addressed one by one: 1. Does NE exist in our game? 2. Does there exist a unique or multiple NE points? 3. How does the network behave at NE? It is known that in general NE point does not necessary exist. Therefore, our first interest turns to investigate the existence and uniqueness of NE in our game. Lemma 3.2. A Nash equilibrium exists in our game G.

Proof. Since Pm is convex, closed, and bounded for each m; um (pm, p-m) is continuous in both pm and p-m; and um (pm, p-m) is concave in pm for any set p-m, at least one Nash equilibrium point exists for G [START_REF] Rosen | Existence and Uniqueness of Equilibrium Points for Concave N-Person Games[END_REF], [START_REF] Fudenberg | Game Theory[END_REF].

NE existing under the assumption of equal crosstalk channel coefficients. This implies that in general our game G does not have unique equilibrium point. Nevertheless, we will show in the following context that there exists a single NE almost surely for random channel gain matrix G with i.i.d. continuous entries.

For any player m, given all other players' strategies p-m, the best-response power strategy pm can be found by solving the following optimization problem, max pm um (pm, p-m)

s.t. n pm,n ≤ Pm (7) pm,n ≥ 0
which is a convex optimization, since the objective function is concave in pm and the constraint set is convex. Therefore, the Karush-Kuhn-Tucker (KKT) condition of the optimization is sufficient and necessary for the optimality [START_REF] Boyd | Convex Optimization[END_REF].

To derive the KKT conditions, we form the Lagrangian for each player m,

Lm (p, λ) = N n=1 log 1 + gm,npm,n σ 2 + j =m gj,npj,n -λm N n=1 pm,n -Pm + N n=1 νm,npm,n.
The corresponding KKT conditions are gm,n σ 2 + M j=1 gj,npj,n

-λm + νm,n = 0, ∀n (8) λm
N n=1 pm,n -Pm = 0 (9)
νm,npm,n = 0, ∀n,

where λm ≥ 0, ∀m and νm,n ≥ 0, ∀m ∀n are dual variables associated with the constraints of limited power and positive power, respectively. The solution to this problem is wellknown as the waterfilling algorithm [START_REF] Cover | Elements of Information Theory[END_REF][START_REF] Goldsmith | Capacity of Fading Channels with Channel Side Information[END_REF][START_REF] Tse | Fundamentals of Wireless Communication[END_REF] pm,n = 1 λm -σ 2 + j =m gj,npj,n gm,n

+ , ∀n (11)
where (x) + = max{0, x} and λm satisfies

N n=1 1 λm - σ 2 + j =m gj,npj,n gm,n + = Pm. ( 12
)
Lemma 3.3. The following conditions are sufficient and necessary for the Nash equilibrium in OFDM game G.

gm,n σ 2 + M j=1 gj,npj,n

-λm + νm,n = 0, ∀m ∀n (13)
λm N n=1 pm,n -Pm = 0, ∀m (14)
νm,npm,n = 0, ∀m ∀n.

Proof. For a certain index m, the single player KKT conditions (8)- [START_REF] Altman | Closed Form Solutions for Symmetric Water Filling Games[END_REF] are necessary and sufficient for the best response condition with index m in [START_REF] Boyd | Convex Optimization[END_REF]. Therefore, as a collection of single player KKT conditions from index 1 to M , ( 13)-( 15) are necessary and sufficient for all the best response conditions in [START_REF] Boyd | Convex Optimization[END_REF].

From [START_REF] Yu | Competition and cooperation in multi-user communication environements[END_REF], it is easy to observe λm > 0. Then from ( 14), we have

N n=1 pm,n = Pm, ∀m (16)
Note that ( 16) has an intuitive meaning: each player at NE must dedicate its total available power on all carriers, due to their "selfish instinct". However, it is still difficult to find the analytical solution for ( 13)-( 15), since the solution form [START_REF] Song | Dynamic Spectrum Management for Next-generation DSL Systems[END_REF] and (12) construct a system of nonlinear equations.

The idea to simplify this problem is therefore to consider linear equations instead of nonlinear ones, we then introduce another lemma, as follows Lemma 3.4. For any realization of channel matrix G ∈ R M ×N ++ , there exist unique values of the Lagrange dual variables λ and ν for any Nash equilibrium of the game G. Furthermore, there is a unique vector s = [s1, . . . , sn] T such that any vector p corresponding to a Nash equilibrium of the game satisfies M m=1 gm,npm,n sn, ∀n.

(

) 17
The proof can be found in Appendix A. Now, let Z be the following (M + N ) by M N matrix

Z =        IM IM • • • IM g T 1 0 T M • • • 0 T M 0 T M g T 2 • • • 0 T M . . . . . . . . . . . . 0 T M 0 T M • • • g T N        (M +N )×M N
where gn is the n th column of G, IM is the M by M identity matrix, and 0M is zero vector of length M . Let c be the following vector of length M + N c = P1 P2 . . . PM s1 s2 . . . sN T Then, ( 16) and ( 17) can be written in the form of linear matrix equation

Zp = c. (18)
Define the following sets:

X {(m, n) : νm,n = 0}, N {n : ∃m such that (m, n) ∈ X }.
From [START_REF] John | Non-cooperative Games[END_REF], if an index (m, n) / ∈ X we must have pm,n = 0. Without loss of generality, we assume that N = {1, . . . , Ñ } for Ñ ≤ N . Let Z be the M + Ñ by M Ñ matrix formed from the first M + Ñ rows and first M Ñ columns of Z, p is formed from the first M Ñ elements of p, and c is formed from the first M + Ñ elements of c, then any NE solution must satisfy

Zp = c. ( 19
)
Let Ẑ be the M + Ñ by |X | (the size of X ) matrix formed from the columns of Z that correspond to the elements of X . Similarly, let p be the vector of length |X | with the values of pm,n with (m, n) ∈ X (and in the same order as they were in p). Then any NE solution must satisfy Proof. When νm,n = 0 , from (8) we have

λm -gm,ndn = 0, ∀(m, n) ∈ X (21)
where dn 1 σ 2 +sn . From Lemma 3.4, we know that all the Nash equilibria must satisfy [START_REF] Meyer | Matrix Analysis and Applied Linear Algebra[END_REF], with the same λm and dn. In [START_REF] Meyer | Matrix Analysis and Applied Linear Algebra[END_REF], the number of independent linear equations is |X |, while the number of unknown parameters is M + Ñ (since the rest of dn, n / ∈ N is known to be dn = 1 σ 2 ). It is well known that the solution to the system of linear equations is the empty set, if the number of independent equations is larger than the number of variables [START_REF] Meyer | Matrix Analysis and Applied Linear Algebra[END_REF]. Since gm,n is chosen randomly from a continuous distribution, it is obvious that, with probability 1, the equations in ( 21) are independent from each other, therefore, we must have |X | ≤ M + Ñ . Lemma 3.6.

1. If M Ñ > M + Ñ and |X | ≤ M + Ñ , rank( Ẑ) = |X |
with probability 1.

If

M Ñ ≤ M + Ñ , rank( Z) = M Ñ with probability 1.
Proof. We only give the proof for case 1) M Ñ > M + Ñ , case 2) M Ñ ≤ M + Ñ can be proved in a similar way. Matrix Ẑ can be transformed into a 2×2 block matrices, by applying some elementary column and row operations, as follows,

Ẑ column -→ Iτ A τ ×ξ 2 B ξ 1 ×τ C ξ 1 ×ξ 2 column -→ Iτ 0 τ ×ξ 2 B ξ 1 ×τ Ĉξ 1 ×ξ 2 row -→ Iτ 0 τ ×ξ 2 0 ξ 1 ×τ Ĉξ 1 ×ξ 2 where τ = min{M, Ñ }, ξ1 = M + Ñ -τ ≥ τ, ξ2 = |X | -τ .
Ĉ is a ξ1 × ξ2 matrix, where each column contains one or two random variables, and each row contains at least one random variable. Again we can transform Ĉ in row echelon form, denoted as Ĉr. Note that the rank of Ĉr is ξ2 with probability 1, since each leading coefficient of a row is a random variable or the linear combination of two continuously distributed random variables. So, with probability 0, any leading coefficient takes the value of 0. Therefore, we have rank( Ẑ) = τ + ξ2 = |X | with probability 1.

Theorem 3.7. For any realization of random channel matrix G with i.i.d. continuous entries, the OFDM game G has exactly one Nash equilibrium.

Proof. If M Ñ > M + Ñ , we have from Lemma 3.6 that, with probability 1, rank( Ẑ) = |X |. Any NE must satisfy [START_REF] Fudenberg | Game Theory[END_REF]; assume that two different power strategies p and p′ are both solutions to [START_REF] Fudenberg | Game Theory[END_REF]. Then Ẑ (p -p′ ) = 0. By the rank-nullity theorem [START_REF] Meyer | Matrix Analysis and Applied Linear Algebra[END_REF], since the rank of Ẑ is equal to the number of its columns, this implies p -p′ = 0, which means there must be exactly one NE.

If M Ñ ≤ M + Ñ , we have from Lemma 3.6 that, with probability 1, there is at most one solution to [START_REF] Rosen | Existence and Uniqueness of Equilibrium Points for Concave N-Person Games[END_REF]. Since any NE must satisfy [START_REF] Rosen | Existence and Uniqueness of Equilibrium Points for Concave N-Person Games[END_REF] and we know that there is at least one NE solution, so the NE must be unique.

PARETO OPTIMALITY

To measure the inefficiency of Nash equilibrium, we consider in this section the Pareto boundary for the total network rate. The total network rate maximization problem can be formulated as max

p m M m=1 um (p) s.t. m pm,n ≤ Pm, m = 1, . . . , M (22)
pm,n ≥ 0 which unfortunately is a difficult problem, since the objective function in non-convex in p. However, a relaxation of this optimization problem (see in [START_REF] Papandriopoulos | Low-Complexity Distributed Algorithms for Spectrum Balancing in Multi-User DSL Networks[END_REF]) can be considered as a geometric programming problem [START_REF] Chiang | Geometric Programming for Communication Systems[END_REF], therefore, can be transformed into a convex optimization problem. A low complexity algorithm was proposed in [START_REF] Papandriopoulos | Low-Complexity Distributed Algorithms for Spectrum Balancing in Multi-User DSL Networks[END_REF] to solve the dual problem by updating dual variables through a gradient descent. Note that this approach enable us to find a tight lower-bound for the Pareto boundary that is sufficient for the performance measurement presented in the next section.

NUMERICAL EVALUATION

In this part, numerical results are provided to demonstrate the network performance at the unique Nash equilibrium (outcome of decentralized networks). To be precise, we are interested in comparing the "average total network rate" (average over the distribution of fading channel gains, and we will use the short term "total network rate" to represent it) instead of the instantaneous one. We denote by ū(M, N ) this total network rate,

ū(M, N ) = E G M m=1 N n=1 log 1 + pm,ngm,n σ 2 + j =m pj,ngj,n
As a basis for comparison, the Pareto-maximum total network rate will also be provided and considered as a upper bound for the decentralized settings. Parameters are selected as: the number of transmitters is M ∈ [START_REF] Debbah | Mobile Flexible Networks: the route towards autonomic communications[END_REF]25]; the number of receivers N takes several representative values as 5, 10 and 15; the power constraint of each AP is Pm = 1, ∀m; the variance of additive white Gaussian noise is set to σ 2 = 0.1 and 1, respectively. Fig. 1 and Fig. 2 show the total network rate for σ 2 = 0.1 and 1, respectively. As expected, the curves of centralized networks always outperform the decentralized ones. More preciously, for a fixed number of receivers N , as the number of transmitters M increases, the performance loss of decentralized networks (compared to centralized networks) becomes more and more apparent. This phenomenon can be easily understood as: when there are a great number of selfish players, the hostile competition turns the multi-user communication system into an interference-limited environment, where interference begins to dominate the performance efficiency.

In Fig. 1 and Fig. 2, we also find that the total network rate of centralized network is an increasing function of M (for a fixed value of N ), and the total network rates of decentralized networks corresponding to Nash equilibrium show an increasing slope before diminishing and reaching convergence. For some typical values of N , i.e., N = 5, 10 and 15, 1, when σ 2 = 0.1 the total network rate of decentralized networks are maximized approximately at M = 4, 9, 14, respectively; in Fig. 2, when σ 2 = 1 the total network rate of decentralized networks are maximized approximately at M = 6, 11, 16, respectively. It simply shows that different noise variance (in general channel condition) has a different impact on the decentralized network performance. This observation is fundamentally important for improving the energy efficiency in such a multi-user decentralized network: for a given area (given the number of receivers N and the current channel condition), there exists an optimal choice for the number of transmitters (denoted as M ⋆ ) to be put in the network. Roughly saying: when M > M ⋆ , the network is overloaded due to the increase of competition over limited spectrum resources; when M < M ⋆ , the network is operated at an unsaturated state, since the spectrum resources are not fully exploited.

CONCLUSIONS AND FUTURE WORKS

In this paper we described the scenario of multiple transmitters communicating with multiple receivers through inde-pendent parallel sub-channels as a strategic non-cooperative game. Each transmitter is modeled as a player in the game who decides, in a distributed way, the strategy of how to allocate its total power through these sub-channels. We studied the existence and uniqueness of Nash equilibrium which represents a natural outcome of the game. For any realization of a random channel matrix with i.i.d. continuous entries, we proved that there exists almost surely a unique Nash equilibrium. This result is a fundamental step to understand the resource allocation conflicts in a decentralized small-cell multi-user OFDM network, and moreover, it offers the possibility to predict the network performance outcomes. Finally, in our simulations results, we show how the average performance of the decentralized network behaves, and we compared it with the centralized network. Future works shall focus on the scaling analysis of the decentralized network performance. + (nonnegative real vector of length M N ), from Lemma 3.3, we have the following equation

φ (p) + ν -λ = 0 where φ (p) =        g 1,1 σ 2 + j p j,1 g j,1 g 1,2 σ 2 + j p j,1 g j,1 . . . g M,N σ 2 + j p j,N g j,N        ν =      ν1,1 ν1,2 . . . νM,N      λ =      λ1 λ2 . . . λM     
Now, assume that there exist two different Nash equilibria, e.g., p 0 , p 1 (p 0 = p 1 ), the following equation must also hold First, from (24) one can easily find that the value of sn = m gm,npm,n is unique for any Nash equilibrium point. Second, from [START_REF] Popescu | Water Filling May Not Good Neighbors Make[END_REF] we know that for any player m there must be a (positive) power allocated on a certain sub-channel n ′ , e.g., p 0 m,n ′ > 0, and we have two important observations:

p 1 -p 0 T p 0 -p 1 T α T      φ p 0 φ p 1 β + ν 0 -λ 0 ν 1 -λ 1 γ      = 0 (
1. Using (15), we have p 0 m,n ′ ν 0 m,n ′ = 0 ⇒ ν 0 m,n ′ = 0.

2. Using (25), we have p 0 m,n ′ ν 1 m,n ′ = 0 ⇒ ν 1 m,n ′ = 0. Then (13) yields

λ 0 m = λ 1 m = - g m,n ′ σ 2 + s n ′ , ∀m (26)
which shows that the dual variable λ is unique for any Nash equilibrium. Furthermore, by using (26) into (13), we derive ν 0 m,n = λ 0 m -gm,n σ 2 + sn = λ 1 m -gm,n σ 2 + sn = ν 1 m,n , ∀m ∀n which confirms that the dual variable ν is also unique for any Nash equilibrium. This completes the proof.

Ẑp = c. ( 20 ) 3 . 5 .

 2035 Lemma For any realization of random channel matrix G with i.i.d. continuous entries, if M Ñ > M + Ñ , |X | ≤ M + Ñ with probability 1.

Figure 1 :

 1 Figure 1: Total network rate, centralized vs. decentralized, σ 2 = 0.1

Figure 2 :

 2 Figure 2: Total network rate, centralized vs. decentralized, σ 2 = 1

2 σ 2 1 =

 221 23)from where we haveα T β = p 1p 0 T φ p 0 + p 0p 1 T φ(p 1 ) + M j=1 p 0 j,n gj,n σ 2 + M j=1 p 1 j,n gj,n ≥ 0 and α T γ = p 1p 0 T ν 0λ 0 + (p 0p 1 ) T ν 1λ ν 1 m,n + p 1 m,n ν 0 m,n ≥ 0From above, we find that (23) holds if and only if the two equalities are satisfied: α T β = 0 and α T γ = 0, which are equivalent to the following two equations, respectively: ∀m ∀n (25)

Once existence is established, it is natural to consider the characterization of the equilibrium set. Ideally, we would prefer there to be a unique equilibrium, but unfortunately this is not true for many game problems[START_REF] Fudenberg | Game Theory[END_REF]. A specific case of our game G, namely symmetric waterfilling game is studied in[START_REF] Altman | Closed Form Solutions for Symmetric Water Filling Games[END_REF], where the authors show infinite number of

ACKNOWLEDGMENTS

This work is partially supported by the "L'Agence Nationale de la Recherche (ANR)"via the program Inter Carnot, project TERROP.

APPENDIX

A. Proof of Lemma 3.4

Proof. Consider a Nash equilibrium p ∈ R M N ×1