# Coverage Probability and Achievable Rate Analysis of FFR-Aided Multi-User OFDM-Based MIMO and SIMO Systems

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IEEE TRANSACTIONS ON COMMUNICATIONS 1

Coverage Probability and Achievable Rate Analysis

of FFR-Aided Multi-User OFDM-Based

MIMO and SIMO Systems

1

2

3

Suman Kumar, Sheetal Kalyani, Lajos Hanzo, Fellow, IEEE, and K. Giridhar, Member, IEEE4

Abstract—Expressions are derived for the coverage probability5

and average rate of both multi-user multiple input multiple output6

(MU-MIMO) and single input multiple output (SIMO) systems7

in the context of a fractional frequency reuse (FFR) scheme. In8

particular, given a reuse region of

1

3

(FR3) and a reuse region of9

1 (FR1) as well as a signal-to-interference-plus-noise-ratio (SINR)10

threshold S

th

, which decides the user assignment to either the FR111

or FR3 regions, we theoretically show that: 1) the optimal choice12

of S

th

which maximizes the coverage probability is S

th

= T,where13

T is the target SINR required for ensuring adequate coverage, and14

2) the optimal choice of S

th

which maximizes the average rate is15

given by S

th

= T

,whereT

is a function of the path loss exponent,16

the number of antennas and of the fading parameters. The impact17

of frequency domain correlation amongst the OFDM sub-bands18

allocated to the FR1 and FR3 cell-regions is analysed and it is19

shown that the presence of correlation reduces both the coverage20

probability and the average throughput of the FFR network.21

Furthermore, the performance of our FFR-aided MU-MIMO and22

SIMO systems is compared. Our analysis shows that the (2 × 2)23

MU-MIMO system achieves 22.5% higher rate than the (1 × 3)24

SIMO system and for lower target SINRs, the coverage probability25

of a (2 × 2) MU-MIMO system is comparable to a (1 × 3) SIMO26

system. Hence the f ormer one may be preferred over the latter.27

Our simulation results closely match the analytical results.28

Index Terms—Author, please supply index terms/keywords for29

your paper. To download the IEEE Taxonomy go to http://www.30

ieee.org/documents/taxonomy_v101.pdf.31

I. INTRODUCTION32

O

RTHOGONAL frequency division multiple access

AQ1

33

(OFDMA) based systems maintain orthogonality among34

the intra-cell users, but the radical OFDMA system deploy-35

ments relying on a frequency reuse factor of unity suffer from36

inter-cell interference. As a remedy, inter-cell interference coor-37

dination (ICIC) schemes have been designed for minimizing the38

co-channel interference [1]. Fractional frequency reuse (FFR)39

[2] constitutes a low complexity ICIC scheme, which has been40

proposed for OFDMA based wireless networks such as IEEE41

WiMAX [3] and 3GPP LTE [4].42

Manuscript received January 18, 2015; revised June 5, 2015; accepted

August 1, 2015. The associate editor coordinating the re view of this paper and

approving it for publication was O. Oyman.

S. Kumar, S. Kalyani, and K. Giridhar are with the Indian Institute of

Technology Madras, Chennai 600 036, India (e-mail: ee10d040@ee.iitm.ac.in;

AQ2

skalyani@ee.iitm.ac.in; giri@ee.iitm.ac.in).

L. Hanzo is with the School of Electrical and Computer Science, University

of Southampton, Southampton SO17 1BJ, U.K. (e-mail: lh@ecs.soton.ac.uk).

Color versions of one or more of the ﬁgures in this paper are available online

at http://ieeexplore.ieee.org.

Digital Object Identiﬁer 10.1109/TCOMM.2015.2465907

Fig. 1. Frequency allocation in FFR for three neighbouring cells with δ = 3.

The cell-centre users of all the cells rely on a common frequency band F

0

, while

the cell-edge users of the three cells occupy different frequency bands, namely

F

1

, F

2

and F

3

.

Explicitly, FFR is a combination of frequency reuse 1 (FR1) 43

and frequency reuse

1

δ

(FRδ). FR1 allocates all the frequencies 44

to each cell, leading to a unity spatial reuse, hence results in 45

a low-quality coverage due to the excessive inter-cell interfer- 46

ence. On the other hand, FRδ allocates a fraction of

1

δ

of the 47

frequencies to each cell and therefore reduces the area-spectral- 48

efﬁciency, but improves the SINR. FFR strikes an attractive 49

trade-off by exploiting the advantages of both FR1 and FRδ by 50

relying on FR1 for the cell-centre users i.e. for those users who 51

would experience less interference from the other cells, because 52

they are close to their serving base station (BS). By contrast, 53

FRδ is invoked for the cell-edge users i.e. for those users who 54

would experience high interference afﬂicted by the co-channel 55

signals emanating from the neighbouring cells in case of FR1, 56

because they are far from their serving BS. Typically, there 57

are two basic modes of FFR deployment: static and dynamic 58

FFR [1]. In this paper, we consider the more practical static 59

FFR scheme, where all the parameters are conﬁgured and kept 60

ﬁxed over a certain period of time [5]. Fig. 1 depicts a typical 61

frequency allocation in the context of the FFR scheme for three 62

adjacent cells, where F

1

, F

2

and F

3

each use x% of the total 63

spectrum, hence F

0

uses (100 − 3x)% of the spectrum. 64

FFR schemes have been lavishly studied using both system 65

level simulations and theoretical analysis [6]–[11]. The optimiz- 66

ation of FFR relying on a distance threshold

1

or SINR threshold

2

67

1

Based on a pre-determined distance from the BS, the subscribers are divided

into cell-centre as well as cell-edge users and hence here the design parameter

is a distance threshold (R

th

).

2

Based on a pre-determined SINR, the subscribers are divided into cell-

centre as well as cell-edge users and here the design parameter is the SINR

threshold (S

th

).

0090-6778 © 2015 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.

See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

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Proof

2 IEEE TRANSACTIONS ON COMM UNICATIONS

has been studied using graph theory in [6] and convex optimiza-68

tion in [7]. Speciﬁcally, it h as been shown in [7] that the optimal69

frequency reuse factor is FR3 for the cell-edge users. The av-70

erage cell throughput of an FFR system was derived in [8] as a71

function of the distance threshold. It was shown in [9] that there72

exists an optimal radius threshold for which the average rate be-73

comes maximum. The performance of FFR and soft frequency74

reuse (SFR) has been studied in [12] under both fully loaded75

and partially loaded scenarios. An algorithm was proposed76

in [13] for enhancing the network capacity and the cell-edge77

performance for a dynamic SFR deployment relying on re-78

alistic irregularly shaped cells. A fuzzy logic based generic79

model was proposed for deriving different frequency reuse80

schemes in [14]. As a further development, an FFR based 3-cell81

network-MIMO based tri-sector BS architecture was presented82

in [15]. FFR and SFR are compared in the presence of corre-83

lated interferers in [16]. The optimal conﬁguration o f FFR is84

determined in [17] for a high-density wireless cellular network.85

The authors of [18] have proposed a distributed and adaptive86

solution for interference coordination based on the center of87

gravity of users in each sector. An optimal FFR and power88

control scheme which can coordinate the interference among89

the heterogeneous nodes is proposed in [19].90

An analytical framework of calculating both the coverage91

probability (CP

r

) and the average rate of FFR schemes was92

presented in [10] and [11] for homogeneous single input single93

output (SISO) and MIMO heterogeneous networks, respec-94

tively, using a Poisson point process (PPP). However, the au-95

thors of [10], [11] assumed having an unplanned FFR network,96

where the cells using the same frequency set are randomly97

allocated. Hence, two cells using the same frequency for the98

cell-edge users may in fact be co-located [10], [11]. However,99

in case of FFR based deployments the regions using the same100

frequency are typically planned to be as far apart as possible101

and our focus is on these types of deployments. An FFR-aided102

distributed antenna system (DAS) and an FFR-aided picocell103

was studied in [20] and [21]. While, an FFR-aided femtocell104

has been extensively studied in [22]–[26].105

However, most of the work based on FFR has considered the106

conventional SISO case. To the best of our knowledge, no prior107

work has analytically derived the optimal SINR threshold for108

FFR, when the number of antennas is high at the transmitter109

and/or at the receiver. Hence, in this work, we derive both the110

CP

r

and the average achievable rate expressions of FFR in the111

presence of both MU-MIMO as well as of SIMO systems and112

derive the optimal SINR threshold corresponding to the desired113

CP

r

and throughput. Furthermore, the performance of FFR-114

aided MU-MIMOs is compared to that of FFR in the presence115

of a SIMO system.116

The key beneﬁt of MU-MIMO is their ab ility to improve117

the spectral efﬁciency, which has b een extensively studied in118

a single-cell context in the presence of AWGN [27]–[29].119

However, it has been shown in [30], [31] with the h elp of120

simulation, that the efﬁciency of MU-MIMOs is signiﬁcan tly121

eroded in a multi-cell environment due to interference, es-122

pecially in the cell-edge region. FFR is capable of signiﬁ-123

cantly improving the cell-edge coverage since it uses FR3 for124

the cell-edge users. Hence we study FFR-aided MU-MIMOs125

and quantify their average throughput as well as coverage 126

probability. 127

Furthermore, we carefully examine the correlation of the sub- 128

bands F

0

, F

1

, F

2

and F

3

in Fig. 1 used in the FFR system 129

considered. All prior work on FFR has assumed that the sub- 130

bands experience independent fading, which is mathematically 131

convenient, but practically not realisable. Indeed, when we 132

consider practical transmission block based modulation such as 133

OFDM, the channel’s delay spread is assumed to b e conﬁned to 134

the cyclic preﬁx of the OFDM symbol. Such a limited-duration 135

(typically less than 20% of the useful OFDM symbol duration) 136

impulse response will result in correlation amongst the adjacent 137

freque ncy domain OFDM sub-channels. More explicitly, unless 138

the sub-bands F

0

···F

3

are spaced apart by more than the recip- 139

rocal of the delay spread, correlation will exist. Since the delay 140

spread experienced in the downlink is user-dependent, it is vir- 141

tually impossible to ensure that the sub-bands F

i

in Fig. 1 are in- 142

dependent for each user scheduled in the downlink. Therefore, 143

in our analysis we will speciﬁcally take into account the corre- 144

lation of the sub-bands. For FFR-aided MU-MIMO and SIMO 145

systems, the expressions of CP

r

and average rate are derived 146

and the following new results are presented: 147

(a) The optimal SINR threshold that m aximizes th e CP

r

of 148

FFR is derived for a given T. We show that the optimal 149

S

th

(denoted by S

opt,C

)isS

th

= T for both the MU-MIMO 150

and SIMO system, and if we choose the SINR threshold 151

to be S

opt,C

, then the achievable CP

r

of FFR is higher 152

than that of FR3. The improvement of the FFR CP

r

over 153

that of FR3 is due to the resultant sub-band diversity gain 154

achieved by the systems when a user is classiﬁed as either 155

a cell-centre or a cell-edge user. 156

(b) The optimal SINR threshold that maximizes the average 157

rate of FFR is derived. We show that the optimal S

th

(de- 158

noted by S

opt,R

) is equal to T

for both MU-MIMO and 159

SIMO systems, where T

is a ﬁxed SINR value, which de- 160

pends on the system p arameters such as the path loss expo- 161

nent, the number of antennas, the fading parameters, etc. 162

(c) The correlation of the sub-bands always degrades both the 163

CP

r

and the average rate of the FFR-aided MU-MIMO 164

and SIMO systems. 165

(d) The performance of FFR-aided MU-MIMO and SIMO 166

systems is compared. It is shown that system designer 167

may choose the (2 × 2) MU-MIMO system over (1 × 3) 168

SIMO system of FFR scheme as MU-MIMO achieves 169

signiﬁcant gain in average rate over SIMO. 170

We will demonstrate that our analytical results are in close 171

agreement with the simulation results. Moreover, it is shown 172

that at optimal S

th

, the FFR achieves signiﬁcantly high gain in 173

CP

r

than that of average rate with respect to FR1 and hence this 174

scheme would be more useful when coverage gain is essentially 175

required. Therefore, FFR-aided MU-MIMO provides both high 176

average rate and satisfactory CP

r

foralowervalueofN

a

. 177

II. SYSTEM MODEL 178

A homogeneous macrocell network relying on hexagonal 179

tessellation and on an inter cell site distance of 2R is considered, 180

IEEE

Proof

KUMAR et al.: COVER AGE PROBABILITY AND ACHIEVABLE RATE ANALYSIS OF MU-MIMO AND SIMO SYSTEMS 3

Fig. 2. Hexagonal structure of 2-tier macrocell. Interference for 0th cell in

FR1 system is contributed form all the neighbouring 18 cells, while in a FR3

system it is contributed only from the shaded cells.

as shown in Fig. 2. Both a MU-MIMO and a SIMO system is181

considered. We assume that in the MU-MIMO case each user182

is equipped with N

r

receive antennas, while the BS is equipped183

with N

t

transmit antennas and that N

t

= N

r

. Our focus is on the184

downlink and hence N

t

transmit antennas are used for transmis-185

sion, while the N

r

receive antennas at the UE are u sed for re-186

ception. We also assume that all N

t

transmit antennas at the BS187

are u tilized to transmit N

t

independent data steams to its own N

t

188

users. A linear minimum mean-square-error (LMMSE) receiver189

[32] is considered. In order to calculate the p ost-processing190

SINR of this LMMSE r eceiver, it is assumed that the (N

r

− 1)191

closest interferers can be completely cancelled using the anten-192

nas at the receiver.

3

For example, in the MU-MIMO case, the193

user will not experience any intra-tier interference emanating194

from the serving BS as N

t

= N

r

. In the SIMO case each user195

is equipped with N

r

antennas. The SINR η

t

(r) of a user in the196

MU-MIMO system and the SINR η

r

(r) of a user in the SIMO197

system located at r meters from its serving BS are given by198

η

t

(r) =

gr

−α

σ

2

P

+ I

t

, I

t

=

i∈ψ

N

t

j=1

h

ij

d

−α

i

(1)

and

199

η

r

(r) =

gr

−α

σ

2

P

+ I

r

, I

r

=

i∈ψ

r

h

ij

d

−α

i

, (2)

respectively, where the transmit power of a BS is denoted by P.

200

Here ψ is the set of interfering BSs in the FR1 network and ψ

r

201

denotes all the interfering BSs, excluding the nearest (N

r

− 1)202

interferers, while N

t

denotes the number of transmit antennas.203

The standard path loss model of x

−α

is assumed, where204

α ≥ 2 is the path loss exponent and x is the distance of a user205

from the BS. We assumed that the users are at least at a distance206

of d away from the BS.

4

The noise power is d enoted by σ

2

.207

Here, r and d

i

are the distances from the user to the serving BS208

andtothei

th

interfering BS, respectively, while g and h

i

denote209

3

It is widely exploited that using the LMMSE receiver (N

r

− 1) interferers

can be mitigated, where N

r

is the number of receive antennas [32]. Ho wever,

for simplicity, we assume that the N

r

− 1 closest interferers can be completely

cancelled.

4

Typically , the path loss model is assumed to be max{d, x}

−α

.

the corresponding channel fading power, which are independent 210

and identically exponentially distributed (i.i.d.) with a unit 211

mean, i.e., g ∼ exp(1) and h

i

∼ exp(1)∀ i. In MU-MIMO case, 212

h

ij

is the channel’s fading power from the j

th

antenna of the 213

i

th

interfering BS to the user and it is i.i.d. with a unit mean. 214

Without loss of generality we have considered a user in the 0

th

215

cell of Fig. 2 in our analysis. 216

Similar to [10], the subscribers are classiﬁed as cell-centre 217

users and cell-edge users based on the SINR at the mobile sta- 218

tion. If the calculated SINR of a user is lower than the speciﬁed 219

SINR threshold S

th

, the user is classiﬁed as a cell-edge user. 220

Otherwise, the user is classiﬁed as a cell-centre user. Typically, 221

FFR divides the whole frequency band into a total of (1 + δ) 222

parts, where F

0

is allocated to all the cells for the cell-centre 223

users, as seen in Fig. 1. One of the {1 , ··· ,δ} parts is assigned 224

to the cell-edge users in each cell in a planned fashion. The 225

users are assumed to be uniformly distributed in a cell and all r e- 226

source blocks are uniformly shared among the users. The trans- 227

mit power is assumed to b e ﬁxed. If we have η

t

(r)(or η

r

(r)) ≥ 228

S

th

for a user, then the user will continue to experience the same 229

fading power, i.e., g and h

i

from the user to the serving BS 230

andtothei

th

interfering BS, respectively. However, if we have 231

η

t

(r)(or η

r

(r)) < S

th

for a user, the user is allocated another 232

sub-band (from the set of sub-bands assigned to cell-edge users) 233

and it experiences a new fading power, i.e., ˆg and

ˆ

h

i

from the 234

user to the serving BS and to the i

th

interfering BS, respectively. 235

Based on the coherence bandwidth of the OFDM system, and 236

the bands associated with F

0

to F

3

in Fig. 1 is is possible that ˆg 237

and

ˆ

h

i

are either correlated with or independent of g and h

i

,re-238

spectively. Note that g, ˆg, h

i

,and

ˆ

h

i

are the channel gains in the 239

frequency domain and the term correlation is used for referring 240

to frequency domain correlation in this paper. The correlation 241

depends both on the particular user’s channel conditions and 242

on the instantaneous coherence bandwidth with respect to the 243

FFR frequency bands. To better understand the impact of corre- 244

lation among the sub-bands on the FFR system’s performance, 245

in this paper, we consider the following two extreme cases: 246

Case 1: g and ˆg are independent and also h

i

as well as

ˆ

h

i

,are247

independent for all i. 248

Case 2: g and ˆg are fully correlated and also h

i

as well as

ˆ

h

i

, 249

are fully correlated for all i. 250

In reality these channel output powers may be partially corre- 251

lated, but the analysis of partial (arbitrary) corr e lation is quite 252

complicated and hence it is beyond the scope of this work. 253

However, the analysis of the above two extreme cases we be- 254

lieve, is sufﬁcient for understanding the impact of correlation 255

among the sub-bands. 256

III. COVERAGE PROBABILITY ANA LY S I S O F FFR 257

In this section, we ﬁrst derive the CP

r

of both the 258

MU-MIMO and SIMO system considered, which is deﬁned 259

as the probability that a randomly chosen user’s instantaneous 260

SINR η

t

(r) is higher than T. This deﬁnes, the average fraction 261

of users are having an SINR higher than the target SINR. The 262

coverage prob ability is determined by the complementry cumu- 263

lative distribution function of the SINR over the network. The 264

IEEE

Proof

4 IEEE TRANSACTIONS ON COMM UNICATIONS

CP

r

of a user who is at a distanc e of r meters from the BS in a265

FR1-aided MU-MIMO scenario is given by266

P

1

(T, r) = P [η

t

(r)>T] = P

g > Tr

α

I

t

+ Tr

α

σ

2

P

, (3)

where I

t

is deﬁned in (2). Since g ∼ exp(1), h

ij

∼ exp(1),and267

h

ij

are i.i.d., P

1

(T, r) is given by268

P

1

(T, r) = E

h

ij

e

−Tr

α

I

t

−Tr

α

σ

2

P

=

i∈ψ

N

t

j=1

E

h

ij

e

−Tr

α

h

ij

d

−α

i

× e

−Tr

α

σ

2

P

=

i∈ψ

1

1 + Tr

α

d

−α

i

N

t

e

−Tr

α

σ

2

P

, (4)

where ψ is the set of interfering BSs in a FR1 network.

269

Similarly, the CP

r

of a user located at a distance of r meters270

from the BS in a FR3 network can be formulated as271

P

3

(T, r) =

i∈φ

1

1 + Tr

α

d

−α

i

N

t

e

−Tr

α

σ

2

P

(5)

where φ is the set of interfering cells in the FR3 scheme, w hich

272

is a function of the frequency reuse plan. Also, the CP

r

of a user273

in the SIMO-based FR1 network and in a FR3 network can be274

expressed as275

P

1

(T, r) =

i∈ψ

r

1

1 + Tr

α

d

−α

i

e

−Tr

α

σ

2

P

and

P

3

(T, r) =

i∈φ

r

1

1 + Tr

α

d

−α

i

e

−Tr

α

σ

2

P

. (6)

Here φ

r

denotes the set of interfering cells in the FR3 scheme276

excluding the nearest (N

r

− 1) interferers. Let us now derive277

the CP

r

of FFR for both the independent and correlated cases.278

A. Case 1: g and ˆg are Independent as Well as h

i

and

ˆ

h

i

are279

Also Independent for all i280

The CP

r

P

F,c

(r) of a cell-centre user who is at a distance of281

r meters from the 0

th

BS in a FFR-aided MU-MIMO scenario282

is given by283

P

F,c

(r)

(a)

= P [η

t

(r)>T|η

t

(r)>S

th

]

= P

gr

−α

I

t

+

σ

2

P

> T

gr

−α

I

t

+

σ

2

P

> S

th

,

where (a) follows from the fact that a cell-centre user has SINR

284

≥ S

th

. Upon applying Bayes’ rule, one can rewrite P

F,c

(r) as285

P

F,c

(r) =

P

gr

−α

I

t

+

σ

2

P

> T,

gr

−α

I

t

+

σ

2

P

> S

th

P

gr

−α

I

t

+

σ

2

P

> S

th

=

i∈ψ

1

1+max{T,S

th

}r

α

d

−α

i

N

t

e

− max{T,S

th

}r

α

σ

2

P

j∈ψ

1

1+S

th

r

α

d

−α

j

N

t

e

−S

th

r

α

σ

2

P

. (7)

Similarly, the CP

r

of a cell-edge user who is at a distance of r 286

meters from the BS in the FFR-aided MU-MIMO case P

F,e

(r) 287

is given by 288

P

F,e

(r) = P

ˆη

t

(r)>T|η

t

(r)<S

th

=

P

ˆgr

−α

ˆ

I

t

+

σ

2

P

> T,

gr

−α

I

t

+

σ

2

P

< S

th

P

gr

−α

I

t

+

σ

2

P

< S

th

.

Here, the cell-edge user will experien ce the new interference

289

term of

ˆ

I

t

=

i∈φ

N

t

j=1

ˆ

h

ij

d

−α

i

and the new channel power ˆg,i.e.a290

new SINR ˆη(r) due to the fact that th e cell-edge user is now a 291

FR3 user. Basically, ˆη(r) denotes the SINR experienced by the 292

user at a distance of r meters from the BS in a FR3 system and 293

is given by 294

ˆη(r) =

ˆgr

−α

ˆ

I

t

+

σ

2

P

,

ˆ

I

t

=

i∈φ

N

t

j=1

ˆ

h

ij

d

−α

i

. (8)

Since both g and ˆg as well as h

i

and

ˆ

h

i

areassumedtobei.i.d,295

P

F,e

(r) can be simpliﬁed to 296

P

F,e

(r) = P

ˆgr

−α

ˆ

I

t

+

σ

2

P

> T

= P

3

(T, r). (9)

Let us now derive the CP

r

P

f

(r) of a user in the FFR-aided 297

MU-MIMO system, which can be written as 298

P

F

(r) =P

F,c

(r)P [η

t

(r)>S

th

] + P

F,e

(r)P [η

t

(r)<S

th

] . (10)

Here, the ﬁrst term denotes the CP

r

contributed by the cell- 299

centre users, while the seco nd term denotes the contribution of 300

the cell-edge users. By using the expression in (7) for P

F,c

(r) 301

and the expression in (9) for P

F,e

(r), (10) can be simpli- 302

ﬁed to 303

P

F

(r) =

i∈ψ

1

1 + max{T, S

th

}r

α

d

−α

i

N

t

e

− max{T,S

th

}r

α

σ

2

P

+ P

3

(T, r) − P

3

(T, r)P

1

(S

th

, r). (11)

Lemma 1: The optimum S

th

(denoted by S

opt,C

)thatmaxi-304

mizes the FFR-aided coverage probability is S

th

= T,andwhen305

the SINR threshold is set to S

opt,c

, the coverage probability of 306

FFR becomes higher than that of FR3. 307

Proof: See Appendix A for the proof. 308

B. Case 2: g and ˆg are Completely Correlated as Well as h

i

309

and

ˆ

h

i

are Also Completely Correlated for all i 310

Note th at the centre CP

r

is the same for both the above 311

Case 1 and for this case, since a user does not change its sub- 312

band, when it becomes a cell-centre user because if η

t

(r) ≥ S

th

313

for a user, then it will continue to experience the same fading 314

power. However, the edge CP

r

is different in Case 1 as well as 315

Case 2, and in this scenario the CP

r

P

F,e

(r) of a cell-edge user, 316

IEEE

Proof

KUMAR et al.: COVER AGE PROBABILITY AND ACHIEVABLE RATE ANALYSIS OF MU-MIMO AND SIMO SYSTEMS 5

who is at a distance of r meters from the BS in our FFR network317

is given by318

P

F,e

(r) =P

ˆη

t

(r)>T|η

t

(r)<S

th

=

P

ˆη

t

(r)>T,η

t

(r)<S

th

P [η

t

(r)<S

th

]

.

(12)

Substituting the value of P

F,c

and P

F,e

from (7) and (12) into319

Eq. (10), the CP

r

P

f

(r) in our FFR network can be written as320

P

F

(r) =

i∈ψ

1

1 + max{T, S

th

}r

α

d

−α

i

N

t

e

− max{T,S

th

}r

α

σ

2

P

+ P

ˆη

t

(r)>T,η

t

(r)<S

th

. (13)

Recall that η

t

(r) and ˆη

t

(r) represent the SINR experienced by a321

user in an FR1 and an FR3 system, respectively. Note that even322

though g and ˆg as well as h

i

and

ˆ

h

i

are completely correlated,323

η

t

(r) is not the same as ˆη

t

(r), because the set of interferers are324

different in the denominator of the η

t

(r) and ˆη

t

(r) expressions325

given in (2) and (8), respectively, i.e., ψ corresponds to the326

set of interferers in the FR1 network, while φ corresponds to327

the set of interferers in the FR3 network. Since g and ˆg are328

completely correlated and h

i

and

ˆ

h

i

are also completely corre-329

lated for all i, we use the following transformation to further330

simplify P

F

(r):331

P

ˆη

t

(r)>T,η

t

(r)<S

th

=P

ˆη

t

(r)>T, ˆη

t

(r)<

ˆ

S

th

. (14)

Basically instead of marking a user as a cell-edge user based

332

on the FR1 SINR η

t

(r), we mark them on the basis of the FR3333

SINR ˆη

t

(r) by introducing a new SINR threshold

ˆ

S

th

.Inother334

words, we introduce a new SINR threshold

ˆ

S

th

for ensuring that335

if for any user we have η

t

(r)<S

th

, then for the same user we336

have ˆη

t

(r)<

ˆ

S

th

and vice-versa. The threshold

ˆ

S

th

is computed337

using the relationship of P[η

t

(r)<S

th

]=P[ˆη

t

(r)<

ˆ

S

th

].This338

ensures that the same user is marked as a cell-edge user for both339

reuse patterns FR1 and FR3. Now, using the transformation340

given in (14), P

F

(r) can be simpliﬁed to341

P

F

(r) =

i∈ψ

1

1 + max{T, S

th

}r

α

d

−α

i

N

t

e

− max{T,S

th

}r

α

σ

2

P

+ P

ˆη(r)>T

− P

ˆη(r)>max{

ˆ

S

th

, T}

. (15)

In this case, to obtain the optimum S

opt,C

, we co nsider the342

following two possibilities: (i) S

th

≥ T, (ii) S

th

< T.343

(i) S

th

≥ T: In this scenario, CP

f

(r) can be expressed in344

terms of T as:345

P

F

(r, S

th

≥ T) =

i∈ψ

1

1 + S

th

r

α

d

−α

i

e

−S

th

r

α

σ

2

P

+ P

3

(T, r) − P

3

(

ˆ

S

th

, r). (16)

Since we have P

3

(

ˆ

S

th

, r) = P

1

(S

th

, r) and P

1

(S

th

, r) =346

i∈ψ

1

1+S

th

r

α

d

−α

i

N

t

e

−S

th

r

α

σ

2

P

, hence347

P

F

(r, S

th

≥ T) = P

3

(T, r). (17)

(ii) S

th

< T: In this case P

f

(r) can be formulated in terms 348

of T as: 349

P

F

(r, S

th

< T) =

i∈ψ

1

1 + Tr

α

d

−α

i

N

t

e

−Tr

α

σ

2

P

+ P

3

(T, r) − P

3

max{

ˆ

S

th

, T}, r

. (18)

Note that when S

th

< T,

ˆ

S

th

may be higher or lower than T. 350

When

ˆ

S

th

> T, 351

P

3

max{

ˆ

S

th

, T}, r

=P

3

(

ˆ

S

th

, r) =P

1

(S

th

, r)>P

1

(T, r) (19)

since S

th

< T.Andwhen

ˆ

S

th

< T,wehave: 352

P

3

max{

ˆ

S

th

, T}, r

= P

3

(T, r)>P

1

(T, r). (20)

Hence, we arrive at:

353

P

F

(r, S

th

< T) =

i∈ψ

1

1 + Tr

α

d

−α

i

N

t

e

−Tr

α

σ

2

P

+ P

3

(T, r) − P

3

max{

ˆ

S

th

, T}, r

< P

3

(T, r). (21)

Comparing the FFR CP

r

for S

th

≥ T and S

th

< T given by (17) 354

and (21), respectively, it becomes apparent that P

F

(r, S

th

≥ 355

T)>P

F

(r, S

th

< T). I n other words, when the fading is fully 356

correlated across the sub-bands, the optimal choice of the SINR 357

threshold is S

th

≥ T and at the optimal SINR threshold the FFR 358

scheme succeeds in achieving the FR3 CP

r

. Unlike for Case 1, 359

the FFR CP

r

is not better than the FR3 CP

r

since there is no sub- 360

band diversity gain, when a user moves from the cell-centre to 361

the cell-edge region. 362

In order to ﬁnd the CP

r

for a typical user, we have to calculate 363

the probability density function (pdf) of r, which is the distanc e 364

between the 0

th

BS (serving BS) and the desired user. To 365

calculate this pdf, we model the cell shape by an inner circle 366

within a hexagonal cell [33], and assume that the users are 367

uniformly distribu ted. Therefore, the pdf f

R

(r) of r is given by 368

f

R

(r) =

2r

R

2

, r R

0, r > R.

(22)

IV. A

VERAGE RATE 369

In this section, we derive the average rate of both the FFR- 370

aided MU-MIMO as well as of its SIMO counterpart and ﬁnd 371

the optimum value of S

th

(denoted by S

opt,R

) for which the 372

average rate is maximum. The average rate of the system is 373

given by R = E[ln(1 + SINR)]. In order to derive the average 374

rate

5

for the FFR system, we have to consider its sub-band al- 375

location. Since the users are uniformly distributed, the speciﬁc 376

sub-band allocated to the cell-centre users and cell-edge users 377

are given by [9], [10] N

c

= N

t

P

F,c

and N

e

=

N

t

−N

c

3

,whereP

F,c

378

denotes the speciﬁc fraction of cell-centre users, while N

t

, N

c

379

and N

e

denote the total band, cell-centre sub-band and cell-edge 380

5

An interference limited system is assumed for simplicity, which implies

ignoring the effects of noise. However, the derivation of the average rate can be

readily e xtended to the case, where the thermal noise is also considered.

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