Performance Analysis of Threshold-based Relaying with Partial Relay Selection over Rayleigh Fading Channels

Performance Analysis of Threshold-based Relayingwith Partial Relay Selection over Rayleigh Fading

ChannelsVo Nguyen Quoc Bao∗, Le Quoc Cuong∗, Hyung Yun Kong †

∗School of Telecommunications,Posts and Telecommunications Institute of Technology,

11 Nguyen Dinh Chieu Str., District 1, Ho Chi Minh City, VietnamEmail: {baovnq,lequocuong}@ptithcm.edu.vn

†School of Electrical EngineeringUniversity of Ulsan, San 29 of MuGeo Dong, Nam-Gu, Ulsan, Korea 680-749

Email: [email protected]

Abstract—In this paper, we investigate threshold-based dual-hop relaying schemes in conjunction with partial relay selectionin terms of outage probability and bit error probability. The bestrelay chosen by partial relay selection will transmit jointly withthe source using either amplify-and-forward (AF) or decode-and-forward (DF) if the received signal-to-noise ratio (SNR) exceeds agiven threshold; otherwise, it will keep silent. Simulation resultsare in excellent agreement with numerical results and they showthat the proposed protocol provides a good compromise betweendirect transmission and conventional dual hop relaying withpartial relay selection schemes.

Index Terms—Threshold-based relaying, partial relay selec-tion, decode-and-forward, amplify-and-forward, Rayleigh fading.

I. INTRODUCTION

It is well known that achieving spatial diversity through theuse of relaying transmission is a promising technology thatprovides the high data-rate coverage required in future cellularand ad-hoc wireless communication [1], [2], [3]. The mainidea is that one or more intermediate nodes are used to supportsignals transmission by taking advantage of the broadcastingnature of the wireless networks, especially when transmittingor receiving from multiple antennas is infeasible, when thedirect communication between the source and the destinationis in a deep fade.

In most recent publications on the cooperative diversitynetworks [4], [5], a distributed relay selection is proposed for atwo-hop AF (or DF) system that can obtain full diversity order,where the selected criterion is the best instantaneous SNRcomposed of the SNR across the two-hops. The only disadvan-tage of this system is the need for perfect time synchronizationand centralized processing approach [6]. In addition, in someresource-constrained wireless systems (especially, ad-hoc orwireless sensor networks), time synchronization and monitor-ing the connectivity among nodes requires feedback channelswhich mean frequent update, an extra computation burden anda high power consumption. To over such problems, Krikidis

et al. [7] proposed a partial relay selection for amplify-and-forward protocol. However, the authors also showed that thecomplexity reduction of partial relay selection comes at theprice of diversity loss.

In cooperative relaying, the threshold-based relaying is alsoan important protocol in which the relay will help the sourceif the source-relay signal-to-noise ratio (SNR) is greater thana pre-determined threshold [8], [9], [10], [11], [12], [13].In particular, in [8], Herhold et al. proposed a thresholddigital relaying for one relaying node where the decodingthreshold is adjusted jointly with power fraction used by therelay and resource. In [9], the SNR-based selective digitalrelaying protocol was investigated where relaying decisionsare made based on the state of the wireless links amongthe source, relay and the destination. In addition, the optimalthreshold for the proposed protocol was derived to minimizethe average bit error rate. In [10], the asymptotic end-to-end bit error rate of a threshold digital relaying with threenode networks over independent Rayleigh fading was studied,and it is shown that the optimal threshold that minimizesthe end-to-end bit error rate increases as log(SNR). In [11],the performance of un-coded, threshold-based opportunisticrelaying and selection cooperation in terms of outage prob-ability and bit error probability was provided. In [12], [13],a threshold-based differential decode-and-forward scheme fortwo-user cooperative communications system was proposedand analyzed in terms of bit error rate performance. Moreover,the authors showed that by allowing the relay to forwardonly the correctly decoded symbols and introducing a decisionthreshold at the destination, the proposed scheme efficientlycombines the signals from the direct and the relay links.

In this paper, motivated by all of the above, we proposeand analyse a relaying protocol, using either Amplify-and-Forward or Decode-and-Forward, which simultaneously ex-ploits two potentials offered by partial relay selection andthreshold-based relaying. The proposed protocol is useful forpractical ad-hoc systems where the relay selection is based

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on neighbourhood channel knowledge (1 hop) and provides acompromise between direct transmission and static relayingsystem with partial relay selection. We derive a compactexpression for the probability density function (pdf) of theend-to-end SNR which is then used to derive the outageprobability and bit error probability for 𝑀 -QAM over bothindependent identically distributed (i.i.d.) and independent butnot identically distributed (i.n.d.) Rayleigh fading channels.Simulations are also performed to verify the analytical results.The numerical results show that the end-to-end performanceof the system at high SNR regime depends neither on numberof relays nor on the pre-determined threshold.

II. SYSTEM MODEL

We consider a wireless relay networks consisting of onesource (𝑆), 𝑁 relays 𝑅𝑖 with 𝑖 = 1, 2, . . . , 𝑁 and onedestination (𝐷). Each node is equipped with single antennaand operates in half-duplex mode. We assume that the relaysare grouped as clusters based on their geographical proximity.

The communication occurs in two time slots. In the firsttime slot, the source broadcasts the information to 𝑁 relays aswell as the destination. In the second time slot, partial selectiondiversity is applied, i.e., only the best relay whose link in thefirst hop has highest SNR and is greater than the predeterminedthreshold 𝛾𝑡ℎ is selected to forward the source information tothe destination. Recall that 𝛾𝑡ℎ is associated with a target raterequired by the quality of services the networks provide. Thedestination will combine two received signals from the sourceand the best relay by using combining technique. Otherwise,none of relays transmits in the second time slot. In practice,the idle time slots can be skipped by using a limited feedbacksignal transmitted from relays and the source continue with thenext data block until a change occurs in the channel states.

It is assumed that every channel between the nodes experi-ences slow, flat, Rayleigh fading. Due to Rayleigh fading, thechannel powers in two time slots, denoted by 𝛼0 = ∣ℎ𝑆𝐷∣2,𝛼1,𝑖 = ∣ℎ𝑆𝑅𝑖

∣2 and 𝛼2,𝑖 = ∣ℎ𝑅𝑖𝐷∣2 are independent andexponential random variables. Their means are 𝜆0, 𝜆1,𝑖 and𝜆2,𝑖, respectively. The average transmit powers for the sourceand the relays in two hops are denoted by 𝜌1 and 𝜌2,respectively. Let us define the effective instantaneous SNRsfor 𝑆 → 𝐷, 𝑆 → 𝑅𝑖 and 𝑅𝑖 → 𝐷 links as 𝛾0 = 𝜌1𝛼0,𝛾1,𝑖 = 𝜌1𝛼1,𝑖 and 𝛾2,𝑖 = 𝜌2𝛼2,𝑖, respectively.

In this paper, we deal with a network in which relays aregrouped into a cluster due to their proximity where the chosencriterion is based on average SNR. Furthermore, the clusterhas been selected by a long-term routing process to performthe communication between the source and the destination.Hence this system model ensures that all channels from 𝑆 →𝑅𝑖 and 𝑅𝑖 → 𝐷 have the same average channel power, i.e.,𝛾1,𝑖 = 𝛾1 and 𝛾2,𝑖 = 𝛾2 for all 𝑖. We further assume that thereceivers at the destination and relays have perfect channelstate information but no transmitter channel state informationis available at the source and relays. In addition, the details ofrouting protocols are beyond the scope of this paper.

III. PERFORMANCE ANALYSIS

A. Statistics of the end-to-end SNR

To evaluate the performance of threshold-based relayingwith partial relay selection schemes, we need to first char-acterize the statistics of the end-to-end SNR. Let us define 𝛽1

as the instantaneous SNR of the link from the source to thebest relay, which is defined as follows:

𝛽1 = max𝑖=1...𝑁

𝛾1,𝑖 (1)

If the branches from the source are independently faded,then the joint probability density function of 𝛽1 is given by[19, p. 246]

𝑓𝛽1(𝛾) =

𝑁∑𝑖=1

(−1)𝑖−1

(𝑁

𝑖

)𝑖

𝛾1𝑒−

𝑖𝛾𝛾1 (2)

Making use the fact that both dual hop AF and DF relayingtend to be dominated by the weakest hop, which accuratelydescribe the behavior of those relaying schemes at mediumand high SNR. More specifically, since un-coded modulationis used, the overall output SNR of the relaying link (usingeither AF or DF), when the SNR of the link from the sourceto the best relay is above 𝛾𝑡ℎ, can be tightly approximated inthe medium and high SNR regimes as follows [14], [15], [16],[17], [18]:

𝛽 = min{𝛽1∣𝛽1 > 𝛾𝑡ℎ, 𝛽2} (3)

where 𝛽2 denotes the instantaneous SNR of the link from thebest relay to the destination. Recalling the independence of 𝛽1

and 𝛽2, the joint pdf of 𝛽 is given by

𝑓𝛽(𝛾) =

⎧⎨⎩

1𝛾2

𝑒−𝛾𝛾2 𝛽 < 𝛾𝑡ℎ

𝑁∑𝑖=1

(−1)𝑖−1(𝑁𝑖 )𝜔𝑖𝑒−𝛾𝜔𝑖

𝑁∑𝑖=1

(−1)𝑖−1(𝑁𝑖 )𝑒− 𝑖𝛾𝑡ℎ

𝛾1

𝛽 ≥ 𝛾𝑡ℎ(4)

where 𝜔𝑖 = 𝑖/𝛾1 + 1/𝛾2.Depending on the signal processing technique used at the

relays, i.e., amplify-and-forward or decode-and-forward, thebest relay will actively scale or decode and then re-encodethe received signal from the source before forwarding to thedestination. It is interesting to note that employing maximal-ratio-combining (MRC) for AF systems and cooperative MRCfor fixed DF systems at the destination will provide the sameinstantaneous combined SNR, which can be written by

𝛾Σ = 𝛾0 + 𝛽 (5)

Under the assumption of independence of 𝛾0 and 𝛽, andwithout getting into detail of the derivation, the joint proba-

173

bility density function of 𝛾Σ is given by

𝑓𝛾Σ(𝛾) =

⎧⎨⎩

𝑓1 , 𝛾 < 𝛾𝑡ℎ

𝑓2 +

𝑁∑𝑖=1

(−1)𝑖−1(𝑁𝑖 )𝑓3

𝑁∑𝑖=1

(−1)𝑖−1(𝑁𝑖 )𝑒− 𝑖𝛾𝑡ℎ

𝛾1

, 𝛾 ≥ 𝛾𝑡ℎ(6)

where 𝑓1, 𝑓2 and 𝑓3 are defined, respectively, as follows:

𝑓1 =

⎧⎨⎩

(𝛾0

𝛾0−𝛾2

)1𝛾0

𝑒−𝛾𝛾0 +

(𝛾2

𝛾2−𝛾0

)1𝛾2

𝑒−𝛾𝛾2 𝛾0 ∕= 𝛾2

𝛾𝛾20𝑒−

𝛾𝛾0 𝛾0 = 𝛾2

𝑓2 =

⎧⎨⎩

𝛾0

𝛾0−𝛾2

[1− 𝑒

−𝛾𝑡ℎ

(1𝛾2

− 1𝛾0

)]1𝛾0

𝑒−𝛾𝛾0 𝛾2 ∕= 𝛾0

𝛾𝑡ℎ

𝛾20𝑒−

𝛾𝛾0 𝛾2 = 𝛾0

𝑓3 =

⎧⎨⎩

𝜔𝑖𝑒−𝛾𝜔𝑖

1−𝜔𝑖𝛾0−[𝜔𝑖𝛾0𝑒

−𝛾𝑡ℎ(𝜔𝑖−1/𝛾0)

1−𝜔𝑖𝛾0

]1𝛾0

𝑒−𝛾𝛾0 𝜔𝑖

−1 ∕= 𝛾0𝛾𝛾20𝑒−

𝛾𝛾0 − 𝛾𝑡ℎ

𝛾20𝑒−

𝛾𝛾0 𝜔𝑖

−1 = 𝛾0

Equation (6) indicates that the joint pdf of 𝛾Σ is expressedunder a mathematic tractable form, which offers a convenientway to derive some performance metrics of the system includ-ing outage probability and average bit error probability of thesystem.

B. Outage Probability

The outage probability of the system can be derived asfollows:

𝑃𝑜 = Pr(𝛽1≤𝛾𝑡ℎ) Pr(𝛾0≤𝛾𝑡ℎ)+Pr(𝛽1>𝛾𝑡ℎ) Pr(𝛾Σ≤𝛾𝑡ℎ)

= Pr(𝛽1 ≤ 𝛾𝑡ℎ) Pr(𝛾0 ≤ 𝛾𝑡ℎ) +

[1− Pr(𝛽1 ≤ 𝛾𝑡ℎ)] Pr(𝛾Σ ≤ 𝛾𝑡ℎ) (7)

From (2), it is straightforward to arrive at

Pr (𝛽1≤𝛾𝑡ℎ) =

∫ 𝛾𝑡ℎ

0

𝑓𝛽1(𝛾)𝑑𝛾

=𝑁∑𝑖=1

(−1)𝑖−1

(𝑁

𝑖

)(1−𝑒−

𝑖𝛾𝑡ℎ𝛾1

)(8)

It is known that the SNR associated to 𝛾0 in a Rayleigh fadingenvironment is exponentially distributed; hence, the pdf of 𝛾0can be expressed as

𝑓𝛾0(𝛾) =

1

𝛾0𝑒−

𝛾𝛾0 (9)

where 𝛾0 = 𝐸{𝛾0} with 𝐸{.} denotes the expectation opera-tor. From (9), the probability that the direct link between thesource and the destination is in outage is given by

Pr(𝛾0 ≤ 𝛾𝑡ℎ) =

𝛾𝑡ℎ∫0

1

𝛾0𝑒−

𝛾𝛾0 𝑑𝛾 = 1− 𝑒−

𝛾𝑡ℎ𝛾0 (10)

Based on (6), the probability that the combined SNR 𝛾Σ (forthe case of cooperative transmission) falls below the giventhreshold 𝛾𝑡ℎ is derived as follows:

Pr(𝛾Σ ≤ 𝛾𝑡ℎ) =

𝛾𝑡ℎ∫0

𝑓𝛾Σ(𝛾)𝑑𝛾 =

𝛾𝑡ℎ∫0

𝑓1(𝛾)𝑑𝛾 (11)

=

⎧⎨⎩

⎡⎣(

𝛾0

𝛾0−𝛾2

)(1− 𝑒−

𝛾𝑡ℎ𝛾0

)+(

𝛾2

𝛾2−𝛾0

)(1− 𝑒−

𝛾𝑡ℎ𝛾2

)⎤⎦ 𝛾0 ∕= 𝛾2

1−(1 + 𝛾𝑡ℎ

𝛾0

)𝑒−

𝛾𝑡ℎ𝛾0 𝛾0 = 𝛾2

Substituting (8),(10) and (11) into (7) yields the desired resultof 𝑃𝑜.

C. Bit Error Probability (BEP)

The average BEP for threshold-based relaying with partialrelay selection schemes can be derived using the law of totalprobability as follows:

𝑃𝑏 = Pr (𝛽1 ≤ 𝛾𝑡ℎ)𝑃1𝐷 + Pr (𝛽1 > 𝛾𝑡ℎ)𝑃

2𝐷

= Pr (𝛽1 ≤ 𝛾𝑡ℎ)𝑃1𝐷 + [1− Pr (𝛽1 ≤ 𝛾𝑡ℎ)]𝑃

2𝐷 (12)

where 𝑃 1𝐷 denotes the average bit error probability at the

destination given that 𝛽1 ≤ 𝛾𝑡ℎ and 𝑃 2𝐷 denotes the average

bit error probability at the destination given that 𝛽1 > 𝛾𝑡ℎ.Having analytical expressions for the pdf of 𝛾0 and 𝛾Σ

allows ones to derive the bit error probability for the system inslow and flat Rayleigh fading channel by averaging the errorprobability for the AWGN channel over the pdf of the SNR inRayleigh fading. In particular, the bit error probability of thesystem provided that 𝛽1 < 𝛾𝑡ℎ over Rayleigh fading channelsfor 𝑀 -ary square quadrature amplitude modulation (𝑀 -QAM,𝑀 = 4𝑚, 𝑚 = 1, 2, . . .) with Gray mapping can be first givenas [20]:

𝑃 1𝐷 =

∞∫0

log2

√𝑀∑

𝑗=1

𝜐𝑗∑𝑛=0

𝜙𝑗𝑛erfc

(√Ω𝑛𝛾

)𝑓𝛾0

(𝛾)𝑑𝛾 (13)

where 𝜐𝑗 = (1−2−𝑗)√𝑀−1, 𝜙𝑗

𝑛 = (−1)⌊𝑛.2𝑗−1/√𝑀⌋(2𝑗−1−⌊

𝑛.2𝑗−1/√𝑀 + 1/2

⌋)/(

√𝑀 log2

√𝑀), Ω𝑛 = (2𝑛 +

1)23 log2 𝑀/(2𝑀−2). Furthermore, we define ⌊.⌋ and erfc(.)as the floor and complementary error function, respectively.

Substituting (9) into (13) and taking the integral with respectto 𝛾, we achieve the closed-form expression for 𝑃 1

𝐷 after somemanipulation as follows [21, p.149, eq.(5A.2)]:

𝑃 1𝐷 =

log2

√𝑀∑

𝑗=1

𝜐𝑗∑𝑛=0

⎡⎣ ∞∫

0

𝜙𝑗𝑛erfc

(√Ω𝑛𝛾

) 1

𝛾0𝑒−

𝑖𝛾𝛾1 𝑑𝛾

⎤⎦

=

log2

√𝑀∑

𝑗=1

𝜐𝑗∑𝑛=0

[𝜙𝑗𝑛

(1−

√Ω𝑛𝛾0

1 + Ω𝑛𝛾0

)](14)

Similar to 𝑃 1𝐷, from (6), 𝑃 2

𝐷 can be derived as follows:

𝑃 2𝐷 =

∞∫0

log2

√𝑀∑

𝑗=1

𝜐𝑗∑𝑛=0

𝜙𝑗𝑛erfc

(√Ω𝑛𝛾

)𝑓Σ(𝛾)𝑑𝛾 (15)

174

By interchanging the order of the summation and theintegral, we can rewrite 𝑃 2

𝐷 as

𝑃 2𝐷 =

log2

√𝑀∑

𝑗=1

𝜐𝑗∑𝑛=0

⎡⎢⎢⎢⎢⎣

𝐾1(𝛾0, 𝛾2) +𝐾2(𝛾0, 𝛾2)+𝑁∑

𝑖=1

(−1)𝑖−1(𝑁𝑖 )𝐾3(𝛾0,𝜔𝑖−1)

𝑁∑𝑖=1

(−1)𝑖−1(𝑁𝑖 )𝑒− 𝑖𝛾𝑡ℎ

𝛾1

⎤⎥⎥⎥⎥⎦ (16)

where 𝐾1(𝛾0, 𝛾2), 𝐾2(𝛾0, 𝛾2) and 𝐾3(𝛾0, 𝜔𝑖−1) are defined,

respectively, as follows:

𝐾1(𝛾0, 𝛾2) =

𝛾𝑡ℎ∫0

𝜙𝑗𝑛erfc

(√Ω𝑛𝛾

)𝑓1𝑑𝛾 (17)

=

⎧⎨⎩

⎡⎣(

𝛾0

𝛾0−𝛾2

)𝐼1(𝜙

𝑗𝑛,Ω𝑛, 𝛾0, 𝛾𝑡ℎ)

+(

𝛾2

𝛾2−𝛾0

)𝐼1(𝜙

𝑗𝑛,Ω𝑛, 𝛾2, 𝛾𝑡ℎ)

⎤⎦ 𝛾0 ∕= 𝛾2

𝐼2(𝜙𝑗𝑛,Ω𝑛, 𝛾0, 𝛾𝑡ℎ) 𝛾0 = 𝛾2

𝐾2(𝛾0, 𝛾2) =

∞∫𝛾𝑡ℎ

𝜙𝑗𝑛erfc

(√Ω𝑛𝛾

)𝑓2𝑑𝛾 (18)

=

⎧⎨⎩

⎡⎣ 𝛾0

𝛾0−𝛾2

[1− 𝑒

−𝛾𝑡ℎ

(1𝛾2

− 1𝛾0

)]×𝐼3(𝜙

𝑗𝑛,Ω𝑛, 𝛾0, 𝛾𝑡ℎ)

⎤⎦ 𝛾0 ∕= 𝛾2

𝛾𝑡ℎ

𝛾0𝐼3(𝜙

𝑗𝑛,Ω𝑛, 𝛾0, 𝛾𝑡ℎ) 𝛾0 = 𝛾2

𝐾3(𝛾0, 𝜔𝑖−1) =

∞∫𝛾𝑡ℎ

𝜙𝑗𝑛erfc

(√Ω𝑛𝛾

)𝑓3𝑑𝛾 (19)

=

⎧⎨⎩

⎡⎢⎢⎣

𝐼3(𝜙𝑗𝑛,Ω𝑛,𝜔

−1𝑖

,𝛾𝑡ℎ)

1−𝜔𝑖𝛾0−(

𝜔𝑖𝛾0𝑒−𝛾𝑡ℎ(𝜔𝑖−1/𝛾0)

1−𝜔𝑖𝛾0

)×𝐼3(𝜙

𝑗𝑛,Ω𝑛, 𝛾0, 𝛾𝑡ℎ)

⎤⎥⎥⎦ 𝜔𝑖

−1 ∕= 𝛾0

[𝐼4(𝜙

𝑗𝑛,Ω𝑛, 𝛾0, 𝛾𝑡ℎ)

−𝛾𝑡ℎ

𝛾0𝐼3(𝜙

𝑗𝑛,Ω𝑛, 𝛾0, 𝛾𝑡ℎ)

]𝜔𝑖

−1 = 𝛾0

with 𝐼1(𝑎, 𝑏, 𝑐, 𝛾𝑡ℎ), 𝐼2(𝑎, 𝑏, 𝑐, 𝛾𝑡ℎ), 𝐼3(𝑎, 𝑏, 𝑐, 𝛾𝑡ℎ) and𝐼4(𝑎, 𝑏, 𝑐, 𝛾𝑡ℎ)

1 are defined, respectively, as follows:

𝐼1(𝑎, 𝑏, 𝑐, 𝛾𝑡ℎ) =𝛾𝑡ℎ∫0

𝑎 erfc(√

𝑏𝛾)

1𝑐 𝑒

− 𝛾𝑐 𝑑𝛾

= 𝑎[1− 𝑒−

𝛾𝑡ℎ𝑐 erfc

(√𝑏𝛾𝑡ℎ

)−√𝑏𝑐

1+𝑏𝑐erf(√

𝛾𝑡ℎ

𝑐 (1 + 𝑏𝑐))]

𝐼2(𝑎, 𝑏, 𝑐, 𝛾𝑡ℎ) =𝛾𝑡ℎ∫0

𝑎 erfc(√

𝑏𝛾)

𝛾𝑐2 𝑒

− 𝛾𝑐 𝑑𝛾

= 𝑎

⎡⎢⎣ 1− 𝑒−

𝛾𝑡ℎ𝑐

(1 + 𝛾𝑡ℎ

𝑐

)erfc

(√𝑏𝛾𝑡ℎ

)+√

𝑏𝛾𝑡ℎ

𝜋(𝑏𝑐+1)2𝑒−

(𝑏𝑐+1)𝛾𝑡ℎ𝑐

−√

(𝑏𝑐+3/2)2𝑏𝑐

(𝑏𝑐+1)3erf

(√(𝑏𝑐+1)𝛾𝑡ℎ

𝑐

)⎤⎥⎦

1They can easily be verified using standard mathematical software packagessuch as Mathematica, Maple. Furthermore 𝐼4(.) is evaluated with the help of[21, p.149, eq.(5A.2)]

𝐼3(𝑎, 𝑏, 𝑐, 𝛾𝑡ℎ) =∞∫

𝛾𝑡ℎ

𝑎 erfc(√

𝑏𝛾)

1𝑐 𝑒

− 𝛾𝑐 𝑑𝛾

= 𝑎[𝑒−

𝛾𝑡ℎ𝑐 erfc

(√𝑏𝛾𝑡ℎ

)−√𝑏𝑐

1+𝑏𝑐erfc(√

𝛾𝑡ℎ

𝑐 (1 + 𝑏𝑐))]

𝐼4(𝑎, 𝑏, 𝑐, 𝛾𝑡ℎ) =∞∫

𝛾𝑡ℎ

𝑎 erfc(√

𝑏𝛾)

𝛾𝑐2 𝑒

− 𝛾𝑐 𝑑𝛾

=∞∫0

𝑎 erfc(√

𝑏𝛾)

𝛾𝑐2 𝑒

− 𝛾𝑐 𝑑𝛾 −

𝛾𝑡ℎ∫0

𝑎 erfc(√

𝑏𝛾)

𝛾𝑐2 𝑒

− 𝛾𝑐 𝑑𝛾

= 𝑎[1−

√𝑏𝑐

1+𝑏𝑐

(3+2𝑏𝑐2+2𝑏𝑐

)]− 𝐼2(𝑎, 𝑏, 𝑐, 𝛾𝑡ℎ)

where erf(.) denotes the error function. Consequently, aclosed-form expression for the BEP of dual hop relaying withpartial relay selection for a certain threshold is derived bycombining (8),(12),(14) and (16).

IV. NUMERICAL RESULTS AND DISCUSSION

0 5 10 15 20 25 3010

−5

10−4

10−3

10−2

10−1

100

Average SNR per Bit [dB]

Ou

tag

e P

rob

abili

ty

DT − SimulationDT − AnalysisN=1 − SimulationN=1 − AnalysisN=2 − SimulationN=2 − AnalysisN=3 − SimulationN=3 − AnalysisN=4 − SimulationN=4 − AnalysisN=5 − SimulationN=5 − Analysis

Direct Transmission (DT)

γth

=8

Fig. 1. Effect of number of relays on the outage probability. Channel setting:𝜆0 = 1, 𝜆1,𝑖 = 2 and 𝜆2,𝑖 = 3 with 𝑖 = 1, . . . , 𝑁

0 5 10 15 20 25 3010

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Average SNR per Bit [dB]

Ou

tag

e P

rob

abili

ty

Simulation − γth

=1

Analysis − γth

=1

Simulation − γth

=3

Analysis − γth

=3

Simulation − γth

=6

Analysis − γth

=6

Simulation − γth

=9

Analysis − γth

=9

Simulation − γth

=12

Analysis − γth

=12

Increasing γth

Fig. 2. Effect of the threshold on the outage probability. Channel setting:𝜆0 = 1, 𝜆1,𝑖 = 2 and 𝜆2,𝑖 = 3 with 𝑖 = 1, . . . , 𝑁

175

0 5 10 15 20 25 30 35 4010

−10

10−8

10−6

10−4

10−2

100

Average SNR per Bit [dB]

Bit

Err

or

Rat

e

Direct Trans. − SimulationDirect Trans. − AnalysisN=1 − SimulationN=1 − AnalysisN=2 − SimulationN=2 − AnalysisN=3 − SimulationN=3 − AnalysisN=4 − SimulationN=4 − Analysis

Fig. 3. Effect of number of relays on the average bit error probability for 4-QAM. Channels setting: 𝜆0 = 1, 𝜆1,𝑖 = 2 and 𝜆2,𝑖 = 3 with 𝑖 = 1, . . . , 𝑁and 𝛾𝑡ℎ = 10.

0 5 10 15 20 25 3010

−7

10−6

10−5

10−4

10−3

10−2

10−1

Average SNR per Bit [dB]

Bit

Err

or

Rat

e

DT − SimulationDT − AnalysisN =1 Simulation − γ

th=0

N =1 Analysis − γth

=0

N =3 Simulation − γth

=0

N =3 Analysis − γth

=0

N =5 Simulation − γth

=0

N =5 Analysis − γth

=0

N =1 Simulation − γth

=5

N =1 Analysis − γth

=5

N =3 Simulation − γth

=5

N =3 Analysis − γth

=5

N =5 Simulation − γth

=5

N =5 Analysis − γth

=5

Fig. 4. Effect of the threshold on the average bit error probability for 4-QAM. Channel setting: 𝜆0 = 1, 𝜆1,𝑖 = 2 and 𝜆2,𝑖 = 3 with 𝑖 = 1, . . . , 𝑁and the given threshold 𝛾𝑡ℎ = 10.

In this section, several numerical examples are presented toinvestigate the system performance and to validate the theoret-ical results. Assuming that the channel state information (CSI)is not known at the transmitting terminals, the uniform powerallocation is used as a natural choice, that is 𝜌1 = 𝜌2 = 𝜌/2where 𝜌 denotes the transmit power of the source in case ofdirect transmission. Here, we first investigate the performanceof AF dual hop relaying with partial relay selection illustratedin Figs. 1- 6 and then provide the comparison between AFand DF in terms of outage probability and bit error probabilityin Fig. ?? and Fig. 7, respectively.

Figs. 1 and 2 show the outage probability in Rayleighfading versus average SNR for a different number of relayand the predetermined thresholds. In particular, Fig. 1 showsthat by increasing the number of relays, the outage probability

0 5 10 15 20 25 3010

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Average SNR per Bit [dB]

Bit

Err

or

Rat

e

4−QAM − Simulation 4−QAM − Analysis 16−QAM − Simulation 16−QAM − Analysis 64−QAM − Simulation 64−QAM − Analysis256−QAM − Simulation256−QAM − Analysis1024−QAM − Simulation1024−QAM − Analysis

Fig. 5. Effect of modulation level on the average bit error probability. Channelsetting: (𝜆0 = 𝜆1,𝑖 = 𝜆2,𝑖 = 1 with 𝑖 = 1, ⋅ ⋅ ⋅ , 𝑁 ), the given threshold:𝛾𝑡ℎ = 10, number of relays: 𝑁 = 2.

0 5 10 15 20 25 30 35 40 45 5010

−12

10−10

10−8

10−6

10−4

10−2

100

Average SNR per Bit [dB]

Bit

Err

or

Rat

e

i.i.d. Simulation − γth

=0

i.i.d. Analysis − γth

=0

i.n.d. Simulation − γth

=0

i.n.d. Analysis − γth

=0

i.i.d. Simulation − γth

=10

i.i.d. Analysis − γth

=10

i.n.d. Simulation − γth

=10

i.n.d. Analysis − γth

=10

i.i.d. Simulation − γth

=100

i.i.d. Analysis − γth

=100

i.n.d. Simulation − γth

=100

i.n.d. Analysis − γth

=100

i.i.d.

i.n.d.

Fig. 6. The average bit error probability for dual hop relaying with partialrelay selection for 4-QAM under i.i.d. channels (𝜆0 = 𝜆1,𝑖 = 𝜆2,𝑖 = 2) andi.n.d. channels (𝜆0, 𝜆1,𝑖 and 𝜆2,𝑖 are uniformly distributed between 0 and 1)with 𝑖 = 1, ⋅ ⋅ ⋅ , 𝑁 and number of relays: 𝑁 = 2.

does not linearly decrease except 3 dB gain as 𝑁 > 1. Fur-thermore, both these figures clearly indicate the high accuracyof the analytically obtained OP for wide range of SNR. Forcomparison purpose, we also plot the performance of directtransmission as a reference. It is seen that the OP for thesystem with more than one relay converges together as SNRincreases regardless additional relays and performs better thandirect transmission.

In Figs. 3 , we present a BEP for 4-QAM assuming 𝛾𝑡ℎ =10. As the number of available relays increases from 1 to 4, thesystem error performance improves only 5 dB and convergeeventually in a limit at high SNR regime since number ofrelays in the network is greater than one.

In Fig. 4, the effect of threshold on BEP is investigatedsince 𝛾𝑡ℎ = 0 and 5. In general, the performance of thesystem is highly affected by the threshold at low SNR regime.We may notice that the proposed protocol with a lower

176

0 5 10 15 20 25 3010

−6

10−5

10−4

10−3

10−2

10−1

100

Average SNR per Bit [dB]

Bit

Err

or

Rat

e

N=1 − AF − SimulationN=1 − DF − SimulationN=1 − AnalysisN=3 − AF − SimulationN=3 − DF − SimulationN=3 − Analysis

Fig. 7. Effect of signal processing technique equipped at relays on thebit error probability for 16-QAM. Channel setting: 𝜆0 = 1, 𝜆1,𝑖 = 2 and𝜆2,𝑖 = 3 with 𝑖 = 1, . . . , 𝑁 , the given threshold: 𝛾𝑡ℎ = 5.

threshold value will provide a better performance. Moreover, itis straightforward to see that with threshold values of zero andinfinity, the system are akin to basic fixed relaying scheme inconjunction with partial relay selection and direct transmissionscheme, respectively. In addition, we can see that the impactof the threshold imposes stronger on the system at SNR below10dB. This results can be explained by considering the factthat when the SNR relative to 𝛾𝑡ℎ is high, it is likely that therelaying link always involve in the transmission leading to theindependence of BEP on the threshold.

In Fig. 5, we study the average BEP for different levelsof 𝑀 -QAM. As expected, the results from theory and fromsimulation are in excellent at medium and high SNR regime.In Fig. 6, the performance of the system under both i.i.d andi.n.d. channels is examined. The results obtained for i.n.d. casehave the same form with those for i.i.d. case and seem to beshifted from those for i.i.d case to the right around 5 dB.

In Figs. 7, we study the effect of signal processing tech-nique at relays on the dual hop relaying with partial relayselection, i.e., amplify-and-forward and decode-and-forward.As expected, partial-based DF relay selection and partial-basedAF relay selection have the same medium and high SNRperformance, especially considering their average BEP.

V. CONCLUSION

We analyzed the performance of threshold-based dual-hoprelaying employing partial relay selection. This scheme issuitable for adaptive systems in which the threshold can beadjusted to adapt the required quality of the end signals aswell as to adjust the level of cooperation. We derived the close-form expression for outage probability and bit error probabilityfor square 𝑀 -QAM. Monte-Carlo simulations validated ouranalytical results.

ACKNOWLEDGMENTS

This research was supported by Basic Science ResearchProgram through the National Research Foundation of Korea

(NRF) funded by the Ministry of Education, Science andTechnology (No. 2009-0073895).

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