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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
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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
3.3 Numerical result of a∗, γ = 3 . . . . . . . . . . . . . . . . . . . . . . . 17
3.4 Numerical result of a∗, γ = 2 . . . . . . . . . . . . . . . . . . . . . . . 18
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
4.9 Cell coverage extension, γ = 3 . . . . . . . . . . . . . . . . . . . . . . 31
4.10 Cell coverage extension, γ = 2 . . . . . . . . . . . . . . . . . . . . . . 32
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
sdPs + α2rdPr + 2αsdαrd
√(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
sdPs + α2rdPr + 2αsdαrd
√(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
Distance Ratio kAngle θ [rad]
a
(a)Numerical result of 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∗
Figure 3.3: Numerical result of a∗, γ = 3
17
0
0.5
1
0
2
40.96
0.98
1
1.02
1.04
Distance Ratio kAngle θ [rad]
a
(a)Numerical result of 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∗
Figure 3.4: Numerical result of a∗, γ = 2
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
(a)Distance ratio k vs. Coverage range r2
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
(b)Distance ratio k vs. Number of required relays NR
Figure 4.9: Cell coverage extension, γ = 3
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
(a)Distance ratio k vs. Coverage range r2
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
(b)Distance ratio k vs. Number of required relays NR
Figure 4.10: Cell coverage extension, γ = 2
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
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38
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