Optimal FSO Relay Nodes Placement With Link ... - IEEE Xplore

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Optimal FSO Relay Nodes Placement With Link Obstacles and Infeasible Regions Bingcheng Zhu∗† , Julian Cheng† and Lenan Wu∗ of Information Science and Engineering, Southeast University Nanjing, Jiangsu, China, Emails: {230109122, wuln}@seu.edu.cn † School of Engineering, The University of British Columbia Kelowna, BC, Canada Email: [email protected]

∗ School

Abstract—Optimal relay placement is studied for free-space optical multi-hop communication when obstacles and infeasible regions exist. We propose an optimization framework to find the optimal relay locations in the specified available regions and make laser links bypass obstacles of various shapes. A grouping optimization technique is proposed to reduce the optimization time when the number of relays is large, and we numerically demonstrate that this technique can provide solutions close to the optimal solutions, and its average searching time grows linearly with the number of relays. Simulation results show that our proposed optimization framework can effectively provide good solution to the problem of optimal relay placement in limited available regions, and enable the laser links to bypass obstacles in order to achieve minimum end-to-end outage probability.

I. I NTRODUCTION Free-space optical (FSO) communication can achieve data transmission rate at Gbps, and it is a promising complementary technology to its radio-frequency (RF) counterparts. Unlike RF communication, FSO communication does not suffer multipath fading as laser beams have high directionality. The main adverse factor in FSO systems is heavy weather conditions such as snow, rain and fog [1]. Under a clear weather condition, the performance of FSO systems is mainly impaired by turbulence-induced fading, known as scintillation [2]. When the link distance is less than 600 meters, the atmospheric turbulence channel is weak and can be modeled by the lognormal distribution [3]. Unlike RF systems, the severity of the fading in an FSO system can depend on the link distance [4]. Therefore, relays can be employed to convert a long link into multiple short links in order to mitigate the fading. In this paper, we consider an intensity-modulation direct-detection (IM/DD) FSO multihop decode-and-forward (DF) system, and each relay node is equipped with a single aperture. When a relay node receives the incoming signal, it remodulates the signal and forwards it to the next node, and the forwarding process continues until the signal reaches its destination. This multi-hop DF relaying scheme has been studied extensively in the literature because it has the benefit of broadening the signal coverage with limited transmission power. Besides, in an FSO system, relays can be introduced to realize none-line-of-sight transmission when there is an obstacle in the direct path between the transmitter and the receiver. The performance of multi-hop FSO system has been studied recently. In [5], an integral outage probability expression was 978-1-4799-3512-3/14/$31.00 ©2014 IEEE

derived for amplify-and-forward (AF) multi-hop relaying over the Gamma-Gamma channels, and a closed-form outage probability expression for DF multi-hop relaying was derived using the Meijer G-function. In [6], closed-form outage probability expression for DF multi-hop relaying and approximate outage probability expression for AF multi-hop relaying were derived for the lognormal channels. Based on that, the authors in [7] recently showed that in the absence of link obstacles an optimal relay placement scheme should place the consecutive relay nodes equidistant along the direct path from the source to the destination. However, in practical scenarios, the direct links may not always be available. Firstly, direct laser links can be obstructed by obstacles. For example, mountains, tall building clusters and industrial parks that constantly emit smog can block the direct laser path. Secondly, certain regions can be unavailable to place FSO relays. For example, there may exist high mountains, lakes, and unpowered wild fields where it is infeasible to place the relay nodes; and for logistical reasons, one may not have the freedom to place the relay nodes anywhere he desires due to licensing issues. These obstacles and infeasible regions are permanent and can come in different shapes. Therefore, it is of practical importance to search for the best relay node locations according to a specified optimal criterion. We consider the relay placement problem on a 2dimensional (2D) plane instead of the 3-dimensional (3D) space because the height of the buildings, where FSO devices are typically placed, are usually insignificant comparable to the distance of one FSO hop; therefore, the 3D relay placement problem can be well approximated by a 2D relay placement problem. In this paper, we propose a general optimization framework to study the problem of relay placement in limited available regions when obstacles of various shapes exist, and to the best of the authors’ knowledge, no prior papers have studied such problem. Using this optimization framework, one can resort to numerical schemes to find the best relay locations in designing an FSO relay network.

II. S YSTEM MODEL We consider an IM/DD FSO multi-hop system shown in Fig. 1, where the number of relays is K; the source is denoted by node 0; and the destination is denoted by node K + 1.

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f n -1 ( xk -1 , yk -1 , xk , yk ) £ 0, "k = 1,

f n +1 ( xk -1 , yk -1 , xk , yk ) £ 0,

, K +1

gk ( xk , yk ) £ 0

Obstacle n-1

"k = 1,

, K +1 Obstacle n+1

Relay k

( xk , yk )

K +1 0

Obstacle n

Ă Destination

Ă Relay k - 1 f x , y , x , y £ 0, ( ) ( xk -1 , yk -1 ) n k -1 k -1 k k "k = 1, , K + 1

Source

Obstacles

Available Region

Source or destination

Relay k + 1 ( xk +1 , yk +1 )

Relay

Laser Link

Fig. 1. A multi-hop FSO system where fn (·) models the obstacle n and gk (·) models the available regions for relay k.

non-signal slots; N0 is the density of the Gaussian white noise; and γth is the electrical threshold signal-to-noise ratio (SNR). 2) Asymptotic Outage Probability for Multi-hop DF Relaying: In the multi-hop system, we assume the (kmax )th hop is the longest with the corresponding link distance dkmax , and it follows that this single link outage probability is ln(L(dkmax )PM /(K + 1)) + 2µkmax Pout,kmax (γth ) = Q 2σkmax (5) where µkmax and σkmax are the associated lognormal parameters of the (kmax )th hop. Note that, from (2), σkmax is larger than the other σk values as long as the scintillation on each link is not saturated1 . Using L’Hopital’s rule, we calculate

A. Channel Model

lim

PM →∞

The aggregate channel power gain can be expressed as 2

hk = αk L (dk )

(1)

where αk is the fading amplitude of the kth link; L(dk ) is the normalized path loss calculated as L(dk ) = l(dk )/l(dSD ); dSD is the distance between the source and the destination and dk is the distance of the kth hop; l (dk ) = −2 AT X ARX e−σdk (λdk ) ; σ, AT X , ARX and λ are, respectively, the visibility-dependent attenuation coefficient, the transmitter aperture area, receiver aperture area, and the optical wavelength [6]. The lognormal parameters of αk , denoted by (µk , σk ), are also determined by the length of the kth link, and its scintillation level σk is calculated as [8] n o 7/6 2 11/6 σk2 = min 0.124kw Cn dk , 0.5 (2)

where kw and Cn2 are the wave number and the refractive index structure constant, respectively, and the minimum operation is due to the consideration of saturated scintillation. To normalize the fading amplitude, we make E[αk 2 ] = 1 so that µk = −σk2 . B. Outage Probability at the Destination 1) Outage Probability for DF Multi-hop Relaying: For multi-hop DF relaying, an outage event at the destination happens when any individual link fails, and the outage probability for a multi-hop DF FSO system can be expressed in terms of the kth hop outage probability Pout,k as [7, eq. (6)] Pout,DF = 1 −

K+1 Y

(1 − Pout,k )

k=1

=1−

K+1 Y k=1

1−Q

ln(L(dk )PM /(K + 1)) + 2µk 2σk

(3)

where Q(·) is the Gaussian Q-function, and PM is the power margin that is given by s Pt2 R2 Tb2 (4) PM = N0 γth where Pt is the total transmitted power; R is the responsitivity of the photodetector; Tb is the duration of the signal and the

=

=

Pout,DF Pout,kmax PK+1

∂ k=1 ∂PM

lim

PM →∞

lim

PM →∞

= 1.

∂ ∂PM

K+1 X

Q

Q

ln(L(dk )PM /(K+1))+2µk 2σk

ln(L(dkmax )PM /(K+1))+2µkmax 2σkmax

exp

k=1

1 8σk2max

1 − 2 8σk

!

(ln PM )

2

!

σkmax σk (6)

The significance of (6) is that not only the asymptotic relative diversity order is determined by the longest hop (which was already established in [7]), but also the asymptotic end-toend outage probability is determined by the longest hop. As a consequence of (6), one can alternatively choose Pout,kmax instead of Pout,DF to optimize an FSO relay network when PM is sufficiently large. III. O PTIMIZATION F RAMEWORK In this section, we introduce an optimization framework that models the problem of relay placement in limited available regions when obstacles exist. This framework consists of three parts: objective function, constraint functions, and feasible region functions. A. Objective Function We can take the exact outage probability in (3) as the objective function to be minimized, and its arguments are the coordinates of each relay location, i.e. (xk , yk ) where k = 1, . . . , K. The distance dk between the (k − 1)th node and the kth node can be calculated as dk = p (xk − xk−1 )2 + (yk − yk−1 )2 . In an FSO communication system with relays, the length of any hop is usually short, and due to the high directionality of laser beams, the SNR at the receiver is typically sufficiently large. Therefore we can alternatively use the asymptotic outage probability as the criterion for relay placement. As discussed in Section II, the DF asymptotic outage probability is mainly determined by the longest hop. As long as we can minimize

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1 Without

loss of generality, we assume there exists only one longest link.


Round Obstacle 0.9

Outage probability of the longest hop

0.8

rn PO ,C

0.7 0.6 0.5

Pk -1 Relay k - 1 ( xk -1 , yk -1 )

0.4

P´

P^

Relay k

PO , B

0.3 0.2

PO , A

0.1 0

3

4

5 6 7 8 Distance of the longest hop (km)

9

Pk

( xk , yk )

Line-segment Obstacle

Fig. 3. Relative positions of round and line-segment obstacles with respect to the (k − 1)th relay node and the kth relay node. Here, rn is the radius of the round obstacle; PO,C is the center of the obstacle; Pk−1 Pk is the kth laser link; and PO,A PO,B is the line-segment obstacle.

10

Fig. 2. The outage probability of the longest hop Pout,kmax as a function of its distance dkmax .

the longest hop by properly placing the relay, the asymptotic outage probability can be minimized. Therefore, our aim is to solve the following optimization problem min max {dk } , k = 1, 2, · · · , K. k

(7)

Figure 2 shows that Pout,kmax in (5) monotonically increases with dkmax , thus the objective function in (7) is equivalent to the asymptotic outage probability in (5) for the optimization problem. If there are no obstacles or infeasible regions, the relay nodes can be placed along the direct path and it is easy to verify that (7) is minimized when d1 = d2 = . . . = dK+1 . This agrees with the result derived in [7] where their results were derived using the exact end-to-end outage probability in (3) as the objective function. B. Constraint Functions Note that the kth link connects node k − 1 and node k, which respectively have coordinates (xk−1 , yk−1 ) and (xk , yk ) on a 2-D plane. We can use an indicator function fn (xk−1 , yk−1 , xk , yk ) to denote whether the kth link is blocked by obstacle n. Specifically, if obstacle n blocks the kth link, then fn (xk−1 , yk−1 , xk , yk ) > 0; otherwise, fn (xk−1 , yk−1 , xk , yk ) ≤ 0. In optimization literatures, such functions are also known as the constraint functions. If the nth obstacle has a round shape as shown in Fig. 3, we can then use the distance between its center PO,C and the kth laser link, whose endpoints are Pk−1 and Pk , to construct an indicator function. If the distance between PO,C and the kth laser link is shorter than the radius of the nth obstacle rn , then the link is blocked by the nth obstacle; otherwise it is not blocked. Note that there are only three candidate points on the kth laser link that can be the nearest to PO,C , including P⊥ which is the projection point of PO,C onto the straight line Pk−1 Pk , and the two endpoints Pk−1 and Pk . Specifically, when P⊥ lies within

line-segment Pk−1 Pk , the distance between Pk−1 Pk and PO,C is the length of PO,C P⊥ , which is denoted by |PO,C P⊥ |; otherwise, the distance between Pk−1 Pk and PO,C is the smaller one between |PO,C Pk−1 | and |PO,C Pk |. Thus, the function indicating whether the kth link is blocked by obstacle n can be obtained from the geometry as fn (xk−1 , yk−1 , xk , yk ) −−−−−→ −−−→ = rn − u(hPk−1 P⊥ , P⊥ Pk i)|PO,C P⊥ | (8) −−−−−→ −−−→ − u(−hPk−1 P⊥ , P⊥ Pk i)min {|PO,C Pk−1 |, |PO,C Pk |} −−→ where AB denotes the vector from point A towards point B; u(·) is the step function, which is used here to determine −−→ −−→ whether P⊥ lies within Pk−1 Pk ; hAB, CDi denotes the −−→ −−→ inner product between vectors AB and CD. Noting that the coordinates of PO,C is a constant, and once (xk−1 , yk−1 ) and (xk , yk ) are fixed, we can obtain the coordinates of P⊥ by solving the following set of two linear equations    (x⊥ − xO,C ) (xk − xk−1 ) + (y⊥ − yO,C ) (yk − yk−1 ) = 0   (y⊥ − yk ) (xk − xk−1 ) − (x⊥ − xk ) (yk − yk−1 ) = 0 (9) where (x⊥ , y⊥ ) is the coordinates of P⊥ , and (xO,C , yO,C ) is the coordinates of PO,C . The first equation in (9) holds because PO,C P⊥ is perpendicular to Pk−1 Pk , and the second equation in (9) holds because Pk−1 P⊥ is collinear with Pk−1 Pk . If the nth obstacle is a line-segment with endpoints PO,A and PO,B , the blocking happens when PO,A PO,B and Pk−1 Pk intersect. As shown in Fig. 3, P× is the common point of the straight line across PO,A and PO,B , and the straight line across Pk−1 and Pk . If P× is on both PO,A PO,B and Pk−1 Pk , we obtain  D−−−−−→ −−−→E  Pk−1 P× , Pk P× < 0 D −−−−→ −−−−−→E (10)  − PO,A P× , PO,B P× < 0

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which implies that the laser link is blocked. Therefore, the indicator function for the kth hop blocked by the nth linesegment obstacle is fn (xk−1 , yk−1 , xk , yk ) nD−−−−−→ −−−→E D−−−−−→ −−−−−→Eo (11) = − max Pk−1 P× , Pk P× , PO,A P× , PO,B P×

where the coordinates of the common point P× , which is denoted by (x× , y× ), can be obtained from    (x× − xO,A ) (yO,A − yO,B ) − (y× − yO,A ) (xO,A − xO,B ) = 0   (x× − xk ) (yk − yk−1 ) − (y× − yk ) (xk − xk−1 ) = 0. (12) In general, if the nth obstacle has a polygonal shape, we can use line-segment obstacles to construct such polygonal obstacle, where each edge of the polygonal obstacle is described by a line-segment obstacle having its indicator function in the form of (11). We comment that irregular shapes can also be approximated by polygonal shape.

Fig. 4. A half-plane feasible region defined by functions g(x, y) = x+y−1.

C. Feasible Region If the relay nodes cannot be placed arbitrarily, we can use the feasible region to model the available areas for the relays. The feasible region is defined by another function gk (xk , yk ) for the kth relay, where (xk , yk ) is the coordinates of the the kth relay node. Specifically, when gk (xk , yk ) > 0, the kth relay is outside its feasible region, and when gk (xk , yk ) ≤ 0, the kth relay is placed within its feasible region. If the feasible region for the kth relay is a disk with radius rk,f and center coordinates (xk,f , yk,f ), we can describe the feasible region with a function 2 gk (xk , yk ) = (xk − xk,f )2 + (yk − yk,f )2 − rk,f .

(13)

The complementary set of a feasible region defined by gk (xk , yk ) can be described by gk′ (xk , yk ) = −gk (xk , yk ). If the feasible region is half of the whole plane, we can describe it using a linear function x1 − x2 y1 − y2 xk + yk − 1. (14) gk (xk , yk ) = y1 x2 − y2 x1 x1 y2 − x2 y1 Substituting (x1 , y1 ) and (x2 , y2 ) into (14), we obtain gk (x1 , y1 ) = 0 and gk (x2 , y2 ) = 0, which implies that the straight line across (x1 , y1 ) and (x2 , y2 ) is the boundary of the feasible region. We can also verify that gk (0, 0) = −1, which implies that the origin is contained inside the feasible region defined by (14). Therefore, the feasible region described by (14) is the half plane associated with the straight line across (x1 , y1 ) and (x2 , y2 ), and this half plane contains the origin. Figure 4 shows the function gk (xk , yk ) = xk + yk − 1, whose boundary is the straight line across (−1, 2) and (2, −1), and the feasible region contains the origin. If the feasible region is the other half of the plane which does not contain the origin, we can use gk′ (xk , yk ) = −gk (xk , yk ) from (14) to describe it. Based on (14), if the feasible region is the area within a convex polygon, we can use a set of linear functions to define

it as   gk,1 (xk , yk ) =         .. .     gk,V (xk , yk ) =     

y1,1 −y1,2 y1,1 x1,2 −y1,2 x1,1 xk x −x1,2 + x1,1 y1,1 yk − 1,2 −x1,2 y1,1

1 (15)

yV,1 −yV,2 yV,1 xV,2 −yV,2 xV,1 xk x −xV,2 yk − + xV,1 yV,1 V,2 −xV,2 yV,1

1

where gk,v (xk , yk ) corresponds to the vth edge of the convex polygon2 , and V is the number of edges necessary to describe the polygonal obstacle. When the coordinates of a point (xk , yk ) make every function in (15) less than zero, the point is within the feasible region. Note that the equation set in (15) has an alternative form as gk (xk , yk ) = max {gk,1 (xk , yk ), . . . , gk,V (xk , yk )} .

(16)

If the feasible region is the area outside the convex polygon, we can use the function gk′ (xk , yk ) = − max {gk,1 (xk , yk ), . . . , gk,V (xk , yk )} (17) to describe the complementary region defined by (15) or (16). We can also use min {·} to construct functions describing the concave polygonal feasible region. For example, if a concave polygonal region is the sum set of two convex regions, which are respectively defined by gka (xk , yk ) and gkb (xk , yk ), the concave polygonal feasible region is defined by gk (xk , yk ) = min gka (xk , yk ), gkb (xk , yk ) . (18) Similarly, we can also use min {·} to define a sum set of several feasible regions. 2 A polygon is convex if each of its internal angle is less than or equal to 180 degrees.

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TABLE I C OMPARISON OF THE C OST T IME OF O PTIMIZATION METHODS

Multi-variate Optimization Grouping Optimization

K=3 1.812 14.448

K=5 4.752 17.244

K=7 27.177 20.457

m=0

K=9 52.842 25.838

We make L=2 and M = 10 for the grouping optimization, and the cost time is in seconds.

k =0

IV. GROUPING OPTIMIZATION

Fixing node k and node k + L, optimize nodes k + 1 to k + L - 1

When the number of relays to be placed is large, searching for the optimal relay locations can be time-consuming since the solution space of the problem increases exponentially with the number of variables. In this section, we propose a grouping optimization technique that provides suboptimal locations but its cost time increases linearly with the number of variables. The basic idea is to optimize the relays in groups in order to bypass the optimization process involving too many variables. Suppose there are K relays to be located and we denote the source node as node 0 and the destination as node K + 1. We can optimize the first group including relays 1 to L − 1 by fixing node 0 and node L, where L − 1 is the number of nodes to be optimized in each group and 2 ≤ L ≤ K. After it is done, we can go on to optimize the second group, including relays 2 to L, by fixing node 1 and node L + 1. Note that a node may be optimized several times because it may be a member of different groups. The ith group contains nodes i − 1 to L + i − 1, and the locations of nodes i to L + i − 2 are optimized in the ith group. This process continues until the last group is optimized, i.e. nodes K −L+2 to K are optimized. To achieve better results, we can also redo the iteration M times, i.e. start over back from the first group and do the grouping optimization M − 1 times more. A flowchart is shown in Fig. 5 to describe the entire grouping optimization process. If the average cost time for optimization in each group is Tg (L), which grows as L grows, then the cost time for the entire optimization process is approximately (K −L+2)M Tg (L), which is a linear function of the number of relay nodes K. A comparison between the cost time of multi-variate optimization and grouping optimization is shown in Table I. Note that the searching time is also dependent on the initial points of the relay nodes and the efficiency of the codes, therefore the searching time of grouping optimization in Table I may not be exactly linear. We can observe that the searching time for multi-variate optimization increases much faster than that of grouping optimization when the number of relays increases. In general, a large L or large M can result in an improved solution; however, in practice, when L is below 3 and M = 10 we can obtain a suboptimal solution close to the optimal solution for most cases. The idea behind the grouping optimization is that every individual group optimization leads to improved performance of the end-to-end multi-hop system. Therefore we can expect the suboptimal locations to be close to the optimal locations after several iterations. A numerical comparison between the optimal relay locations and the suboptimal relay locations is shown in Section V.

k = k +1

N

k + L -1 = K ?

Y

m = m +1 m=M?

Y

N

End Fig. 5. Grouping optimization flowchart, where k is the number of groups that have been optimized in this round and m is the number of rounds that have been finished.

V. NUMERICAL RESULTS The hardware platform we use is an Intel Core i7-2670QM laptop with 4 gigabytes memory. The software platform we use is the embedded fmincon function in MATLAB R2009b, and the iteration algorithm is active-set, which is a type of optimization techniques that remove the inactive constraints in order to reduce the complexity of the search. In Fig. 6, we vary the transmit power and study how it influences the optimal relay location in terms of the exact outage probability in (3). We set the system parameters as Cn2 = 10−14 m−2/3 , σ = 0.1 and λ = 1550 nm. The source has coordinates (−5, 0) and the destination has coordinates (5, 0). The obstacle is round with radius 1.8 km and its center is (3.2, 0.5). The initial point is (0, 0). When the distance of the longest hop in (7) is the objective function, the optimal relay location is (2.340, −4.419). When PM = 25 dB, the optimal relay location is (2.375, −4.360), which is close to the optimal location associated with the asymptotic outage probability. As long as PM > 20 dB, the coordinates of the optimal relay nodes are close within a circle of 0.162 km centered at (2.340, −4.419). Noting that 20 dB power margin is relatively low in this scenario because it results in a 8.7 × 10−3 outage probability. We can therefore conclude that the optimal relay locations with respect to the exact outage probability and the asymptotic outage probability are close to each other when the transmission power is in practical range. For such a scenario, it can also be proved that the relayto-destination link must be touching to the obstacle, and the

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6

For the optimal relay locations achieved by multi-variate optimization, the longest hop is 0.7266 km. When L = 2, which implies that only one relay is optimized in each group, we can observe that the associated suboptimal locations have noticeable discrepancies with the optimal locations and the longest hop for the suboptimal relays is 0.7520 km. When L = 3, which implies that two relays are jointly optimized in each group, we can observe the suboptimal locations are much closer to the optimal locations, and the longest hop is 0.7308 km. Therefore, we can conclude that the parameter L has significant impact on the performance of grouping optimization. By properly choosing the parameters of group optimization, the suboptimal locations can closely agree with the optimal locations. In general, larger L leads to better performance, but it also increases the optimization time. For example, when L = 2, the searching time is 19.9314 seconds and when L = 3, the searching time is 100.1921 seconds.

N Source or destination Optimal relay locations Laser P =∞ dB

4

W

M

E

Laser PM=25 dB Laser PM=20 dB

S

Laser PM=15 dB

2

Laser PM=10 dB Obstacle

km

Source 0

Destination

−2

PM gets larger −4

−6 −6

−4

−2

0 km

2

4

6

VI. C ONCLUSIONS

Fig. 6. Optimal relay locations in terms of exact DF outage probability for a single-relay single-round-obstacle scenario.

6

5

Obstacles Laser Source or destination Grouping optimization L=2 Grouping optimization L=3 Multi−variate optimization

Destination

4 Obstacle

We developed an optimization framework to solve the problem of relay placement in multi-hop FSO system, where the relays can only be located in limited regions and there are obstacles blocking the direct link. We proposed a grouping optimization technique to reduce the cost time for optimization when the number of relays to be located is large. Its searching time grows linearly with the number of relay nodes. Simulation results show that the optimization framework is practical and that the grouping optimization can reduce the searching time significantly especially when the number of relay nodes to be placed is large.

3 km

R EFERENCES

2 Obstacle Optimized relays N

1

W

E

0 Source S −1 −1

0

1

2

3

4

5

6

km

Fig. 7. Comparison between the relay locations achieved by grouping optimization and multi-variate optimization.

detailed discussion are omitted here due to space limitation. In Fig. 7, a comparison is made between the optimal relay locations obtained by the typical multi-variate optimization, and the suboptimal relay locations obtained by the proposed grouping optimization in Section IV. The objective function used here is the distance of the longest hop in (7). The initial points for the relay nodes are (1.00, 2.50), (1.30, 2.60), (1.66, 2.72), (2.02, 2.84), (2.38, 2.96), (2.74, 3.08), (3.10, 3.20), (3.46, 3.32) and (4.00, 3.50). We make group optimization parameters M = 10, L = 2 or 3, and nine relay nodes are to be placed.

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