geometric analysis for the cell coverage extension with

Geometric Analysis for the Cell Coverage

Extension with Wireless Relay

Seon-Yeong Park

The Graduate School

Yonsei University

Department of Electrical and Electronic

Engineering

Geometric Analysis for the Cell CoverageExtension with Wireless Relay

by

Seon-Yeong Park

A Thesis Submitted to the

Graduate School of Yonsei University

in Partial Fulfillment of the

Requirements for the Degree of

Master of Science

Supervised by

Professor Hong-Yeop Song, Ph.D.

Department of Electrical and Electronic EngineeringThe Graduate School

YONSEI University

December 2007

This certifies that the thesis ofSeon-Yeong Park is approved.

Thesis Supervisor: Hong-Yeop Song

Sanghoon Lee

Jang-Won Lee

The Graduate SchoolYonsei UniversityDecember 2007

ÇÔ��+ ;³�

��$� ��H õ�&ñ�¦ îîß�y� ��}9� Ãº e��2�¤ ��K� ÅÒ�� ÅÒ_��a� y��\�¦ ×¼wn�m�

��.

¢ô�Ç �<ÆÂÒ M:ÂÒ'� $�\�¦ ]j��Ð �� t��K� ÅÒr��¦ C��9\�¦ K�ÅÒ�� 5Åx<�ª\P� �§

Ãº_��a� U�·�Ér y��\�¦ ×¼wn�m��. �7Hë�H\� @/ô�Ç ��Ðü< ��¦ K�ÅÒ�� s��©�ô ¥ �§Ãº

_��õ� s��©�"é¶ �§Ãº_��a�� d��Ü¼�Ð y��×¼wn�m��.

Õªo��¦ �<Êa� ��½z�\�"f Òqt�Ö ��"f �H �¹¡§�¦ ÅÒ�� #��Q ��ÊêC�_��[þta� U�·

�Ér y��\�¦ ×¼wn�m��.

��t�}��Ü¼�Ð#î×�æ\�� $�\�¦0AK�l��K�ÅÒr��¦��|½Ó�¦��z�t�·ú§Ü¼��#Q Qm�,

��!Qt�, Õªo��¦ 1lxÒqt ��\�>� y��\�¦ ��½+Ëm��.

2007�� 12�Z4

¹ÇÐ ¤n> ÆZ� ×¼aË>

Contents

List of Figures iv

List of Tables iv

Abstract v

1 Introduction 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Capacity Theorems for Relay Channel 3

2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1.1 Propagation Model . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1.2 Capacity of AWGN channel . . . . . . . . . . . . . . . . . . . 4

2.1.3 Power Assignment Model . . . . . . . . . . . . . . . . . . . . 4

2.2 Relay System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2.1 The Gaussian Degraded Relay Channel . . . . . . . . . . . . . 5

2.2.2 Coherent Relaying with Interference Subtraction . . . . . . . . 6

i

3 The Geometric Model and Capacity Analysis 9

3.1 The Geometric Model for three-terminal single relay system . . . . . . 9

3.2 Capacity Theorem over the Geometric Model . . . . . . . . . . . . . . 12

3.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4 Cell Coverage Extension with Multiple Relays 19

4.1 Effective Coverage Angle of Relay . . . . . . . . . . . . . . . . . . . . 19

4.2 Source-Relay Channel Allocation Scheme . . . . . . . . . . . . . . . . 20

4.2.1 Network Model and Exclusion Region . . . . . . . . . . . . . . 22

4.2.2 Source-Relay Channel Assignment . . . . . . . . . . . . . . . 23

4.3 Estimation of Cell Coverage Extension . . . . . . . . . . . . . . . . . . 26

4.3.1 Coverage Range and Coverage Angle . . . . . . . . . . . . . . 26

4.3.2 Approximation of Coverage Range and Coverage Angle . . . . 26

4.3.3 Extended Cell Coverage and Relations of Distance Ratio, Cov-

erage Angle, and Coverage Range . . . . . . . . . . . . . . . . 28

5 Concluding Remarks 33

5.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

5.2 Future Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

Bibliography 35

Abstract (in Korean) 39

ii

List of Figures

2.1 Degraded Gaussian relay channel . . . . . . . . . . . . . . . . . . . . . 5

2.2 Three-terminal single relay system . . . . . . . . . . . . . . . . . . . . 6

3.1 Geometric model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.2 Numerical result of a∗, γ = 4 . . . . . . . . . . . . . . . . . . . . . . . 16



4.1 Relations between parameters . . . . . . . . . . . . . . . . . . . . . . 20

4.2 Example of effective coverage angle, k = 0.8, γ = 4 . . . . . . . . . . 21

4.3 The ∆ri/2 neighborhood of a transmitter-receiver pair . . . . . . . . . 23

4.4 Exclusion region (grey area) for four transmitter-receiver pairs . . . . . 24

4.5 Overlap of single exclusion region and extended coverage region . . . . 25

4.6 Extended cell coverage by multiple relays . . . . . . . . . . . . . . . . 27

4.7 Approximation of coverage range . . . . . . . . . . . . . . . . . . . . 28

4.8 Cell coverage extension, γ = 4 . . . . . . . . . . . . . . . . . . . . . . 30



iii

List of Tables

3.1 Description of optimized parameters . . . . . . . . . . . . . . . . . . . 15

4.1 Maximum coverage range . . . . . . . . . . . . . . . . . . . . . . . . 29

iv

ABSTRACT

Geometric Analysis for the Cell Coverage Extensionwith Wireless Relay

Seon-Yeong ParkDepartment of Electricaland Electronic Eng.The Graduate SchoolYonsei University

In this thesis, we propose the method to extend the original cell coverage covered by

single source node from the viewpoint of geometric analysis. We propose the capacity

theorem over our geometric model.

In cell environment, we deploy multiple relays and draw the relations of the required

number of relays NR, distance ratio k, and effective coverage angle θ′ to guarantee

maximum achievable data rate. Also the condition to achieve the maximum coverage

range is presented.

Key words : Wireless relay, Gaussian degraded relay channel, coherent relaying

with interference subtraction (CRIS), relay capacity theorem, cell coverage, exclu-

sion region

v

Chapter 1

Introduction

1.1 Motivation

As the communication systems evolves into fourth-generation (4G) wireless system,

wireless relay attracts tremendous interest in both industry and academia. Since 4G

systems use higher band than 3G’s, the cell radius in these bands is significantly shrunk

from 2-5km to 200-500m. Also, since higher data rates are required for 4G wireless

system, serious power problems can be caused. It is expected that wireless relay can be

a good solution of these problems. Deploying wireless relays can expend these shrunk

cell coverage and reduce required transmission power more economically than densely

deploying base stations (BS) [1], [3].

The topic of relay channel was introduced by van der Meulen and has also been

studied by T. M. Cover and A. El Gamal [5]. In [5], capacity of relaying channel was

established for Gaussian degraded case. In recent years, there have been research about

relaying strategy. L. Xie and P. R. Kumar propose the coherent multistage relaying

with interference subtraction (CRIS) [6]. Also, P. Razaghi and W. Yu describe practical

implementation of the decode-and-forward (DF) strategy for the relay channel [7].

1

In recent years, there have been research about deploying relay for extending cell

coverage in [8], [9], [10], [11], and [12]. Most of the research assume channel reuse

strategy to ignore the impact of interference caused by deploying relay and analyze the

coverage extension.

In this paper we consider the Gaussian degraded relay channel and CRIS relaying

strategy. We derive the optimal cooperation rate a in the geometric model of three-

terminal single relay system. Based on the optimized a, we estimate the extended cell

coverage region. Also, we bound the number of required source-relay channels to mini-

mize the intracell interference among multiple relay channel and maximize the efficiency

of channel resource.

1.2 Overview

The remainder of the paper is organized as follows. In Chapter 2, we give preliminaries

to develop the main issue of paper. Also, we consider the three-terminal single relay

system and define the relaying strategy. In Chapter 3, the geometric model for three-

terminal single relay system is defined. We draw the relations of cooperation ratio a,

distance ratio k, and θ. In Chapter 4, we deploy multiple relays to extend the coverage

of original cell covered by single s in cell environment. We also propose the source-relay

channel allocation scheme to minimize the intracell interference among multiple relay

channel and maximize the efficiency of channel resource. We estimate the extended cell

coverage. Conclusions are drawn in Chapter 5.

2

Chapter 2

Capacity Theorems for RelayChannel

In this chapter we give preliminaries to develop the main issue of paper. Also, we con-

sider the three-terminal single relay system and define the relaying strategy.

2.1 Preliminaries

2.1.1 Propagation Model

For system design the following simplified model [22] for path loss as a function of

distance is commonly used:

Pr = PtK

(d0

d

)γ

, (2.1)

where Pt is transmitted power from transmitter and Pr is received power at receiver. The

path loss in dB is then

PL = −KdB + 10γ log(

d

d0

). (2.2)

d0 is a reference distance for the antenna far field and is set to 1 m, γ is the path-loss

exponent. K is a unitless constant that depends on the antenna characteristics and the

3

average channel attenuation. In the free space

KdB = 20 logλ

4πd0. (2.3)

2.1.2 Capacity of AWGN channel

The amount of interference experienced by MS is captured by signal-to-interference-

plus-noise power ratio (SINR) [22], defined as

SINR =Pr

NB + PI, (2.4)

where Pr is the received signal power and PI is the received power associated with

interference. B is the system bandwidth and N is noise variance.

We assume additive white Gaussian noise (AWGN) channel and the capacity of the

AWGN channel [23] is

CAWGN =12

log2(1 + SINR) bits/s/Hz. (2.5)

2.1.3 Power Assignment Model

In this paper we use the power assignment model introduced in [13]. It considers in-

terference from different nodes and makes reception possible between transmitter and

receiver. Let the system requires signal to interference plus noise ratio (SINR) of β for

successful receptions between transmitter and receiver. We consider that nodes are lying

in a circle of area A. Then the transmission power is assigned as

P = c∆2d2 (2.6)

with c = Ncγ

(∆√

2A)γ−2 suffices, where cγ = 24γ−22γ−2, d is the distance between

transmitter and receiver, γ is the path loss exponent, and N is the ambient noise power.

4

Relay

Encoder

1X

1Y 2X

Y

1Z2Z

sP

rP

Power

Power

Figure 2.1: Degraded Gaussian relay channel

The quantity ∆ > 0 quantifies a guard zone required around the receiver to ensure that

there is no destructive interference from neighboring nodes transmitting on the same

channel at the same time. For given β, ∆ > (482α−2

α−2 β)1/γ .

2.2 Relay System Model

2.2.1 The Gaussian Degraded Relay Channel

First we define the model for discrete time AWGN degraded relay channel [5] as shown

in Figure 2.1. Let Z1 = (Z11, · · · , Z1n) be a sequence of independent identically

distributed (i.i.d.) normal random variables with mean zero and variance N1, and let

Z2 = (Z21, · · · , Z2n) be i.i.d. normal random variables independent of Z1 with mean

zero and variance N2. At the ith transmission the real numbers x1i and x2i are sent and

y1i = x1i + z1i

yi = x2i + y1i + z1i + z2i (2.7)

are received. Thus the channel is degraded.

5

Figure 2.2: Three-terminal single relay system

2.2.2 Coherent Relaying with Interference Subtraction

Based on the degraded Gaussian relay channel introduced in Section 2.2.1 we consider

three-terminal single relay system as shown in Figure 2.2. The system consists of source

s, relay r, and destination d. The source node can only transmitt with transmission power

Ps. The relay node help the same downstream with transmission power Pr. The distance

between source and relay is denoted as dsr. Similarly, dsd and drd can be defined. Let

αsr, αsd, and αrd denote the corresponding signal attenuation factors. Since we use

simple propagation model introduced in Section 2.1.1, attenuation factors can be written

as α2sr = Kd−γ

sr , α2sd = Kd−γ

sd , and α2rd = Kd−γ

rd .

Now we present a coherent relaying and interference subtraction (CRIS) strategy

[6]. The basic idea of CRIS is that relay node divide its energy into two part for helping

different downstream nodes in forwarding messages. The whole transmission time is

divided equally into blocks of the same size. In each block except the first and the last,

s divides its power Ps into two parts, aPs and (1 − a)Ps with 0 ≤ a ≤ 1. These two

parts are used for different purposes. The part aPs is used to inform relay r so that it

6

help coherently in the next block. By the capacity formula in Section 2.1.2 any rate R

satisfying

R < S

(α2

sraPs

N1

), (2.8)

where S(x) = 1/2 log2(1 + x), is achievable for this task. The part (1 − a)Ps is used

to collaborate with relay r, which will transmit coherently with s to send a signal to

receiver d, using its full power Pr.The collaboration information has already arrived at

r at the end of the previous block.

At the end of this block, d receivers the addition of three components. The first one is

the signal due to the coherent cooperation between s and r with power(αsd

√(1− a)Ps+

αrd

√Pr

)2. The second part is the bonus signal sent by s intended mainly for r for

preparing the next-block cooperation with power α2sdaPs. The last one is the noise with

power N1 + N2.

Then the decoding procedure at d is as follows. At the end of each block, it decodes

based on signal from both this block and the previous one. The information bearing parts

for this decoding are the first part aPs of this block and the second part (1 − a)Ps of

the previous block. They both represent the same information. Since the first part of the

previous block becomes known after the decoding at the end of the previous block, it

can be removed before decoding. Therefore, the following rate is achievable.

R < S

((αsd

√(1− a)Ps + αrd

√Pr)2

α2sdaPs + N1 + N2

)+ S

(α2

sdaPs

N1 + N2

)

= S

(α2

sdPs + α2rdPr + 2αsdαrd

√(1− a)PsPr

N1 + N2

)(2.9)

7

a =2AB − 1 +

√(2AB − 1)−A2(B2 − 1)

2A2,

where A =2α2

srPs

2αsdαrd

√PsPr

, B =α2

sdPs + α2rdPr

2αsdαrd

√PsPr

.

Together with the constraint (2.8), this leads to the following end-to-end achievable

rate:

Theorem 2.1 The capacity R of the Gaussian degraded relay channel with attenuation

factor is given by

R < max0≤a≤1

min{

S

(α2

sraPs

N1

),

S

(α2


√(1− a)PsPr

N1 + N2

)}(2.10)

If Pr/N2 ≥ Ps/N1 it can be seen that R = S(α2

srPs/N1

)and this is achieved by

a = 1. The channel appears to be noise free after the r, and the capacity S(α2

srPs/N1

)

from s to r can be achieved. Thus the rate without the relay S(α2

sdPs/(N1 + N2))

is

increased to S(α2

srPs/N1

).

For Pr/N2 ≤ Ps/N1, it can be seen that the maximizing a = a∗ is strictly less than

one, and is given by solving for a in

S

(α2

sraPs

N1

)= S

(α2


√(1− a)PsPr

N1 + N2

)(2.11)

yielding R = S(α2

sra∗Ps/N1

).

8

Chapter 3

The Geometric Model and CapacityAnalysis

In this chapter we define the geometric model for three-terminal single relay system. We

propose the capacity theorem over proposed geometric model and present the numerical

results.

3.1 The Geometric Model for three-terminal single relay sys-tem

For simple analysis, we consider a geometric model as depicted in Figure 3.1a. It is a

physical model that it considers the concrete locations of three nodes. We assume that

two dashed circles are the coverage boundary of s and r, respectively. The distance

between s and r is dsr and d is located around r with distance drd = kdsr (0 < k ≤ 1),

where k is the distance ratio of dsr and drd. Then the dsd can be calculated as

dsd =√

d2sr + (kdsr)2 − 2kd2

sr cos(π − θ), (3.1)

where θ is the angle between the line of link sr and the line of link rd. If we assume

that d is in s’s coverage and there is no obstacle between link sr, r need not help d.

9

Therefore, the range of θ can be restricted within

0 ≤ θ ≤ π − arccos(12k), 0 ≤ k ≤ 1. (3.2)

Under this restriction d is located nearer r than s. Figure 3.1b shows the relation between

k and range of θ.

10

kdsr

dsrs r

d

(a)The geometric model

0 0.2 0.4 0.6 0.8 11.5

1.6

1.7

1.8

1.9

2

2.1

2.2

2.3

Ang

le θ

[ra

d]

Distance ratio k

(b)Range of θ

Figure 3.1: Geometric model

11

3.2 Capacity Theorem over the Geometric Model

Theorem 3.1 The capacity R of the geometric model is given by

R∗ =α2

srPs

N1, (3.3)

if there exists the values of k and θ such that

A−AB + B ±√

AB(A− 2)(B − 2)2

= 1, (3.4)

where A =√

1 + k2 − 2k cos(π − θ)−γ

and B = k−γk2.

Proof: In the case of Pr/N2 ≥ Ps/N1, R = S(α2

srPs/N1

)and this is achieved by

a = 1, which is the case when the optimal strategy is not to allocate any portion of the

transmitter’s power to cooperate with the relay message. Thus, no coherent transmission

is needed between the r and d [6]. Since the parameter αsr = Kd−γsr only reflects the

geometric model. It achieves maximum capacity as r closes to s. This case, however,

should use more relaying power than s’s transmitting power to support condition of

Pr/N2 ≥ Ps/N1. Also, since we use a relay to extend the coverage of source, this case

does not coincide with our purpose. Therefore, we consider the case of Pr/N2 ≤ Ps/N1.

In the case of Pr/N2 ≤ Ps/N1, coherent transmission is beneficial. The optimal

a = a∗ is strictly between 0 and 1. Its numerical value can be found by equating

α2sraPs

N1=

α2sdPs + α2

rdPr + 2αsdαrd

√(1− a)PsPr

N1 + N2. (3.5)

We now find the optimal a∗ in our geometric model. To calculate the lower bound of

capacity, we assume that r is located at the border of s’s and d is located at the border of

12

r’s coverage. Then, r covers the circular area with radius kdsr. Since we use the power

model introduced in Section 2.1.3, the relation between Ps and Pr is

Pr = k2Ps. (3.6)

For simple analysis we assume that N1 = N2 = N . Then (3.5) can be rewritten as

α2sraPs

N=

α2sdPs + α2

rdk2Ps + 2αsdαrd

√(1− a)Psk2Ps

2N. (3.7)

The attenuation factors are parameters of distances and path loss exponents and ex-

pressed as α2sr = Kd−γ

sr , α2sd = Kd−γ

sd , and α2rd = Kd−γ

rd . For simplicity let K = 1 and

distances can be expressed as the parameters of the geometric model. Then combining

(3.1) and (3.7) we get

2d−γsr a =

√d2

sr + (kdsr)2 − 2kd2sr cos(π − θ)

−γ+ (kdsr)−γk2

+2√√

d2sr + (kdsr)2 − 2kd2

sr cos(π − θ)−γ

(kdsr)−γ k√

(1− a).(3.8)

for fixed Ps. For fixed dsr (3.9) is rewritten as

2a =√

1 + k2 − 2k cos(π − θ)−γ

+ k−γk2

+2√√

1 + k2 − 2k cos(π − θ)−γ

k−γ k√

(1− a). (3.9)

For simplicity let A =√

1 + k2 − 2k cos(π − θ)−γ

and B = k−γk2, then

2a = A + B + 2√

AB√

1− a. (3.10)

Therefore, we get a expression of a

a =A−AB + B ±

√AB(A− 2)(B − 2)2

(3.11)

13

under constraint that

√1 + k2 − 2k cos(π − θ)

−γk−γk2 ·

(√1 + k2 − 2k cos(π − θ)

−γ − 2)(

k−γk2 − 2) ≥ 0 (3.12)

and it is a function of distance ratio k and θ. It is clear that a∗ = A−AB+B+√

AB(A−2)(B−2)

2 .

If there exists the values of k and θ such that a∗ = 1, capacity R∗ = α2srPs

N1can be

achieved.

If there exists the specific values of k and θ, say k∗ and θ∗ such that satisfy (3.4),

then we don’t need to change a∗ = 1 for the smaller value of θ than θ∗. As θ closes to

0◦, d is only influenced by the signal from r. We discuss more about this property in

Chapter 4.

14

3.3 Numerical Results

In the following we present the numerical results of a∗ with the change of value of k

and θ when γ = 2, γ = 3, and γ = 4. In the graphs of contour of a∗, blue line is the

maximum value of θ.

In numerical results, three cases show very different aspects. Figure 3.2 shows the

case of γ = 4. Since the attenuation is severe that we cannot select optimal value of a∗

when the value of k is less than 0.7. In this case value of a∗ varies with the values of k

and θ.

When γ = 3, the value of a∗ appears in Figure 3.3. Since the attenuation is not less

severe than the case of γ = 4 that we can select optimal value of a∗ when the value of k

is more than 0.5. In this case value of a∗ varies with the values of k and θ.

When γ = 2, the channel is in good condition of each link. The value of a in Figure

3.4 closes to 1 in most cases of k and θ. Therefore, the optimal strategy is not to allocate

any portion of the transmitter’s power to cooperate with the relay message.

The optimized values of k and θ are described in Table 3.1.

Table 3.1: Description of optimized parameters

γ = 4 γ = 3 γ = 2

k k > 0.7 k ≥ 0.5 k > 0

a∗ 0.6 ∼ 1 1

Strategy Cooperation depending on k and θ repetition

15

0.70.8

0.91

0

2

40

0.5

1

Distance Ratio kAngle θ [rad]

a

(a)Numerical result of a∗

0.40.50.60.7 0.8

0.8

0.9

0.9

0.9

1

1

1

1

1

Distance Ratio k

Ang

le θ

0.7 0.75 0.8 0.85 0.9 0.95 1

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

(b)Contour of a∗

Figure 3.2: Numerical result of a∗, γ = 4

16

0.40.6

0.81

0

2

40.2

0.4

0.6

0.8

1


a


0.4

0.5

0.6

0.7

0.7

0.8

0.8

0.9

0.9

0.9

0.9

1

1

1

1

1

Distance Ratio k

Ang

le θ

0.5 0.6 0.7 0.8 0.9 1

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

(b)Contour of a∗


17

0

0.5

1

0

2

40.96

0.98

1

1.02

1.04


a


0.9750.98 0.9850.985 0.990.99 0.9950.995

1

1

1

1

Distance Ratio k

Ang

le θ

0 0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

(b)Contour of a∗


18

Chapter 4

Cell Coverage Extension withMultiple Relays

In this chapter we consider the cellular downlink environment. We deploy multiple

relays to extend the coverage of original cell covered by single s based on the results of

Chapter 3. Also, we assume that interference caused by mobiles is ignored (e.g. TDMA

environment).

4.1 Effective Coverage Angle of Relay

Based on the results from previous chapter, we draw the relations between k, θ, and a.

Figure 4.1 describes the relations. The power of relay Ps is the function of k. Coop-

eration ratio a represents the dependency on relay and also relates to data rate. we can

choose the optimal value of a (a∗ = 1) based on the given value of k to maximize the

extended coverage and achievable data rate.

After the specific value of a∗ or k is decided, we find the optimal value of θ′ to

maximize the extended coverage.

19

Distance ratio k

Cooperation ratio aRange of

(Related to the coverage)N

aPSRaP

ssr

s

2

,

Geometric

relation

1a

Given k

Maximum data rate

Repetition

srPkP

2

Figure 4.1: Relations between parameters

Definition 4.1 Effective coverage angle of relay: The maximum value of θ of relay that

guarantees maximum achievable rate, a∗ = 1, for given values of k and γ is said to be

effective coverage angles of relay. We denote this angle as θ′.

Figure 4.2 shows the example of effective coverage angle θ′ when k = 0.8 and

γ = 4. As shown in Figure 4.2a, the value of θ which is lower than 1.65 (rad) achieves

the a∗ = 1. Therefore, the effective coverage angle θ′ is 1.65 (rad). In Figure 4.2b, d

located in grey sector can achieve the maximum data rate.

4.2 Source-Relay Channel Allocation Scheme

To use relays deployed at border of coverage of source effectively, we assign source-

relay channels. In this section we present the concept of exclusion region introduced

at [18]. Based on the property of exclusion region, we propose the source-relay channel

allocation scheme to minimize the intracell interference among multiple relay channel

and maximize the efficiency of channel resource.

20

0.40.5 0.6 0.7 0.8

0.8

0.9

0.9

0.9

1

1

11

1

Distance Ratio k

Ang

le θ

0.7 0.75 0.8 0.85 0.9 0.95 1

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

k=0.8

θ’=1.65

(a)Contour of a∗

19095

65.1,1*a

(b)Effective coverage angle

Figure 4.2: Example of effective coverage angle, k = 0.8, γ = 4

21

4.2.1 Network Model and Exclusion Region

Protocol Model

Consider a network of n nodes in a coverage area. Let Xi, 1 ≤ i ≤ n, denote a note i as

well as its location. Let (Xk, XR(k)) : k ∈ T be the set of all active transmitter-receiver

pairs in some particular slot. The distance between Xk and XR(k) is referred as rk.

Then we describe the Protocol model in [18], [4]. The transmission from node Xi,

i ∈ T , is successfully received by the receiver XR(i) only if

|Xk −XR(i)| ≥ (1 + ∆)|Xi −XR(i)|,

∀k ∈ Activetransmitters, k 6= i. (4.1)

Here ∆ > 0 is a guard zone around the transmission region and it ensures that there

is no destructive interference from neighboring nodes transmitting on the same slot. If

a power assignment of transmitters follows the power assignment model in 2.1.3, the

protocol model with guard zone ∆ can be configured to the physical model with certain

SINR threshold β [13].

Exclusion Region and Interference Region

Now, we define an exclusion region [18].

Definition 4.2 Exclusion region: For a Particular configuration of transmitters and re-

ceivers in a network, an exclusion region of an active transmitter-receiver pair is an

associated area such that, for the transmission to be successful, it must be kept disjoint

from every other exclusion region at that time and over the same sub-channel.

22

iX )(iR

X2/i

r

ir

Figure 4.3: The ∆ri/2 neighborhood of a transmitter-receiver pair

The following Theorem 4.1 shows that there exists a capsule-shaped region around

the transmitter-receiver axis that is an exclusion region for the protocol model.

Theorem 4.1 [18] In a wireless network under the protocol model, if (Xi, Xj) is an

active transmitter-receiver pair, then the ∆dij/2 radius neighborhood of the line joining

them is an exclusion region for (i, j).

Figure 4.3 shows the capsule-shape exclusion region around a transmitter-receiver

pair for the protocol model.

4.2.2 Source-Relay Channel Assignment

In this section we assign source-relay channel by using the capsule-shaped exclusion

region. From the definition of exclusion region, the exclusion regions of transmitter-

receiver pairs communicating on the same slot in network should be kept disjoint. On our

cell layout, however, every pair’s transmitters are the same as source node. Therefore, we

allow the certain amount of overlap around the source node between distinct exclusion

regions as shown in Figure 4.4. In the example of figure 4.4 four transmitter-receiver

pairs can communicate simultaneously on the same channel.

We now derive the number of required source-relay channels to cover the extended

23

s

r

Figure 4.4: Exclusion region (grey area) for four transmitter-receiver pairs

cell. Figure 4.5 shows the overlap of single exclusion region and extended cell coverage

region. Let t and t′ represent the points of intersection of exclusion region and border

line of original cell coverage covered by source node only. Let the angle φ = ∠tst′.

Then φ can be expressed as the function of ∆:

φ = 2 arcsin∆2

. (4.2)

The entire cell coverage is divided by d360◦/φe exclusion regions. As we show an

example in Figure 4.4 certain number of transmitter-receiver pairs can communicate

simultaneously. Let Np be the number of transmitter-receiver pairs can communicate

simultaneously on the same channel. Since we should assign different channel to nodes

in overlap of single exclusion region and extended cell coverage region, we need at

24

s

r

t

t

Figure 4.5: Overlap of single exclusion region and extended coverage region

least dφ/αe channels are required. Therefore, the total required number of source-relay

channels Nc,total to cover the extended is upper bounded as follow:

Nc,total =⌈360◦

φ

⌉ 1Np

⌈φ

α

⌉. (4.3)

25

4.3 Estimation of Cell Coverage Extension

4.3.1 Coverage Range and Coverage Angle

Now we deploy multiple relays at the border of s’s cell coverage to extend the s’s cell

coverage based on the effective coverage angle θ′. We assume that the relays do not

cooperate in transmission and each relay transmits data independently. Also, each d does

not combine signals from different relays. We require a circular cell and we want the new

cell also to be a circular cell. Figure 4.6 shows the example of extended cell coverage by

multiple relays. For clarity, we use the concepts of coverage angle and coverage range

in [10]. Coverage range is the maximum cell radius achieved by deploying of relays. In

Figure 4.6 r1 is the coverage of source node only and r2 is the coverage range. α is an

angle supported by one relays and it is referred as coverage angle. The coverage range

can be extended by placing more relays around the source node. The coverage range of

the system can approximate r3 by deploying infinite relays.

4.3.2 Approximation of Coverage Range and Coverage Angle

The coverage range r2 can be approximate with the given value of θ′ and k. In Figure 4.7

since the coverage of relay is kdsr and r1 = dsr, the coverage rage r2 is approximated

as

r2 =√

d2sr + (kdsr)

2 − 2dsr(kdsr) cos(π − θ′). (4.4)

Then the coverage angle α is approximated as

α = 2arccos(

d2sr + r2

2 − (kdsr)2

2dsrr2

). (4.5)

It means that we need NR = d360◦/αe relays to achieve the coverage range r2.

26

'

r1

r2

r3

s

r

Figure 4.6: Extended cell coverage by multiple relays

27

r1

r2

s

r

Figure 4.7: Approximation of coverage range

4.3.3 Extended Cell Coverage and Relations of Distance Ratio, CoverageAngle, and Coverage Range

In the following we present the extended cell coverage and draw the relations of distance

ratio, coverage angle, and coverage range. It is assume that the fixed value of dsr is set

to 1.

Figure 4.8 shows the the extended coverage range r2 and the relations of distance

ratio k, number of required relays NR, and coverage range r2 when γ = 4. r2 is extended

up to 1.5 with NR ≤ 15. r2 is extended reciprocally to the increase of k. The number of

required relays decreases as k increases.

Figure 4.9 shows the case of γ = 3. It shows similar tendency. r2 is extended up to

1.5 with NR ≤ 16.

In the case of γ = 2 in Figure 4.10 it yields very different result. Since the effective

coverage angle θ′ is close to the maximum value of θ, we cannot get extended cell with

28

circular shape. Therefore, we use 2NR relays to cover circular cell shape. r2 is extended

in proportional to the increase of k.

The maximum extended cell coverage is described in Table 4.1. In all cases r2/r1 is

larger than the case which uses two s (r2/r1 =√

2). In cases of γ = 2 and 3 transmission

power of r, Pr, is lower than s’s, Ps. r uses the same transmission power of s to get the

maximum extended coverage when γ = 2.

From the observation of two examples, we need more number of relays to support

the condition a∗ = 0.8, higher value of required data rate. But, we also obtain much

larger extended cell coverage in this case. Therefore, we conclude that there exists a

proportional relation between a∗ and coverage range r2 for the fixed value of k.

Table 4.1: Maximum coverage range

γ = 4 γ = 3 γ = 2

r2/r1 1.6 1.5 1.7

Pr/Ps[dB] −2.22 −5.23 0

NR 15 16 6

29

0.7 0.75 0.8 0.85 0.9 0.95 11

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

Distance Ratio k

Cov

erag

e ra

nge

r 2, dsr

=1

(a)Distance ratio k vs. Coverage range r2

0.7 0.75 0.8 0.85 0.9 0.95 12

4

6

8

10

12

14

16

Distance Ratio k

Num

ber

of r

equi

red

rela

ys N

R

(b)Distance ratio k vs. Number of required relays NR

Figure 4.8: Cell coverage extension, γ = 4

30

0.5 0.6 0.7 0.8 0.9 11

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

Distance Ratio k

Cov

erag

e ra

nge

r 2, dsr

=1


0.5 0.6 0.7 0.8 0.9 10

5

10

15

20

Distance Ratio k

Num

ber

of r

equi

red

rela

ys N

R



31

0 0.2 0.4 0.6 0.8 11

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

Distance Ratio k

Cov

erag

e ra

nge

r 2, dsr

=1


1 1.2 1.4 1.6 1.80

100

200

300

400

500

600

700

Coverage range r2, d

sr=1

Num

ber

of r

equi

red

rela

ys N

R



32

Chapter 5

Concluding Remarks

5.1 Summary

In this thesis we study the cell coverage extension with wireless relay from the viewpoint

of geometric analysis.

We define three-terminal single relay system and coherent relaying and interference

subtraction (CRIS) strategy. We reinterpreted the Gaussian degrade relay channel capac-

ity in simple path loss environment.

We apply the reinterpreted capacity theorem to proposed geometric model. We find

the relations of cooperation ratio a, distance ratio k, and θ.

In cell environment, we deploy multiple relays and draw the relations of the required

number of relays NR, distance ratio k, and effective coverage angle θ′ to guarantee

maximum achievable data rate.

The condition to achieve the maximum coverage range for a∗ = 1 depends on the

channel condition. In low attenuation regime maximum coverage range is achieved as

Pr increases and NR decreases. On the other hand, in high attenuation regime maximum

coverage range is achieved as Pr decreases and NR increases.

33

5.2 Future Works

In the further research, the following problems are desired to be studied.

• We need to find the practical situation to apply our cell extension scheme.

• In case of γ = 2 in Chapter 4 we use 2NR relays to get circular cell coverage and

this method is ad hocery. We want to propose more specific method for deploying

relays in this case.

34

Bibliography

[1] R. Pabst et al., “Relay-Based Deployment Concepts for wireless and Mobile Broad-

band Radio,” IEEE Communications Magazine, Volume 42, Issue 9, pp. 80-89,

September 2004.

[2] P. Gupta and P. R. Kumar, “The Capacity of Wireless Networks,” IEEE Transac-

tions on Information Theory, vol. 46, no. 2, pp. 388-404, March 200.

[3] H. Yanikomeroglu, “Fixed and Mobile Relaying Technologies for Cellular Net-

works,” Proc. Second Workshop on Applications and Services in Wireless Net-

works, pp. 75-81, Paris, France, July 2002.

[4] F. Xue and P. R. Kumar, “Scaling Laws for Ad Hoc Wireless Networks: An Infor-

mation Theoritic Approach,” Foundations and Trends in Networking, vol. 1, no. 2,

pp. 145-270, 2006.

[5] T. M. Cover and A. El Gamal, “Capacity Theorems for the Relay Channel,” IEEE

Transactions on Information Theory, vol. 25, no. 5, pp. 572-584, September 1979.

[6] L. Xie and P. R. Kumar, “A Network Information Theory for Wireless Communi-

35

cation: Scaling Laws and Optimal Operation,” IEEE Transactions on Information

Theory, vol. 50, no. 5, May 2004.

[7] P. Razaghi and W. Yu, “Bilayer Low-Density Parity-Check Codes for Decoding-

and-Forward in Relay Channels,” IEEE Transactions on Information Theory, vol.

53, no. 10, October 2007.

[8] H. Hu, H. Yanikomeroglu, D. D. Falconer, and S. Periyalwar, “Range extension

without capacity penalty in cellular networks with digital fixed relays”, Proc. IEEE

Global Telecommunications Conference, vol. 5, pp. 3053-3057, 2004.

[9] H. Hu, “Performance Analysis of Cellular Networks with Digital Fixed Relays,”

M.A.Sc. Thesis, Carleton University, 2003.

[10] J. Zhao et al., “Coverage Analysis for Cellular Systems with Multiple Antennas

Using Decode-and-Foward Relays”, Proc. IEEE Vehicular Technology Conference,

pp. 944-948, spring, 2007.

[11] V. Sreng, H. Yanikomeroglu, and D. Falconer, “Coverage Enhancement through

Two-hop Relaying in Cellular Radio Systems,” Proc. Wireless Communications

and Networking Conference, vol. 2, pp. 881-885, March 2002.

[12] C. Muller, A. Klein, and F. Wegner, “Coverage Extension of Wimax Using Multi-

hop in a Low User Density Environment,” Frequenz, vol. 61, pp. 3-6, 2007.

[13] A. Agarwal and P. R. Kumar, “Capacity Bounds for Ad-Hoc and Hybrid Wireless

Networks,” ACM SIGCOMM Computer Communications Review, vol. 34, no. 3,

pp. 71-81, July 2004.

36

[14] V. S. Abhayawardhana, I. J. Wassell, D. Crosby, M. P. Sellars, and M. G. Brown,

“Comparison of Empirical Propagation Path Loss Models for Wireless Access Sys-

tems,” Proc. IEEE Vehicular Technology Conference, vol. 1, pp. 73-77, spring,

2005.

[15] IEEE Standard 802.16, Air Interface for Fixed Broadband Wireless Access Sys-

tems,October 2004.

[16] Electronic Communication Committee (ECC) withing the European Conference of

Postal and Telecommuncations Administration (CEPT), “The Analysis of the Co-

existence of FWA Cells in the 3.4-3.8GHz Band,” tech. rep., IEEE 802.16 Broad-

band Wireless Access Working Group, January 2003.

[17] COST Action 231, “Digital Mobile Radio Towards Future Generation Systems,

final report,” tech. rep., European Communities, EUR 18957, 1999.

[18] A. Agarwal and P. R. Kumar, “Improved Capacity Bounds for Wireless Networks,”

Wireless Communications and Mobile Computing, vol. 4, pp. 251-261, 2004.

[19] M. Turkboylari and G. Stuber, “An Efficient Algorithm for Estimating the Signal-

to-Interference Ratio in TDMA Cellular Systems,” IEEE Transactions on Commu-

nications, vol. 46, No. 6, pp. 728-731, June 1998.

[20] X. Bao and J. Li, “Progressive Network Coding for Message-Forwarding in Ad-

Hoc Wireless Networks,” Proc. IEEE SECON 2006, vol. 1, pp. 207-215, 2006.

[21] A. Fujiwara, S. Takeda, H. Yoshino, and N. Umeda, “Evaluation of Cell Coverage

37

and Throughput in the Multihop CDMA Cellular Network,” Proc. IEEE Vehicular

Technology Conference, vol. 4, pp. 2375- 2378, spring, 2004.

[22] A. Goldsmith, Wireless Communication, Cambridge University Press, 2005.

[23] D. Tse and P. Viswanath, Fundamentals of Wireless Communication, Cambridge

University Press, 2005.

[24] T. M. Cover and J. A. Thomas, Elements of Information Theory, Second Edition,

Wiley, 2006.

38

²DGë�H¹כ��

Áº��×�æ>�l�\�¦s�6 xô�Ç!sq&�!Qo�t�SX��©�\�@/ô�Çl� ��<Æ&h�

ì�r$3�

�:r�7Hë�H\�"f��Hl� ��<Æ&h�ì�r$3�_��'a&h�\�"f×�æ>�l��Ö6 x�¦:�xô�Ç!sqSX��©�~½ÓZO��¦

]jîß� �%i��. Gaussian degraded relay channel\�"f_�H�J�r�w�\�¦ simple path loss 8�

�â\�"f F�K�$3� �%i��. F�K�$3��)a ��õ�\�¦ ]jîß��)a l� ��<Æ&h� �4Sq\� &h�6 x �#� a�?§4�

q�Ö�¦,×�æ>�l�_�5Åx��§4�,×�æ>�l�_�&�!Qo�t�çß�_��'a>�\�¦ �Ø�¦ �%i��. �Ø�¦�)a�'a

>�\�¦s�6 x �#�!sq 8��â\�"f��Ãº_�×�æ>�l�\�¦C�u��<ÊÜ¼�Ð+�!sq&�!Qo�t�_�SX��©� �

%i��.!sqSX��©��¦0Aô�Ç×�æ>�l�_�>hÃº,SX��©��)a!sq_�ß¼l�,×�æ>�l�_�5Åx��§4�çß�_�

�'a>�\�¦ �Ø�¦ �%i�Ü¼ 9þj@/�ÐSX��©��)a!sq�¦%3�l�0Aô�Ç �|��¦]jr� �%i��.

Ùþ�d��÷&��H ú�: Áº�� ×�æ>�l�, Gaussian degraded relay channel, coherent relaying with

interference subtraction (CRIS), relay capacity theorem, cell coverage, exclusion region

39

geometric analysis for the cell coverage extension with

Documents