# Throughput Analysis and Optimization of Relay Selection Techniques for Millimeter Wave Communications

Authors: Raed Alhamad; Hatem Boujemaa
Journal: Journal of Computer Networks and Communications
 (2019)

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## Abstract

In this paper, we optimize the throughput of millimeter wave communications using relay selection techniques. We study opportunistic amplify and forward (OAF), opportunistic decode and forward (ODF), and partial and reactive relay selection (PRS and RRS). Our analysis is valid for interference-limited millimeter wave communications. We suggest a new optimal power allocation (OPA) strategy that offers significant performance enhancement with respect to uniform power allocation (UPA). The proposed OPA offers up to 2 dB gain with respect to UPA. Our analysis is confirmed with extensive simulation results for Nakagami fading channels.


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## Body

## 1. Introduction

Millimeter wave (mmWave) communications offer high data rates of several Gb/s [1–3]. Millimeter wave communication operates on an important bandwidth from 3 to 300 GHz [1–6]. Cooperation is mandatory in mmWave communications because the mmWave signals cannot penetrate through walls [1–7]. Dual relaying can be used where the signal goes from a source S to relay Rk and then to the destination D. Multihop multibranch relaying allows to extend the coverage of mmWave communications [7–12]. Amplify and forward relays use an adaptive or fixed gain. When the gain is adaptive, these are called nonblind relays. Otherwise, relays with fixed gain are less complex and are called blind relays. AF can be implemented with relay selection techniques such as OAF, PRS, and RRS.In decode and forward (DF) relaying, each relay decodes the signal and is allowed to transmit only if it has correctly decoded the packet [13–15]. In opportunistic DF (ODF), the chosen relay offers the best SINR of the second hop between relays and destination D [15–18].Optimal power allocation has not been yet proposed for mmWave communication with an analysis at the packet level. In previous studies, power allocation has been optimized to minimize the symbol or bit error probabilities [19]. The innovations of the paper are as follows:(i)
We derive the throughput and packet error probability (PEP) of millimeter wave communications for OAF, ODF, PRS, and RRS. Previous papers [1–10] studied only the symbol and bit error probabilities. To the best of our knowledge, the PEP has not been yet derived in closed form for mmWave communication with OAF, ODF, PRS, and RRS.(ii)
We propose a new optimal power allocation (OPA) strategy that offers up to 2 dB gains with respect to uniform power allocation (UPA). Our optimization is performed at the packet level.(iii)
Our analysis is valid for quadrature amplitude modulation (QAM), amplitude-shift keying (ASK), and phase-shift keying (PSK) modulations with any number of relays having arbitrary positions.The paper contains seven sections. The signal to interference plus noise ratio (SINR) is analyzed in Section2. Section 3 derives the cumulative distribution function (CDF) of SINR of different relay selection techniques. In Section 4, we derive the throughput of mmWave communications. Section 5 suggests an optimal power allocation strategy, while Section 6 gives some simulation and theoretical results. The last section summarizes the obtained results.


## 2. SINR Statistics

The system model illustrated in Figure1 contains a source S, M relays Rk, and a destination D. It is assumed that the received signal k−th relay Rk is affected by PRk interferers as shown in Figure 1. In interference-limited millimeter wave communications, the SINR at k-th relay is equal to [9](1)ΓSRk=EShSRk2N0+IRk,where ES is the transmitted energy per symbol of the source S, hSRk is the channel coefficient between S and Rk, N0 is the power spectral density (PSD) of additive white Gaussian noise (AWGN), and IRk is the interference term at Rk written as follows [9]:(2)IRk=∑p=1PRkEpgp,Rk2,where PRk is the number of interferers at Rk, Ep is the transmitted energy per symbol of p-th interferer, and gp,Rk is the channel coefficient between p-th interferer and relay Rk.Figure 1
Network model.We derive the statistics of an upper bound of SINR:(3)ΓSRk=EShSRk2N0+IRk<ΓSRkup=EShSRk2IRk.For Nakagami fading channels,Uk=EShSRk2 follows a gamma distribution Gα1,k,β1,k defined as(4)fUkx=xα1,ke−x/β1,kΓα1,kβ1,kα1,k,where α1,k>0.5 is the m-fading figure of S−Rk link (α1,k=1 corresponds to Rayleigh channels),(5)β1,k=ESEhSRk2α1,k=Yk¯α1,k.The interference term is expressed as(6)IRk=∑p=1PRkEpgp,Rk2=∑p=1PRkIp,k,where Ip,k=Epgp,Rk2.It is assumed thatIRk is the sum of independent and identically distributed (i.i.d) gamma random variables (r.v) Ip,k that follows a Gak,bk where ak>0.5 and(7)bk=EIp,kak.The sum ofPRk i.i.d gamma r.v. Ip,k is a gamma r.v. GPRkak,bk. Therefore, ΓSRkup in (3) is the quotient of two gamma r.v. that follows a general prime distribution with PDF [20]:(8)fΓSRkupx=xα1,k−1wkα1,kBα1,k,Ak1+xwk−Bk,where(9)Bα1,k,Ak=Γα1,kΓAkΓα1,k+Ak,(10)Ak=PRkak,(11)Bk=Ak+α1,k,(12)wk=β1,kbk.The CDF of SINRΓSRk is obtained by a simple primitive of PDF. We use the following result [21]:(13)F21a,b;c;z=1Bb,c−b∫01xb−11−xc−b−11−zx−adx,where b>0, c>0, and F21a,b;c;z is the hypergeometric function.We use (13) to express the CDF of SINR as(14)FΓSRkupx=xα1,kBα1,k,Akwkα1,kF21Bk,α1,k;α1,k+1,−xwk.Using (8), the asymptotic PDF is expressed as(15)fΓSRkupx≈xα1,k−1Bα1,k,Akwkα1,k.By a primitive, we deduce the asymptotic CDF expressed as(16)FΓSRkupx≈xα1,kα1,kBα1,k,Akwkα1,k.These asymptotic expressions will be used to derive the optimal power allocation (OPA) strategy for cooperative mmWave communications.It is assumed that the interference term atD is the sum of PD i.i.d gamma r.v. with distribution Gc,d. The CDF of SINR of the second hop between Rk and D is written similarly to (8)(17)fΓRkDupx=xα2,k−1ukα2,kBα2,k,E1+xuk−Dk,where α2,k>0.5 is the m-fading figure of Rk−D link,(18)E=PDc,(19)Dk=E+α2,k,(20)uk=β2,kd,(21)β2,k=ERkEhRkD2α2,k=Z¯kα2,k.


## 3. Performance Analysis of Relay Selection Techniques

In this section, we assume that a single relay is selected. There is no interference between the signals transmitted by different relays as a single relay is selected to amplify or decode the source packet. Therefore, our analysis is valid for any number of relays.


### 3.1. Opportunistic AF Relaying

For AF relaying, the SINR between the sourceS, relay Rk, and destination D is expressed as [22](22)ΓSRkD=ΓSRkΓRkDΓSRk+ΓRkD+1.The SINR can be tightly upper bounded by [22](23)ΓSRkD<ΓSRkDup=minΓSRk,ΓRkD.The CDF of SINR is expressed as(24)FΓSRkDx>FΓSRkDupx,where(25)FΓSRkDupx=1−PminΓSRk,ΓRkD>x.Assuming thatΓSRk and ΓRkD are independent, we have(26)FΓSRkDupx=1−1−FΓSRkx1−FΓRkDx.In OAF, the chosen relayRsel offers the best end-to-end SINR(27)ΓSRselD=maxk∈1,…,MΓSRkD<ΓSRselDup=maxk∈1,…,MΓSRkDup.Using (27) and assuming that the SINRs for different relays are independent, the CDF of SINR is expressed as(28)FΓSRselDx>FΓSRselDupx=∏n=1MFΓSRnDupx=∏n=1M1−1−FΓSRnx1−FΓRnDx.


### 3.2. Partial Relay Selection

In PRS, the selected relay offers the best SINR of the first hop. Letpk be the probability that relay Rk is selected and Γk the corresponding SINR is expressed as(29)Γk=ΓmaxΓRkDΓmax+ΓRkD+1,where Γmax is the SINR at relay Rk which is the highest SINR between S and relays,(30)Γmax=maxj∈1,…,MΓSRj.The SINRΓk can be tightly upper bounded by [22](31)Γk<Γkup=minΓmax,ΓRkD.IfΓmax and ΓRkD are independent, we have(32)FΓkx>FΓkupx=1−PΓRkD>xPΓmax>x.We have(33)PΓmax>x=1−PΓmax≤x=1−∏j=1MFΓSRjx.Using (32) and (33), we have(34)FΓkx>FΓkupx=1−1−∏j=1MFΓSRjx1−FΓRkDx.The CDF of SINR is expressed as(35)FΓSRselDx=∑k=1MpkFΓkx.The probability to select relayRk is expressed as follows:(36)pk=PΓSRk>maxj≠kΓSRj.LetX=maxj≠kΓSRj, we can write(37)pk=∫0+∞fXxPΓSRk>xdx=∫0+∞fXx1−FΓSRkxdx,where fXx is the PDF of X.Assuming that the SINRs of the first hop are independent, the CDF ofX is written as(38)FXx=∏q=1,q≠kMFΓSRqx.We have(39)fXx=∑q=1,q≠kMfΓSRqx∏m=1,m≠k,m≠qMFΓSRmx.For PRS, the CDF of SINR atD is written as (35) with pk given in (37).


### 3.3. Reactive Relay Selection

In RRS, the chosen relay offers the best SINR between relays and destination. Letrk be the probability that relay Rk is selected and Γk the corresponding SINR is expressed as(40)Γk=ΓSRkΓmaxΓmax+ΓSRk+1,where Γmax is the highest SINR of the second hop(41)Γmax=maxj∈1,…,MΓRjD.The SINRΓk can be tightly upper bounded by [22](42)Γk<Γkup=minΓmax,ΓSRk.IfΓmax and ΓSRk are independent, we have(43)FΓkx>FΓkupx=1−PΓSRk>xPΓmax>x.We have(44)PΓmax>x=1−∏j=1MFΓRjDx.Using (43) and (44), we have(45)FΓkx>FΓkupx=1−1−∏j=1MFΓRjDx1−FΓSRkx.The CDF of SINR is expressed as(46)FΓSRselDx=∑k=1MrkFΓkx.The probability to select relayRk is expressed as follows:(47)rk=PΓRkD>maxj≠kΓRjD.LetY=maxj≠kΓRjD, we can write(48)rk=∫0+∞fYyPΓRkD>ydy=∫0+∞fYy1−FΓRkDydy,where fYy is the PDF of Y.Assuming that the SINRs of the second hop are independent, the CDF ofY is written as(49)FYy=∏q=1,q≠kMFΓRqDy.We have(50)fYy=∑q=1,q≠kMfΓRqDy∏m=1,m≠k,m≠qMFΓRmDy.For RRS, the CDF of SINR atD is written as (46) with rk given in (48).


### 3.4. Performance Analysis of Opportunistic DF

In ODF, we activate the relay with largest SNR of the second hop. This relay should correctly decodeS packet. The PEP is written as(51)PEP=∑v∈1,…,MPvPEPv,where(52)Pv=∏n∈v1−PEPn∏q∉vPEPq,where PEPq is the PEP at relay Rq.PEP
v is the PEP when v is the set of relays having correctly decoded expressed as [23](53)PEPv<∏k∈vPΓRkDT0,where T0 is a waterfall threshold(54)T0=∫0+∞txdx,where tx is the PEP for SINR x. It is detailed in the next section for different modulations.


## 3.1. Opportunistic AF Relaying

For AF relaying, the SINR between the sourceS, relay Rk, and destination D is expressed as [22](22)ΓSRkD=ΓSRkΓRkDΓSRk+ΓRkD+1.The SINR can be tightly upper bounded by [22](23)ΓSRkD<ΓSRkDup=minΓSRk,ΓRkD.The CDF of SINR is expressed as(24)FΓSRkDx>FΓSRkDupx,where(25)FΓSRkDupx=1−PminΓSRk,ΓRkD>x.Assuming thatΓSRk and ΓRkD are independent, we have(26)FΓSRkDupx=1−1−FΓSRkx1−FΓRkDx.In OAF, the chosen relayRsel offers the best end-to-end SINR(27)ΓSRselD=maxk∈1,…,MΓSRkD<ΓSRselDup=maxk∈1,…,MΓSRkDup.Using (27) and assuming that the SINRs for different relays are independent, the CDF of SINR is expressed as(28)FΓSRselDx>FΓSRselDupx=∏n=1MFΓSRnDupx=∏n=1M1−1−FΓSRnx1−FΓRnDx.


## 3.2. Partial Relay Selection

In PRS, the selected relay offers the best SINR of the first hop. Letpk be the probability that relay Rk is selected and Γk the corresponding SINR is expressed as(29)Γk=ΓmaxΓRkDΓmax+ΓRkD+1,where Γmax is the SINR at relay Rk which is the highest SINR between S and relays,(30)Γmax=maxj∈1,…,MΓSRj.The SINRΓk can be tightly upper bounded by [22](31)Γk<Γkup=minΓmax,ΓRkD.IfΓmax and ΓRkD are independent, we have(32)FΓkx>FΓkupx=1−PΓRkD>xPΓmax>x.We have(33)PΓmax>x=1−PΓmax≤x=1−∏j=1MFΓSRjx.Using (32) and (33), we have(34)FΓkx>FΓkupx=1−1−∏j=1MFΓSRjx1−FΓRkDx.The CDF of SINR is expressed as(35)FΓSRselDx=∑k=1MpkFΓkx.The probability to select relayRk is expressed as follows:(36)pk=PΓSRk>maxj≠kΓSRj.LetX=maxj≠kΓSRj, we can write(37)pk=∫0+∞fXxPΓSRk>xdx=∫0+∞fXx1−FΓSRkxdx,where fXx is the PDF of X.Assuming that the SINRs of the first hop are independent, the CDF ofX is written as(38)FXx=∏q=1,q≠kMFΓSRqx.We have(39)fXx=∑q=1,q≠kMfΓSRqx∏m=1,m≠k,m≠qMFΓSRmx.For PRS, the CDF of SINR atD is written as (35) with pk given in (37).


## 3.3. Reactive Relay Selection

In RRS, the chosen relay offers the best SINR between relays and destination. Letrk be the probability that relay Rk is selected and Γk the corresponding SINR is expressed as(40)Γk=ΓSRkΓmaxΓmax+ΓSRk+1,where Γmax is the highest SINR of the second hop(41)Γmax=maxj∈1,…,MΓRjD.The SINRΓk can be tightly upper bounded by [22](42)Γk<Γkup=minΓmax,ΓSRk.IfΓmax and ΓSRk are independent, we have(43)FΓkx>FΓkupx=1−PΓSRk>xPΓmax>x.We have(44)PΓmax>x=1−∏j=1MFΓRjDx.Using (43) and (44), we have(45)FΓkx>FΓkupx=1−1−∏j=1MFΓRjDx1−FΓSRkx.The CDF of SINR is expressed as(46)FΓSRselDx=∑k=1MrkFΓkx.The probability to select relayRk is expressed as follows:(47)rk=PΓRkD>maxj≠kΓRjD.LetY=maxj≠kΓRjD, we can write(48)rk=∫0+∞fYyPΓRkD>ydy=∫0+∞fYy1−FΓRkDydy,where fYy is the PDF of Y.Assuming that the SINRs of the second hop are independent, the CDF ofY is written as(49)FYy=∏q=1,q≠kMFΓRqDy.We have(50)fYy=∑q=1,q≠kMfΓRqDy∏m=1,m≠k,m≠qMFΓRmDy.For RRS, the CDF of SINR atD is written as (46) with rk given in (48).


## 3.4. Performance Analysis of Opportunistic DF

In ODF, we activate the relay with largest SNR of the second hop. This relay should correctly decodeS packet. The PEP is written as(51)PEP=∑v∈1,…,MPvPEPv,where(52)Pv=∏n∈v1−PEPn∏q∉vPEPq,where PEPq is the PEP at relay Rq.PEP
v is the PEP when v is the set of relays having correctly decoded expressed as [23](53)PEPv<∏k∈vPΓRkDT0,where T0 is a waterfall threshold(54)T0=∫0+∞txdx,where tx is the PEP for SINR x. It is detailed in the next section for different modulations.


## 4. Throughput of Cooperative mmWave Communications

The PEP is expressed as(55)Pbloc=∫0+∞fΓxtxdx,where fΓx is the PDF of SINR Γ and tx is the PEP for SINR x.For K-QAM, we have(56)tx=1−1−21−1Kerfcx3log2K2K−1N,where N is packet length in symbols and K is the constellation size.For K-ASK, we have(57)tx=1−1−1−1Kerfcx3log2KK2−1N.For K-PSK modulation, we have(58)tx=1−1−erfcxlog2Ksin2πKN.The PEP can be upper bounded by [23](59)PEP≤∫0T0fΓxdx=FΓT0,where FΓx is the CDF of SINR and T0 is a waterfall threshold [23]:(60)T0=∫0+∞txdx.In order to compute the PEP, we have to only derive the CDF of SINR and use (51). The throughput is written as(61)Thr=log2K21−PEP,where coefficient 0.5 is due to the fact that half the frame duration is used for transmission by the source and the other half by the selected relay.The throughput of OAF, PRS, and RRS are given in (61) where the PEP is deduced from CDF of SINR as (59). The CDF of SINR of OAF, PRS, and RRS are respectively given in (28), (35), and (46). The throughput of ODF is given in (61) with PEP expressed as (51).


## 5. Optimal Power Allocation

For each candidate relay, we suggest an optimal power allocation that minimizes the asymptotic packet error probability expressed as(62)FΓSRkDupT0≃FΓSRkT0+FΓRkDT0≈T0α1,kα1,kBα1,k,Akwkα1,k+T0α2,kα2,kBα2,k,Eukα2,k.Using the definitions ofuk (20) and wk (12), we have(63)FΓSRkDupx≈T0α1,kbkα1,kα1,kα1,kα1,kBα1,k,AkY¯kα1,k+T0α2,kdα2,kα2,kα2,kα2,kBα2,k,EZ¯kα2,k.LetaS=ES/Es be the fraction of power allocated to the source. ES is the transmitted energy per symbol of S and Es=ES+ERk is the transmitted energy per symbol. ERk is the transmitted energy per symbol of Rk. Similarly, aRk=ERk/Es is the fraction of power allocated to Rk. We deduce(64)Y¯k=aSEsEhSRk2,Z¯k=aRkEsEhRkD2.Therefore, the outage probability (63) to be minimized is expressed as(65)FΓSRkDupx≈λkaSα1,k+μkaRkα2,k,under the constraint that aS+aRk=1 and where(66)λk=T0α1,kbkα1,kα1,kα1,k−1Bα1,k,AkEsα1,kEhSRkα1,k,μk=T0α2,kdα2,kα2,kα2,k−1Bα2,k,EEsα2,kEhRkDα2,k.The Lagrangian of the problem is expressed as(67)J=λkaSα1,k+μkaRkα2,k+ρaS+aRk−1.The derivative ofJ is expressed as(68)∂J∂aS=−λkaSα1,k+1+ρ=0,(69)∂J∂aRk=−μkaRkα2,k+1+ρ=0.When the fading figure ofS−Rk link is the same as that of Rk−D, i.e., α1,k=α2,k=αk, we obtain the OPA strategy(70)aS=λk1/αk+1μk1/αk+1+λk1/αk+1,aRk=μk1/αk+1μk1/αk+1+λk1/αk+1.Ifα1,k≠α2,k, we deduce from (68) and (69) that we have to solve with the Newton algorithm the following equation:(71)aS+aSα1,k+1/α2,k+1μkα2,kλkα1,k1/α2,k+1=1.We deduce the fraction of power allocated to relayaRk=1−aS.


## 6. Theoretical and Simulation Results

We plot theoretical curves with MATLAB and made some simulations for 64-QAM modulation. We have varied the number of relaysM=2,3,4. The path loss exponent is equal to 3. The results are valid for Nakagami channels with α1,k=2 and α2,k=2. The fading figure of interference terms is ak=2.Figures2 and 3 show the PEP and throughput of OAF for different number of relays M. There are two interferers. The distance between S and Rk is dSRk=0.3. The distance between Rk and D is dRkD=1−dSRk=0.7. We notice that the throughput improves as M increases due to cooperative diversity. A good accordance between theoretical and simulation results is observed.Figure 2
PEP of OAF for Nakagami fading channels.Figure 3
Throughput of OAF for Nakagami fading channels.Figure4 shows the throughput of ODF in the same context as Figures 2 and 3. We notice a good accordance between theoretical and simulation results. Also, the throughput improves as the number of relays is increased.Figure 4
Throughput of ODF for Nakagami fading channels.Figure5 compares the throughput of OAF, PRS, and RRS for dRkD=1−dSRk=0.7. There are three relays and two interferers. We observe that OAF offers the highest throughput. RRS offers better performance than PRS because the relays are close to S. Figure 6 shows that PRS offers a higher throughput than RRS when relays are close to D, dRkD=1−dSRk=0.2. OAF offers the highest throughput since it selects the relay with the largest end-end SINR.Figure 5
Throughput of OAF, PRS, and RRS for Nakagami fading channels: relays close to the source.Figure 6
Throughput of OAF, PRS, and RRS for Nakagami fading channels: relays close to the destination.Figures7–10 show the throughput of OAF, PRS, RRS, and ODF for optimal or uniform power allocation (OPA or UPA) with two interferers and three relays. Figures 7–10 show that the proposed OPA allows up to 2 dB gain with respect to UPA.Figure 7
Throughput of OAF with UPA and OPA for Nakagami fading channels.Figure 8
Throughput of PRS with UPA and OPA for Nakagami fading channels.Figure 9
Throughput of RRS with UPA and OPA for Nakagami fading channels.Figure 10
UPA and OPA for ODF: Nakagami fading channels.


## 7. Conclusion

In this paper, we have optimized the throughput of millimeter wave communications using OAF, ODF, PRS, and RRS. Our analysis is valid for interference-limited millimeter wave communications with any number of relays with arbitrary positions. The proposed optimal power allocation (OPA) strategy offers up to 2 dB gain with respect to uniform power allocation (UPA). Our analysis was confirmed with extensive simulation results in the presence of Rayleigh fading channels. The main contribution of our paper is to suggest an optimal power allocation to enhance the throughput at the packet level, while previous studies perform the optimization to minimize bit or symbol error probabilities [19]. Besides, our results showed that OAF offers higher throughput than PRS and RRS as it selects the relay with highest SINR at destination, while PRS and RRS use only the SINR of the first and second hops.



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*Source: 7523181-2019-10-01.xml*