Optimal Relay Locations for Minimizing the Outage

and

dADF= lim

g!•

log(P_out^{ADF, Cor}) log(g)

= lim

g!•

⇢ log

 ln(4) 1

g³(1 r_t²)+4ln(2) +2ln(4) 6 2g⁴(1 r_t²) +

✓

1 1

g

◆ ln(4) 1

g²(1 r_s²)+ 2ln(2) 3

2g³(1 r_s²)+ ln(4) ³₂ g³(1 r_s²)

!#

/log(g) )

=2,

r_s6=1,r_t6=1, (70)

whereg_SD,g_RD, andg_SRare represented by a generic symbolgunder the equilateral triangle nodes location assumption. It is shown that the full diversity gain can be achieved by LF, DF, and ADF relaying as long asr_s6=1 for LF and DF, andr_s6=1 and rt6=1 for ADF. Whenrs=1 andrt=1, i.e., in fully correlated fading, the diversity order reduces to one.

2.2.5 Optimal Relay Locations for Minimizing the Outage Probability

independent fading with respect to the position of R as P_out^{LF, Ind}= 1

g_SD g_SD+10lg(₁¹_d)^a 1 1 g_SD+10lg(¹_d)^a

!

·

✓

ln(4) 1+4ln(2) 3 2g_SD

◆

+ 1

g_SD g_SD+10lg(¹_d)^a

· ln(4) 1+ 4ln(2) 3 2 g_SD+10lg(¹_d)^a

!

, (71)

P_out^{DF, Ind}= 1

g_SD g_SD+10lg(₁¹_d)^a 1 1 g_SD+10lg(_d¹)^a

!

·

✓

ln(4) 1+4ln(2) 3 2g_SD

◆

+ 1

g_SD g_SD+10lg(¹_d)^a , (72) and

P_out^{ADF, Ind}= 1

g_SD g_SD+10lg(₁¹_d)^a 1 1 g_SD+10lg(_d¹)^a

!

·

✓

ln(4) 1+4ln(2) 3 2g_SD

◆

+ 1

g_SD g²_SD+10lg(¹_d)^a

·

✓

ln(4) 1+4ln(2) 3 2g²_SD

◆

, (73)

under the assumption that R moves along the line between S and D, with whichdSR=d anddRD=1 d.

The general optimization problem with regard todcan be formulated as d^⇤=arg min

d Pout(d) subject to: d 1<0,

d<0.

(74)

The Karush-Kuhn-Tucker (KKT) condition for (74) can be written as

∂Pout(d)

∂d +µ1 µ2, µ1>0,

µ2>0, µ1(d 1)<0, µ2( d)<0, d 1<0,

d<0,

(75)

whereµ1andµ2are the constraint coefficients.

Proposition 4. Outage probability expressions of LF, DF, and ADF relaying in(71), (72), and(73), are convex with respect to d2(0, 1).

Proof. See Appendix 1.

Taking the first-order derivative ofP_out^{LF, Ind},P_out^{DF, Ind}, andP_out^{ADF, Ind}, respectively, in (71), (72), and (73) with respect todand setting the derivative results to zero, we have

∂P_out^{LF, Ind}

∂d =

∂_g ¹

SD(^gSD+10lg(₁¹_d)^a)

✓

1 _g ¹

SD+10lg(¹_d)^a

◆⇣ln(4) 1+^4ln(2)_2g ³

SD

⌘

∂d

+

∂_g ¹

SD(^gSD+10lg(¹_d)^a)

✓

ln(4) 1+₂(^gSD^4ln(2)+10lg(³¹_d)^a)

◆

∂d =0, (76)

∂P_out^{DF, Ind}

∂d =

∂_g ¹

SD(^gSD+10lg(₁¹_d)^a)

✓

1 _g ¹

SD+10lg(_d¹)^a

◆⇣ln(4) 1+^4ln(2)_2g ³

SD

⌘

∂d

+

∂_g ¹

SD(^gSD+10lg(¹_d)^a)

∂d =0, (77)

and

∂P_out^{ADF, Ind}

∂d =

∂_g ¹

SD(^gSD+10lg(₁¹_d)^a)

✓

1 _g ¹

SD+10lg(¹_d)^a

◆⇣ln(4) 1+^4ln(2)_2g ³

SD

⌘

∂d

+

∂_g ¹

SD(^g2SD+10lg(¹_d)^a)

⇣ln(4) 1+^4ln(2)_2g2 ³

SD

⌘

∂d =0. (78)

It may be excessively complex to derive the explicit expression ford^⇤from (76), (77), and (78). Hence, the Newton-Raphson method [80, Chapter 9] is used to numerically calculate the solution ofd^⇤.

Optimal Relay Locations in Correlated Fading

Similar to the independent fading case, the outage probability expressions of LF, DF and ADF relaying in correlated fading with respect to the position of R can be expressed as (79), (80), and (81).

P_out^{LF, Cor}= 1 1 g_SD+10lg(¹_d)^a

! ln(4) 1

g_SD g_SD+10lg(₁¹_d)^a (1 rs²)

+ 2ln(2) 3

2g²_SD g_SD+10lg(₁¹_d)^a (1 r_s²)+ ln(4) ³₂

g_SD g_SD+10lg(₁¹_d)^{a 2}(1 rs²) 1 A

+ 2ln(4) 3

2g_SD g_SD+10lg(¹_d)^{a 2}

+ 2ln(4) 3

2g²_SD g_SD+10lg(¹_d)^a + ln(4) 1 2g_SD g_SD+10lg(_d¹)^a

+ 4ln(16) 11

4g²_SD g_SD+10lg(¹_d)^{a 2}, (79)

P_out^{DF, Cor}= 1 1 g_SD+10lg(¹_d)^a

! ln(4) 1

g_SD g_SD+10lg(₁¹_d)^a (1 rs²)

+ 2ln(2) 3

2g²_SD g_SD+10lg(₁¹_d)^a (1 rs²)+ ln(4) ³₂

g_SD g_SD+10lg(₁¹_d)^{a 2}(1 rs²) 1 A

+ 1

g_SD g_SD+10lg(_d¹)^a , (80)

P_out^{ADF, Cor}= 1 1 g_SD+10lg(¹_d)^a

! ln(4) 1

g_SD g_SD+10lg(₁¹_d)^a (1 rs²)

+ 2ln(2) 3

2g²_SD g_SD+10lg(₁¹_d)^a (1 r_s²)+ ln(4) ³₂

g_SD g_SD+10lg(₁¹_d)^{a 2}(1 rs²) 1 A

+ 4ln(2) 3

2g²_SD g_SD+10lg(₁¹_d)^a g_SD+10lg(_d¹)^a (1 rt²)

+ ln(4) ³₂

g_SD g_SD+10lg(₁¹_d)^{a 2} g_SD+10lg(¹_d)^a (1 rt²)

. (81)

The optimization problem for the correlated fading channels’ case can also be formulated by (74) for LF, DF, and ADF relaying.

Proposition 5. Outage probability expressions of LF, DF, and ADF relaying, respectively, given by(79),(80), and(81), are convex with respect to d2(0, 1).

! !

! !

!

!

! !

! !

! !

! !

! !

! !

! !

! !

!

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

10⁻² 10⁻¹ 10⁰

x−coordinate of relay

Outage Probablity

P^LF_out P^DF_out P^ADF_out P^LF_out, Approx P^DF_out, Approx P^ADF_out, Approx

ρs=0.99

ρs=0

Fig. 18. Outage probabilities versus relay locations for LF, DF, and ADF relaying over spa-tially independent and correlated channels, wherert=0.

Proof. The convexity of (79), (80), and (81) can be proven by taking second-order partial derivative ofP_out^{LF, Cor},P_out^{DF, Cor}, andP_out^{ADF, Cor}in (79), (80), and (81) with respect tod, and showing that the derivative results are positive in the ranged2(0,1)⁹.

Take the first-order derivative ofP_out^{LF, Cor},P_out^{DF, Cor}, andP_out^{ADF, Cor}, respectively, in (79), (80), and (81) with respect todand set the derivative result to zero, ^{∂PoutLF, Cor}_∂d =0,

∂P_out^{DF, Cor}

∂d =0, and^{∂PoutADF, Cor}_∂d =0. As in the independent fading case, the optimal relay locationd^⇤for LF, DF, and ADF relaying over correlated fading channels can also be obtained by utilizing an iterative root-finding algorithm [80].

Fig. 18 depicts the impact of the relay location on outage performances of LF, DF and ADF relaying in spatially independent(rs=0)and correlated (rs=0.99) cases.

The relay is assumed to move along the line between S(x=0)and D(x=1). Both the numerically calculated outage probabilities and approximated outage probabilities are plotted. It is found from the figure that the approximated and numerically calculated outage probability curves are consistent with each other. This observation indicates the

9The details of the proof are straightforward but lengthy, and therefore are omitted here for brevity.

! !

! !

! !

! !

!

!

! !

! !

!

! !

!

ρt ρs

ADF

LF

DF

Fig. 19. Impact of temporal correlationrtand spatial correlationrson optimal relay location d^⇤, whereg_SD=2(dB).

accuracy of the approximations. It is found from Fig. 18 that in spatially and temporally independent fading (rs=0,rt=0) the optimal relay locations for LF, DF, and ADF that achieve the lowest probabilities ared^⇤=0.5,d^⇤<0.5 and,d^⇤>0.5, respectively.

In spatially correlated fading (rs=0.99), the optimal relay locations for LF, DF, and ADF relaying shift closer to the destination compared to the case in independent fading.

This is because the fading correlation reduces the diversity gain provided by the R-D link transmission, which results in the optimal position of the relay shifting close to the destination to keep the average SNR of the R-D link high.

The impact of the temporal correlationrt and the spatial correlationrs on the optimal relay locationd^⇤of LF, DF, ADF relaying is depicted in Fig. 19. For DF, LF, and ADF relaying, the higher ther_s, the larger thed^⇤, which indicates that the optimal relay locations shift close to the destination as the spatial correlation increases. This is because the average SNR of the R-D link need to be kept high to compensate the loss caused by the fading correlation. Obviously,rthas no impact ond^⇤in LF and DF relaying. With ADF, it is found that the value ofd^⇤becomes small asrtincreases. The

time diversity gain achieved by using the S-D link twice is reduced by the temporal correlation, which leads to that the relay should be located close to the source in order to reduce the S-R transmission error. This decreases the probability of retransmission from the source.

2.2.6 Optimal Power Allocation for Minimizing the Outage Probability In this section, our aim is to minimize the outage probabilities for LF, DF, and ADF relaying by adjusting power allocated to S and R. We consider a transmit power sharing strategy. The source (or the relay) can use the transmit power that is allocated to the system when the power is not used by the relay (or the source). The source and the relay can control its transmit power properly under total transmit power constraint. In this work, we do not consider the practical implementation of the transmit power sharing strategy and only focus on the theoretical analysis. We assume that the CSI is only available at the receiver side. Let the power allocated to S and R be denoted asPTkand PT(1 k), respectively, wherePT represents the total transmit power andk(0<k<1) is the power allocation ratio. With the noise variance of each link being normalized to the unity, the geometric gain times transmit power is equivalent to their corresponding average SNR. Then the average SNRs of each link can be expressed as functions ofkas

g_SD=PTkGSD, (82)

g_RD=PT(1 k)GRD, (83)

g_SR=PTkGSR. (84)

Optimal Power Allocation in Independent Fading

By substituting (82), (83), and (84) into (45), (46), and (47) in section 2.2.3, the outage expressions of LF, DF, and ADF relaying in independent fading with respect tokcan be written as

P_out^{LF, Ind}= 1

P_T²k(1 k)GSDGRD

✓

1 1

PTkGSR

◆✓

ln(4) 1+4ln(2) 3 2PTkGSD

◆

+ 1

P_T²k²GSDGSR

✓

ln(4) 1+4ln(2) 3 2PTkGSR

◆

, (85)

P_out^{DF, Ind}= 1

P_T²k(1 k)GSDGRD

✓

1 1

PTkGSR

◆✓

ln(4) 1+4ln(2) 3 2PTkGSD

◆

+ 1

P_T²k²GSDGSR, (86)

and

P_out^{ADF, Ind}= 1

P_T²k(1 k)GSDGRD

✓

1 1

PTkGSR

◆✓

ln(4) 1+4ln(2) 3 2PTkGSD

◆

+ 1

P_T³k³GSRGSD

✓

ln(4) 1+4ln(2) 3 2PTkGSD

◆

, (87)

respectively.

The optimization problem with regards tokcan be formulated as k^⇤=arg min

k Pout(k)

subject to: k 1<0, k<0, . (88)

The KKT condition for (88) can be written as

∂Pout(k)

∂k +µ₃ µ₄, µ3>0,

µ4>0, µ3(k 1)<0, µ4( k)<0, k 1<0,

k<0,

(89)

whereµ3andµ4are the constraint coefficients.

Proposition 6. Outage probability expressions of LF, DF, and ADF relaying, respectively, in(85),(86), and(87), are convex with respect to k2(0, 1).

Proof. See Appendix 2.

Taking the first-order derivative ofP_out^{LF, Ind},P_out^{DF, Ind}, andP_out^{ADF, Ind}, respectively in (85), (86), and (87) with respect tokand setting the derivative results to zero, we have,

∂P_out^{LF, Ind}

∂d =

∂_P2 ¹

Tk(1 k)G_SDGRD

⇣1 _PTkG¹_SR⌘⇣

ln(4) 1+^4ln(2)_2PTkGSD³⌘

∂k

+

∂_P2 ¹

Tk²G_SDG_SR

⇣ln(4) 1+^4ln(2)_2PTkGSR³⌘

∂k =0, (90)

P_out^{LF, Cor}=

✓

1 1

PTkG_SD

◆ ln(4) 1

PTkG_SDPT(1 k)G_RD(1 rs²)+ 2ln(2) 3

2PTkG²_SDPT(1 k)G_RD(1 rs²)

+ ln(4) ³₂

PTkG_SDPT(1 k)G²_RD(1 rs²)

!

+ 2ln(4) 3

2PTkG_SDP_T²k²G²_SR+ 2ln(4) 3 2P_T²k²G²_SDPTkG_SD

+ ln(4) 1

2PTkG_SDPTkG_SD+ 4ln(16) 11

4P_T²k²G²_SDP_T²k²G²_SR, (93)

P_out^{DF, Cor}=

✓

1 1

PTkG_SD

◆ ln(4) 1

PTkG_SDPT(1 k)G_RD(1 rs²)+ 2ln(2) 3

2PTkG²_SDPT(1 k)GRD(1 r_s²)

+ ln(4) ³₂

PTkG_SDPT(1 k)G²_RD(1 rs²)

!

+ 1

PTkG_SDPTkG_SD, (94)

∂P_out^{DF, Ind}

∂d =

∂_P2 ¹

Tk(1 k)G_SDGRD

⇣1 _PTkG¹_SR⌘⇣

ln(4) 1+^4ln(2)_2PTkGSD³⌘

∂k +

∂_P2 ¹

Tk²G_SDG_SR

∂k =0, (91)

and

∂P_out^{ADF, Ind}

∂d =

∂_P2 ¹

Tk(1 k)G_SDG_RD

⇣1 _PTkG¹_SR⌘⇣

ln(4) 1+^4ln(2)_2PTkG ³

SD

⌘

∂k

+

∂_P3 ¹

Tk³G_SRG_SD

⇣ln(4) 1+^4ln(2)_2PTkGSD³⌘

∂k =0. (92)

The optimal power allocation ratiok^⇤can be obtained by numerically solving (90), (91), and (92).

Optimal Power Allocation in Correlated Fading

The outage expressions of LF, DF, and ADF in correlated fading can be written as (93), (94), and (95), respectively. The optimization problem in correlated fading can also be formulated by (88) for LF, DF, and ADF relaying.

Proposition 7. Outage probability expressions of LF, DF, and ADF relaying, respectively, in(93),(94), and(95), are convex with respect to k2(0, 1).

P_out^{ADF, Cor}=

✓

1 1

PTkG_SD

◆ ln(4) 1

PTkG_SDPT(1 k)G_RD(1 rs²)+ 2ln(2) 3

2PTkG²_SDPT(1 k)G_RD(1 rs²)

+ ln(4) ³₂

PTkG_SDPT(1 k)G²_RD(1 rs²)

!

+ ln(4) 1

PTkG_SDPT(1 k)G_RDPTkG_SD(1 rt²)

+ 4ln(2) 3

2P_T²k²G²_SDPT(1 k)G_RDPTkG_SD(1 rt²)+ ln(4) ³₂

PTkG_SDP_T²(1 k)²G²_RDPTkG_SD(1 rt²). (95)

Proof. The convexity of (93), (94), and (95) can be proven by taking second-order partial derivative ofP_out^{LF, Cor},P_out^{DF, Cor}, andP_out^{ADF, Cor}in (93), (94), and (95) with respect tok, and showing that the derivative results are positive in the rangek2(0,1)¹⁰.

By taking the first-order derivative ofP_out^{LF, Cor},P_out^{DF, Cor}, andP_out^{ADF, Cor}, respectively, in (93), (94), and (95) with respect tok and setting the derivative results to zero,

∂P_out^{LF, Cor}

∂k =0, ^{∂PoutDF, Cor}_∂k =0, and^{∂PoutADF, Cor}_∂k =0. Similar to the independent fading case, the optimal power allocation ratiok^⇤of LF, DF, and ADF relaying over correlated fading channels can be numerically obtained.

Fig. 20 presents the impact of the power allocation ratio to S and R on outage probability. We normalize the geometric gain of the S-D link to the unity under the assumption that the relay is located at the midpoint of the S-D link. The total transmit power is set at 2 (dB). It can be observed from Fig. 20 that, the optimal power ratio k^⇤of LF is larger than that of ADF, and smaller than that of DF, which indicates that LF needs more power for the source than ADF, and needs less power than DF. This is because in DF relaying, the source needs more power to keep the quality of the S-D links for reducing the error probability. With ADF relaying, even though error occurs in the S-R link due to the deep fade of the channel, the source still can retransmit the information sequence to the destination, yielding the time diversity gain. It is also found that when more transmit power is allocated to the source (largerk), the outage curves of DF, LF, and ADF relaying with the same spatial correlationr_sasymptotically merge. This indicates that with more transmit power for S, the probability of error occurring in the S-R link reduces, resulting in that R always forwards the received information sequence. Therefore, the outage curves with DF, LF, and ADF become almost the same. Moreover, the larger the spatial correlationrs, the higher the outage

10The details of the proof are straightforward but lengthy, and therefore are omitted here for brevity.

! !

! !

! !

! !

! !

! !

!

! !

! !

! !

! !

! !

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

10⁻² 10⁻¹ 10⁰

Transmit Power Ratio k

Outage Probablity

s=0

s=0.9 DF

ADF LF

Fig. 20. Outage probabilities versus power allocations for LF, DF, and, ADF relying over independent and correlated channels, wherePT=2(dB) andrt=0.

probability with the LF, DF, and ADF relaying schemes, however, more transmit power needs to be allocated to R (smallerk) for achieving the lower outage probabilities. This is because correlations reduce the diversity gain provided by the R-D link and R needs more transmit power to maintain the quality of the R-D link. In practice, to keep a certain outage, one can allocate larger range of the transmit power with LF than DF under a total power constraint.

Fig. 21 shows the impact ofr_tandr_son optimal power allocation ratiok^⇤for LF, DF, ADF relaying, where the total transmit power for S and R is set atPT=2 (dB). It can be seen from Fig. 21 that the larger the spatial correlationrs, the larger the optimal k^⇤value with DF, LF, and ADF. Since the spatial correlation reduce the diversity gain provided by the R-D link, more power needs to be allocated to the source to increase the quality of the S-D transmission. It is also found that, as the temporal correlationr_tin ADF increases, the value of optimalk^⇤decreases, in order to achieve the smallest outage probability. This is because the higher the temporal correlation, the lower the diversity gain provided by the S-D retransmission, which results in that more power should be allocated to the relay to increase the quality of the R-D transmission.

! !

! !

ρt ρs

ADF

LF DF

Fig. 21. Impact of temporal correlationrt and spatial correlationrson optimal power alloca-tion ratiok^⇤, wherePT=2(dB).

2.3 Impact Analysis of Line-of-Sight Components in Lossy-Forward

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