Romain Couillet
email: romain.couillet@supelec.fr

Mérouane Debbah
email: merouane.debbah@supelec.fr

Uplink capacity of self-organizing clustered orthogonal CDMA networks in flat fading channels

In this paper, we derive a deterministic equivalent of the Shannon-transform of certain type of large unitary random matrices. This approximation is exploited to evaluate the uplink channel capacity of clustered orthogonal CDMA network. When non-uniform power allocation among the users of each cluster is allowed, we derive an explicit iterative waterfilling algorithm which, upon convergence, achieves the multiuser decoding capacity. In particular, we show that, in a selforganizing clustered orthogonal CDMA network, each cluster can optimize its power allocation policy independently of the other clusters at the expense of a small feedback overhead. Simulations corroborate the theoretical derivations.

I. INTRODUCTION

Previous releases of the universal mobile telecommunications system (UMTS) aimed to ensure quality of service for all users in the cells. The recent developments of 3G protocols however push more and more towards the maximization of the user sum rates in the cell. 3G systems are based on code division multiple access (CDMA) which use random i.i.d. codes in the uplink and incur interference among the users. Orthogonal codes (which reduce drastically the interference) are unfortunately not used as it would require some important overhead signaling to synchronize the users. One way of dealing with this issue, which is part of the recent cognitive radio incentive [START_REF] Mitola | Cognitive radio: making software radios more personal[END_REF], proposes to allow users to self-organize in small clusters (e.g. here CDMA-based clusters) or coalitions [START_REF] Coso | Virtual MIMO channels in cooperative multi-hop wireless sensor networks[END_REF], [START_REF] Saad | A distributed merge and split algorithm for fair cooperation in wireless networks[END_REF] to optimize their uplink communications to a central base station [START_REF] Jindal | Capacity of ad-hoc networks with node cooperation[END_REF], [START_REF] Cai | Allocate fair payoff for cooperation in wireless ad hoc networks using shapley value[END_REF]. Due to the very short distance between the users within the cluster, non-interfering transmissions can be performed through the use of orthogonal codes per cluster. Those networks are referred to as self-organizing networks, which we shall consider in the following.

Among the past contributions in the multi-cell CDMA case, a performance study of multi-cell orthogonal CDMA networks in frequency selective channels was carried out in [START_REF] Debbah | Downlink CDMA: to cell or not to cell[END_REF]. The authors suggest that frequency selectivity has a stronger negative impact on CDMA performance than the path loss; therefore multiple small CDMA cells might turn out more advantageous than a large single CDMA cell. The same conclusions were given in a similar uplink random CDMA study in [START_REF] Somekh | Spectral Efficiency of Joint Multiple Cell-Sites Processors for Randomly Spread DS-CDMA[END_REF]. In [START_REF] Shamai | The Impact of Frequency-Flat Fading on the Spectral Efficiency of CDMA[END_REF], the performance of single-cell CDMA in flat-fading channels with random codes was analyzed for different types of decoders when power optimization is performed at the transmitter. Most of these studies derive results under the assumption that the number of users per cell is large, since simpler derivations are possible using the theory of large random matrices [START_REF] Tulino | Random Matrix Theory and Wireless Communications[END_REF].

In this paper we derive results for the uplink capacity of a clustered CDMA self-organizing network when the CDMA code matrices are large and isometric. 1 Those results are based on tools from random matrix theory [START_REF] Tulino | Random Matrix Theory and Wireless Communications[END_REF] and free probability [START_REF] Voiculescu | Free random variables[END_REF], [START_REF] Biane | Free probability for probabilists[END_REF]. We will in particular derive two theorems which relate respectively the η-transform and the Shannon-transform (see definitions hereafter) of such a channel model to deterministic approximations independent of the code matrix realization. Another deterministic equivalent of the η-transform has been considered recently in the frequency selective case by Peacock et al. [START_REF] Peacock | Eigenvalue distributions of sums and products of large random matrices via incremental matrix expansions[END_REF] using a different approach, referred to as "incremental matrix expansions". However, the form of their deterministic approximations does not allow for a trivial calculus of the Shannon-transform, which is provided here. Up to the authors' understanding, both our and Peacock's results are different, though they are shown to tend to the same asymptotic expression. Our result on the Shannon-transform allows us first to derive explicit expressions for the uplink clustered orthogonal CDMA capacity and more importantly to produce an iterative water-filling algorithm for the cell users to achieve network capacity under per-cluster sum power constraint (where the power within a cluster is seen as a long term exchangeable utility, see [START_REF] Saad | A distributed merge and split algorithm for fair cooperation in wireless networks[END_REF] for more justifications). We will also propose an algorithm for each cluster to automatically reconfigure their power allocations, independently of the other clusters, when a user connects or disconnects to the cell.

The remainder of this paper unfolds as follows: in Section II, the fundamental mathematical results of this paper are provided. In Section III, the clustered CDMA self-organizing network model is introduced. In Section IV, the capacity achieving power allocation algorithm is presented. In Section V, numerical simulations are carried out. Finally, in Section VI, we draw our conclusions.

Notation: Capital boldface characters denote matrices (I N is the N × N identity matrix). Hermitian transpose is denoted (•) H . The operator det X represents the determinant of the matrix X. The function δ(x) is the Kronecker Dirac function such that δ(x) = 1 if x = 0 and equals 0 otherwise. The first derivative of the function f is denoted f ′ .

II. MATHEMATICAL PRELIMINARIES

Our main results rely on tools of random matrix theory analysis, the η-transform and the Shannon-transform, which we introduce in the following.

Definition 1 (η-transform): Let µ be a probability density function. The η-transform η(x) of µ is defined as

η(x) = 1 1 + xt µ(dt) (1)
Note that the η-transform is linked to the better known Stieltjes transform S(x) via the relation S(-1/x) = xη(x). Theorem 1: Let K, N be positive integers, {W i } i=1,...,K be K independent N × N Haar distributed complex random matrices and {D i } i=1,...,K be K diagonal N ×N non-negative matrices. Denote µ i the empirical distribution of the entries of D i . Then, for N large, K fixed, and for some x ≥ 0, the η-transform η(x) of the empirical eigenvalue distribution of

B N = K k=1 W k D k W H k satisfies approximately η(x) = 1 + x K k=1 β k (x) -1 (2)
where the functions β k (x), k ∈ {1, . . . , K} satisfy the K fixed-point equations

β k (x) = t 1 -xη(x)β k (x) + xη(x)t µ i (dt) (3)
Proof: The proof relies mostly on the asymptotic freeness [START_REF] Tulino | Random Matrix Theory and Wireless Communications[END_REF] of the matrices 1) and ( 5), we have

W k D k W H k when N grows to infinity. In this case, the R-transform R(x) of B N is asymptotically equal to the sum of the R-transform R k (x) of W k D k W H k . This R- transform is defined with respect to the η-transform as 2 R(-xη) = - 1 x (1 - 1 η ) (4) η(- 1 R + 1 x ) = xR + 1 (5) Denote η k and R k the η-and R-transform of W k D k W H k . Then, from (
xR k + 1 = 1 1 -t R k + 1 x µ k (dt) which leads to R k (x) = 1 x t R k (x) + 1 x -t µ k (dt) (6)
and, in particular, defining β k (x) = R k (-xη), we have

β k (x) = t 1 -xηβ k + xηt µ k (dt) (7)
which is exactly (3). Now, since R = K k=1 R k , using (4), this gives

η(x) = 1 + x K k=1 R k (-xη) -1 = 1 + x K k=1 β k -1 (8)
which completes the proof. The η-transform is of no direct practical use in our current analysis (though it might be essential in the analysis of MMSE decoders for CDMA downlink, see e.g. [START_REF] Peacock | Eigenvalue distributions of sums and products of large random matrices via incremental matrix expansions[END_REF]). To establish the capacity of uplink transmissions in CDMA networks, we need a deterministic approximation of the so-called Shannontransform, defined as follows.

Definition 2 (Shannon-transform): Let µ be some probability density function. We define the Shannon-transform V(x) of µ as

V(x) = log(1 + xt)µ(dt) (9)
The Shannon-transform is intimately linked to the capacity of a multi-dimensional channel whose matrix empirical eigenvalue density function is µ. Note also that the Shannon-transform is an integral of 1/x(1 -η(x)). From this observation, equivalently to Theorem 1, we derive in the following theorem a deterministic approximation for V(x). Theorem 2: Let B N be an N × N matrix as defined in the conditions of Theorem 1. We have the following asymptotic approximation of the Shannon-transform V(x) of B N as [START_REF] Dumont | On the Capacity Achieving Covariance Matrix for Rician MIMO Channels: An Asymptotic Approach[END_REF] where β k (x) and η(x) are given by ( 2) and [START_REF] Debbah | Downlink CDMA: to cell or not to cell[END_REF].

V(x) = log 1 + x K k=1 β k (x) + K k=1 log (1 + xη(x)[t -β k (x)]) µ k (dt)
Proof: The Shannon-transform satisfies [START_REF] Tulino | Random Matrix Theory and Wireless Communications[END_REF] V

(x) = x 0 1 u (1 -η(u))du
we therefore seek an integral form for the η-transform. Note first that

1 x (1 -η) = 1 x (1 -(1 + x K k=1 β k ) -1 ) = K k=1 β k η (11)
Also note from (3) that

1 -xηβ k = 1 -xηβ k 1 -xηβ k + xηt µ k (dt) (12)
and therefore that

1 = 1 1 -xηβ k + xηt µ k (dt) (13)
Now, for any k, the derivative along Recalling now ( 13) and ( 3), this yields

x of C k = log(1 - xηβ k + xηt)µ k (dt) is C ′ k = [-ηβ k -xη ′ β k -xηβ ′ k ] + [η + xη ′ ]t 1 -xηβ k + xηt µ k (dt) (14)
C ′ k = (-ηβ k -xη ′ β k -xηβ ′ k )•1+(η+xη ′ )•β k = -xηβ ′ k (15)
We also have

log(1 + x K k=1 β k ) ′ = η K k=1 β k + x K k=1 ηβ ′ k (16)
Adding this last expression to k C ′ k , we end up with the desired K k=1 β k η. Verifying that V(0) = 0, we finally obtain [START_REF] Dumont | On the Capacity Achieving Covariance Matrix for Rician MIMO Channels: An Asymptotic Approach[END_REF].

III. SYSTEM MODEL

A. Model

Consider a network composed of K clusters 3 and a single base station, each cluster being composed of a maximum of N users each. The users of cluster k ∈ {1, . . . , K} are assigned orthogonal spreading codes forming the columns of a unitary N × N Haar distributed matrix W k . Each cluster operates independently regarding the allocation of spreading codes, and therefore the W k matrices are considered independent. Depending on its position in cluster k, user n ∈ {1, . . . , N } induces a path-loss λ kn in the uplink transmission to the base station 4 . The path-losses in cluster k are gathered in the diagonal matrix Λ k = diag(λ k1 , . . . , λ kN ). Moreover we assume that the transmission channel is frequency flat, i.e. the propagation channel fade is constant over the transmission bandwidth. User n in cluster k transmits with power p kn , the collection of which is gathered in the matrices P k = diag(p k1 , . . . , p kN ). We additionally assume the base station experiences a background noise distributed as CN(0, σ 2 ). This situation is depicted in Figure 1.

B. Uplink capacity

Denoting D k = Λ k P k , the uplink (per-time sample) capac- ity C reads C(σ 2 ) = 1 N log det I N + 1 σ 2 K k=1 W k D k W H k ( 17
)
3 we assume that the clusters are formed in a self-organized manner based on the distance, see [START_REF] Saad | A distributed merge and split algorithm for fair cooperation in wireless networks[END_REF] 4 when there are less than N users in a specific cluster, we can freely write λ kn = 0 for the virtual users.

Denoting µ k (t) = 1 N N n=1 δ(t -λ kn p kn ), from Theorem 2, when N is large, we have approximately5

C(σ 2 ) = log 1 + 1 σ 2 K k=1 β k (σ -2 ) + K k=1 log 1 + σ -2 η(σ -2 )[t -β k (σ -2 )] µ k (dt) (18)
with η(x) and β k (x) satisfying the implicit Equation (3). We therefore have an expression of the uplink capacity which only depends on the path losses λ kn , k ∈ {1, . . . , K}, n ∈ {1, . . . , N }. This suggests that the choice of the spreading codes W k 's has asymptotically no impact on the achievable uplink sum rate. We will now seek a power allocation strategy which maximizes the capacity under cluster power constraints.

IV. SELF-ORGANIZED POWER ALLOCATION

In the following, based on relative fairness among clusters and on individual mobile terminal capabilities, cluster k is allocated a total allowed transmit P k ; therefore N n=1 p kn = P k .

A. Optimal power allocation

To maximize the system capacity with respect to the KN scalars p kn , we shall first show the following lemma, Lemma 1: The power allocation policy p kn = p ⋆ kn optimizing the deterministic capacity approximation (18) satisfies, for all k, n,

p ⋆ kn = α k - σ 2 -η ⋆ β ⋆ k λ kn η ⋆ + ( 19
)
where η ⋆ , β ⋆ k are the respective values of η and β k when C achieves its maximum, and α k is such that k p ⋆ kn = P k . Proof: To prove this lemma, we follow Prop. 4 in [START_REF] Dumont | On the Capacity Achieving Covariance Matrix for Rician MIMO Channels: An Asymptotic Approach[END_REF]. We need to ensure successively that, denoting C(σ 2 ) = V (p 11 , . . . , p KN , β 1 , . . . , β K , η), 6(i) ∂V /∂β k = ∂V /∂η = 0. (ii) C is strictly concave in the p kn . Part (ii) is in fact obvious once (i) is shown. Indeed, in this case, the successive derivatives of C along p kn equal those of V along p kn plus other terms (the derivatives of V along η and β k ) which are null. Therefore, since V is the sum of strictly concave functions of the p kn , C is strictly concave.

To show (i), we simply differentiate and observe that

∂V ∂η = k tσ -2 -β k σ -2 1 -ηβ k σ -2 + ησ -2 t µ k (dt) = 0 (20) ∂V ∂β k = ησ -2 + -ησ -2 1 -ηβ k σ -2 + ησ -2 t µ k (dt) = 0 (21)
both equalities stemming from the remarks ( 13) and (3).

From (ii), C admits a unique maximum and from (i), when β k and η have reached this maximum, the power allocation of the p kn 's reduces to maximizing

log 1 + σ -2 K k=1 β k + K k=1 log 1 + σ -2 η[t -β k ] µ k (dt) (22)
independently of η and the β k 's. This is, for each k, it reduces to maximizing log 1 + σ -2 η[t -β k ] µ k (dt), whose solution is the water-filling solution (19).

To achieve the capacity optimizing power allocation policy, similarly to [START_REF] Dumont | On the Capacity Achieving Covariance Matrix for Rician MIMO Channels: An Asymptotic Approach[END_REF], we propose the following iterative waterfilling algorithm, At initialization, for all k, p kn = P k N , η = 1, β k = 1. while the p kn 's have not converged do for k ∈ {1, . . . , K} do Set (η, β k ) as solution of ( 2), ( 3)

for n = 1 . . . , N do Set p kn = α k -σ 2 -ηβ k λ kn η +
, with α k such that n p kn = P k . end for end for end while This algorithm cannot be proved to converge. However, from the proof of Lemma 1 and following the arguments in [START_REF] Dumont | On the Capacity Achieving Covariance Matrix for Rician MIMO Channels: An Asymptotic Approach[END_REF], upon convergence, the algorithm is proved to converge to the optimal solution.

The power allocation algorithm allows now the base station to indicate to all users the uplink transmission powers, provided that it is aware of all the λ kn 's. We will show in the following that such a centralized approach is unnecessary to achieve optimal power allocation; precisely, we will show that each cluster k can independently perform power allocation regardless of the λ jn , j = k, provided that each cluster feedbacks a single real parameter to a neighboring cluster after the individual optimization process.

B. Self-organized capacity optimization

The main interest of the system model under consideration lies in the independence of every cluster in terms of code allocation: each cluster can behave autonomously, without the need for the base station to intervene in the code allocation policy. Now, in order to achieve the optimal uplink capacity throughout the K clusters, user n in cluster k must be allocated a transmit power p kn = p ⋆ kn . Those p kn 's can be computed explicitly at the base station or by any central entity aware of all the λ kn 's. However, this requires a large amount of overhead data to be transferred back and forth through the network; this comes at a non-negligible cost and goes against the philosophy of self-organized networks.

It is actually possible to circumvent this issue by performing successive local optimizations in every cluster and by successively transmitting a single parameter to the other clusters: this parameter is the updated evaluation of the ηtransform of B N . More explicitly, we consider the following algorithm,

end for end while

This algorithm only requires for the clusters to transmit the updated value of η at each step. Note that computing η requires only to know the updated value of k β k . This value does not need be additionally transmitted if we assume that each cluster stores the last value of η and infers from it the updated k β k .

This way, when a new user registers or a current user disconnects from a cluster, all the transmitted powers can be quickly updated throughout the network at a low feedback cost. Moreover, as will be observed in Section V, the convergence time of the algorithm is in general so fast that a single round of optimization over the K clusters is sufficient (and therefore the outer while loop is not needed) to achieve a satisfying uplink rate.

In the next section, the theoretical results discussed so far are confronted to simulations.

V. SIMULATION AND RESULTS

In this section, we verify the theoretical formula derived in Theorem 2, the power allocation algorithm derived from Lemma 1 and the self-organizing successive power allocation proposed in Section IV-B. Throughout this section, we

= -α k ([n -1]/[N -1]
) dB, with α 1 = 10 dB, α 2 = 20 dB, α 3 = 30 dB; this allows to have in each cluster a linear range (in dB) of users with relative path losses varying from 0 dB to (-α k ) dB.

In Figure 2, we present simulation results based on the averaging of 100 Monte Carlo simulations when uniform power allocation is applied or when the allocated powers are issued from random Monte Carlo trials (but satisfy the power constraints); for the latter, for each of the 100 simulations, we define the system capacity as the maximal rate achieved over 10, 000 random realizations of vectors {p 11 , . . . , p KN } ordered in such a way that p kj < p k,j+1 . 7 Those are compared against the solutions given by Theorem 2 with uniform power allocation and the iterative water-filling algorithm, respectively. We observe a perfect fit, even for this low N = 8 value, between the simulated and theoretical results in the uniform power allocation case. As for the optimal power allocation, we observe an almost perfect fit, which will surely be more accurate if one increases the number of random power allocations of the Monte Carlo method.

In Figure 3, we apply the self-organizing power allocation algorithm under the same constraints as in Figure 3, when 0, 1 or an infinite number of rounds (at least a sufficient number of rounds to ensure convergence of η) are performed). We observe surprisingly enough that a single round is enough to already achieve the optimal capacity limit of the complete algorithm. As a consequence, the algorithm is extremely fast at providing high performance power allocation among the K clusters. For the sake of comparison, we also propose in Figure 3 the so-called 'local power allocation' policy applied 7 this allows to significantly reduce the number of required simulations by discarding obviously wrong solutions. by each cluster as though it were alone in the network, which is maximized by the classical water-filling solution, i.e. with powers

p kn = α ′ k - σ 2 λ kn + ( 23
)
where α ′ k ensures that n p kn = P k . We observe that, while in the low SNR region, this local strategy is reasonable, it is no longer the case in the high SNR regime. Since the cost of the transmission of the datum η is obviously very low in this regime, the self-organizing power allocation policy is even more interesting.

VI. CONCLUSION

In this paper, we provided two novel theorems relating the uplink capacity of a large clustered orthogonal frequency flat CDMA network communicating to a single base station to deterministic capacity approximations. Those actually very accurate approximations only depend on the path losses of the users' uplink signals. Moreover, we provided an efficient algorithm for the clusters to perform local power allocation in order to maximize the system throughput at the cost of a single datum exchange in the network. Simulation results show a perfect fit between Monte Carlo and theoretical results even when the system dimension is not very large.

Fig. 1 .

 1 Fig. 1. Self-organizing clustered CDMA network

2 )

 2 Fig. 2. Uplink capacity of a K = 3-clustered orthogonal CDMA, uniform and optimal power allocation, N = 8. For user n in cluster k, λ kn = -α k ([n -1]/[N -1]) dB, α 1 = 10 dB, α 2 = 20 dB, α 3 = 30 dB.

2 )

 2 Fig. 3. Uplink capacity of a K = 3-cluster orthogonal CDMA, iterative water-filling power allocation, N = 8. For user n in cluster k, λ kn = -α k ([n -1]/[N -1]) dB, α 1 = 10 dB, α 2 = 20 dB, α 3 = 30 dB.

by isometric, we mean a non-necessarily square matrix with orthogonal columns.

for readability, we will use the shortcut notation f = f (x) when no confusion is possible.

note that the capacity is the Shannon-transform of σ -2 and not of σ 2 .

V can be seen as a function of (N + 1)K + 1 independent variables.


Les différentes catégories de cellules souches et la notion de plasticité cellulaire

Une cellule souche est une cellule indifférenciée capable:

• d'auto-renouvellement (multiplication à l'identique pour produire de nouvelles cellules souches)

• de se différencier en cellules matures (phénotypiques, morphologiques et fonctionnelles) ii) une meilleure compréhension des dérégulations affectant les progéniteurs myogéniques chez les espèces de production ;

iii) l'étude de l'impact de la vaccination sur la croissance et la composante inflammatoire chez les animaux d'élevage iv) la mise à disposition d'un modèle d'étude pour le processus de maturation de la viande.

La viande in vitro

Scénario 1, les MS (micro supports) sont utilisés comme substrats temporaires pour l'expansion des Cellules Souches et doivent être retirés à la fin du processus. Il existe deux conditions préalables importantes dans ce cas, à savoir la nécessité pour les MS de fournir un rendement de détachement élevé et une séparation facile des cellules.

Scénario 2, les MS servent également de substrat temporaire pour l'expansion des CS mais ils peuvent être dégradés lors d'une étape préalable. Dans ce cas, l'étape de dissociation peut être remplacée par une étape de dégradation du MS pour obtenir une suspension cellulaire unique.

Scénario 3, les MS sont composés de matériaux comestibles et peuvent donc être intégrés dans le produit final. Ils doivent être ici conformes aux réglementations pour une utilisation en tant qu'ingrédient ou additif alimentaire.

Bodiou V, Moutsatsou P, Post MJ (2020) Microcarriers for upscaling cultured meat production. Front Nutr 7:10. doi: 10.3389/fnut.2020.00010

  Assurer l'homéostasie des organes/tissus grâce à leurs propriétés d'auto-renouvellement et de différenciation FONCTION Quelques méthodes pour étudier les cellules souches: et Raff, 2000, Science, 289:1754 La plasticité Etude de la plasticité mammaire pour l'identification de biomarqueurs d'une meilleure persistance de la lactation chez la vache laitière Les sein de la glande mammaire Taux moyen de prolifération / jour: 0.3% Taux moyen d'apoptose / jour: 0.56% Taux de renouvellement cellulaire au cours d'une lactation: 50% Mésenchyme Cellules endodermiques Qu'est ce qui détermine le devenir des cellules ? pas seulement liés à la CEM en activité !!! Paramètres déterminant la quantité de lait Importance de la plasticité cellulaire au sein de la glande dans le contrôle de la persistance de la lactation Objectifs du projet 1-Caractérisation de marqueurs de surface spécifiques des différents types cellulaires présents au sein de la glande mammaire chez la vache laitière. 2-Phénotyper ces différents types cellulaires chez la vache laitière au cours d'un cycle de lactation afin d'étudier la répartition de ces cellules mammaires en lien avec un stade physiologique et un statut hormonal précis. Seules les populations CD49 f high CD24 pos and the CD49 f high CD24 neg sont capables de former des mammosphères in vitro (Finot et al., 2018) Tri cellulaire et test de fonctionnalité des populations cellulaires Test de fonctionnalité des cellules après tri Transplantation de mammosphères dans des souris nude Balb/c AnNRj-Foxn1nu/nu Utilisation des mammosphères Utilisation de culture primaire Reflet du stade de lactation Fonctionnalité proche de l'in vivo Utilisation des modèles in vitro de tissus musculaires pour l'étude des animaux d'élevage et perspectives d'application i) la caractérisation approfondie des progéniteurs myogéniques dans un large éventail d'espèces, notamment le Porc, la volaille et le Poisson pour étudier la notion de diversité cellulaire résidente dans les tissus ;


Romain Couillet

Mérouane Debbah

Outage performance of flexible OFDM schemes in packet-switched transmissions

In this paper, α-OFDM, a generalization of the OFDM modulation, is proposed. This new modulation enhances the outage capacity performance of bursty communications. The α-OFDM scheme is easily implementable as it only requires an additional time symbol rotation after the IDFT stage and a subsequent phase rotation of the cyclic prefix. The physical effect of the induced rotation is to slide the DFT window over the frequency spectrum. When successively used with different angles α at the symbol rate, α-OFDM provides frequency diversity in block fading channels. Interestingly, simulation results show a substantial gain in terms of outage capacity and outage BER in comparison with classical OFDM modulation schemes. The framework is extended to multiuser/multi-antenna/multicellular OFDM based standards. Practical simulations, in the context of 3GPP-LTE, called hereafter α-LTE, sustain our theoretical claims.

I. INTRODUCTION

With the recent growth of wireless communications and the increasing demand for high transmission rates, orthogonal frequency division multiplexing (OFDM) is being considered the desirable modulation scheme of most future wireless communication technologies. Many wireless standards [START_REF]ANSI/IEEE Std[END_REF]-[3] have already rallied in favour of OFDM. The attractive features of OFDM are numerous; a key advantage over other classical modulation schemes is that OFDM can be designed to reach a high spectral efficiency. The main advantage in practice lies in the flat fading aspect of the channel that facilitates equalization at the receiver side. This property originates from the cyclic prefix (CP) addition, prior to signal transmission, that allows to model the channel 1 ST-NXP Wireless -505 Route des Lucioles, 06560 Sophia Antipolis 2 Chair Alcatel Lucent, Supélec, 2229 Route des Cretes, 91192 Gif-sur-Yvette as a circulant matrix in the time-domain [START_REF] Bingham | Multicarrier modulation for data transmission: An idea whose time has come[END_REF]. Those circulant matrices are diagonalizable in the Fourier basis, hence the seemingly flat fading aspect of the channel in the frequency domain.

The analysis of circulant matrices, and their generalization, is the starting point of the present work.

In addition to the demand for high transmission rates, recent wireless standards have also steadily moved from the connected circuit-switched transmission mode to the bursty packetswitched transmission mode. The main drawback of the packet-switched mode arises when the transmission time is less than the channel coherence time [START_REF] Biglieri | Fading channels: Information-theoretic and communications aspects[END_REF], as the channel is then static over the communication duration. Indeed, for any target transmission rate R, there exists a non-null probability that the channel is so ill-conditioned that the resulting achievable rate is less than R. This outage probability is especially non negligible in OFDM when the channel delay spread is small, or equivalently when the channel coherence bandwidth [START_REF] Biglieri | Fading channels: Information-theoretic and communications aspects[END_REF] is large. These observations have led to consider methods which provide channel diversity to protect the transmitted symbols from deep channel fading. Among those methods, [START_REF] Viswanath | Opportunistic beamforming using dumb antennas[END_REF] proposed a dynamic beamforming scheme using multiple antennas, known as dumb antennas, which induces fast channel variations over time. The relevant effect of this method is to increase the channel diversity during the transmission period. Recently, [START_REF] Müller | Compact MIMO Receive Antennas[END_REF] introduced a compact MIMO system which mimic the behaviour of multiple antennas from one single rotating antenna, thus producing additional degrees of freedom. A desirable method to cope with the outage problem in OFDM is to provide diversity in the frequency domain; this is accomplished by the cyclic delay diversity method [START_REF] Dammann | Performance of low complex antenna diversity techniques for mobile OFDM systems[END_REF] by introducing different symbol delays on an array of transmit antennas. However, all those methods require additional antennas to provide channel diversity, or are still difficult to implement in practice.

In the following work, we first introduce a new single antenna modulation scheme called α-OFDM, which produces frequency diversity with neither a tremendous increase in complexity, nor additional antenna requirement. The main idea is to successively transmit data on different sets of frequency carriers, with no need for an higher-layer frequency allocation scheduler. In particular, we show that α-OFDM allows to flexibly reuse adjacent frequency bands by properly adjusting a single (rotation) parameter: α.

This work is then extended to the multi-user OFDMA (multiple access OFDM) case, where 3GPP-Long Term Evolution (LTE) is used as benchmark for comparison against classical OFDM.

Although both α-LTE and LTE with frequency hopping techniques seem to be similar, the α-OFDM algorithm is more flexible and can be adapted on a per-OFDM symbol rate without advanced scheduling methods. α-OFDM can also be seen as yet another technique applicable for software defined radios [START_REF] Mitola | Cognitive radio: making software radios more personal[END_REF], [START_REF] Haykin | Cognitive Radio: Brain-Empowered Wireless Communications[END_REF] for it enables flexible frequency management by means of digital processing.

The rest of the paper unfolds as follows: in Section II, we study the mathematical extension of circulant matrices, which is at the heart of the classical OFDM modulation and we introduce the novel α-OFDM scheme. Practical communication systems, based on α-OFDM, are then introduced in Section III and are shown to benefit from the frequency diversity offered by α-OFDM in terms of outage capacity. The theoretical claims are validated by simulations in Section IV. We then introduce in Section V some practical applications and quantify the outage performance gain. In particular we propose an extension for the 3GPP LTE standard, called hereafter α-LTE, which is further extended into a general α-OFDMA scheme. Finally, conclusions are provided in Section VI.

Notations: In the following, boldface lower and capital case symbols represent vectors and matrices, respectively. The transposition is denoted (•) T and the Hermitian transpose is (•) H . The operator diag(x) turns the vector x into a diagonal matrix. The symbol det(X) is the determinant of matrix X. The symbol E[•] denotes expectation. The binary relation symbol X|Y means that Y is divisible by X.

II. SYSTEM MODEL

The basic idea of this work relies on a mathematical generalization of the diagonalization property of circulant matrices. More specifically we introduce in the following the so-called (ρ, α)-circulant matrices which can be diagonalized in a modified Fourier basis, hereafter referred to as the (ρ, α)-Fourier basis. This will entail the introduction of the α-OFDM scheme, which in turn generalizes the OFDM concept and whose key physical property is to provide dynamic subcarrier frequency shift controlled by the parameter α.

A. Mathematical Preliminaries

Let us first recall the diagonalization property of circulant matrices.

Definition 1: A circulant matrix H 0 of size N with base-vector h = {h 0 , . . . , h L-1 } ∈ C L (L ≤ N) is the N × N Toeplitz matrix H 0 =                  h 0 0 . . . 0 h L-1 . . . h
. . . 0 h L-1 . . . h 1 h 0                  . ( 1
)
Circulant matrices can be diagonalized by F N , the discrete Fourier transform (DFT) matrix of size N. The eigenvalues in their spectral decomposition are formed by the DFT of their first column [h 0 , . . . , h L-1 , 0, . . . , 0] T [START_REF] Gray | Toeplitz and Circulant Matrices: A Review[END_REF].

Those circulant matrices actually enter a broader class of matrices, which we will call (ρ, α)circulant matrices. Those are defined as follows,

Definition 2: For z = ρe iα ∈ C, (ρ, α) ∈ R + × R, we call an N × N matrix H (ρ,α) (ρ, α)- circulant if it is of the form H (ρ,α) =                  h 0 0 . . . 0 ρe iα h L-
. . . 0 h L-1 . . . h 1 h 0                  . (2)
This is a matrix with first column [h 0 , . . . , h L-1 , 0, . . . , 0] T , and subsequent columns successive cyclic shifts of this column. The upper triangular part of the matrix is multiplied by ρe iα .

Proposition 1: All N × N (ρ, α)-circulant matrices are diagonalizable by the (ρ, α)-Fourier matrix F N,(ρ,α) , defined as

F N,(ρ,α) = F N • diag 1, ρ 1 N e iα 1 N , . . . , ρ N-1 N e iα N-1 N (3)
Hence we denote

diag (H α (0), . . . , H α (N -1)) = F N,(ρ,α) H (ρ,α) F -1 N,(ρ,α) (4)
where the diagonal elements are given by the (ρ, α)-DFT of the first column of H (ρ,α)

[H α (0), . . . , H α (N -1)] T = F N,(ρ,α) [h 0 , . . . , h L-1 , 0, . . . , 0] T (5)

Proof: The proof is an adaption of a proof from Gray [START_REF] Gray | Toeplitz and Circulant Matrices: A Review[END_REF], Sec. 3.1, where the author characterizes the eigenvectors and eigenvalues of a circulant matrix. Given a (ρ, α)-circulant matrix H, the eigenvalues H α (m) and the eigenvectors v m of H are the solutions of

Hv = φv (6)
This can be written in scalar form as the system of equations, for m = 0, . . . , N -1,

min(m,L-1) k=0 h k v m-k + ρe iα L-1 k=m+1 h k v N -(k-m) = φv m (7)
In [START_REF] Bingham | Multicarrier modulation for data transmission: An idea whose time has come[END_REF], we use the convention that the second summation is zero if

L -1 < m + 1.
Let us assume v k = ̺ k and replace it in [START_REF] Bingham | Multicarrier modulation for data transmission: An idea whose time has come[END_REF]. Cancellation of ̺ m yields min(m,L-1)

k=0 h k ̺ -k + ρe iα ̺ N L-1 k=m+1 h k ̺ -k = φ (8)
Thus, choosing ̺ such that ρe iα ̺ N = 1, we obtain the eigenvalue

φ = L-1 k=0 h k ̺ -k (9)
and the associated (normalized) eigenvector

v = β 1, ̺, ̺ 2 , . . . , ̺ N -1 T (10) with β = 1/ √ N if ρ = 1 and β = 1-1 |α| 1/N 1-1 |α|
otherwise. Then we choose ̺ m as the complex

N th root of ρe -iα , ̺ m = 1 ρ 1/N e -iα
/N e 2πim/N , obtaining the eigenvalue

H α (m) = L-1 j=0 h j ρ j N e iα j N e -2πij m N ( 11
)
and eigenvector

v m = β 1, e -iα 1 N ρ 1 N e 2πi m N , . . . , e -iα N-1 N ρ N-1 N e 2πim N-1 N T (12)
such that

Hv m = H α (m)v m , m = 0, . . . , N -1 (13)
From ( 11), we can deduce immediately an inverse transform (analog to the inverse Fourier transform) to obtain the elements of the first column of H from the eigenvalues

h ℓ = 1 ρ ℓ N e -iα ℓ N 1 N N -1 m=0 H α (m)e 2πim ℓ N , ℓ = 0, . . . , L -1 (14)
Therefore, (ρ, α)-circulant matrices only differ from circulant matrices by (i) an additional rotation matrix turning the discrete Fourier transform matrix F N into a (ρ, α)-Fourier matrix F N,(ρ,α) and (ii) an additional rotation of the upper-triangular part of the circulant matrix.

When applied to the OFDM communication chain (both at the transmit and receive sides), these alterations entail interesting physical properties which we introduce in the following.

B. OFDM

Let us first recall the classical OFDM modulation scheme and introduce our notations, before we present the more general α-OFDM framework. Consider a regular OFDM transmission

r = H 0 F H s + n (15)
where F N is rewritten F for the sake of readability. Therefore H 0 is diagonalizable by the Fourier matrix F, with diagonal elements the discrete Fourier transform of the first column [h 0 , . . . , h L-1 , 0, . . . , 0] T . This is simply obtained by multiplying r in (15) by F. The distribution of the noise does not change, since a unitary transformation of a Gaussian i.i.d. (independent and identically distributed) vector is still a Gaussian vector of same mean and variance. Thus,

F • r = diag (H 0 (0), . . . , H 0 (N -1)) s + n (16) with H 0 (•) the DFT of the first column of H 0 , H 0 (m) = L-1 j=0 h j e -2πij m N (17) F -1 α x N x N-D+1 x 1 x 2

DAC

To Channel

e iα x N-D+1 e iα x N s 1 s 2 s N-1 s N P/S

C. α-OFDM

Due to the mathematical generalization of circulant matrices previously proposed, the classical OFDM scheme can in turn be generalized in form of the α-OFDM scheme which we define as follows. At the transmitter side, the α-OFDM scheme is similar to the classical OFDM modulation, except that it requires

• the frequency-domain OFDM signal vector s to be first multiplied by the matrix diag 1, e -iα/N , . . . , e -iα(N -1)/N (18) after the inverse DFT (IDFT) stage in the OFDM transmission chain,

• the time-domain symbols of the CP to be multiplied by the constant z = ρe iα where we set here ρ = 1.

Note that, if ρ were chosen different from 1, F N,(ρ,α) would not be unitary, which would generate noise amplification at the receiver side.

Hence, the time-domain received signal r reads

r = H α F H α s + n (19)
where F α is a simplified notation for F N,(1,α) and H α is the transmission channel matrix. The α-OFDM transmission chain is depicted in Figure 1. Considering the mathematical development previously presented, the channel matrix H α associated to this model is (1, α)-circulant (for short, this will now be referred to as α-circulant) and therefore diagonalizable in the α-Fourier domain.

At the receiver side, the CP is discarded like in the classical OFDM scheme and the receive useful data are multiplied by the matrix F α . Mathematically, this is

F α • r = diag (H α (0) , . . . , H α (N -1)) s + n (20) with H α (•) the α-DFT of the first column of H α : H α (m) = L-1 j=0 h j e -2πi j N (m-α 2π ) (21)
≃ H 0 ({⌈m - α 2π ⌉} N ) (22)
where {•} N denotes the modulo N operation and ⌈x⌉ denotes the closest integer larger than x.

The approximation (22) deserves some more comments: classical OFDM systems are designed such that the typical channel coherence bandwidth is greater or equal to one subcarrier spacing.

Therefore, the channel frequency gain decimal at decimal frequency {m -α 2π } N is usually well approximated by that at frequency ⌈{m -α 2π } N ⌉. Remark 1: Note that H α (m) is a frequency shifted version of H 0 (m). Thus α-OFDM introduces a fractional frequency shift α/2π to the channel {H 0 (0), . . . , H 0 (N -1)} as shown in

III. OUTAGE CAPACITY ANALYSIS

The transmission rates achievable in bursty communications cannot be evaluated with Shannon's ergodic capacity which involves infinite delay transmission. Instead, the transmission capabilities of a bursty system are usually measured through the rate achievable (100-q)% of the time. This rate C 0 is known as the q%-outage capacity and verifies P (C > C 0 ) = (100 -q)/100, with C the Shannon's capacity [START_REF] Shannon | A Mathematical Theory of Communication[END_REF] for fixed channels.

A. α-OFDM capacity

The normalized, i.e. per subcarrier, capacity C of a regular OFDM system 1 for the fixed frequency-domain channel {H 0 (0), . . . , H 0 (N -1)} reads

C = 1 N N -1 m=0 log 1 + |H 0 (m)| 2 σ 2 (23)
while the capacity for α-OFDM is

C α = 1 N N -1 m=0 log 1 + |H α (m)| 2 σ 2 (24)
In the rest of this document, we refer to capacity as the normalized system capacity. The system-wide capacity will be referred to as total capacity.

Remind that the channel coherence bandwidth of an OFDM system is at least as large as the subcarrier spacing 2 . Therefore, as already mentioned,

H α (m) ≃ H 0 ( m -α 2π N ).
From the expressions (23) and (24), we then conclude that C ≃ C α in some sense. As a consequence, α-OFDM does not bring any gain, either in terms of ergodic or outage capacity in this simple setting.

Nonetheless, for various reasons, such as the introduction of frequency guard bands or oversampling at the receiver, many OFDM systems with a size-N DFT use a limited number N u < N of contiguous subcarriers to transmit useful data. Those are referred to as useful subcarriers. In such schemes, the OFDM capacity for the fixed channel H 0 trivially extends to

C = 1 N u Nu-1 m=0 log 1 + |H 0 (m)| 2 σ 2 (25)
while the equivalent for α-OFDM is

C α = 1 N u Nu-1 m=0 log 1 + |H α (m)| 2 σ 2 (26)
The sizes of the useful bandwidths of both OFDM and α-OFDM are the same but their locations differ and therefore C = C α . However, using different frequency bands is only acceptable in practice if these bands are not protected, which is rarely the case in classical OFDM systems. In the subsequent sections, we will propose practical schemes to overcome 1 also called spectral efficiency 2 otherwise the channel delay spread would be longer than the OFDM symbol duration.

this limitation at the expense of a small bandwidth sacrifice: this is, we will propose to discard subcarriers on the sides of the available bandwidth to allow the useful transmission band to 'slide' over the total valid bandwidth using different values for α. In the remainder of this paper, we therefore consider OFDM systems with N subcarriers transmitting data over N u contiguous useful subcarriers whose locations are subject to different constraints.

B. α-OFDM-based systems 1) α-OFDM#1: First consider a single-user OFDM system with the N -N u non-useful subcarriers gathered into a contiguous subband, up to a circular rotation over the total bandwidth.

This constraint allows to use α-OFDM for any α ∈ R as long as the N u useful subcarriers are gathered in a contiguous band. We then naturally introduce the α-OFDM#1 scheme as follows.

Let M ∈ N and M be a set of cardinality M defined as

M = 0, 2π N M , . . . , 2π N(M -1) M (27)
where α ∈ R is a priori known both to the transmitter and to the receiver. The consecutive time-domain symbols are transmitted successively using a (2πkN/M)-OFDM modulation, for k ranging from 0 to M -1, i.e. denoting s (k) the k th transmitted symbol, s (1) is sent using 0-OFDM, then s (2) is sent using (2πN/M)-OFDM and so on. At the receiver side, the corresponding (2πkN/M), k ∈ N, rotation values are used to align to the transmission band and to decode the received symbols.

The per-subcarrier capacity C #1 of the α-OFDM#1 scheme is

C #1 = 1 MN u Nu-1 m=0 α∈M log 1 + |H α (m)| 2 σ 2 (28)
Note that the total capacity C (Wu) for a system of bandwidth W and useful bandwidth W u = W • N u /N scales with N u , or equivalently with W u ,

C (Wu) = W u • C #1 (29)
2) α-OFDM#2: Assuming perfect channel state information at the transmitter (CSIT), an improved scheme, α-OFDM#2, can be derived from α-OFDM#1, which selects among the M values of M the one that maximizes the instantaneous capacity. Its capacity C #2 reads

C #2 = 1 N u max α∈M Nu-1 m=0 log 1 + |H α (m)| 2 σ 2 . ( 30
)
This scheme can prove useful when the channel coherence time is sufficiently long for the receiver to feed back channel information. The transmitter will then be allowed to transmit over one particular subchannel among the M subchannels provided by the different bandwidth shifts.

3) α-OFDM#3: Consider now that the protected unused subcarriers can only be placed on the lowest frequency and highest frequency sides of the bandwidth, as is the case of most practical systems. Following the same structure as the α-OFDM#1 scheme, we introduce α-OFDM#3, based on a set M of M values for α which are constrained by N u -N ≤ α 2π ≤ N -N u . The introduction of this particular scheme is relevant as the basis for some practical applications studied in Section V. The capacity C #3 is identical to Equation (28) with the additional constraint on the set of rotations M. Therefore, C #3 ≤ C #1 . This upper-band on the capacity C #3 is needed for further analysis in the following.

C. Capacity Gain

The effect of the α-OFDM#k schemes is to successively slide the OFDM DFT window by different frequency shifts. This generates channel diversity that is highly demanded in outage scenarios. The following lemma shows that the (per subcarrier) fixed-channel capacity limit of α-OFDM#1, for any N u < N, equals the (per subcarrier) fixed-channel capacity of an OFDM system, for N u = N. The ratio between the total α-OFDM#1 capacity and the total OFDM capacity is then N u /N for a proper choice of M. In contrast, this ratio is on average less than N u /N when a single value for α is used (or similarly when a pure OFDM scheme with N u < N subcarriers is used). This claim is the most important result of this paper and is summarized in the following lemma.

Lemma 1: Consider a single user OFDM system with N u useful subcarriers and N total subcarriers without CSIT. Apply α-OFDM#1 with a pattern M of cardinal M defined in (27) with the constraint N|{M • gcd(N, N u )}, gcd(x, y) being the greater common divider of x and y. Assuming the channel coherence time is more than M times the OFDM symbol duration, the

α-OFDM#1 capacity C #1 satisfies C #1 = C (31) with C = 1 N N -1 m=0 log 1 + |H 0 (m)| 2 σ 2 (32)
the capacity of the equivalent OFDM system under the fixed channel H 0 .

Proof: Recall that the α-OFDM#1 capacity reads

C #1 = 1 MN u Nu-1 m=0 α∈M log 1 + |H α (m)| 2 σ 2 (33)
Since all α ∈ M are multiple of 2π, the frequency shifts correspond to subcarrier jumps.

Therefore, after M symbols, every subcarrier indexed by i ∈ {1, . . . , N} has been used an integer number of times λ i ≤ M. This allows to rewrite C #1 :

C #1 = 1 MN u N -1 m=0 λ m log 1 + |H 0 (m)| 2 σ 2 (34) with N -1 m=0 λ m = MN u . The variable change β m = N λm M Nu leads to C #1 = 1 N N -1 m=0 β m log 1 + |H 0 (m)| 2 σ 2 (35) ≤ 1 N log N -1 m=0 β m (1 + |H 0 (m)| 2 σ 2 (36) = 1 N log N + N -1 m=0 β m |H 0 (m)| 2 σ 2 (37) ( 38
)
where the inequality (37) stems from the concavity of the log function.

Without CSIT, one has to assume equal gains at all subcarriers. The best β m allocation strategy requires then to maximize all β m with sum constraint m β m = N. This leads to ∀m ∈ [0, N -1], β m = 1. This is the equality case of Equation (37) which is achievable when N|{M • GCD(N, N u )}.

Note that this proof cannot be adapted to the α-OFDM#3 scheme for the β m 's are restricted by the constraint on M. This means that the achievable outage capacity for α-OFDM#3 being less than the outage capacity for α-OFDM#1, Lemma 1 provides an upper bound on the achievable capacity for α-OFDM#3. As a consequence, if an OFDM system is constrained to use N u consecutive subcarriers with sidebands, using α-OFDM allows to reach at most the same outage spectral efficiency as the classical OFDM scheme using all N subcarriers, which, on average, is not achievable with classical OFDM over a fixed set of N u subcarriers. In the following section, simulations are carried out for different scenarios to analyze the potential outage gains of α-OFDM-based schemes, before practical applications are discussed. It will in particular be observed that the upper-bound on C #3 is actually very tight in non-degenerated scenarios.

The consequence of Lemma 1, in terms of performance in block fading channels, is that the outage capacity of α-OFDM#1 equals the outage capacity of the classical OFDM scheme, might all subcarriers in the bandwidth be useful (i.e. N u = N). The close behaviour of α-OFDM#3 compared to α-OFDM#1 implies that this scheme also achieves outage capacity performances close to that of the OFDM modulation with all subcarriers in use. On the contrary, when classical OFDM is used on a restricted set N u < N, the outage channels may lead to tremendous losses in outage capacity performance; in particular, if the channel coherence bandwidth is close to N u subcarrier spacings, then the outage scenarios correspond to deep fades in the exact position of the N u subcarriers, which entails considerable performance loss.

IV. SIMULATION AND RESULTS

The 3GPP-LTE OFDM standard is considered in most simulations. We present results for the 1.4 MHz bandwidth (N u = 76, N = 128) and the 10MHz bandwidth (N u = 602, N = 1024). In LTE, the null subcarriers on the bandwidth sides do not correspond to guard bands but arise from oversampling at the receiver; as a result, the N -N u empty subcarriers belong to adjacent users.

We study the outage capacity and bit error rate (BER) gain assuming that we were allowed to slide the spectrum over those bands while still sending data on N u consecutive subcarriers. This seemingly awkward assumption will prove necessary for the concrete applications in Section V, in which all users will be allowed to jointly slide their useful bandwidths; this will turn the adjacent illegal spectrum into dynamically usable frequencies. Channels are modeled either as exponential decaying with mean zero and unit variance or as LTE standardized channels [START_REF] Ericsson | Proposal for LTE channel models[END_REF] with the following characteristics,

• Extended Pedestrian A model (ETA), with RMS delay spread 43ns

• Extended Vehicular A model (EVA), with RMS delay spread 357ns

• Extended Typical Urban model (ETU), with RMS delay spread 991ns Outage Capacity of α-OFDM#1 vs OFDM 3GPP-LTE EPA Channel (N=1024, Nu=602)

Plain OFDM α-OFDM#1 (M=2) α-OFDM#1 (M=8)
Fig. 3. α-OFDM#1 Outage Capacity in 3GPP-LTE EPA over all available subcarriers, providing already a high diversity gain 3 . Note also that this gain is extremely dependent on the channel length. The shorter the channel delay spread, the more significant the capacity gain.

A. Single User with multiple antennas

Figure 4 depicts the outage capacity gain of α-OFDM schemes versus OFDM in Rayleigh channels using multiple antennas at the transmitter. For N t transmit and receive antennas, the ergodic capacity for the classical OFDM schemes (in plain lines in the Figure ) reads

C = 1 N u log det I Nu + 1 σ 2 H (Nu) 0 (39)
3 however, since a part of the subcarriers are used twice in both transmissions while the others are only used once, the optimal bandwidth usage of the proof in lemma 1 is not achieved; this requires to transmit as much data on all subcarriers and then demands larger M sets.

where H (Nu) 0

is the N u × N u matrix extracted from the N u useful rows/columns of H 0 . While the respective capacity expressions of the α-OFDM1 and α-OFDM2 schemes are

C #1 = 1 MN u α∈M log det I Nu + 1 σ 2 H (Nu) α (40)
C #2 = 1 N u max α∈M log det I Nu + 1 σ 2 H (Nu) α (41)
where

H (Nu) α
is similarly the N u × N u matrix taken from the N u useful rows/columns of H α .

In the simulations, we use N t = 1, 2, 4 transmit and receive antennas.

It is observed that the usage of additional diversity antennas can be partially or fully replaced by α-OFDM schemes. Channel diversity from the space domain can then be traded off with diversity in the time-domain thanks to α-OFDM, reducing then the cost of extra antennas.

However, particular care is demanded to appreciate those results. Indeed, the performance gains in terms of outage capacity and BER are highly dependent on the exact definition of the outage conditions and the channel delay spread. In mobile short-range wireless communications, bursty transmissions using α-OFDM can therefore be performed at higher rates.

B. Multi-cell systems

In multi-cell systems, inter-cell interference engenders outage situations whenever the terminal's channel to the base station is in outage and the channels to the interferers in adjacent cells are strong. Using α-OFDM in its own cell, not only will the terminal diversify its own channel but it will also face different interference patterns. Therefore, it is even less likely to simultaneously be confronted to bad channel conditions in its own cell and strong interference over M consecutive α-OFDM symbols. Figure 5 provides this analysis, in which a receive user faces interference from a single adjacent cell at varying signal to interference ratios (SIR). The SNR is fixed to SNR = 15 dB. The channel lengths at both user's cell and the interfering cell is set to L = 3 while N u = 76 and the DFT size is N = 128. At high SIR, one finds again the α-OFDM capacity gain observed in Figure 3. Around SIR = 20 dB, a level for which interference becomes a relevant factor, the outage gain due to α-OFDM#1 is more than 3 dB, which is twice the capacity gain obtained in single-cell OFDM. As discussed in the previous section, those gains are even larger if the considered outage is less than 1% but are less important for more frequency selective channels. In this study, on the specific example of the 1.4 MHz LTE frequency band, we assumed that one were allowed to transmit data on side-bands to capitalize on channel diversity. This requires that those bands are not in use. In the following we propose schemes for service providers to overcome this problem by sacrificing a small part of the total bandwidth.

A. α-LTE

In the LTE context, service providers are allowed to use up to 20 MHz bandwidth that they can freely subdivide in several chunks. Consider that one decides to split the available bandwidth in 16 chunks of 1.25 MHz each. Those chunks are composed of 76 subcarriers which are oversampled to 128 for ease of computation at the transmitter and at the receiver. We propose to sacrifice 4 subcarriers per chunk that then results in 4 × 16 = 64 free subcarriers in total.

Those 64 subcarriers are gathered in two subbands of 32 subcarriers, placed on both sides of Figure 7 illustrates a simplified version of the afore described scheme, with 4 chunks instead of 16. In this particular example, an M = 3 α-OFDM-based scheme is used that synchronously exploits the left subcarriers of the dotted part of each chunk, then the central subcarriers and finally the right subcarriers in any three consecutive OFDM symbols (s (3k) , s (3k+1) , s (3k+2) ).

Therefore, data will always be sent on non-overlapping bands. In our particular example, we use a 136-DFT for a signal occupying the central 72-subcarrier band.

The gain of α-LTE lies in outage BER and also, at a low-to-medium SNR, in outage capacity.

Indeed, the lack of 4 subcarriers induces a factor 72/76 on the total outage capacity C which scales like N u log(SNR) at high SNR. But at low-to-medium SNR, the gain discussed in Section III appears and overtakes the loss in outage capacity introduced by the factor 72/76. The loss in outage capacity per chunk at high SNR is then fairly reduced while the gain in outage BER per chunk is kept constant independently of (N, K). In this asymptotic scenario, α-OFDM does not require to sacrifice any frequency band but provides diversity and outage capacity gain.

C. Cognitive Radios in Unlicensed Bands

Advanced techniques of channel sensing allow cognitive radios [START_REF] Mitola | Cognitive radio: making software radios more personal[END_REF] to figure out the spectral occupation of the neighbouring frequencies. For bursty systems, it could be convenient to reuse these free bandwidths W e . By increasing the rotation pattern M accordingly, for instance M = {0, 2πNW e /W}, it is possible to dynamically gain in channel diversity. The terminal can be informed of the communication mode, i.e. of the set M, to be used in a few bits through a dedicated control channel. This especially allows for software-defined dynamic spectrum

  scheme. Denote s ∈ C N the transmitted OFDM symbol, n ∈ C N some additive white Gaussian noise (AWGN) sensed by the receiver with entries of variance E[|n i | 2 ] = σ 2 and H 0 the circulant time-domain channel matrix, as in (1). The time-domain received signal r ∈ C N reads

Fig. 1 .

 1 Fig. 1. α-OFDM transmission scheme

Figure 2 .

 2 Figure 2. Particularly, if α/2π is an integer, then α-OFDM merely remaps the OFDM symbol onto a circular-shifted version of the subcarriers.Therefore α-OFDM introduces only a minor change compared to OFDM and has for main incidence to circularly shift the OFDM subcarriers by a decimal value α/2π, α ∈ R. Note that the classical OFDM modulation is the particular α-OFDM scheme for which α = 0. By controlling the rotation angle α and allowing the use of adjacent frequency bands, α-OFDM enables a computationally inexpensive frequency diversity, which in turn is known to increase outage capacity performance. The main results are introduced in the following.

  Fig. 2. α-OFDM

Figure 3

 3 Figure 3 compares the outage capacity gain of α-OFDM#1 against OFDM in LTE EPA channels for a set M of length M = 2 and M = 8. The bandwidth is 1.4 MHz. A strong SNR gain is provided by α-OFDM#1 already for M = 2 (+1.1 dB), while growing M does not bring significant improvement. This is explained by the fact that M = 2 suffices to transmit

Fig. 4 .

 4 Fig. 4. Outage Capacity gain for α-OFDM MISO

Fig. 5 .

 5 Fig. 5. α-OFDM#1 with intercell interference

Figure 6

 6 Figure 6 provides the simulation results obtained in 1% outage capacity for a transmission through 3GPP-EVA channels with N u = 72, N = 136 in the low-to-medium SNR regime. For fair comparison, we plotted the outage capacity cumulated over a bandwidth of 76 subcarriers (therefore, when N u = 72, 4 subcarriers are left unused) that we normalize by 76. At low-tomedium SNR, a gain in capacity is observed, despite the loss of 4 subcarriers, while at high SNR the classical OFDM fills the gap with our improved method.

  outage capacity gain EVA Channel -(ratio over 72 subcarrier band) SNR (dB) 1% outage capacity (bit/s/Hz)

Fig. 6 .Fig. 7 .

 67 Fig. 6. LTE & α-LTE outage capacity

  1 . . . ρe iα h 1

	h 1 h 0	. . .		. . .	. . .	. . .
	. . .		. . .	. . .		. . . ρe iα h L-1
	h L-1			. . .	. . .	0
	0	. . .			. . .	. . .	. . .
	. . .	. . .	. . .			h 0	0
	0

September 7, 2009 DRAFT