B Hildebrand

T Jaouen

C Didiot

E Razzoli

G Monney

M.-L Mottas

A Ubaldini

H Berger

Céline Barreteau

H Beck

D R Bowler

P Aebi

Short-range phase coherence and origin of the 1T -TiSe 2 charge density wave

Keywords: numbers: 71.45.Lr, 68.37.Ef, 71.15.Mb, 74.70.Xa

published or not. The documents may come

The understanding of broken-symmetry ground states is facing the complexity of the many-body problem arising from the multiplicity of coupled electron-electron, electron-hole, and electron-lattice interactions. A typical material exhibiting this close entanglement is the quasi-two dimensional, layered transition-metal dichalcogenide 1T -TiSe 2 consisting of hexagonal 1T -stacked Se-Ti-Se layers separated by a van der Waals (vdW) gap. At T CDW ∼200 K, it undergoes a phase transition towards a 2x2x2 commensurate CDW phase at which a weak periodic lattice distortion (PLD) develops and it shows a broad maximum in the temperature dependent resistivity in the vicinity of T CDW [1]. 1T -TiSe 2 is also superconducting upon Cu intercalation [2], and under pressure [3], thus offering the way for the study of the interplay between ordered CDW phase and the superconducting state [4].

However, the origin of the CDW transition itself is not yet unambiguously determined. Although many propositions have been made since the 70's, there still remain two competing hypotheses for the mechanism driving the CDW formation. Several studies have highlighted strong electron-phonon [5][6][7] coupling suggesting a Jahn-Teller (JT) mechanism of the CDW transition (first order JT [8] or pseudo-JT(PJT) [9]). Electron-hole correlations leading to the excitonic insulator [10,11] or inducing fluctuations responsible for phonon softening [12,13] have been also invoked as well as a cooperative combination of both effects [14,15].

Very recently, significant insights favoring the JT scenario has been obtained by using ultra-broadband terahertz pulses to separately trace the coexisting lattice and electronic orders [16]. Taking profit of the selectivity in the dynamics of elementary electronic and structural processes [17,18], it has been demonstrated that JTlike CDW can exist in a metastable non-thermal phase without excitonic correlations. However, time-resolved experiments [16][17][18][19], probe the CDW out-of-equilibrium and thus cannot be straightforwardly extended to the thermal equilibrium states. Also, most of the studies devoted to this controversy used experimental techniques non-local in essence and therefore not probing directly local, real-space characteristics of the CDW.

Using scanning tunneling microscopy (STM) it is possible to gain insight on the impact of defects on the local CDW properties [20][21][22]. In 1T -TiSe 2 , Ti selfdoping is known to occur depending on the crystal growth temperature [1], with a dramatic effect on resistivity. Intercalated-Ti atoms lead to electron-donor impurity states close to the Fermi energy [23], enhance the Coulomb screening and tend to reduce electron-hole correlations. Therefore, the real-space STM investigation of the CDW in Ti self-doped crystals can in principle reveal the microscopic nature of the phase transition.

The present STM study of Ti self-doped 1T -TiSe 2 shows that accumulation of intercalated-Ti atoms act as preferential nucleation centers for the CDW. The CDW breaks up in randomly phase-shifted nanodomains with subsisting commensurate 2x2 charge modulation and separated by atomically sharp phase slips. We find a close relationship between the CDW domain size and the Tidoping density and demonstrate that the nucleationgrowth mechanism lying behind the CDW formation is unique, regardless of the intercalated-Ti density. Our observations together with density functional theory (DFT) calculations support a local origin of the CDW as driven by a PJT instability, not necessitating long-range correlations. We interpret the dramatic influence of Ti-doping on the anomalous resistivity peak as a confinement effect of excitons within CDW domains.

The 1T -TiSe 2 single crystals were grown at 770 • C , 860 • C and 900 • C by iodine vapor transport, therefore containing increasing concentrations of Ti doping atoms [1]. Resistivity measurements were performed by a standard four-probe method using a lock-in as current source and voltage meter. The samples were cleaved in-situ below 10 -7 mbar at room temperature. Constant current STM images were recorded at 4.7 K using an Omicron LT-STM, with bias voltage V bias applied to the sample. Base pressure was better than 5×10 -11 mbar.

DFT model calculations were performed using the plane-wave pseudopotential code VASP [24,25], version 5.3.3. Projector augmented waves [26] were used with the Perdew-Burke-Ernzerhof (PBE) [27] exchange correlation functional. The cell size of our model was 28.035 Å × 28.035 Å. The 1T -TiSe 2 surface was modeled with two layers and the bottom Se layer fixed. A Monkhorst-Pack mesh with 2×2×1 k points was used to sample the Brillouin zone of the cell. The parameters gave an energy difference convergence of better than 0.01 eV. During structural relaxations, a tolerance of 0.03 eV/ Å was applied.

Figure 1(a) presents filled-state STM images recorded at -1 V of the 1T -TiSe 2 surface of a high Ti-doped 1T -TiSe 2 crystal grown at 900 • C . Bright defects are clearly seen and correspond to Ti atoms intercalated in the vdW gap of the pristine 1T -TiSe 2 atomic structure [23]. The defect density is 2.57±0.22% as extracted from statistics made on large STM images and similar to bulk measurements by Di Salvo et al. on a 900 • C -grown crystal [1]. Thus, even if the densities of Ti defects estimated from our STM images mainly account for Ti intercalation in the first vdW gap, they are representative of the bulk Ti-doping density.

In a recent study, it has been shown that the defect density associated with low-doped 1T -TiSe 2 samples does not affect the long-range CDW order [28]. For the highly doped crystal and looking at regions with high density of intercalated Ti atoms, atomically-resolved STM images (representing the electronic density) near the Fermi energy [Fig. 1(b)] show that the CDW is locally completely absent (bright regions). Approximately 50% of the 1T -TiSe 2 surface clearly exhibits short-range CDW charge modulation following the well-known 2x2 commensurate pattern [see inset Fig. 1(b)] in regions associated with low defect density.

Locating the CDW maxima with respect to the underlying Se atomic layer [Fig. 1(c)] now reveals a CDW patterning of nanometer size. Several CDW patterns which differ in their phase relationship with the 1T -TiSe 2 lattice are observed and correspond to the four possible configurations of a 2x2 charge modulation. Interestingly, the CDW boundaries in surface areas free of accumulation of intercalated-Ti consist of atomically-sharp phase slips between adjacent domains [see inset Fig. 1(c)] and appear to occur at random positions of the surface with the charge modulation on either side of the phase slip staying commensurate with the underlying lattice. Isolated Ti impurities are mainly observed within domains suggesting that the CDWs emerged around the bright regions [see bottom, left of Fig. 1(c)] associated with an accumulation of intercalated-Ti defects.

Therefore, our results strongly indicate that the CDW domains patterning originates in a CDW nucleation- growth process from intercalated-Ti agglomerates. Upon temperature lowering, the nucleating CDWs grow and meet at phase-slip boundaries whose sharpness reflects the very short CDW coherence length [6,29].

At first sight, it may be surprising that a commensurate CDW can break into short-range phase coherent domains in the presence of impurities. Indeed, as previously pointed out by Wu and Lieber who observed commensurate CDW domains on another (Ti-doped TaSe 2 ) [30], a commensurate CDW phase that is already strongly coupled to the crystal lattice tempts to remain unperturbed by impurity pinning according to a seminal work of McMillan [31]. However, in contrast to McMillan's Landau theory calculations that describe how a preexisting charge modulation reacts to the random field introduced by impurities [31], we are concerned here by the emergence of the CDW in with defects already present. Then, the size and arrangement of the CDW domains as well as the domain boundary locations are fully determined by the density of intercalated-Ti nucleation centers, their distribution and the cooling kinetics.

STM measurements carried on 1T -TiSe 2 crystals grown at lower temperature further support our CDW nucleation-growth scenario. Figure 2 Comparing both cystals with the 900 • C -grown one for V bias = ±1V, we clearly see that the density of randomly distributed Ti defects [highlighted with triangles on Fig. 1(a) and Fig. 2(a),(c)] increases with growth temperature. From statistics made on large STM images (not shown), we obtain 0.75±0.10% and 1.21±0.14% for the crystals grown at 770 • C and 860 • C , respectively compared to the 2.57±0.22% of the 900 • C -grown 1T -TiSe 2 .

Identically to the highly-doped crystal, STM images near Fermi level [Fig. 2(b),(d)] now reveal for both crystals the presence of atomically-sharp phase slips of the CDW modulation. Whereas for the lowest doped crystal 60×60nm 2 images are needed to observe a shift of the CDW phase [inset [Fig. 2(b)], every 17×17nm 2 images of the 860 • C -grown crystal show at least one phase slip. This not only demonstrates the close relationship between the CDW domain sizes and the Ti-doping density but especially reveals that the nucleation-growth mechanism lying behind the CDW formation in the three Ti-doped crystals is the same.

At this stage, we have shown that the CDW can well exist on a very small area and that its pinning to the nucleation centers is energetically more favorable than maximizing a domain size or avoiding phase-slips. Also, the coherence length of the CDW is definitely small suggesting that the strong-coupling theory and a local chemical bonding picture apply [29]. This naturally indicates a local origin of the CDW as driven by a PJT instability [9,[START_REF] Bersuker | The Jahn-Teller Effect[END_REF], i.e., not necessitating long-range electronic correlations.

Let us now consider the impact of the intercalation of Ti on the 1T -TiSe 2 atomic structure. Table I presents DFT-calculated, relaxed interatomic distances of the normal phase of 1T -TiSe 2 with and without one intercalated-Ti atom in the vdW gap. The impact of the intercalation is rather strong and of the order of magnitude of the PLD. As exemplified in Fig. 3, the major effect of the intercalation is the strong decrease (4.3%) of the dis-TABLE I. DFT-calculated structural parameters of 1T -TiSe2 with and without intercalated Ti atom in the vdW gap. Two TiSe6 octahedra separated by one vdW gap are considered: Interatomic distances between Se atoms of the topmost layer (dSe-Se), between the Se neighboring the intercalated-Ti defect (dSe-Se(vdW)), between Ti atoms along the c-axis (dT i-T i), and between Ti and Se atoms of the TiSe6 octahedra (dT i-Se).

Ti-doped 1T -TiSe2

1T -TiSe2 ( Å) ( Å)

dSe-Se Therefore, the intercalation of one Ti atom in the vdW gap does not favor the CDW instability since it acts against the PLD where 3/4 of the Ti-Se bonds are shortened [1]. This explains why one isolated Ti defect does not act as a nucleation center for the CDW and also why the CDW charge modulation is highly perturbed just above the intercalated-Ti atom. However, at intercalated-Ti agglomerate edges, the coherent superposition of symmetric Se displacements are expected to induce shortening of the Ti-Se bonds favorable for driving the nucleation of multiple phase-shifted domains in a PJT scenario [9].

Finally our findings provide insight on the anomalous resistivity peak known to appear around T CDW . Because of a favorable band configuration with energetically close positioned electron-and hole-like bands, it has been interpreted as a reduction of the density of free charge carriers due to electron-hole pairing [10,11]. Its disappearance at lower temperatures is then related to the development of the full CDW gap with unfavorable conditions for the excitonic pairing. the mean domain size L 0 .

The abrupt but continuous change of the resistivity maximum with doping can now be understood in terms of exciton confinement within the CDW nanodomains, as discussed for Wannier-Mott excitons in three-dimensional quantum wells [START_REF] Kayanuma | [END_REF]. As the typical size of the CDW domain is reduced to 1.5-2 times the effective Bohr diameter of the bulk exciton (d exc. =4.2 nm [36]) [dashed-blue curve Fig. 4(b)], the electron-hole pairs initially confined as quasiparticles start to break up into individual charge carriers [START_REF] Kayanuma | [END_REF]. The result is the disappearance of the exciton-related resistivity maximum for sufficient doping or sufficiently small CDW domains.

The resistivity peak attributed to excitons thus reflects sufficiently long-range electronic correlations in the CDW state. In that sense, the excitonic pairing is an epiphenomenon of the CDW phase transition driven by electron-phonon interaction that allows for the direct coupling of electron and hole bands. Exciton-phonon coupling may thus simply reinforce the CDW state [37], without playing the predominant role in driving the phase transition itself.

To summarize, we report the first observation of shortrange phase coherent CDW nano domains in self-doped 1T -TiSe 2 . Our results support a PJT instability as the origin of the CDW patterning. Evidence for confinement effects of electron-hole pairs within CDW domains further provide a coherent picture of the 1T -TiSe 2 CDW phase transition and its relation to excitons.

This project was supported by the Fonds National Suisse pour la Recherche Scientifique through Div. II. We would like to thank C. Monney

FIG. 1 .

 1 FIG. 1. (Color online) (a) 17×17nm 2 constant current STM images of the 900 • C -grown 1T -TiSe2 crystal showing the intercalated-Ti defects (black triangles), V bias = -1V, I= 0.2nA. (b) Constant current STM image at -0.1V bias voltage and 0.2 nA current set point of the same region as (a) showing the CDW charge modulation and its perturbation by the presence of Ti defects. The inset presents an eye guide of the 2x2 and 1x1 charge modulations. Green spheres correspond to Se atoms of the topmost layer (Se up) and the blue ones to the maxima of the 2x2 charge modulation (max CDW). (c) Same STM image as (b) with added meshes highlighting the different CDW domains. Four colors have been used to differentiate between the four possible kinds of modulations that are phase shifted to each other. The gray lines show examples where CDW phase-slips occur. The inset shows a height profile through an atomically abrupt CDW phase-slip.

FIG. 2 .

 2 FIG. 2. (Color online) (a) 17×17nm 2 constant current STM image of the 770 • C -grown 1T -TiSe2 crystal showing the intercalated-Ti defects (black triangles), V bias = +1V, I= 0.2nA. (b) Constant current STM image at -0.1V bias voltage and 0.2 nA current set point of the same region as (a). (c), (d) same as (a), (b) for the 860 • C -grown 1T -TiSe2 sample. V bias = -1V for (c). The gray lines on (b) and (d) show surface regions where a CDW phase-slip occurs. The inset in (b) is a 7.8×4.5nm 2 constant current STM image showing the presence of a CDW phase-slip on the 770 • C -grown sample.

  shows 17×17nm 2 constant current STM images of a 770 • C [Fig. 2(a), (b)] and 860 • C -grown [Fig. 2(c), (d)] 1T -TiSe 2 crystals.

FIG. 3 .

 3 FIG. 3. (Color online) Structural models of 1T -TiSe2 as obtained by DFT. Two TiSe6 octahedra separated by one vdW gap are considered (a) Pristine 1T -TiSe2 (b) Ti-doped 1T -TiSe2. The amplitude of displacements induced by Ti intercalation (Tii) have been enhanced.

Figure 4 (

 4 a) shows the relative resistivity ρ/ρ 300K of the three measured crystals. A maximum is clearly observed for the 770 • C and 860 • C -grown crystals whereas it is almost absent for 900 • C. In the evolution of the anomalous resistivity peak amplitudes [33], as a function of the average domain sizes L 0 extracted from our STM measurements [34] plotted in [Fig. 4(b)] there is a dramatic change between the sample grown at 860 • C and 900 • C despite the only slight diminution of the domain size from 10 nm to 2.5 nm. It is also interesting to note in Fig. 4(b) that a small doping increase from 0.75 % (770 • C) to 1.21 % (860 • C) is accompanied by a small change in the resistivity but induces a huge reduction of

FIG. 4 .

 4 FIG. 4. (Color online) (a) Temperature dependent resistivity curves measured on 770 • C (blue curve), 860 • C (dark blue curve), and 900 • C (black curve) grown 1T -TiSe2 crystals. The curves are normalized to the resistivity measured at room temperature. b) Evolution of the intensity of the resistivity peak as a function of the CDW domains size. The dashed blue curve indicates the typical value of the effective Bohr diameter of excitons in 1T -TiSe2.

  , C. Renner, and A. M. Novello for motivating discussions. Skillful technical assistance was provided by F. Bourqui, B. Hediger and O. Raetzo. B.H. and T.J. contributed equally to this work.


E V Belmega

B Djeumou

S Lasaulce

PERFORMANCE ANALYSIS FOR THE AF-BASED FREQUENCY DIVISION COOPERATIVE BROADCAST CHANNEL

This paper considers a system where one transmitter broadcasts a single common message to two receivers. These receivers can cooperate through a bidirectional channel that is assumed to be orthogonal to the downlink channel. For the case where the assumed cooperation protocol is amplify-and-forward we calculated the final equivalent SNR in the MRC output at each receiver for an arbitrary number of cooperation exchanges. The corresponding analytical expressions can then be used for evaluating different performance criteria in order to discuss issues such as: Which receiver should start cooperating first? Is there an optimum number of cooperation exchanges? What is the difference between asymmetric and symmetric cooperations?

INTRODUCTION

In the conventional broadcast channel (BC) introduced by [START_REF] Cover | Broadcast channels[END_REF], one transmitter sends independent messages to several receivers. The channel under investigation in this paper differs from the original BC for at least two reasons. First, the receivers are allowed to cooperate in order to enhance the overall network performance. Second, the users want to decode the same message. We will refer to this situation as the cooperative broadcast channel (CBC) with a single common message. For sake of simplicity a 2-user CBC will be assumed. Note that the considered channel is different from the original relay channel (RC) defined in [START_REF] Cover | Capacity theorems for the relay channel[END_REF] because each terminal not only acts as a relay but also as a receiver, which means that ultimately, the information message has to be decoded by both terminals. Additionally the cooperation channel between the two receivers is assumed to be bidirectional (versus unidirectional for the RC) and orthogonal to the downlink (DL) channels. Although their sub-optimality, orthogonal channels are often assumed for practical reasons (e.g. it is difficult to implement relay-receivers that receive and transmit at the same time in the same frequency band).

To the author's knowledge the most significant contributions1 concerning the situation under investigation are [START_REF] Draper | Interactive decoding of a broadcast message[END_REF] [START_REF] Dabora | Broadcast channels with cooperating decoders[END_REF][6] [START_REF] Lasaulce | Gaussian broadcast channels with cooperating receivers: The single common message case[END_REF] [START_REF] Azarian | On the achievable diversity-multiplexing tradeoffs in half-duplex cooperative channels[END_REF]. The discrete broadcast channel with a bidirectional conference link and a single common message was originally studied by Draper et al. in [START_REF] Draper | Interactive decoding of a broadcast message[END_REF]. The authors proposed a way of decoding the message in multiple rounds and applied their scheme to the binary erasure channel. This channel has also been analyzed by [START_REF] Dabora | Broadcast channels with cooperating decoders[END_REF] where the authors essentially proposed achievable rates based on the use of estimate-and-forward (EF) at both receivers and two-round cooperation schemes. The Gaussian counterpart of this channel has been studied in [START_REF] Lasaulce | Gaussian broadcast channels with cooperating receivers: The single common message case[END_REF]. Showing the optimality of decode-and-forward for a unidirectional cooperation the authors evaluated the exact loss due to orthogonalization of the cooperation channel. For the bidirectional case, the proposed achievable rate is based on a combination of EF and decode-andforward and shown to always outperform the pure EF-based solution (always for the 2-round decoding). The fading case has been partially treated in [START_REF] Azarian | On the achievable diversity-multiplexing tradeoffs in half-duplex cooperative channels[END_REF]. The diversity-multiplexing trade-off, achieved by using a "dynamic" version of decode-and-forward, is derived for the unidirectional cooperation case.

While the authors of [START_REF] Dabora | Broadcast channels with cooperating decoders[END_REF][7] [START_REF] Azarian | On the achievable diversity-multiplexing tradeoffs in half-duplex cooperative channels[END_REF] addressed situations where only one or two cooperation exchanges (or decoding rounds) are allowed, this paper focuses on the case where the number of exchanges is arbitrary. For erasure channels [START_REF] Draper | Interactive decoding of a broadcast message[END_REF] and [START_REF] Khalili | Broadcasting a message over erasure channels with cooperating receivers[END_REF] have shown that the higher the number of exchanges the better the performance in terms of information rate. However the discrete channel analysis (including erasure channels) does not take into consideration the spectral resources aspect. As it will be seen the point is in fact crucial and accounting for it can lead to markedly different conclusions from [START_REF] Draper | Interactive decoding of a broadcast message[END_REF] [START_REF] Khalili | Broadcasting a message over erasure channels with cooperating receivers[END_REF] concerning the optimum number of cooperation exchanges. Additionally [START_REF] Draper | Interactive decoding of a broadcast message[END_REF] and [START_REF] Khalili | Broadcasting a message over erasure channels with cooperating receivers[END_REF] only considered the information rate as a performance criterion whereas other criteria of interest can also be considered. This paper aims precisely at taking into account these two aspects for providing some insights to the following issues. If the total bandwidth is fixed what is the optimal number of cooperation exchanges? What is the most efficient way to cooperate: The symmetric or asymmetric way? For the asymmetric cooperation, which user should start cooperating first? What about these issues if the total bandwidth constraint is relaxed?

In order to provide elements of response to these questions we will use a simple approach. After presenting the used system model (sec. 2), we will evaluate the exact equivalent SNR (signal-to-noise ratio) for each user (sec. 3) in the case where amplify-and-forward (AF) is assumed at both receivers. Based on the choice of different system performance criteria (sec. 4) numerical and simulation analyses will be conducted (sec. 5). Concluding remarks and possible extensions of the present work will be given in section 6.

SYSTEM MODEL

As mentioned in the introduction the system under investigation comprises one transmitter (source) and two receivers (destinations). The transmitted signal is denoted by X and subject to a power constraint: E[|X 2 |] ≤ P . Its bandwidth is denoted by BDL. For sake of simplicity X will be assumed to be a scalar quantity (e.g. a complex Gaussian input or QAM symbol) in the whole paper. As we will see this simplifying assumption will not prevent us from capturing the main issues related to the channel under investigation. Assuming an AWGN model for the DL channels, the received signals write:

Y1 = X + Z1 Y2 = X + Z2 (1)
where for all j ∈ {1, 2}, Z j ∼ N (0, n j B DL ) and I(Z 1 ; Z 2 ) = 0. We will assume that orthogonality between the DL and cooperation channels is implemented by frequency division (FD). The bandwidth allocated to the cooperation channel between the two receivers is denoted by BC . The cooperation channel is then divided into several sub-channels, each of them having a bandwidth equal to ∆B. The two receivers cooperate by applying the same relaying strategy namely the AF protocol, which imposes ∆B = BDL. Additionally we will assume zero-delay relaying. In real situations, this can be implemented for instance by re-synchronizing the DL and cooperation signals at each receiver. In this paper two main ways to cooperate are distinguished: The symmetric cooperation (figure 1) and asymmetric cooperation (figure 2).

For the symmetric cooperation the two receivers cooperate as follows. Each receiver (e.g. receiver 1) amplifies and forwards his received DL signal (Y (0) I = Y1 for receiver 1) to his partner (receiver 2). This is done simultaneously. Then each receiver (e.g. receiver 2) applies a maximum ratio combining to the cooperation signal (Y The first case BDL + BC = const. corresponds to the situation where the system total bandwidth is fixed, which is generally assumed to compare two systems before implementation. The second case BDL = const. does not lead to fair comparisons in terms of bandwidth since the total bandwidth B = (2Ks + 1)∆B = (2Ks + 1)BDL increases with Ks. In fact it is still of practical interest. For instance consider the case where one wants to assess the benefits of cooperation by coupling two existing communication systems such as a DVB (digital video broadcasting) system and a cellular system. As modifying the DVB system downlink signal bandwidth would be a difficult/an impossible task, the second assumption, which amounts to extending the available bandwidth is more appropriate for comparing a DVB system with its terrestrial cooperation-based counterpart. To denote the signals of interest for a given cooperation round or iteration i, with i ∈ {1, ..., Ks}, we will use the following notations:

         Y (i) I = α (i) I X + Z (i) I Y (i) II = α (i) II X + Z (i) II Y (i) 12 = a (i) 12 Y (i-1) I + Z (i) 12 Y (i) 21 = a (i) 21 Y (i-1) II + Z (i) 21
where a

(i) 12 , a (i)
21 are AF amplification gains, which are determined by the total cooperation powers available: P 12 at receiver 1 and P21 at receiver 2.

For the asymmetric case, the cooperation is not simultaneous. Assume that receiver 1 starts relaying. For example, receiver 1 sends an amplified version of his DL signal (Y (0) I = Y1) to receiver 2. Receiver 2 applies an MRC and amplifies and forwards the MRC output (Y We will keep the same notations for the signals of interest as for the symmetric case but in contrast with the symmetric case combining operations take place at receiver 2 for i odd only and at receiver 1 for i even only (under the assumption that receiver 1 starts relaying).

SNR ANALYSIS

As the DL and cooperation signals are AWGN signals using an MRC for combining these signals is the best choice (in terms of mutual information, of raw BER at the combiner output, etc.). In this section we want to determine the equivalent SNR at the MRC output and use this knowledge to evaluate different performance criteria. It turns out that it is not possible, in general, to express the equivalent SNR as a function of the sole channel parameters (P, P 12, n1, ...). In fact the equivalent SNR has to be determined through a recursive formula. The purpose of the following theorem (which proof does not require any specific mathematical tool) is precisely to provide this relationship both for asymmetric and symmetric cooperations.

Theorem 3.1 (General expression for the equivalent SNRs).

Assume that n1 < n2 and receiver 2 performs the MRC task in the first place if asymmetric cooperation is considered. For iteration i the corresponding weights are denoted by w (i) 2 (weighting the MRC output at iteration i -1) and w (i) 12 (weighting the cooperation signal). For receiver 1 the weights are denoted by w

(i) 1 , w (i) 21 . Denote by Y (i) I = α (i) I X + Z (i) I (resp. Y (i) II = α (i) II X + Z (i)
II ) the signal at MRC output for receiver 1 (resp. receiver 2) and iteration i, with Z (i)

I ∼ N (0, N (i) I ) (resp. Z (i) II ∼ N (0, N (i) II )). Let ρ (i) I (resp. ρ (i)
II ) be the signal-to-noise ratio associated with the signal Y

S (i) II = α (i-1) I α (i-1) II e (i-1) + e (i-1), * ρ (i-1) I ρ (i-1) II ρ 12 -α (i-1) I α (i-1) II 2 P ρ (i-1) II 1 + ρ (i-1) I + ρ12 ρ (i-1) I + ρ (i-1) II T (i) II =
e (i-1) e (i-1), * P ρ

(i-1) I ρ (i-1) II ρ12 -α (i-1) I α (i-1) II 2 P (1 + ρ 12 ) -α (i-1) I 2 N (i-1) II ρ (i-1) I ρ (i-1) II where ρ12 = P12 n12∆B
, k is a constant depending on the cooperation scheme (asymmetric or symmetric), (.) * denotes the conjugate, e (0) = 0, N

I = N 1 , N (0)
II = N 2 , ρ (0)
I = P/N 1 , ρ (0)
II = P/N 2 , α (0) I = α (0) II = 1. (0)
The cooperative channels gains are defined as: a II depend on the cooperation type. Expressing these quantities is the purpose of the following two propositions.

(i) 12 = P (i) 12 (α (i-1) I ) 2 P +N (i-1) I , a (i) 21 = P (i) 21 (α (i-1) II ) 2 P +N

Proposition 3.2 (MRC weights for the symmetric cooperation). For the symmetric cooperation the MRC weights can be shown to be:

           w (i) 12 = a (i) 12 α (i-1) I N (i-1) II -a (i) 12 α (i-1) II e (i-1) w (i) 2 = (a (i) 12 ) 2 N (i-1) I + N (i) 12 α (i-1) II -(a (i) 12 ) 2 α (i-1) I e (i-1) w (i) 21 = a (i) 21 α (i-1) II N (i-1) I -a (i) 21 α (i-1) I e (i-1) w (i) 1 = (a (i) 21 ) 2 N (i-1) II + N (i) 21 α (i-1) I -(a (i) 21 ) 2 α (i-1) II e (i-1)
where • e (i-1) E Z 1) ,

(i-1) I Z (i-1) II with e (i) = w (i) 12 a (i) 12 w (i) 1 N (i-1) I + w (i) 21 a (i) 21 w (i) 2 N (i-1) II + w (i) 1 w (i) 2 + w (i) 12 a (i) 12 w (i) 21 a (i) 21 e (i-
• for all i ∈ {1, ..., Ks} the useful signal coefficients are given by

α (i) I = w (i) 21 a (i) 21 α (i-1) II + w (i) 1 α (i-1) I α (i) II = w (i) 12 a (i) 12 α (i-1) I + w (i) 2 α (i-1) II ,
• the cooperation powers per subchannel are for all i ∈ {1, ..., Ks} given by     

P (i) 12 = P 12 Ks P (i) 21 = P21 Ks ,
• the equivalent noise powers N (i)

I , N (i)
II are determined through

Z (i) I = w (i) 21 a (i) 21 Z (i-1) II + w (i) 21 Z (i) 21 + w (i) 1 Z (i-1) I Z (i) II = w (i) 12 a (i) 12 Z (i-1) I + w (i) 12 Z (i) 12 + w (i) 2 Z (i-1) II .
• for all i ∈ {1, ..., Ks}: N

(i) 12 = n12∆B and N (i) 21 = n21∆B
• the constant k of Theorem 3.1 equals 2Ks + 1 Ks .

Proposition 3.3 (MRC weights for the asymmetric cooperation). For the asymmetric cooperation the MRC weights can be shown to coincide with that of Proposition 3.2 where

• e (i-1) E Z

(i-1) I Z (i-1) II with e (i) = w (i) 1 e (i-1) + w (i) 21 a (i) (21) N (i-1) II f or i even w (i) 2 e (i-1) + w (i) 12 a (i) (12) N (i-1) I f or i odd,
• the useful signal coefficients are given by α (i)

I = w (i) 21 a (i) 21 α (i-1) II + w (i) 1 α (i-1) I f or i even α (i-1) I f or i odd, α (i) II = α (i-1) II f or i even w (i) 12 a (i) 12 α (i-1) I + w (i) 2 α (i-1) II f or i odd,
• the cooperation powers per subchannel are for all i ∈ {1, ..., Ka}

P (i) 12 = 2P12 Ka f or Ka even 2P12 Ka + 1 f or K a odd, P (i) 21 = 2P21 Ka f or K a even 2P 21 K a -1 f or K a odd,
• the equivalent noise powers N (i)

I , N (i)
II are determined through

Z (i) I = w (i) 21 a (i) 21 Z (i-1) II + w (i) 21 Z (i) 21 + w (i) 1 Z (i-1) I i even Z (i-1) I i odd, Z (i) II = Z (i-1) II i even w (i) 12 a (i) 12 Z (i-1) I + w (i) 12 Z (i) 12 + w (i) 2 Z (i-1) II i odd,
• for all i ∈ {1, ..., Ka}: N These results will be used in section 5 for evaluating different performance criteria and provide insights to the issues mentioned in the introduction of this paper, in particular to the issue of the optimal number of cooperation exchanges, which is equivalent to a bandwidth allocation issue for the AF protocol.

PROPOSED SYSTEM PERFORMANCE CRITERIA

In the simulation section we will dedicate most of our attention to the mutual information criterion but also consider other performance criteria presented here. In order to compare the different cooperation schemes one needs to select suited system performance criterion. For instance, if we fix the information rate/spectral efficiency at the transmitter and obtain the pair of BERs (BER1, BER2) for the coding scheme C and (BER ′ 1 , BER ′ 2 ) for the coding scheme C ′ with BER1 < BER ′ 1 and BER2 > BER ′ 2 one cannot easily conclude. From now on we will denote by K the number of cooperation exchanges with K equals Ka or Ks depending on the cooperation scheme. For comparing the AF protocol-based cooperation strategies we propose 4 performance criteria (eq. ( 2)-( 5)).

R (K) AF = BDL min log 1 + ρ (K) I , log 1 + ρ (K) II (2)
where ρ

(K) I , ρ (K)
II are the SNRs at the end of the cooperation procedure. One can notice that R (K)

AF represents the maximum information rate possible for a reliable transmission achieved by the proposed cooperation schemes and a Gaussian channel input. Following the proof of [START_REF] El Gamal | Bounds on capacity and minimum energy-per-bit for AWGN relay channels Gamal[END_REF] for the capacity of the FD relay channel with linear relaying one can show that maximizing R (K) AF over K, choosing the best cooperation order and using a convexification procedure with respect to the transmit power (the capacity is a concave function of P ), one can obtain the CBC capacity for the relaying protocol under consideration.

P (K)

e,max = max P (K)

e,I , P

(K) e,II (3)
where

P (K)
e,I and P

(K)
e,II are the raw BERs at the MRC outputs at the end of the cooperation procedure. This criterion is useful in a broadcasting system for which one wants every user to have a minimum transmission quality, which requires P (K) e,max ≤ Pe,0 where Pe,0 is the minimum quality target. Of course our goal is to minimize P (K) e,max with respect to K. P (K) e,sum = P

This criterion gives an image of the average transmission quality of the broadcasting system. The system probability of errors P (K) e,sys , which is defined as the probability that receiver 1 or (inclusive or) receiver 2 makes a decision error. This probability is generally not easy to explicit but can be bounded by using the last two performance indices: P (K) e,max ≤ P (K) e,sys ≤ P (K) e,sum .

Note that the capacity of the channel under consideration is precisely defined with respect to the system error probability, which means that communicating at a rate less than the capacity insures the existence of a code such that P Kopt e,sys → 0.

SIMULATION RESULTS

Asymmetric cooperation: Which user should start cooperating first?

Here, the total bandwidth is fixed. Figure 3 P 21 ∈ {-30 dB, -10 dB, 0 dB, 10 dB, 30 dB}. The different curves delimit the decision regions that allows us to determine the best cooperation order in terms of information rate. When the cooperation powers are equal (i.e. P 12 P 21 = 0 dB) and the pair (n1, n2) is above the line n2 = n1 receiver 1 has to start first and conversely. If there is an asymmetry in terms of cooperation powers the answer is less trivial. We represented the decision frontiers for the 5 values for the ratio P 12 P 21 . We clearly see that both the DL and cooperation SNRs have to be considered to optimize the overall performance since the frontier can be quite different from the the line n 2 = n 1 . In practice it can happen that the cooperation powers can be quite close. For instance this is the case if the cooperation powers are chosen to be a fraction of the mobile transmit power in a cellular system. The cooperation order is then less critical. Asymmetric cooperation vs symmetric cooperation. First we assume the total bandwidth to be limited. Figure 4 represents the information rate as the number of cooperation exchanges for the asymmetric (- * -) and symmetric (-o-) cases for two different scenarios: ( P n 1 B , P n 2 B , P 12 n 12 B , P 21 n 21 B , ) = (10 dB, 0 dB, 30 dB, 30 dB) and ( P n 1 B , P n 2 B , P 12 n 12 B , P 21 n 21 B , ) = (-1 dB, -4 dB, 30 dB, 30 dB). It can be seen that the rate always decreases for K ≥ 2. This is not surprising since a system with K > 2 is a special case of the system for which K = 2. However note that the system with K = 2 is not a special of the system K = 0 or K = 1, which means that cooperating can compensate/overcompensate for the performance loss due do orthogonalizing the DL channel. We also see that the asymmetric system performs better than its symmetric counterpart. We observed from other simulations not presented here that most of the cooperation benefits are captured with one cooperation exchange: In contrast with the discrete CBC with a conference channel [START_REF] Draper | Interactive decoding of a broadcast message[END_REF][5] [START_REF] Khalili | Broadcasting a message over erasure channels with cooperating receivers[END_REF] the performance can decrease with K. Now we look at two scenarios where the downlink bandwidth is fixed: ( P n 1 B , P n 2 B , P 12 n 12 B , P 21 n 21 B ) = (10 dB, 10 dB, 30 dB, 30 dB) and ( P n 1 B , P n 2 B , P 12 n 12 B , P 21 n 21 B , ) = (10 dB, 10 dB, 6 dB, 6 dB). We see that in the high cooperation regime the SIMO bound is rapidly attained that is for K = 2. When less cooperation powers are available the performance still decreases with K. This time this is not due to the orthogonalization loss but to the fact that the cooperation power per exchange decreases in ∼ 1 K whereas the gain brought by increasing the number of recombinations increases more slowly. Note that now the symmetric system performs better than the asymmetric one because nothing is lost in terms of bandwidth by increasing K (while for the case where the total bandwidth was limited the DL bandwidth was decreasing according to propositions 3.2 and 3.3).

  DL signal (Y (0) II = Y2). As a further step, each receiver amplifies and forwards his MRC output (Y (1) I ). The latter is then received (Y recombined with the previous MRC output (Y (1) II ) and so on. If one denotes by Ks the number of pairs of cooperation exchanges we have ∆B = B 2K s + 1 when B DL + BC = const. B B when BDL = const. B

( 1 )

 1 II ) to receiver 1. Receiver can now combine his DL signal (Y (0) I = Y1) and cooperation signal (Y (1) 21 ) to form the new signal Y (2) I . If one denotes by Ka the number of cooperation exchanges we have ∆B = B Ka + 1 when B DL + BC = const. B B when B DL = const. B

Fig. 1 .

 1 Fig. 1. AF-based symmetric cooperation

Fig. 2 .

 2 Fig. 2. AF-based asymmetric cooperation

2 .

 2 available cooperation powers per subchannel. For the SNR ρ (i) I do the following changes for the indices: I ↔ II and 1 ↔ The expressions of the signals coefficients α

  the constant k of Theorem 3.1 equals 1.

Fig. 3 .

 3 Fig. 3. Decision regions: Who starts cooperating first?

  represents the plane (n1, n2) with linear scales: n1 ∈ [10 -2 , 10 2 ], n2 ∈ [10 -2 , 10 2 ] for different ratios

	P 12

For example the authors note that[START_REF] Liang | Cooperative relay broadcast channels[END_REF] also addressed the CBC but did not focus on the common message case.

Influence of the performance criterion. Simulations such as figure 6 (system BER vs K for a 2 2(K+1) -QAM modulation) have shown that the previous observations are generally confirmed. In the final version a complete commented analysis of this issue will be provided instead of fig. 6.

CONCLUDING REMARKS

In this paper we treated the bidirectional CBC with a single common message when power and spectral resources are limited, which cannot be considered through the discrete approach [START_REF] Draper | Interactive decoding of a broadcast message[END_REF][5] [START_REF] Khalili | Broadcasting a message over erasure channels with cooperating receivers[END_REF]. The proposed approach, though very simplified, allowed us to capture the main implementation issues posed by the bidirectional cooperation. In contrast with [START_REF] Draper | Interactive decoding of a broadcast message[END_REF][5] [START_REF] Khalili | Broadcasting a message over erasure channels with cooperating receivers[END_REF] we saw that the best performance is generally achieved for one or two cooperation exchanges. The asymmetric cooperation appears as the best choice for a system where the total bandwidth is fixed. The symmetric system performs better than its asymmetric counterpart where only the DL bandwidth is fixed. We insist on the fact that this assumption, although unfair to design a system, still corresponds to real situations (e.g. an existing DVB system coupled with a cellular system). The present work can be extended by considering other relaying protocols (which do not impose the cooperation channel bandwidth to be equal to the DL channel bandwidth) and most importantly fading channels.