Jean-Marc Kelif
email: jeanmarc.kelif@orange-ftgroup.com

Marceau Coupechoux
email: coupecho@enst.fr

Frédéric Marache
email: frederic.marache@orange-ftgroup.com

Limiting Power Transmission of Green Cellular Networks: Impact on Coverage and Capacity

Reducing power transmission is of primary importance in future green cellular networks. First of all, the induced reduction of the interference encourages the deployment of opportunistic radios in the same spectrum. Then, it directly implies a reduction of the energy consumption. At last, electric field radiations reduction mitigates the potential risks on health. From a technical point of view, power control is however likely to degrade network performance. In this paper, we evaluate the impact of power reduction on the coverage and the capacity of cellular networks. We establish closed form formulas of outage probability by taking into account shadowing, thermal noise and base stations (BS) transmitting power impacts. We quantify the transmitting power needed for different kinds of environments (urban, rural) and frequencies and we show that the transmitting power can be optimized according to networks characteristics without decreasing the quality of service. We show at last that increasing the BS density results in a reduction of the global power density in the network.

I. INTRODUCTION

An important challenge of future wireless networks consists in decreasing the base station (BS) transmitting powers with a limited impact on the quality of service. This limitation allows indeed to decrease (1) the amount of interference in the considered spectrum; (2) the energy consumption with aim of preserving the environment and reducing the cost of the network; and (3) the electromagnetic pollution in order to mitigate potential risks on human health. For these reasons, power control appears to be the main feature of green cellular networks.

However, a BS transmitting power limitation may have a negative impact on the network performance and quality of service. Both interference and useful signal strength indeed decrease. In this paper, we propose an analytical approach to quantify at a click speed the impact of transmitting power on capacity and coverage.

The estimation of cellular networks capacity and coverage mainly depends on the characterization of interference. In this paper, we focus on the downlink, but the proposed framework can easily be extended to the uplink. An important parameter for estimating interference is the other-cell interference factor f (OCIF). The precise knowledge of the OCIF allows the derivation of outage probabilities and Signal to Interference plus Noise Ratio (SINR) distributions, needed for coverage and capacity evaluations. In this paper, we define OCIF as the ratio of total other-cell received power to the total inner-cell received power.

Working on the uplink, [START_REF] Evans | Effective Bandwidth-Based Admission Control for Multiservice CDMA Cellular Networks[END_REF] derived the distribution function of a ratio of path-losses, which is essential for the evaluation of external interference. For that, authors approximate the hexagonal cell with a disk of same area. On the downlink, [START_REF] Viterbi | Other-Cell Interference in Cellular Power-Controlled CDMA[END_REF] [2] aimed at computing an average OCIF over the cell by numerical integration in hexagonal networks. In [START_REF] Elayoubi | Admission Control in the Downlink of WCDMA/UMTS[END_REF], othercell interference is given as a function of the distance to the base station (BS) thanks to Monte-Carlo simulations. Chan and Hanly [START_REF] Chan | Calculating the Outage Probabability in CDMA Network with Spatial Poisson Traffic[END_REF] precisely approximate the distribution of the other-cell interference. They however provide formulas that are difficult to handle in practice. Reference [START_REF] Abu-Dayya | Outage Probabilities in the Presence of Correlated Lognormal Interferers[END_REF] computes the outage probability in presence of multiple log-normal interferers, but assumes that all interference signals have the same statistics and mobile location influence is not studied.

Coverage studies in cellular networks, especially when frequency reuse is one, often neglect the impact of noise. This is the case in [START_REF] Chan | Calculating the Outage Probabability in CDMA Network with Spatial Poisson Traffic[END_REF], [START_REF] Elayoubi | Admission Control in the Downlink of WCDMA/UMTS[END_REF], [START_REF] Abu-Dayya | Outage Probabilities in the Presence of Correlated Lognormal Interferers[END_REF]. In [START_REF] Stuber | Principles of Mobile Communications[END_REF] also, outage probability calculation does not consider thermal noise. This is a reasonable assumption for typical output powers and in urban environments. It is however questionable at high frequencies, in rural environments or when BS output power is decreased.

In this paper, we give a new perspective on the influence of BS transmission power and thermal noise in different environments and for different frequencies. We derive closed form formulas for the outage probability with and without shadowing and we analyze the effect of decreasing BS power on coverage and capacity. We use the expressions of the OCIF without shadowing established in [START_REF] Kelif | Downlink Fluid Model of CDMA Networks[END_REF] and with shadowing established in [START_REF] Kelif | Impact of Topology and Shadowing on the Outage Probability of Cellular Networks[END_REF] to calculate analytically the outage probability by using a Gaussian approximation.

In the next section, we first introduce the model and the notations. In section III, we establish outage probability formulas. In sections IV and V, we analyze the interference factor and the thermal noise contribution. In section VI, we propose an application to green networks and show that it is possible to decrease the transmitting power of a wireless network with a limited impact on the quality of service.

II. INTERFERENCE MODEL AND NOTATIONS

We consider a cellular radio system and we focus on the downlink. BS have omni-directional antennas, so that a BS covers a single cell. If a mobile u is attached to a station b (or serving BS), we write b = ψ(u).

The following power quantities are considered:

• P b,u is the transmitted power from station b towards mobile u (for user's traffic), • P b = P cch + Σ u P b,u is the total power transmitted by station b, • P cch represents the amount of power used to support broadcast and common control channels, • g b,u designates the path-gain between station b and mobile u, • p b,u is the power received at mobile u from station b, we can write p b,u = P b g b,u , • S b,u = P b,u g b,u is the useful power received at mobile u from station b (for traffic data); since we do not consider soft handover, we can write S b,u = S ψ(u),u = S u . We consider a reuse 1 radio system such as OFDMA or CDMA. The total amount of power experienced by a mobile station u in a cellular system can be split up into several terms: useful signal (S u ), interference and noise (N th ). It is common to split the system power into two terms: I u = I int,u + I ext,u , where I int,u is the internal (or own-cell) received power and I ext,u is the external (or other-cell) interference. Notice that we made the choice of including the useful signal S u in I int,u , and, as a consequence, it has to be distinguished from the commonly considered own-cell interference.

With the above notations, we define the interference factor in u, as the ratio of total power received from other BS to the total power received from the serving BS ψ(u): f u = I ext,u /I int,u . The quantities f u , I ext,u , and I int,u are location dependent and can thus be defined in any location x as long as the serving BS is known.

In downlink, a coefficient α, may be introduced to account for the lack of perfect orthogonality in the own cell. In a CDMA system 0 < α < 1, and in an OFDMA system α = 0.

In this paper, we use the signal to interference plus noise ratio (SINR) as the criteria of radio quality: γ * u is the SINR target for the service requested by MS u or required to obtain a minimum throughput in case of elastic traffic. This figure is a priori different from the SINR evaluated at mobile station u. However, we assume perfect power control, so SIN R u = γ * u for all users. With the introduced notations, the SINR experimented by u can thus be derivated (see e.g. [START_REF] Lagrange | Principes et évolutions de l'UMTS[END_REF]):

γ * u = S u α(I int,u -S u ) + I ext,u + N th . (1)
In case of OFDMA, this equation is valid per sub-carrier. We can now express S u as:

S u = γ * u 1 + αγ * u I int,u (α + I ext,u /I int,u + N th /I int,u ). (2)
Using relations I int,u = P b g b,u and f u = I ext,u /I int,u we can write

f u = j =b P j g j,u P b g b,u (3)
and the transmitted power for MS u, P b,u = S u /g b,u ,

P b,u = γ * u 1 + αγ * u (αP b + f u P b + N th /g b,u ). (4)
From this relation, the output power of BS b can be computed as follows:

P b = P cch + u P b,u ,
and so, according to Eq.4,

P b = P cch + u γ * u 1+αγ * u N th g b,u 1 -u γ * u 1+αγ * u (α + f u ) . ( 5
)
Note that this equation is valid for multiple SINR targets and thus for multi-service networks since γ * u is specific to user u. In the rest of this paper, we will however consider a single service (γ * u = γ * ∀u) and an homogeneous network (P j = P b ∀j).

In the following sections, we will consider two channel models. Without shadowing, g b,u = Kr -η b,u . With shadowing, g b,u = Kr -η b,u A b , where A b = 10 -ξ b /10 is a lognormal random variable (RV) with logarithm mean and standard deviation 0 dB and σ (dB) respectively and K is constant which depends on the frequency.

Noise power can be expressed as N th = N 0 W , where N 0 is the noise power spectral density and W is the system bandwidth. Index b will be sometimes dropped when no confusion is possible, so that r u = r b,u .

III. OUTAGE PROBABILITIES

For a given number of MS per cell, n, outage probability, P (n) out , is the proportion of configurations, for which the needed BS output power exceeds the maximum output power: P b > P max :

P (n) out = P r n-1 u=0 f u + h u > 1 -ϕ β -nα , (6)
where

ϕ = P cch /P max , β = γ * /(1 + αγ * ) and h u = N th
Pmaxg b,u . The first term of the left hand side of the inequality represents the impact of the interferences due to the other base stations of the system. The second term represents the impact of the thermal noise and BS output power. We denote T u the following term:

T u = f u + h u . (7)
In order to compute the outage probability, we rely on the Central Limit theorem and use the Gaussian approximation.

We have thus to compute the mean and standard deviation of T u .

A. Without shadowing

Let first assume that there is no shadowing. Let µ T and σ T be the mean and standard deviation of T u . The outage probability can be approximated by:

P (n) out = Q 1-ϕ β -nµ T -nα √ nσ T . (8)
Knowing mobile location r u , the interference factor is here a deterministic function. However, mobile locations are random. Hence, variable T u is random and depends on the locations of the n mobiles in the cell. Without shadowing, we denote f u = f 0 (r u ) the interference factor and h u = h 0 (r u ) the thermal factor.

Without shadowing, we have thus:

µ T = µ f0 + µ h0 (9) σ 2 T = σ 2 f0 + σ 2 h0 + 2E[f 0 h 0 ] -2µ f0 µ h0 , (10)
where µ f0 and σ f0 are respectively the spatial mean and standard deviation of f 0 , µ h0 and σ h0 the spatial mean and standard deviation of h 0 and the expectation is taken on the uniform distribution of mobiles over the cell area.

B. Shadowing impact

Let us now consider the shadowing impact. For each location r u , f u is a random variable because of shadowing. Mobiles are still assumed to be randomly located in the cell. Hence, variable T u is random and depends both on shadowing and on mobile location. Using the Gaussian approximation, the outage probability is expressed as:

P (n) out = Q 1-ϕ β -nM T -nα √ nS T , (11)
where M T and S T are respectively the spatial mean and standard deviation of the random variable T u .

With shadowing, we have:

M T = M f + M h , ( 12
) S 2 T = S 2 f + S 2 h + 2E[f u h u ] -2M f M h , (13)
where M f , S f , M h and S h are the spatial mean and standard deviation respectively of random variables f u and h u .

The expectation is taken both on shadowing variations and mobile location. In the next section, we now characterize the interference factor more precisely.

IV. INTERFERENCE FACTOR ANALYSIS

In this section, we compute the mean and standard deviations of f u without shadowing (µ f0 and σ f0 ) and with shadowing (M f and S f ).

A. Without shadowing 1) Interference factor at distance r u : Forgetting in this section the shadowing impact, Eq. 3 can be written:

f 0 = j =b r -η j,u r -η b,u . (14)
For a mobile at the distance r u from its serving base station, the interference factor can be written as f 0 [START_REF] Kelif | Downlink Fluid Model of CDMA Networks[END_REF] [13]:

f 0 = 2πρ BS r η η -2 (2R c -r) 2-η . ( 15
)
where 1/ρ BS = πR 2 e represents the cell area and 2R c the distance between two neighbour BS. We have R e = R c 2 √ 3/π. 2) Integration over the cell area: In order to compute µ f0 and σ f0 , we integrate f 0 on a disk of radius R e . As MS are uniformly distributed over the disk of radius R e , the PDF of

r is: p r (r) = 2r R 2 e . µ f0 = 2πρ BS η -2 Re 0 r η (2R c -r) 2-η 2r R 2 e dr = 2 4-η πρ BS R 2 c η 2 -4 R e R c η × 2 F 1 (η -2, η + 2, η + 3, R e /2R c ), (16)
where 2 F 1 (a, b, c, z) is the hypergeometric function, whose integral form is given by:

2 F 1 (a, b, c, z) = Γ(c) Γ(b)Γ(c -b) 1 0 t b-1 (1 -t) c-b-1 (1 -tz) a dt,
and Γ is the gamma function.

In the same way, the variance of f 0 is given by:

σ 2 f0 = E f 2 0 -µ 2 f0 (17) E f 2 0 = 2 4-2η (2πρ BS R 2 c ) 2 (η + 1)(η -2) 2 R e R c 2η × 2 F 1 (2η -4, 2η + 2, 2η + 3, R e 2R c
).

B. Shadowing impact

1) Random variable at distance r u : Let us now assume the presence of shadowing. Recall that the interference factor is defined as f u = I ext,u /I int,u . On one hand, I ext,u = B j =b P b g j,u is a sum of lognormal RV. Such a sum can be approximated by another lognormal distribution [START_REF] Stuber | Principles of Mobile Communications[END_REF] using the Fenton-Wilkinson [START_REF] Fenton | The Sum of Lognormal Probability Distributions in Scatter Transmission System[END_REF] method.

On the other hand, I int,u = P b g b,u is a lognormal RV. As a ratio of two lognormal RV, the interference factor f u can thus be approximated by a lognormal RV with parameters m f and s f . ln(f u ) is thus a normal RV expressed as N (m f , a 2 s 2 f ), where a = ln(10)/10.

We introduced in [14]

L(r u , η) = f 0 (r u , 2η) f 0 (r u , η) 2 and J(r u , σ) = e a 2 σ 2 /2 L(r u , η)(e a 2 σ 2 -1) + 1 -1 2 ,
and showed that

m f = ln(f 0 (r u , η)J(r u , σ)), (18) a 2 s 2 f = 2(a 2 σ 2 -ln J(r, σ)), (19)
when received powers are uncorrelated. We also showed that

m f,corr = ln(f 0 (r u , η)J(r u , σ g )), (20) a 2 s 2 f,corr = 2(a 2 σ 2 g -ln J(r, σ g )) + 2ta 2 σ 2 , ( 21
)
when received power are correlated with a correlation coefficient t and a correlated variance σ 2 g = σ 2 (1t).

2) Integration over the cell area:

We now integrate the RV f u over the cell area assuming a uniform distribution of the mobiles. The mean E[f u ] of the interference factor can be written:

M f = Re 0 E[f u |r]p r (r)dr = Re 0 f 0 (r)J(r, σ)e a 2 s 2 f /2 2r R 2 e dr. (22)
In the same way, from Eq. 19, the variance of f u is given by:

S 2 f = E f 2 u -E 2 [f u ] = E f 2 u -M 2 f (23) E f 2 u = E f 2 u |r p r (r)dr = Re 0 (f 0 (r)J(r, σ)) 2 e 2a 2 s 2 f 2r R 2 e
dr.

V. NOISE AND POWER ANALYSIS

Recalling Eq. 7, the interference factor term has been fully characterized in the previous section. We now turn to the thermal noise contribution, h u .

A. Without shadowing

In this simple case, h u = h 0 = N th PmaxKr -η u is a deterministic function at a given distance r u . We thus simply need to average this value over the cell area in order to take into account mobile random locations.

µ h0 = N th P max K E[r η u ] = 2N th R η e P max K(η + 2) . ( 24
)
In the same way:

σ h0 = N th P max K 2 E[r 2η u ] = R 2η e η + 1
N th P max K 2 . ( 25
)
According to Eq. 10 and as f 0 and h 0 are not independent, we need to compute:

E[f 0 h 0 ] = N th P max K E[f 0 (r u )r η u ] (26) = 2N th πρ BS P max K(η -2) E r 2η u (2R c -r u ) 2-η , with E[f 0 (r u )r η u ] = Re 0 r 2η (2R c -r) 2-η p r (r)dr.
This integral can be expressed with a formula involving the hypergeometric function but it is not given here for the sake of simplicity.

B. Shadowing impact

The thermal noise factor is now

h u = h 0 /A b = h 0 10 -ξ b /10
. At a distance r u from the serving BS, this is thus a lognormal RV. Assuming uniform random mobile locations, we have:

M h = µ h0 E 10 -ξ b /10 = µ h0 e a 2 σ 2 /2 , (27)
and

S 2 h = E[h 2 0 ]E 10 -2ξ b /10 -M 2 h = (σ 2 h0 + µ 2 h0 )e 2a 2 σ 2 -M 2 h . (28)
According to Eq. 13 and keeping in mind that f u and h u are not independent RV, we still need to compute E[f u h u ]:

f u h u = N th P max K B j =b A j r -η j,u A 2 b r -2η b,u . ( 29
)
With similar arguments as in section IV-B, we see that f u h u can be approximated, at a given distance r u , by a lognormal RV. Using the Fenton-Wilkinson method as in [START_REF] Kelif | Impact of Topology and Shadowing on the Outage Probability of Cellular Networks[END_REF], we show that the average value of f u h u at distance r u is given by:

E[f u h u |r u ] = N th P max K f 0 r η u e 3a 2 σ 2 /2 . ( 30
)
We are now able to integrate this equation over the cell area:

E[f u h u ] = Re 0 E[f u h u |r]p r (r)dr (31) = 4πρ BS N th e 3a 2 σ 2 /2 P max KR 2 e (η -2) × ( 32
) Re 0 r 2η+1 (2R c -r) 2-η dr. ( 33
)
Again, the last integral can be explicitly expressed with the hypergeometric function but the expression is not detailed here for a reason of room.

VI. APPLICATIONS

We present in this section applications of our analysis. We first show the influence of the BS output power on capacity on one hand, on coverage on the other hand. Then, for a typical output power, we give the inter-BS distance from which thermal noise is not negligible anymore. At last, we show that BS densification reduces the global power density in the network.

We consider urban and rural environments (see Tab. I [START_REF][END_REF]) and two frequency bands (920 MHz and 2 GHz). We assume voice service in a CDMA network (γ * = -19 dB) and the following parameters: W = 5 MHz, α = 0.6, ϕ = 0.2, N 0 = -174 dBm/Hz, t = 0.5, η = 3.41.

A. Impact on capacity

Fig. 1 shows the impact of the BS maximum transmitting power on the system capacity. For a given value of P max , the maximum number of MS per cell, n MS , such that the outage probability is less than a target value of P * out of 5% is deduced from Eq. 11. MS density is ρ MS = n MS /πR 2 e . Inter-BS distance R c is set to 1 Km in urban and 5 Km in rural.

This figure quantifies the loss of capacity observed when transmitting power is decreased. For example, when power goes from 20 dBm to 10 dBm, capacity decreases from 5 to 1.5 MS/Km 2 (urban, 920 MHz). It is also shown that output power can be limited without loosing capacity: to 28 dBm (urban, 920 MHz), 39 dBm (urban, 2 GHz), 26 dBm (rural, 920 MHz), or 33 dBm (rural, 2 GHz), depending on the environment and the frequency band.

Increasing the frequency decreases parameter K. For a given inter-BS distance R c (e.g. 1 Km), the interference factor f u is, in this case, unchanged. Only the noise and power contribution h u is affected. This explains why a higher output power is needed to reach the maximum capacity when frequency increases.

In rural environment, several effects appear compared to the urban case. On one hand, since R c is higher, more power is needed in average per MS (see Eq. 4 and the expression of h u ). On the other hand, a higher K (see e.g. Eq. 24) and a lower σ (see e.g. Eq. 27 and 22) implies a higher capacity. With the chosen parameters, the increase of R c has a dominant effect and explains the lower MS density.

B. Impact on coverage

We now quantify the influence of the BS transmitting power on coverage. We consider a fixed MS density of ρ MS = 5.5 MS/Km 2 in urban and ρ MS = 0.4 MS/Km 2 in rural and we set the target outage probability to P * out = 1%. Half inter-BS distance R c (and so ρ BS ) is constant, but the effective coverage range R e is now variable. Integrations over the cell area in sections IV-B2 and V-B are done on a disk of radius R e . Decreasing R e reduces the number of MS per cell and diminishes the average power per user, so that outage probability is decreased.

In our study and for a given output power P max , we search the coverage range for which target outage probability is reached. Results are illustrated in Fig. 2. For example, in urban environment and at 2 GHz, when P max is less than 30 dBm, coverage holes appear and at P max = 20 dBm, the coverage range is only 800 m. This figure thus provides the minimum BS power that ensures service continuity.

We now study the effect of a small degradation of the quality of service while setting the target outage to P * out = 5%. In urban, only 1 and 2 dBm can be saved at 920 and 2 GHz. In rural however, important gains can be made. While keeping service continuity (coverage range of 5 Km), it is possible to reduce the BS power from 14 to 7 dBm at 900 MHz and from 21 to 15 dBm at 2 GHz. This gain is obtained at the price of a small degradation of the quality of service.

Of course, these figures depend on the chosen set of parameters and in particular on the propagation model. But they also show that our equations can provide very quick results on the impact of power reduction on the network performance.

C. Thermal noise impact on outage

We now question the traditional assumption of neglecting thermal noise in outage probability computations. We set P max = 43 dBm, which is a standard figure used in cellular networks and we compare in Fig. 3 the outage probabilities obtained with and without considering the thermal noise impact. We set the number of MS per cell, fixed per environment, such that outage probability is P * out = 5% when noise is neglected. When noise is neglected, outage probability depends neither on K nor on the frequency band (h u = 0 in Eq. 6 and f u does not depend on K). Moreover, it is easy to show with an analysis of Eq. 22 and 23 that varying the inter-BS distance produces homothetic networks, and so, that outage probability does not depend on R c .

On the contrary, when noise is not neglected, its contribution is increasing with R c . If we accept a error of 0.5% on the outage, Fig. 3 shows that noise cannot be neglected for R c > 1 Km in urban and R c > 7 Km in rural at 2 GHz. Limits are higher at 920 MHz.

D. Power density and BS densification

We now analyze the global network power density as the function of the BS density in a urban environment. In Fig. 4, MS density is supposed to be constant and set to ρ MS = 5.5 MS/Km 2 . Full coverage is also assumed. For a given value of R c , we compute the minimum maximum power P max ensuring that outage probability is below P * out = 5%, power density in the network is obtained with P max /πR 2 e . Fig. 4 shows that increasing the BS density results in a decrease of the network power density. This effect is particularly important at 2 GHz. For example, reducing R c from 1 Km to 950 m implies a division by almost two of the power density, from 0.46 W/Km 2 to 0.24 W/Km 2 . This would mean only a 11% increase of the number of BS in the network. This also shows that deploying small and femto cells are good means of reducing electromagnetic pollution provided that transmission power is optimized.

VII. CONCLUSION

In this paper, we derived closed form formulas for outage probabilities in cellular networks, taking into account pathloss, shadowing, thermal noise and random mobile locations. We showed that these formulas depend on one hand on the interference factor and, on the other hand, on the noise and BS power contribution. This analysis allows the study of the capacity and coverage of a cellular network at a click speed.

As an example of it, we studied the influence of the BS power on the network performance. We showed that it is possible to drastically reduce power emissions without loss or with a limited degradation of the quality of service. We also quantified the decrease of the network power density when cell size is reduced.

Figure 1 .

 1 Figure 1. Influence of the output power on capacity.

Figure 2 .

 2 Figure 2. Influence of the output power on coverage.

Figure 3 .

 3 Figure 3. Outage probability vs. half inter-BS distance.

Figure 4 .

 4 Figure 4. Power density vs. half inter-BS distance.


Masood Maqbool
email: masood.maqbool@telecom-paristech.fr

Philippe Godlewski
email: philippe.godlewski@telecom-paristech.fr

Marceau Coupechoux
email: marceau.coupechoux@telecom-paristech.fr

Analytical Performance Evaluation of Various Frequency Reuse and Scheduling Schemes in Cellular OFDMA Networks

Keywords: OFDMA, Fluid Model, SINR, Frequency Reuse, IFR, FFR, Scheduling, Equal data rate scheduling, Equal bandwidth scheduling, Opportunist PACS:

In this paper, we present an analytical solution to carry out performance analysis of various frequency reuse schemes in an OFDMA based cellular network. We study the performance in downlink in terms of signal to interference (SIR) ratio and total cell data rate. The latter is analyzed keeping in view three different scheduling schemes: equal data rate, equal bandwidth and opportunist. Analytical models are proposed for integer frequency reuse (IFR), fractional frequency reuse (FFR) and two level power control (TLPC) schemes. These models are based on a fluid model originally proposed for CDMA networks. The modeling key of this approach is to consider the discrete base stations entities as a continuum. To validate our approach, Monte Carlo simulations are carried out. Validation study show that results obtained through our analytical method are in conformity with those obtained through simulations. We present a comparison between above three frequency reuse schemes and three scheduling schemes. We also propose an optimal tuning of involved parameters (inner cell radius and power ratios).

Introduction

Orthogonal frequency division multiple access (OFDMA) is a promising multiple access technique being proposed for next generation mobile networks. The underlying technology for OFDMA based systems is orthogonal frequency division multiplexing (OFDM). With OFDM, available spectrum is split into a number of parallel orthogonal narrowband subcarriers. These subcarriers can be independently assigned to different users in a cell. Resources of an OFDMA system occupy place both in time (OFDM symbols) and frequency (subcarriers) domains thus introducing both the time and frequency multiple access [START_REF] Kulkarni | Subcarrier Allocation and Bit Loading Algorithms for OFDMA-Based Wireless Networks[END_REF].

Co-channel interference (CCI) limits the spectral efficiency of an integer frequency reuse 1 (IFR1) cellular network (cf. section 3 for an introduction on IFR). The CCI becomes more critical for the users present in the border area of a cell. To combat this problem in an OFDMA based system, fractional frequency reuse (FFR) has been proposed in [START_REF]Mobile WiMAX-Part II: A Comparative Analysis[END_REF]. In FFR, cell is divided into inner (close to base station) and outer (border area) regions. Available bandwidth is divided among inner and outer regions in a way that former employs reuse 1 while the latter applies frequency reuse 3. Hence, users located in cell border mitigate CCI because of frequency reuse 3. By properly adjusting the sizes of inner and outer regions, spectral efficiency can be improved. This paper extends the framework, based on a fluid model, proposed in [START_REF] Kelif | Downlink Fluid Model of CDMA Networks[END_REF] and [START_REF] Kelif | Spatial Outage Probability for Cellular Networks[END_REF]. The model provides a simple closed-form formula for the other-cell interference factor f on the downlink of CDMA networks as a function of the distance to the BS, the path-loss exponent, the distance between two BS, and the network size. The modeling key of this approach is to consider the discrete BS entities of a cellular network as a continuum.

Rest of the paper is organized as follows: section 2 introduces notations used throughout the paper and recalls the main result of the fluid model. Section 3 focuses on IFR and derives SIR and spectral efficiency expressions for both reuse 1 and reuse K. The case of FFR is studied in section 4. A two level power control scheme used in conjunction with FFR is considered in section 5. In section 6, three frequency reuse schemes (IFR, FFR and TLPC) are compared in terms of SINR and total cell data rate. Finally, section 7 discusses the conclusion of this analysis.

Fluid Model and Notations

In this section, we explain the application of fluid model to an OFDMA system. We focus on the downlink and consider a single subcarrier. BS have omnidirectional antennas, such that one BS covers a single cell. If a user u is attached to a station b (or serving BS), we write b = ψ(u).

The propagation path gain g b,u designates the inverse of the pathloss pl between station b and user u, g b,u = 1/pl b,u . In the rest of this paper, we assume that g b,u = Ar -η b,u , where A is a constant, r b,u is the distance between BS b and user u and η (> 2) is the path-loss exponent.

Before presenting the expression of fluid model, we establish the following terms:

• P T x is the transmitted power per subcarrier. We assume that the output power per subcarrier is constant. Only in section 5, we consider two possible values of output power per subcarrier: P i for the inner region of the cell and P o for the outer region. In this paper, we do not consider dynamic power allocation per subcarrier since in current OFDMA systems (WiMAX, Long Term Evolution), output power per subcarrier is constant; • S b,u = P T x g b,u is the useful power received at user u from station b (for traffic data); • W is the total system bandwidth and W u is the bandwidth dedicated to user u; • R, R c and R nw are respectively the cell radius, half distance between base stations and network range (see Fig. 1);

• R e is the radius of a circular region whose area is equal to that of the hexagon with length of each side equal to R. Based on definitions of R e , R and R c , it can be deduced that R e = aR c = a

√ 3 2 R, where a = 2 √ 3
π . • ρ u , ρ BS and N u are respectively the user density, BS density and number of users per cell; • D u is the data rate allocated to a user and D T is the total cell data rate;

• N BS represents the total number of base stations in the network.

The total amount of power experienced by a user u in a cellular system can always be split up into three parts: useful signal (S b,u ), interference and noise (N th ). It is common to split the system power into two terms: I u = I int,u +I ext,u , where I int,u is the internal (or own-cell) received power and I ext,u is the external (or other-cell) interference. We consider that useful signal S b,u is included in the I int,u . It should be noted that this useful signal power has to be distinguished from the commonly considered own-cell interference. In a CDMA network, the lack of orthogonality induces own-cell interference. In a OFDMA network, there is a perfect orthogonality between users and thus I int,u = S b,u .

With the above notations, the signal to interference plus noise ratio (SINR) is given by:

γ u = P T x g b,u j =b P T x g j,u + N th , (1)
where g j,u is the path-gain between BS j and user u.

Reference [START_REF] Kelif | Downlink Fluid Model of CDMA Networks[END_REF] has defined the interference factor in u, as the ratio of total power received from other BS to the total power received from the serving BS ψ(u): f u = I ext,u /I int,u . The quantities f u , I ext,u , and I int,u are location dependent and can thus be defined at any location x as long as the serving BS is known. In an OFDMA network, I ext,u is the total interference, and thus f u is the inverse of the signal to interference ratio (SIR) per subcarrier. Throughout this paper, we shall neglect noise in our analytical calculations. This is a common assumption for macro-cells in dense urban areas. In this case, the SINR, γ u can be approximated by the SIR:

γ u ≈ S b,u I ext,u = 1/f u .
As a consequence, it is clear that the approach developed in [START_REF] Kelif | Downlink Fluid Model of CDMA Networks[END_REF] can be adapted to OFDMA networks, as soon as the orthogonality factor α considered in CDMA networks is zero (details on fluid model are given in appendix). In this case, SIR per subcarrier is simply the inverse of the interference factor considered in [START_REF] Kelif | Downlink Fluid Model of CDMA Networks[END_REF].

γ u = r -η u (η -2) 2πρ BS (2R c -r u ) 2-η .
(

) 2
Note that the shadowing effect is neglected in this paper. An extension of the fluid model has been proposed in [START_REF] Kelif | Effect of Shadowing on Outage Probability in Fluid Cellular Networks[END_REF] to take into account shadowing. The results presented in this paper can thus be extended accordingly.

We now compare the results obtained with Eq. 2 with those obtained numerically by Monte Carlo simulations. The simulator assumes an homogeneous hexagonal network made of several rings around a central cell. Fig. 1 shows an example of such a network with the main parameters involved in the study.

Fig. 2 shows the simulated SINR (using Monte Carlo simulations) as a function of the distance from the base station. Following are the simulation parameters: R = 1 Km, η between 2.7 and 4, ρ BS = (3 √ 3R 2 /2) -1 , the number of rings is 15, and the number of snapshots is 3000. To include the effect of pathloss, Erceg model [START_REF] Erceg | An Empirically Based Path Loss Model for Wireless Channels in Suburban Environments[END_REF] is used:

g b,u = Ar -η b,u , such that A = 4πd 0 f cd η/2 0 2 , d 0 = 100 m, f = 2.
5 GHz and c is the speed of light. Thermal noise density has been taken as -174 dBm/Hz and a subcarrier spacing of 11 KHz is considered [START_REF] Ramadas | WiMAX System Evaluation Methodology[END_REF]. Eq. 2 is also plotted for comparison.

In all cases, the fluid model matches very well the simulations in an hexagonal network for various values of path-loss exponent. Only in a short area around the BS, the fluid model is a little bit pessimistic, but this is not a region of primary interest for operators. It is to be noted that thermal noise was not considered in the fluid model while simulator does include its effect. However, the results of the two (fluid model and simulator) still match. It indicates that value of interference is much more pronounced as compared to that of thermal noise. Hence, neglecting thermal noise in the fluid model is a reasonable

assumption.

For the rest of the paper, all above parameters are used for simulations. However, instead of different values of η, a fixed value of three is used in rest of the simulations except mentioned otherwise. The closed-form formula 2 will allow us to quickly compute performance parameters of an OFDMA network and in particular compare different reuse pattern schemes with different scheduling algorithms.

Integer Frequency Reuse (IFR)

In this section, we consider the application of the fluid model to IFR environments. In integer frequency reuse, all subcarriers allocated to a cell can be used anywhere in the cell without any specification of user's location. However, reutilization of subcarriers in network cells may be one or greater. An example of IFR with frequency reuse 1 is shown in Fig. 3, where W represents the available network bandwidth. For frequency reuse 1, cell bandwidth equals network bandwidth. Two cases, frequency reuse 1 and K, have been considered. We first derive SINR and spectral efficiency expressions as functions of the distance from the BS using the fluid model. We then take into account three scheduling schemes (equal data rate, equal bandwidth and opportunist) and derive the total cell data rate for each of them. Results of analytical expressions are also compared with results of Monte Carlo simulations.

IFR with Reuse 1

Consider Eq. 2 that gives expression for SINR of a subcarrier for a user at distance r u . A user at distance r from BS has a specific value of SINR and spectral efficiency. Hence, in rest of the paper, subscript u is omitted for r, SINR and spectral efficiency. With

ρ BS = 1/(2 √ 3R 2 c
) and introducing the normalized distance x such that x = r/R c , the expression for SINR can be rewritten as:

γ IF R1 (x) = √ 3 π (η -2)(2 -x) -2 (2/x -1) η . (3)
For comparison between simulation and fluid model, Fig. 2 can be consulted in which all curves for fluid model have been drawn using Eq. 3.

By using Shannon's formula, spectral efficiency (in bps/Hz) as a function of variable x is given as:

C IF R1 (x) = log 2 [1 + γ IF R1 (x)], (4)
where γ IF R1 (x) is furnished by Eq. 3.

In following sections, we use this expression of spectral efficiency to calculate total cell data rate for three scheduling schemes.

Equal data rate

While considering equal data rate, users are assigned the bandwidth in a way that resultant data rate for every user, D u , is the same. Higher is the distance of a user from the BS, lower is the available spectral efficiency and thus, higher is the bandwidth (or number of subcarriers) allocated to it. As SINR and spectral efficiency depend on r, let W u (r) be the bandwidth allocated by the scheduler to a user at distance r from the BS.

User data rate, D u , can now be written for any r:

D u = W u (r)C(r), (5)
under the constraint that total cell bandwidth W cannot be exceeded. Total bandwidth used in a cell is thus given as:

W = 12 π/6 0 Rc/ cos θ 0 W u (r)ρ u r dr dθ. (6)
Integration is done over the shaded triangular region in Fig. 3 and multiplied by twelve to obtain the result over the entire hexagon cell.

If N u is the number of users in a cell, user density is

ρ u = N u /(2 √ 3R 2 c
). Using Eq. 6 and 5, value of ρ u and variable transformation r to x, user data rate is given as:

D u = √ 3W/6 N u π/6 0 1/ cos θ 0 x C IF R1 (x) dx dθ ,
where C IF R1 (x) is given by Eq. 4.

Since all users receive same data rate and there are N u users in the cell, total cell data rate is D T,IF R1 = N u D u and can be written using previous result as:

D T,IF R1 = √ 3W/6 π/6 0 1/ cos θ 0 x C IF R1 (x) dx dθ . (7)
A worth noting observation regarding Eq. 7 is that the total cell data rate neither depends upon the number of users in the cell nor upon the value of R c .

The change of variables θ = z and x = y cos z , whose Jacobian is provides the equivalent equation:

∂(θ,x) ∂(z,y) = 1 cos z ,
D T,IF R1 = √ 3W/6 π/6 0 1 0 y/ cos z cos zC IF R1 (y/ cos z) dy dz .
To compare the above results with those of simulations, parameters of section 2 are used. We set the available network bandwidth to W = 10 MHz and the number of users per cell to N u = 30 in simulations. In Tab. 1, total cell data rate D T,IF R1 with both the fluid model (Eq. 7) and simulations, for various values of η, are given. The best agreement is for η = 2.7, while the difference remains below 10% for η between 2.6 and 3.2. The user data rate (D u ) can be easily obtained by dividing the total cell data rate (i.e., D T,IF R1 in this case) by number of users (N u ) in the cell. To avoid the complexity of calculating double integral, it is also possible to integrate W u (r) over a disk, whose area equals the hexagon area. Such a disk has a radius R e = aR c , where a = √ 2 √ 3 π . Using this approach, total cell data rate can be approximated as:

D T,IF R1 ≈ √ 3W/π a 0 x C(x) dx . ( 8
)
For value of η = 3 and W = 10 MHz, data rates obtained with Eq. 7 and 8 are found to be 12.4 Mbps and 12.6 Mbps respectively with a difference of 1.6%.

Equal bandwidth

Equal bandwidth means that all users are assigned equal bandwidth whatever the spectral efficiency is available to them. Since users close to BS benefit from higher spectral efficiency, they will attain a higher data rate as compared to users at cell edge. Let W u denote the bandwidth allocated to each user such that W = N u W u . Data rate of a user at a distance r is then given as:

D u (r) = W u C(r), (9)
total data rate then can be obtained by integrating the user data rates over cell surface:

D T,IF R1 = 12 π/6 0 Rc/ cos θ 0 D u (r)ρ u r dr dθ, (10)
using Eq. 9, user density

ρ u = N u /(2 √ 3R 2 c
), user bandwidth W u = N u /W and transformation of variable r to x, we get:

D T,IF R1 = 6W √ 3 π/6 0 1/ cos θ 0 xC IF R1 (x) dx dθ. (11)
The simulation and fluid model results are compared in Tab. 2.

Opportunist

In opportunist scheduling, user experiencing the greatest SINR is assigned all the resources and rest of the users receive no resources at all. In light of assumptions considered in this paper, user closest to the BS will have the maximum SINR value. To calculate the total cell data rate for opportunist scheduling, we require PDF (probability density function) of the distance, X, of the user nearest to the BS. For N u users in the cell, the CDF of X is given by:

F X (r) = p[X ≤ r] = 1 -p[X > r] = 1 -(1 -πr 2 /2 √ 3R 2 c
) Nu , and its PDF can be obtained by differentiating the CDF:

p X,IF R (r) = πN u r √ 3R 2 c 1 - πr 2 2 √ 3R 2 c Nu-1
, with change of variable from r to x, the PDF for IFR over small distance dx can be rewritten as:

p X,IF R (x)dx = πN u x √ 3 1 - πx 2 2 √ 3 Nu-1 dx, (12)
taking into account the circular disc of radius R e = aR c with a = √ 2 √ 3 π (refer section 2) the average cellular spectral efficiency for opportunist scheduling can be calculated using following equation:

CIF R1 = a 0 C IF R1 (x)p X,IF R (x) dx.
Finally total cell data rate is written as:

D T,IF R1 = W CIF R1 .
To verify this approach, simulations are carried out with N u = 30. The results of simulation and model are given in Tab. 3.

IFR with Reuse K

For IFR with reuse higher than one, analytical study is very similar to the previous one. The difference lies in the fact that only co-channel BS are considered in interference calculation and thus the half-distance between base stations and BS density have to be modified. As a consequence, previous analysis results are still valid provided that R c is replaced by √ KR c and BS density is divided by K, i.e., ρ BS is replaced by ρ BS /K. Hence using Eq.2 and this new half distance between BS, SINR is given as: Using the same distance normalization as before (leading to the transformation of variable r to x) and after few manipulations, SINR can be written as:

γ(r) = r -η (η -2) 2π • ρ BS K (2 √ KR c -r) 2-η . ( 13
)
γ IF RK (x) = K √ 3 π (η -2)(2 √ K -x) -2 (2 √ K/x -1) η . (14)
Hence spectral efficiency (in bps/Hz) for IFR reuse K can be given as:

C IF RK (x) = log 2 [1 + γ IF RK (x)]. ( 15
)
To validate above approach, reuse 3 is considered as an example. Plot of SINR versus distance to BS for reuse 3 case, for both fluid model and simulation, is shown in Fig. 4. As expected SINR is higher than for reuse 1. However, bandwidth per cell equals one third the network bandwidth. Again, both analysis and simulation provide similar results. The fluid model is thus accurate not only for reuse 1 networks but also for higher reuse factors provided the parameters are adjusted.

As far as total cell data rate for three scheduling schemes is concerned, method used for IFR reuse 1 is still valid provided that C IF R1 (x) is replaced by C IF R3 (Eq. 15) and cell bandwidth is divided by 3, in all calculations. Values of total cell data rate for fluid model and simulations are shown in Tab. 4, 5 and 6. In all cases, the difference between simulation and fluid model remains below 10% for η between 2.6 and 3.5. 4 Fractional Frequency Reuse (FFR)

An example of FFR scenario is depicted in Fig. 5. As can be seen in the figure, cell space is divided into two regions, inner and outer. Inner region is a circular disc with radius R 0 ≤ R c and rest of the hexagon space forms the outer region.

Bandwidth is allocated to inner and outer in a way that former incorporates frequency reuse 1 while the latter applies frequency reuse 3. As can be seen in Fig. 5, the network bandwidth W is equal to

W 0 + W 1 + W 2 + W 3 . It is also considered that W 1 = W 2 = W 3 .
SINR versus distance with R 0 = 0.7R c for fluid model and simulation is given in Fig. 6. As expected FFR scheme improves radio quality at cell edge. Moreover, the results follow the previous positive trend, i.e., fluid model and simulations are in conformity.

Now we derive the expressions for total cell data rate assuming schedulers fair in throughput, fair in resource allocation and opportunist. We also estimate the optimized inner region radius.

A Cluster of three cells

W 0 W 0 W 0 W 0 W 0 W 0 W 0 W 1 W 1 W 2 W 3 W 2 W 1 W 2 R c R R 0 W= W 0 +W 1 +W 2 +W 3
Fig. 5. Fractional Frequency Reuse (FFR) case. Bandwidth W 0 is used to deploy reuse 1 between inner regions, while W 1 , W 2 and W 3 are used to deploy reuse 3 between outer regions.

Equal data rate

To carry out total cell data rate, we first consider the inner circular region. Since for this region, frequency reuse is 1, the expression of SINR is given by Eq. 2 or equivalently by Eq. 3. Once again, it is considered that users are uniformly distributed in the cell space with

ρ u = N u /2 √ 3R 2 c .
Since it is a circular region, the elementary area ds of Eq. ?? can be simply taken as 2πrdr and the bandwidth of inner region is given as:

W 0 = R 0 0 W u (r)ρ u 2πr dr. ( 16
)
After replacing ρ u by its value, transformation of variable r to x and using Eq. 5, we get

W 0 = π √ 3 D u N u R 0 /Rc 0 x C IF R1 (x) dx,
where C IF R1 (x) is given by Eq.4. Let I 0 be the integral in the previous expression, so that

W 0 = π/ √ 3I 0 D u N u .
Let us now consider the outer area, which applies reuse 3. SINR for reuse 3 is given by Eq. 13 or equivalently by Eq. 14. In order to calculate the total bandwidth used in the outer region, double integral used in Eq. 6 is applied with change of limits and replacing W u (r) by W 1,u (r) we get:

W 1 = 12 π/6 0 Rc/ cos θ R 0 W 1,u (r)ρ u r dr dθ,
where W 1,u (r), assuming equal data rate, is given as:

W 1,u (r) = D u C IF R3 (r).
After replacing ρ u by its value and transformation of variable r to x, we get

W 1 = 6 √ 3 D u N u π/6 0 1/ cos θ R 0 /Rc x C IF R3 (x) dx dθ. (17)
Let I 1 be the double integral in the previous expression, so that

W 1 = 6 √ 3 I 1 D u N u .
Considering the fact that total network bandwidth is W with W = W 0 + W 1 + W 2 + W 3 and W 1 = W 2 = W 3 we can write: W = W 0 + 3 × W 1 . Finally using Eq. 16 and 17 and keeping in view that D T = D u N u we get expression of the total cell data rate D T,F F R for FFR case:

D T,F F R = √ 3W πI 0 + 18I 1 . ( 18
)
Total cell data rate calculation also shows that fluid model and simulation differ by 5.6% with values of 13.2 Mbps and 12.5 Mbps respectively for 10 MHz of network bandwidth. Fig. 7 shows the total cell data rate as a function of the inner cell radius R 0 . Both fluid model and simulations provide an optimum value of approximately 757 m.

Equal Bandwidth

To calculate cell data rate in this case, we adopt the same approach as was used in section 3.1.2 while considering N u W u = W 0 + W 1 . By integrating the user data rate over inner circular region (using frequency reuse 1), total data rate of inner region is given by Eq. 19.

D T,Inner = π(W 0 + W 1 ) √ 3 R 0 /Rc 0 xC IF R1 (x) dx. (19)
Similarly, assuming frequency reuse 3 for the outer region, total data rate of outer region is:

D T,Outer = 6(W 0 + W 1 ) √ 3 π/6 0 1/ cos θ R 0 /Rc xC IF R3 (x) dx dθ, (20)
and total cell data rate is the sum of inner and outer region data rates:

D T,F F R = D T,Inner + D T,Outer . (21)
Since this scheduling scheme assigns equal resources to all users (whether located in inner or outer region), the ratio of W 0 to W 1 should be proportional to number of users in two regions. But as we have assumed that users are uniformly distributed in cell space, this ratio should be equal to ratio of inner region area to outer region area:

W 1 W 0 = 2 √ 3R 2 c πR 0 -1, (22)
and since W 0 + 3W 1 = W , it can be shown that:

W 1 = 2 √ 3R 2 c -πR 2 0 6 √ 3R 2 c -2πR 2 0 W. (23)
It can be made out from Eq. 23 that for every value of R 0 , there is a specific value of W 1 and hence W 0 . For different values of R 0 , total cell data rate has been plotted in Fig. 8. It can be seen that maximum value of total cell data rate is for R 0 = R c .

Tab. 2 and 5 have shown that IFR1 provides higher data rates than IFR3. In FFR, increasing R 0 and thus the bandwidth W 0 used with reuse 1, makes FFR closer to IFR1. On the contrary, decreasing R 0 makes FFR closer to IFR3. As a consequence, total cell data rate is maximized for the highest value of R 0 .

Opportunist

For opportunist scheduling in FFR case, we assume that two users are simultaneously scheduled in a cell such that one user having the greatest SINR among inner region users and other one having the highest SINR among outer region users. If there is no user in any of the regions, bandwidth of that region goes unallocated. Following the methodology of section 3.1.3, for the inner region, PDF of the user's (nearest to the BS) distance, knowing there are N u,i users located in the inner region is given as:

p X,inner|N u,i (r) = 2rN u,i R 2 0 1 - r 2 R 2 0 N u,i -1
, or with substitution of r by x, the same PDF over small distance dx can be written as:

p X,inner|N u,i (x) dx = 2xN u,i (R 0 /R c ) 2 1 - x 2 (R 0 /R c ) 2 N u,i -1 dx.
Hence, the average spectral efficiency of inner region, given that there are N u,i located inside it, can be expressed by the following integral:

CF F R,inner|N u,i = R 0 /Rc 0 C IF R1 (x)p X,inner|N u,i (x) dx.
Similarly for the outer region, PDF of the user's (nearest to the BS) distance, knowing there are N u,i users located in the inner region is given as:

p X,outer|N u,i (r) = 2π(N u -N u,i )r 2 √ 3R 2 c -πR 2 0 1 - πr 2 -πR 2 0 2 √ 3R 2 c -πR 2 0 Nu-N u,i -1
, where (N u -N u,i ) is the number of users in the outer region.

Once again with change of variable r to x, the PDF over small distance dx for outer region can be expressed as:

p X,outer|N u,i (x) dx = 2π(N u -N u,i )x 2 √ 3 -π(R 0 /R c ) 2 1 - πx 2 -π(R 0 /R c ) 2 2 √ 3 -π(R 0 /R c ) 2 Nu-N u,i -1 dx,
and the average spectral efficiency for the outer region such that there are N u,i users in the inner region is given by following equation:

CF F R,outer|N u,i = a R 0 /Rc C IF R3 (x)p X,outer|N u,i (x) dx.
Note that in this latter equation, we approximate the hexagon by a disk of radius R e . Finally, averaging the sum of data rates (of inner and outer regions) for all possible values of N u,i results in total cell data rate for FFR case:

D T,F F R = Nu N u,i =0 (W 0 CF F R,inner|N u,i + W 1 CF F R,outer|N u,i )P [N u,i ],
where P [N u,i ] is the probability that there are N u,i out of N u users in the inner region:

P [N u,i ] = πR 2 0 2 √ 3R 2 c N u,i 1 - πR 2 0 2 √ 3R 2 c Nu-N u,i N u N u,i .
The total cell data rate (for FFR scheme) as a function of different values of W 1 and R 0 is shown in Fig. 9. Maximum value of total cell data rate can be observed for a value of W 1 = 11 KHz (i.e., one subcarrier) and R 0 = 487 m for both the simulation and fluid model. The maximum difference between simulation and analytical model results for all values of W 1 and R 0 is found to be 6.7%.

Two Level Power Control (TLPC)

In previous section, we discussed the concept of FFR in OFDMA and we have shown that SINR could be improved by using a reuse 3 pattern in cell outer regions. With FFR, it is however not possible to use full network bandwidth in a cell, which reduces the overall cell bandwidth.

To overcome this drawback, it is possible to adopt a reuse 1 pattern while using a two level power control (TLCP) mechanism to improve the radio quality in

= W 2 = W 3
the outer region. The TLPC scheme is shown in Fig. 10. Total bandwidth in a cell (equal to network bandwidth) is divided into three equal parts: two parts allocated to inner region and one to the outer region. The output power per subcarrier in the inner region is P i and that in the outer region is P o . These two values of power are related as: P o = δP i , such that δ ≥ 1. The three spectrum parts W 1 , W 2 and W 3 alternate from cell to cell in such a way that there is a pseudo-reuse 3 scheme between outer regions. Neighboring cells contribute in fact to interference in the outer region but with a reduced power P i . The consequence is that the total network bandwidth is used in every cell but interference is expected to be reduced in outer regions.

Let us calculate SINR for inner and outer region of this two level power control network. For a user in the outer region (using e.g. W 3 in the center cell in Fig. 10), we divide the interference into two categories. One is from the cells using same subcarriers in the outer region and we represent it by I outer . Other is from cells using same subcarriers in the inner region (neighboring cells) and is represented by I inner . Then, SINR for a subcarrier in the outer region can be written as:

γ T LP C,outer (r) = P o Ar -η I inner + I outer .
With slight change of form, the above expression can be rewritten as:

1 γ T LP C,outer = I inner P o Ar -η + I outer P o Ar -η . (24)
In order to find the values of I inner and I outer , consider that B outer represents the set of BS causing I outer . For a user u in outer region of a cell b, I outer is given as:

I outer = P o A N BS j=1,j∈Bouter r -η j,u . (25)
As outer regions of BS in B outer form together a reuse 3 scheme, the second term of right hand side of Eq. 24 is simply 1/γ IF R3 .

Adding up the interference from all network cells, I inner can be written as:

I inner = P i A N BS j=1,j =b r -η j,u -P i A N BS j=1,j∈Bouter r -η j,u . (26)
Thus, considering δ = P o /P i :

I inner P o Ar -η = 1 δ 1 γ IF R1 - 1 δ 1 γ IF R3 . (27)
Combining previous results, we can rewrite Eq. 24 as:

1 γ T LP C,outer = 1 δ 1 γ IF R1 + (1 - 1 δ ) 1 γ IF R3 . (28)
Now we find out SINR expression for inner region. Consider the central cell of Fig. 10, in which W 1 and W 2 are allocated to inner region and W 3 is used in the outer region. A user in inner region will be allocated a subcarrier that will either belong to W 1 or W 2 . If we look at the bandwidth utilized in the six neighboring cells of center cell, we notice that out of six, three are transmitting on the same subcarrier with power P o , while other three with P i . Hence, SINR for this inner region subcarrier can be approximated while considering that neighboring cells transmit with average power (P i + P o )/2 and

γ T LP C,inner (r) ≈ P i Ar -η b,u Po+P i 2 A N BS j=1,j =b r -η j,u = 2 1 + δ γ IF R1 . (29)
Using the values of SINR for outer and inner region, spectral efficiencies for two regions are given in Eq. 30 and 31. To verify the above results, SINR versus distance (with R 0 = 0.7R c and δ = 5) is given in Fig. 11. As expected, radio quality is improved in outer region with TLPC compared to the IFR1 case in a similar way FFR does. Let us now compare total cell data rates.

C T LP C,inner = log 2 (1 + γ T LP C,inner ), (30)
In order to calculate data rate for inner and outer region for three scheduling schemes, we assume that in Fig. 10, W 1 = W 2 = W 3 . We start with equal data rate scheduling scheme.

Equal Data Rate

Using the similar approach of section 4, we can write:

W 1 + W 2 = 2W 3 = π √ 3 D u,inner N u R 0 /Rc 0 xdx C T LP C,inner (x)
,

and

W 3 = W 3 = 6 √ 3 D u,outer N u π/6 0 1/ cos θ R 0 /Rc x C T LP C,outer (x) dx dθ.
Using above two equations, we can write the ratio between data per user for inner and outer region as: one. For a given value of δ there exists a unique value of R 0 for which above condition is satisfied. In Fig. 12, total cell data rate satisfying equal data rate has been plotted for various values of δ. It is clear that for small values of δ, total cell data rate increases with increasing values of δ. The total cell data rate attains its maximum value of 13.68 Mbps for δ = 13.2. Beyond δ = 13.2 (or 11.2 dB), the total cell data rate starts decreasing. The corresponding values of R 0 for these total cell data rates are given in Fig. 13. The figure shows R 0 is a decreasing function of δ. The value of R 0 corresponding to maximum value of cell data rate is approx. 600 m.

Equal bandwidth

With equal bandwidth allocation, the bandwidth of inner and outer regions should be proportional to areas of the two regions (cf. section 4.2) and since for TLPC schemes bandwidth of inner and outer region are fixed, there exists a unique value of R 0 satisfying the two conditions.

W 3 W 1 + W 2 = 2 √ 3R 2 c πR 2 0 -1, with W 1 = W 2 = W 3 = W/3, R 0 = 2 π √ 3 R c .
Now the total data rate of inner region is given by Eq. 32:

D T,Inner = πW √ 3 R 0 /Rc 0 xC T LP C,inner (x) dx. (32)
Similarly total data rate for outer region is given as:

D T,Outer = 6W √ 3 π/6 0 1/ cos θ R 0 /Rc xC T LP C,outer (x) dx dθ. (33)
Total cell data rate is the sum of total data rates of two regions:

D T,T LP C = D T,Inner + D T,Outer . (34)
A comparison of simulation and fluid model D T,T LP C values, as function of δ values, is shown in Fig. 14 which shows a close proximity between the two curves.

Opportunist

The computation of cell data rate is similar to the method used in section 4.3. However, in this case, inner bandwidth is 2W/3, outer bandwidth is W/3 and average spectral efficiency for TLPC is used: schemes while considering three different scheduling types: equal data rate, equal bandwidth and opportunist. In this section, we present a comparison between these three reuse schemes and the three scheduling policies by applying fluid model.

D T,T LP C = Nu N u,i =0 ( 2W 3 CTLP C,inner|N u,i + W 3 CTLP C,outer|N u,i )P [N u,i ].
If we look at Fig. 16, it can be deduced that IFR with reuse 3 shows the best performance in terms of SINR values. IFR reuse 1 is much lower than IFR reuse 3 in terms of radio quality. FFR exactly follows IFR reuse 1 curve until R 0 and IFR reuse 3 onwards. Compared to IFR reuse 1, TLPC improves SINR in outer region at the expense of a degraded radio quality in inner region.

We now compare total cell data rates for all frequency reuse schemes in presence of three scheduling algorithms. Total cell data rates in FFR may depend upon value of R 0 and W 1 . The same applies to TLPC w.r. and δ. For these two schemes, maximum possible value of total cell data rate, based on optimal values of their parameters, has been considered in the comparison. These optimal values are presented aside with values of data rates (see Tab. 7).

The value of network bandwidth is 10 MHz. Number of users per cell N u is considered to be thirty in all cases. Path loss constant η is taken as three. Results of cell data rate are listed in Tab. 7.

With equal data rate, IFR3 touches the lowest performance although SINR values are greater. This is due to the fact that utilization of network bandwidth per cell is lower as compared to other schemes. TLPC has the maximum value with δ = 13.2 and has a comparable performance w.r.t. FFR (with R 0 = 757 m). Hence, by applying TLPC and FFR schemes, we can diminish the problem of reduced radio quality (SINR) in case of IFR1. At the same time, bandwidth is more efficiently utilized than with IFR3. Due to its simplicity, FFR could be preferred to TLPC with equal data rate scheduling.

An equal bandwidth scheduling aims at improving resource utilization (compared to equal data rate scheduling) while ensuring resources to every user (on the contrary to opportunist scheduling). So, for all reuse schemes, total cell data rate is in-between two other schemes. With equal bandwidth, IFR1 achieves the highest cell capacity because it benefits from the usage of total network bandwidth in every cell. Although SINR are higher with IFR3, this scheme allocates only one third of the total network bandwidth to each cell, this explains again the lower achieved performance.

FFR and TLPC cannot do better than IFR1 and reach their maximum value for a parameter set that makes them very close to IFR1. With FFR, setting R 0 = R c makes most of the bandwidth to be used with reuse 1. A higher radio quality at cell border is obtained at the price of a small reduction of the total cell data rate compared to IFR1. With TLPC, setting δ = 1 reduces almost this scheme to IFR1.

Total cell data rate obtained with opportunist scheduling provides an upper bound (for a given number of users) on the cell performance at the price of fairness. Except in TLPC, only the best user is served and gets the whole bandwidth. IFR1 achieves again the highest cell data rate. FFR tends towards IFR1 with very small bandwidth being allocated to the outer region. The choice of scheduling two users (one in the inner region, one in the outer) reduces the performance of TLPC compared to IFR1 since part of the bandwidth is allocated to a user a bit far from the BS.

Note that γ u does not depend on the BS output power. This is due to the fact that we assumed an homogeneous network and so all base stations emit the same power on a given subcarrier. In this model, γ only depends on the distance r from the BS and can be defined in each location, so that we can write γ as a function of r, γ(r). If the network is large, i.e., R nw is big in front of R c , γ u can be further approximated by:

γ u = r -η u (η -2) 2πρ BS (2R c -r u ) 2-η . ( .3)
The fluid model and the traditional hexagonal model are two simplifications of the reality. None is a priori better than the other but the latter is widely used, for dimensioning purposes. That is the reason why comparisons are performed throughout the paper.

Reference [START_REF] Kelif | Spatial Outage Probability for Cellular Networks[END_REF] has shown that the considered network size can be finite and can be chosen to characterize each specific local network's environment. This model thus allows us to do the analysis adapted to each zone while taking into account considered zone's specific parameters. Moreover, it can be noticed that the fluid model can be used even for great distances between the base stations.

Fig. 1 .

 1 Fig. 1. Hexagonal network and main parameters of the study.

Fig. 2 .

 2 Fig. 2. SINR versus distance to the BS; comparison of the fluid model with simulations on an hexagonal network with η = 2.7, 3, 3.5, and 4 (reuse 1).

Fig. 3 .

 3 Fig.3. Integer Frequency Reuse (IFR) case (reuse 1). Shaded triangular region shows the basic integration area.

Fig. 4 .

 4 Fig. 4. SINR versus distance to BS for IFR with reuse 3.

Fig. 6 .

 6 Fig.6. SINR versus distance to BS for FFR with R 0 = 0.7R c .

Fig. 7 .

 7 Fig. 7. D T,F F R (total cell data rate) versus radius of inner region with equal data rate for FFR scheme. Maximum value occurs at R 0 = 757 m approx.

Fig. 8 .

 8 Fig. 8. D T,F F R (total cell data rate) versus radius of inner region with equal bandwidth for FFR scheme. Maximum value occurs at R 0 = R c .

Fig. 9 .

 9 Fig. 9. D T,F F R (total cell data rate) versus radius of inner region and versus versus bandwidth allocated in the outer region with opportunist scheduling for FFR scheme. Maximum value occurs at R 0 = 487 m and W 1 = 11 KHz.

CFig. 11 .

 11 Fig. 11. SINR versus distance to BS for TLPC scheme with R 0 = 0.7R c and δ = 5.

Fig. 12 .

 12 Fig. 12. Total cell data rate versus different values of δ with equal data rate for TLPC scheme.

  Fig.[START_REF] Tarhini | On Capacity of OFDMA-based IEEE802.16 WiMAX Including Adaptive Modulation and Coding (AMC) and Inter-cell Interference[END_REF]. R 0 (radius of inner region that guarantees equal data rate among users) versus different values of δ for TLPC scheme.

Fig. 14 . 1 Fig. 15 .

 14115 Fig. 14. Total cell data rate versus different values of δ with equal bandwidth for TLPC scheme.

Fig. 17 .

 17 Fig. 17. Data rate versus pathloss exponent with opportunist scheduling for three reuse schemes.

Fig. 19 .

 19 Fig. 19. Data rate versus pathloss exponent with equal bandwidth scheduling for three reuse schemes.

FFR 13 .

 13 61 (R 0 = 757 m) 19.5 (R 0 = Rc) 84.28 (W 1 = 11 KHz, R 0 = 487 m)

Table 1

 1 Total cell data rate (D T,IF R1 ) versus η (IFR reuse 1, bandwidth=10 MHz, Equal data rate).

	η	D T,IF R1 [Mbps]	Difference
		Fluid model Simulation
	2.5	6.62	7.43	10.9%
	2.6	7.83	8.26	5.21%
	2.7	9.01	9.08	0.77%
	2.8	10.16	9.89	2.73%
	3	12.4	11.6	6.9%
	3.2	14.5	13.2	9.85%
	3.3	15.5	13.9	11.51%

Table 2

 2 Total cell data rate (D T,IF R1 ) versus η (IFR reuse 1, bandwidth=10 MHz, equal bandwidth).

	η	D T,IF R1 [Mbps]	Difference
		Fluid model Simulation
	2.5	12.2	13	5.92%
	2.6	14.2	14.3	0.76%
	2.7	16.1	15.8	1.86%
	2.8	18	17.3	4.28%
	3	21.6	20.1	7.3 %
	3.2	25.1	22.7	10.33%
	3.3	26.8	24.5	9.43%

Table 3

 3 Total cell data rate (D T,IF R1 ) versus η (IFR reuse 1, bandwidth=10 MHz, opportunist).

	η	D T,IF R1 [Mbps]	Difference
		Fluid model Simulation
	2.5	56.6	56.9	0.48%
	2.6	62.8	61.6	1.95 %
	2.7	68.5	66.5	3%
	2.8	74	71.5	3.6%
	3	84.5	80.3	5.2%
	3.2	94.4	89.2	5.9%
	3.3	99.3	93.8	5.8%

Table 4

 4 Total cell data rate (D T,IF R3 ) versus η (IFR reuse 3, BW=10/3 MHz, equal data rate).

	η	D T,IF R3 [Mbps]	Difference
		Fluid model Simulation
	2.5	6.4	7.2	11.11%
	2.6	7.55	7.95	5.03%
	2.7	8.66	8.73	0.8%
	2.8	9.74	9.55	1.99%
	3	11.8	11.2	5.36%
	3.1	12.8	12.0	6.67%
	3.2	13.8	12.8	7.81%
	3.5	16.7	15.2	9.87%

Table 5

 5 Total cell data rate (D T,IF R3 ) versus η (IFR reuse 3, bandwidth=10/3 MHz, equal bandwidth).

	η	D T,IF R3 [Mbps]	Difference
		Fluid model Simulation
	2.5	8.38	9.04	7.38%
	2.6	9.63	9.91	2.81%
	2.7	10.8	10.76	0.71%
	2.8	12	11.7	3.04%
	3	14.2	13.4	5.9%
	3.1	15.3	14.4	6.55%
	3.2	16.4	15.2	7.46%
	3.5	19.4	17.8	8.8%

Table 6 Total

 6

	η	D T,IF R3 [Mbps]	Difference
		Fluid model Simulation
	2.5	25.4	26	2.2%
	2.6	27.8	28.0	0.7%
	2.7	30	29.6	1.3%
	2.8	32.2	31.5	2%
	3	36.2	34.7	4.3%
	3.1	38.2	36.6	4.4%
	3.2	40.1	38.2	5.1%
	3.5	45.7	43.4	5.4%

cell data rate (D T,IF R3 ) versus η (IFR reuse 3, bandwidth=10/3 MHz, opportunist).

  Fig. 10. Two level power control case. Bandwidth W is partitioned into three equal parts, i.e., W 1

			A Cluster of
			three cells
		W 2 +W 3
		P i
	W 1 +W 3	W 1 , P o	W 1 +W 3
	P i		P i
	W 2 , P o	W 1 +W 2	W 2 , P o
		P i
	W 2 +W 3	W 3 , P o	W 2 +W 3
	P i		P i
	W 1 , P o	W 1 +W 3	W 1 , P o
		P i
	W=W 1 +W 2 +W 3	W 2 , P o

  t. parameters R 0

		22
		20
		18	FFR TLPC
	Data Rate [Mbps]	12 14 16	IFR, Reuse1 IFR, Reuse3
		10
		8
		2.6 6	2.8	3	3.2	3.4	3.6
			Path Loss Exponent
	Fig. 18. Data rate versus pathloss exponent with equal data rate scheduling for
	three reuse schemes.
		35
			FFR
		30	TLPC
			IFR, Reuse1
			IFR, Reuse3
	Data Rate [Mbps]	15 20 25
		10
		2.6 5	2.8	3	3.2	3.4	3.6
			Path Loss Exponent

Table 7

 7 Total cell data rate (D T ) comparison of three reuse schemes and three scheduling policies.

	Frequency Reuse		D T [Mbps]
	Scheme	equal data rate	equal bandwidth	opportunist (Nu =
				30)

Conclusions

In this paper, we have presented an analytical approach, based on the fluid model, for analyzing OFDMA based networks. We have shown that our proposed technique is very flexible and can be used in different frequency reuse scenarios. We have introduced expressions of SINR and cell data rate for IFR, FFR and TLPC schemes and taking into account equal data rate, equal bandwidth and opportunist scheduling types. We have also validated our technique by comparing its results with those obtained from Monte Carlo simulations. Time required to obtain results with our analytical technique is however much shorter. We have shown that our proposed technique gives a fairly good performance for η values between 2.6 and 3.5 which is a range found in most of practical scenarios. A comparison of above three schemes is also provided. For each scheduling type, we have shown which frequency reuse scheme provides the maximum cell data rate.

Acknowledgments

This work was prepared through collaborative participation in the URC: Urban Planning for Radio Communications project, sponsored by the System@tic Paris-Région Cluster.

Authors would also like to thank Trung Dung Nguyen, student at ENST, whose project [11] under the supervision of Philippe Godlewski formed the first step in carrying out this work.

Fluid Model in OFDMA Networks

In this section, we recall the main results of the fluid model and derive the closed-form formula for SIR per subcarrier.

The key modeling step of the fluid model is replacing a given fixed finite number of BS by an equivalent continuum of transmitters which are spatially distributed in the network. This means that the transmitting power is now considered as a continuum field all over the network. In this context, the network is characterized by a user density ρ u and a base station density ρ BS [START_REF] Kelif | Downlink Fluid Model of CDMA Networks[END_REF]. We assume that users and BS are uniformly distributed in the network, so that ρ u and ρ BS are constant. We assume also that all base stations have the same output power per subcarrier P T x . Let's consider a user u at a distance r u from its serving base station b. Each elementary surface zdzdθ at a distance z from u contains ρ BS zdzdθ base stations which contribute to I ext,u . Their contribution to the external interference is thus ρ BS zdzdθP T x Az -η . We approximate the integration surface by a ring with center u, inner radius 2R c -r u , and outer radius R nw -r u (see Fig. .2).

Rc

So, the SINR γ u ≈ S b,u /I ext,u = P T x Ar -η u /I ext,u can be expressed by:


Ben Dorra

Cheikh
email: bencheikh@enst.fr

Jean-Marc Kelif

Marceau Coupechoux
email: coupecho@enst.fr

Philippe Godlewski
email: godlewski@enst.fr

Outage Probability in a Multi-Cellular Network using Alamouti Scheme

Keywords: Multiple inputs single output (MISO), multicellular, Rayleigh fading, Alamouti, outage probability

In this paper we analyze the performance of a downlink single user multi-cellular multiple inputs single output (MISO) system using the Alamouti code. We derive, for the first time, an analytical expression of the cumulative distribution function of the output signal to interference plus noise ratio (SINR) or equivalently the outage probability in flat Rayleigh fading. The system is considered interference limited. Two assumptions are considered: equal received power and unequal received power from the interfering base stations. In the first case, a closedform expression for the outage probability is derived. In the second case, from an approximation of the interference power distribution the outage probability expression is obtained.

I. INTRODUCTION

Multiple antennas systems have attracted many research attention due to their potential to increase capacity through the multiplexing gain [START_REF] Foschini | On limits of wireless communications in a fading environment when using multiple antennas[END_REF], [START_REF] Telatar | Capacity of multi-antenna Gaussian channels[END_REF] and to enhance reliability owning to the diversity gain. However, the capacity gains promised by MIMO techniques degrade drastically in a multi-cellular system. Consequently many transmission and/or reception strategies have been proposed to mitigate this degradation. The performance of these strategies have been evaluated over multiple studies. In [START_REF] Kang | Outage Probability of MIMO Optimum Combining in Presence of Unbalanced Co-channel Interferers and Noise[END_REF], the outage probability of the optimum combining (OC) receiver was studied for unequal co-channel interference plus noise power. In [START_REF] Romero | Receive Antenna Array Strategies in Fading and Interference: An Outage Probability Comparison[END_REF], [START_REF] Pe´na-Martin | Performance Comparison of MRC and IC Under Transmit Dieversity[END_REF] and [START_REF] Shah | Performance Analysis of Maximal Ratio Combining and Comparison with Optimum Combining for Mobile Radio Communications with Cochannel Interference[END_REF], the performance of different reception strategies for MIMO systems in presence of co-channel interference have been compared based on an outage probability study. It was shown that the interference cancellation (IC) receiver yields better performance than the maximum ratio combining (MRC) in an interference limited system and for few number of dominant interferers (less than the number of receive antennas). It was also shown that the OC receiver achieves the best performance. In [START_REF] Tokgoz | Performance Analysis of Maximum Ratio Transmission Based Multi-Cellular MIMO System[END_REF], the performance of maximum ratio transmission (MRT) technique in a multi-cellular multi-user MIMO system was analyzed. A closed form expression of the outage probability was proposed for an arbitrary number of transmit and receive antennas. In [START_REF] Schnurr | The Impact of Different MIMO Strategies on the Network Outage Performance[END_REF], different transmission strategies were compared. The Alamouti [START_REF] Alamouti | A simple transmit diversity technique for wireless communications[END_REF] scheme was studied in a single cell multi-user MISO and MIMO system scenarios for line of sight (LOS) and non line of sight (NLOS) channels. In [START_REF] Chalise | Exact Outage Probability Analysis for a Multiuser MIMO Wireless Communication System With Space-Time Block Coding[END_REF], the Alamouti code performance was examined in the case of multiuser uplink communication. The outage probability was derived for a MIMO system in Rayleigh fading channels. In the present work, we will analyze the performance of the full transmit diversity Alamouti scheme in a multi-cellular system when considering Rayleigh fading. We derive, for the first time, an expression of the SINR cumulative distribution function or equivalently the outage probability for a 2 × 1 multi-cellular MISO system. The remainder of this paper is organized as follows. In the next section, we describe the system model. In section III, we derive an expression for the outage probability in case of equal received interference power and unequal received power for 2 × 1 MISO Alamouti and SISO systems. Section IV includes a comparison between simulated and analytical results and discussions. Finally, in section V we conclude.

II. SYSTEM MODEL

Consider a downlink, multi-cell single user communication. Each BS is equipped with two antennas and each user equipment (UE) with a single antenna as depicted in Fig 1 . A user in a cell receives a useful signal from its serving base station (BS) and an interfering signal from the B neighboring BSs. Each BS uses the Alamouti scheme to code the information symbols. It consists in transmitting the symbols s 1 and s 2 from the two antennas in the first channel use period and -s * 2 , s * 1 in the second channel use period. It is a rate 1 code that achieves full transmission diversity for a two transmit antennas system even without channel knowledge at the transmitter. The Alamouti diversity gain for a 2 × 1 MISO system is equivalent to the receive diversity when using a maximum ratio combining (MRC) receiver [START_REF] Rooyen | DS-CDMA performance with maximum ratio combining and antenna arrays[END_REF]. The Alamouti scheme presents the advantage of keeping the user equipments simple and unburdened. Owing to its implementation simplicity, the Alamouti scheme has been adopted for the W-CDMA, CDMA-2000 and WiMax standards. The Alamouti code matrix is given by:

X = s 1 -s * 2 s 2 s * 1 , (1)
where s (i=1,2) are the transmitted symbols. At the receiver side, the signal can be represented in the following form:

y = P 0 2 h 1,0 -h 2,0 h * 2,0 -h * 1,0 H0 s 1,0 s 2,0 (2)
+ B j=1 P j 2 h 1,j -h 2,j h * 2,j -h * 1,j s 1,j s 2,j + n,
where s i,j is the symbol transmitted from the antenna i of the BS j, h i,j is the flat fading Rayleigh channel gain between the antenna i of the BS j and the user it serves. The flat fading is assumed quasi-static over the two channel use periods and n is the additive white Gaussian noise vector with covariance matrix σ 2 n I. P j is the received power from the j th BS (P 0 is the power received from the serving BS) including path-loss and shadowing terms and is given by:

P j = P T Kd -η j 10 ξ j 10 , (3)
where P T is the transmit power, K is a constant, d j is the distance between the considered user and BS j, η is the path-loss exponent and is characteristic of the propagation environment and ξ j is a Normal random variable with zero mean and standard deviation σ. By pre-multiplying the received signal by the channel transpose conjugate of the channel H 0 , the signal at the receiver becomes:

H0 y = P 0 2 |h 1,0 | 2 + |h 2,0 | 2 0 0 |h 1,0 | 2 + |h 2,0 | 2 x + B j=1 P j 2 h 1,0 -h 2,0 h * 2,0 -h * 1,0 H h 1,j -h 2,j h * 2,j -h * 1,j x j + h 1,0 -h 2,0 h * 2,0 -h * 1,0 H n.
The signal to interference plus noise ratio (SINR) per symbol is, thus, given by:

γ = P0 2 (|h 1,0 | 2 + |h 2,0 | 2 ) B j=1 Pj 2 ( |h * 1,0 h1,j +h2,0h * 2,j | 2 +|h * 1,0 h2,j -h2,0h * 1,j | 2 |h1,0| 2 +|h2,0| 2 ) + σ 2 n . (4)

III. OUTAGE PROBABILITY

The outage probability is an important metric for the evaluation of the performance of a wireless communication system. It measures the probability of failing to achieve an output SINR threshold value required for a desired service. It is given by the cumultaive distribution function (CDF) of the output SINR, i.e. :

P out = P [γ < γ th ], (5)
where γ th is the threshold SINR value.

A. Outage probability for the 2 × 1 MISO Alamouti system

The SINR expressed in (4) can be written as:

γ = X Y + σ 2 n , (6)
where

X = P 0 2 (|h 1,0 | 2 + |h 2,0 | 2 ), (7)
and

Y = B j=1 P j 2 |h * 1,0 h 1,j + h 2,0 h * 2,j | 2 + |h * 1,0 h 2,j -h 2,0 h * 1,j | 2 |h 1,0 | 2 + |h 2,0 | 2 . (8)
We consider an interference limited cellular system where the background noise is assumed to be negligible. The SINR can, thus, be approximated by:

SIN R ≈ X Y . (9)
To calculate the outage probability we need to calculate first, the probability distribution function (PDF) of X and the PDF of Y . We will suppose that the shadowing remains constant during the period of our study. Since |h i,j | are zero mean unit variance Rayleigh distributed channel gains, it can be easily shown [START_REF] Simon | Digital Communication Over Fading Channels[END_REF] that X is Gamma distributed and that the PDF of X is given by:

f X (x) = 4x P 2 0 Γ(2) e -2x P 0 . ( 10
)
1) Equal interference power assumption: In this section, we will assume that P j = P, ∀j = 1, ..., B. This assumption allows us to derive a simple closed form expression of the outage probability. However this scenario is not completely unrealistic. It can correspond to the case where the considered user is close to its serving BS mainly interfered by the first surrounding layer of cells. The interference power Y can be written as:

Y = B j=1 P 2 (|c j | 2 + |d j | 2 ), (11)
where

c j = h0h t j |h0| and d j = h0g t j |h0| .
where

h 0 = [h * 1,0 h 2,0 ], h j = [h 1,j h * 2,j ] and g j = [h 1,j - h * 2,j ].
As proven in [START_REF] Shah | Performance Analysis of Maximal Ratio Combining and Comparison with Optimum Combining for Mobile Radio Communications with Cochannel Interference[END_REF], c j and d j are complex Gaussian random variables independent of h 0 . Hence, Y is the sum of two correlated Gamma distributed random variables. Using the analysis conducted in [START_REF] Holm | Sum and Difference of Two Squared Correlated Nakagami Variates in Connection With the McKay Distribution[END_REF], we can prove that the probability distribution of Y is given by:

f Y (y) = 2 √ π P (1 -ρ) B Γ(B) 2y P 2 δ B-1 2 (12) × exp(- 2y P (1 -ρ) )I B-1 2 (δy),
where ρ is the correlation coefficient between the two correlated variables

Y c = B j=1 |c j | 2 et Y d = B j=1 |d j | 2 and can be expressed as ρ = Cov(Yc,Y d ) √ var(Yc)var(Y d )
. δ = 2 √ ρ P (1-ρ) and I n (.) is the n th order modified Bessel function of first kind. Using the PDF expressions ( 10) and ( 12), the outage probability can be derived as follows:

P ( X Y < γ th ) = ∞ 0 F X (γ th y)f Y (y)dy (13) = ∞ 0 (1-e -2γ th P 0 y (1+ 2γ th P 0 y))f Y (y)dy.( 14
)
Substituting f Y (y) by its expression and using the following formula [START_REF] Holm | Sum and Difference of Two Squared Correlated Nakagami Variates in Connection With the McKay Distribution[END_REF]:

∞ 0 x µ-1 e -ηx I ν (βx)dx = β ν 2 ν η µ+ν Γ(µ + ν) Γ(ν + 1) × (15) 2 F 1 µ+ν 2 , µ+ν +1 2 ; ν +1; β 2 η 2 ,
the outage probability is given by:

P out (γ th ) = 1 - C P 2B-1 ς 2B Γ(2B) Γ(B + 1 2 ) (16)
× 2 F 1 (B, B + 1 2 ; B + 1 2 ; δ 2 ς 2 ) + 4B ς γ th P 0 2 F 1 (B + 1 2 , B + 1; B + 1 2 ; δ 2 ς 2 ) ,
where 2 F 1 (., .; .; .) is the Gauss' hypergeometric function [START_REF] Gradshteyn | Table of integrals series and products[END_REF],

C = 2 √ π P (1-ρ) B Γ(B) and ς = 2γ th P0 + 2 P (1-ρ) .
2) Unequal interference power assumption: Since the equal power assumption does not cover many possible scenarios, we will try to derive an expression for the outage probability in the case of unequal received power. When the received powers from the different interfering BSs are unbalanced, it is more difficult to find a closed form expression for the interference power PDF. However it can be approximated using the central limit theorem for causal functions [START_REF] Papoulis | The Fourier Integral and Its Applications[END_REF] by a Gamma distribution given by:

f Y (y) = y α-1 exp(-y β ) Γ(α)β α , (17)
where

α = E[Y ] 2 var(Y ) and β = var(Y ) E[Y ]
. In order to compute E[Y ] and var(Y ), we will note Y = B j=1 Pj 2 Z j where Z j is given by:

Z j = |c j | 2 + |d j | 2 . ( 18
)
Z j is the sum of two correlated exponentially distributed variables, thus, the PDF of Z j is given by [START_REF] Holm | Sum and Difference of Two Squared Correlated Nakagami Variates in Connection With the McKay Distribution[END_REF]:

f Zj (z) = 1 √ ρ z exp(- z 1 -ρ z ) sinh( √ ρ z 1 -ρ z z), f or z > 0.
(19) where ρ z is the correlation coefficient between the two correlated random variables |c j | 2 and |d j | 2 . E[Z j ] and E[Z 2 j ] can be derived as follows:

E[Z j ] = E[|c j | 2 ] + E[|d j | 2 ] = 2, ( 20
)
and

E[Z 2 j ] = ∞ 0 z 2 √ ρ z exp(- z 1 -ρ z )sinh( √ ρ z 1-ρ z z)dz, (21)
= 1 √ ρ z Γ(3) 2 1 - √ ρ z 1 -ρ z -3 - 1 + √ ρ z 1 -ρ z -3 , = 2(3 + ρ z ).
As Z j for j = 1, ..., B are independent random variables, the mean of the interference power is given by:

E[Y ] = B j=1 P j 2 E[Z j ] = B j=1 P j . ( 22
)
The variance of Y can be expressed as:

var(Y ) = B j=1 P 2 j 4 var(Z j ) (23) = B j=1 P 2 j 2 (1 + ρ z ). (24)
The parameters α and β of Eq.17 are, thus given by:

α = 2 1 + ρ z ( B j=1 P j ) 2 B j=1 P 2 j , (25)
β = 1 + ρ z 2 B j=1 P 2 j B j=1 P j . ( 26
)
The outage probability can be derived using ( 14) in section III-A1. Substituting the interference power PDF f Y (y) by its expression [START_REF] Simon | Digital Communication Over Fading Channels[END_REF], the outage probability is given by:

P out (γ th ) = 1- P 0 2γ th β +P 0 α 1+ 2γ th β 2γ th β +P 0 Γ(α+1) Γ(α) . (27)

B. Outage Probability for a SISO system

In a multi-cellular single input single output (SISO) communication system and assuming a flat fading Rayleigh channel, the SINR is given by:

γ SISO = P 0 |h 0 | 2 B j=1 P j |h j | 2 + σ 2 n . (28)
We consider an interference limited system, the SINR can be written as:

γ SISO = X SISO Y SISO , (29)
where:

X SISO = P 0 |h 0 | 2 , Y SISO = B j=1 P j |h j | 2 . ( 30
)
The PDF of the useful power is given by [START_REF] Simon | Digital Communication Over Fading Channels[END_REF]:

f X SISO (x) = 1 P 0 e -x P 0 . (31)
As in section III, we consider two assumptions: equal received interference powers and unequal received interference power. 1) Equal interference power: Consider P j = P, ∀j = 1, ..., B, in this case, it can be easily proven that the interference power is Gamma distributed [START_REF] Simon | Digital Communication Over Fading Channels[END_REF] and that the PDF of the interference power is given by:

f Y SISO (y) = y B-1 Γ(B)P B e -y P . (32)
Using the useful power PDF (31) and the interference power PDF (32), the outage probability for a SISO system can be derived as explained in section III-A1 using the expression [START_REF] Gradshteyn | Table of integrals series and products[END_REF] and can be written as:

P siso out (γ th ) = 1 - 1 ( P P0 γ th + 1) B .
(33)

2) Unequal interference power: In this section we consider the case of unequal power and considering path-loss and lognormal shadowing in our analysis. As in section III-A2, the interference power PDF can be approximated by a Gamma distribution given by: (36)

f Y SISO (y) = y ν-1 Γ(ν)λ ν e -y λ , (34)
The outage probability can be obtained using the expression [START_REF] Gradshteyn | Table of integrals series and products[END_REF] and is given by:

P siso out (γ th ) = 1 - 1 ( λ P0 γ th + 1) ν . (37)
IV. SIMULATION RESULTS

In Fig. 2, we plot simulated and analytical outage probability in the case of equal received interference power for the Alamouti scheme and for a SISO system. We consider 6 interfering BSs. The cell radius is R c = 1 Km, the mobile station is at a distance d = 0.2 Km from the BS and the interfering BSs are at distance d int = 2R c . The path-loss exponent is taken η = 3.41 and the standard deviation of the log-normal shadowing is taken σ = 6 dB, the shadowing is considered constant over the period of study. Simulated curves are obtained using the Jakes' model [START_REF] Jakes Jr | Microwave Mobile Communications[END_REF] to generate zero mean unit variance Rayleigh fading. The correlation factor ρ is estimated by simulation. We can see that the Alamouti scheme achieves better performance. In fact, as can be seen in Fig. 2 (b), the outage probability of the Alamouti scheme decreases faster owning to a higher slope corresponding to the diversity gain of the scheme G D = 2. Thus, the Alamouti scheme allows a better coverage. It can also be remarked from Fig. 2 (a) that the variance of the SINR output when using the Alamouti scheme is less than the variance of the SISO output SINR. The simulated curves fit very well the analytical ones. Fig. 3 shows a comparison between the performance of the Alamouti scheme of a 2 × 1 MISO system and a SISO system in terms of outage probability in the case of unbalanced received interference powers. Simulated and analytical curves are presented. The considered user is at a distance d = 0.5 Km from its serving BS, the standard deviation of the shadowing is σ = 4 dB and we consider 18 dominant interfering BSs (two rings of BS in a hexagonal network). It can be seen that there is a good match between the simulated results and the analytical approximation of the outage probability. As in the case of equal received interference power (Fig. 2 (b)), the diversity gain of the Alamouti scheme allows for better performance.

Fig. 4 presents a comparison between the simulated per- formance of the Alamouti scheme in a 'point-to-point' communication and in a multi-cellular system with the simulated performance of a SISO system for the same scenarios. As expected, the performance gain of the Alamouti scheme degrades considerably in a multi-cellular system compared to a single cell system. This degradation can be explained by the fact that, as for the useful signal, the Alamouti scheme induces a diversity gain on the interference channel. V. CONCLUSION

In this paper, the performance of multi-cellular network interference limited system using the Alamouti scheme is analyzed. Outage probability expressions for the case of equal received interference power and unequal interference power are derived. In the two cases, a comparison between the performance of the 2 MISO Alamouti scheme and SISO system is illustrated. As a future work, we can use the fluid model [START_REF] Kelif | Downlink Fluid model of CDMA Networks[END_REF] approach to express the outage probability as function of the distance between the considered user and its serving BS. This analysis will allow us to address issues related to network dimensioning.

Fig. 1 .

 1 Fig. 1. Downlink multi-cell (3 BSs) single user communication.

  where the parametersν = E[Y SISO ] 2var(Y SISO ) and λ = var(Y SISO ) E[Y SISO ] are given by:

Fig. 2 .

 2 Fig. 2. Equal received interference power: Pout versus SINR threshold 2×1 MISO Alamouti scheme and SISO systems.

Fig. 3 .

 3 Fig. 3. Unequal received interference power: Pout versus SINR threshold 2 × 1 MISO Alamouti scheme and SISO systems.

Fig. 4 .

 4 Fig. 4. Multi cell versus single cell outage probability for the 2 × 1 MISO Alamouti scheme and SISO systems.


Jorge Zavala Bojorquez

Stéphanie Bricq

Paul M Walker

Alain Lalande
email: alain.lalande@u-bourgogne.fr.

First steps toward the generation of PET/MR attenuation map in the case of prostate cancer

Keywords:

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

First steps toward the generation of PET/MR attenuation map in the case of prostate cancer

I. INTRODUCTION

Prostate cancer is the most commonly diagnosed cancer and a leading cause of cancer death worldwide. Magnetic Resonance Imaging (MRI) is currently the imaging modality of choice for visualization, diagnostic and evaluation of prostate cancer. However, on the MR images it is relatively difficult to detect cancer spread to the lymph nodes and the bones. The Positron Emission Tomography / Computer Tomography (PET/CT) is considered as the gold standard for this end. This will change with the arrival of hybrid PET/MR systems and the advantages that MR represents in contrast to CT [START_REF] Herzog | PET/MRI: Challenges, solutions and perspectives[END_REF]. Nevertheless, the change cannot take place if an accurate method to generate the photon attenuation maps used for the reconstruction of PET images from MR information is not designed. We propose a robust and automatic method based on the intrinsic parameters T1-and T2-relaxation times of tissues and anatomical-knowledge to create accurate attenuation maps at the level of the pelvis with six regions: fat, muscle, prostate, air, background and bones. To the best of our knowledge, most of these studies have been applied in brain and thorax imaging and little efforts have been done to create accurate attenuation maps for prostate imaging. Our method does not require special sequences to extract bones neither depend on voxel intensities, therefore, is scanner independent and can be implemented in any system.

II. MATERIALS AND METHODS

II.1. T1 and T2 measurements

The images were acquired with a 3T Trio TIM clinical scanner (Siemens Medical Solutions, Germany). Two healthy volunteers (22 and 42 years old) participated in this study after they gave informed consent. Four axial slices were acquired for both T1 and T2 measurements with the same field of view, resolution and position at the high of the pelvis in order to avoid registration between T1 and T2 images (Figure 1). An inversion recovery-prepared turbo spin echo (TSE) was acquired at eight inversion times (TI: 50, 200, 400, 700, 1200, 2000, 5000, 7860 ms) to obtain the T1-weighted images. The TI were chosen to assure maximum contrast among tissues and to cover the whole span of T1 relaxation times of tissues present in the slice section. The transverse relaxation time T2 was measured using a 32-echo spin echo sequence acquired with TE from 8.8 to 281.6 ms in 8.8 ms steps (TR = 5000 ms). The T1 and T2 were calculated from a magnitude monoexponential fit to the signal recovery data for each voxel using the bisquare nonlinear least squares fitting. For T1, a three-parameter model was used with T1, the equilibrium magnetization So and the inversion efficiency α (variable fitting parameters):

The used of a long TR ( > 5×T1 of the longest T1 of the tissues) allowed the fit of the equation, since the longitudinal magnetization fully recovers before another repetition of the sequence. For T2, a three parameter model was used with T2, the equilibrium magnetization So and the y-offset C as fitting parameters:

Volunteer 1

To automatically extract the background, we kept the region with the highest number of voxels on the binary image obtained from the longest TI image with a threshold of 4% of highest voxel intensity. The obtained maps are showed on the figure 2.

Figure 2: T1 and T2 maps for volunteer 2.

II.2. Segmentation and classification

Segmentation was performed using weighted Fuzzy Cmeans (wFCM) [START_REF] Chuang | Fuzzy c-means clustering with spatial information for image segmentation[END_REF]. The method performs a fuzzy partition and takes as inputs the T1 and T2 maps, the number of clusters (c=100 to do firstly an oversegmentation), the fuzzification constant (m=2) and the weights. To define these weights, the measure adopted here was entropy, to take advantage of the good spatial resolution of MR images, and because it permits to estimate the randomness of the information which is a common reference in both maps. The centers of clusters were biased towards these measures:

where pT represent the probabilities of the different measures T1 or T2. T1 and T2 values were normalized to avoid bias. The centroid values of each cluster were compared with the references T1 and T2 values of the tissues using the euclidean norm for a two dimensional point:

where ce represents T1 and T2 values of the centroids, Tref are those of the reference tissues. These reference values were calculated from the average values of different regions of interest, set on the T1 and T2 maps of volunteer 1 for the tissues: fat (T1 = 405 ± 34ms, T2 =147 ± 20ms), muscle (T1 = 1290 ± 83ms, T2 = 39 ± 3ms) and prostate (T1 = 1742 ± 175ms, T2 = 75 ± 34ms). For air, zero values were used for both T1 and T2 relaxation times. However cortical bone does not produce signal with conventional spin echo sequences used in clinical practice. To detect bones, firstly, the fat's regions are localized. Then a morphological opening operation is performed to remove small regions and narrow segments of fat connecting big regions. Regions in contact with the background are removed and regions having an area smaller than a predefined threshold (100 mm 2 ) are discarded. For bones, the boundaries of the fat's regions are surrounded by cortical bone, but since this tissue produces no signal with the sequences implemented, it is labeled as air. Therefore, the boundaries of the fat's regions are localized thanks to a dilation with a kernel size of 5x5, and their labels analyzed.

If the labels do not correspond to an area of at least 22.5 mm 2 of air, they are rejected. Finally, only symmetric regions are classified as bones.

III. RESULTS

The obtained relaxation maps are homogeneous (Figure 2), showing that the fitting model correctly compensates for the inhomogeneities of the B1-field yielding exact values.

The figure 3 displays the obtained classification. Fat, muscle, prostate, air and background were correctly segmented and classified.

Figure 3: Classification of the different tissues at the level of the pelvis.

IV. DISCUSSION -CONCLUSION

In this work, we have proposed a new methodology for a first step to generate the attenuation correction map of PET/MRI systems for prostate imaging. Our approach correctly segments and classifies the attenuation-relevant structures used for this purpose (fat, air, background, bones and soft tissues). Big bones presented no problem and were accurately identified. However, small bones structures severely affected by partial volume effect were not efficiently detected. Nevertheless, the current results clearly demonstrate that the method could provide a first step to generate the attenuation map of PET/MRI systems in prostate imaging. The validation on a big data set is planned.

Figure 1 :

 1 Figure 1: Example of an axial slice at the level of the pelvis.


Noureddine-Yassine Nair-Benrekia

Pascale Kuntz

Frank Meyer

Interactive multi-label classification for data personalization

Keywords: Interactive learning, multi-label classification, interactivity constraints

come

Interactive multi-label classification for data personalization N.Y. Nair Benrekia 1,2 , P. Kuntz 2 , and F. Meyer 1 1 Orange Labs, AV. Pierre Marzin 22307 Lannion cedex France (yacinenoureddine.nairbenrekia, franck.meyer)@orange.com 2 LINA, la Chantrerie-BP 50609, 44360 NANTES cedex France pascale.kuntz@univ-nantes.fr

Abstract. Recent interactive machine learning solutions have shown that embedding the end-user into the system might be the best way to fit his/her individual preferences. We here focus on a user-centered classification process which allows the user to interact with pre-computing results via a friendly visual interface. Recent experiments showed that interactive classification yields accurate results when classifiers are associated with a sufficient number of user-presented examples [START_REF] Drucker | Helping Users Sort Faster with Adaptive Machine Learning Recommendations[END_REF]. However, the efficient interactive learning solutions generally limit users to mono-labeling which may be not expressive enough in real-life situations; for instance, in some organization tasks, such as text labeling or multi-criteria recommendation where the user will naturally seek to handle multiple labels. In parallel, multi-label classification has received significant attention over the past few years [START_REF] Madjarov | An extensive experimental comparison of methods for multi-label learning[END_REF]. But, as far as we know, integrating multi-label approaches into an interactive learning framework still set open questions. In this communication, we propose a state-of-the-art of the multi-label classification algorithms capable of withstanding the interactivity constraints. We complete the presentation with first experimental results on literature datasets of various sizes (e.g. Music and IMDB ).