asymptotic sep analysis of two-way relaying networks with distributed alamouti codes

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IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, ACCEPTED FOR PUBLICATION 1

Asymptotic SEP Analysis of Two-way RelayingNetworks with Distributed Alamouti Codes

Feng-Kui Gong,Member, IEEE, Jian-Kang Zhang,Senior Member, IEEE, and Jian-Hua Ge

Abstract—Asymptotic symbol error probability (SEP) perfor-mance with the optimal maximum likelihood (ML) receiver andsquare quadrature amplitude modulation (QAM) constellationsis investigated for two-way amplify-and-forward (AF) half-duplexrelaying networks employing distributed Alamouti space-timeblock codes recently proposed by Duong, Yuen, Zepernick andLei. By developing a novel strategy to particularly deal with somespecific Gaussian integrals, two asymptotic SEP formulae arederived for both fixed-gain AF and variable-gain AF protocols.These analytic results reveal that when signal to noise ratio (SNR)is large, the diversity gain for the variable-gain AF is 2, whereasthe diversity gain function for the fixed-gain AF is proportionalto SNR−2 ln SNR.

Index Terms—SEP, distributed Alamouti space-time blockcode, amplify-and-forward, half-duplex two-way relaying net-works.

I. I NTRODUCTION

OVER the past several years, cooperative diversity hasbeen developed [1], [2] to enhance error performance

or increase the range of mobile wireless communications, inwhich the in-cell mobile users share the use of their antennasto create a virtual array through distributed transmission andsignal processing. When channel state information is availableat the receiver, a diversity gain for one-way cooperative relaysystem with product flat fading channels is characterized bythe diversity gain function [3]–[5] and achieved by utilizingwell-designed precoders [5] or distributed space-time blockcodes [3], [4], [6], [7]. However, the one-directional dual-hophalf-duplex relaying networks are not able to transmit andreceive simultaneously and thus, lose half of the throughputcompared to the direct communication. To improve bandwidth

Copyright (c) 2012 IEEE. Personal use of this material is permitted.However, permission to use this material for any other purposes must beobtained from the IEEE by sending a request to [email protected].

Manuscript received March 9, 2012; revised May 31, 2012; accepted July15, 2012. The work of F.-K. Gong was supported by National NaturalScience Foundation of China (NSFC) (61001207, 61101145 and 61101144),National Science and Technology Major Project of the Ministry of Science andTechnology of China (2012ZX03001027), the Fundamental Research Fundsfor the Central University (K50510010008) and the 111 Project (B08038). Thework of J.-K. Zhang was funded in part by National Science and EngineeringResearch Council of Canada. This work was performed while F.-K. Gong wasa visitor in the Department of Electrical and Computer Engineering, McMasterUniversity. The review of this paper was coordinated by Prof. A. Tonello.

Feng-Kui Gong and Jian-Hua Ge are with the State Key Lab. of ISN,Xidian University, Shaanxi Province (710071), China (email:fkgong,[email protected]).

Jian-Kang Zhang is with the Department of Electrical and Computer Engi-neering, McMaster University, 1280 Main Street West, L8S 4K1, Hamilton,Ontario, Canada (email: [email protected]).

Digital Object Identifier XXXXXXX

efficiency, a two-way relaying network has been recently pro-posed in [8], where two source nodes transmit simultaneouslyto the relay node and the relay forwards its received signalsto both source nodes in the consecutive two time slots. Sincethen, various forms of two-way relaying protocols have beenpresented, including employing physical network coding in therelay [9]–[13]. Furthermore, the Alamouti coding scheme [4],[14]–[16] has been smartly and successfully applied fromone-way relaying systems to two-way relaying systems [17]–[23]. More recently, Duong, Yuen, Zepernick and Lei [24]have proposed a distributed Alamouti space-time block cod-ing transmission scheme for a two-way AF relay networkwith two sources and one relay terminals, where each nodehas only one antenna and the direct link communication isconsidered. For such a system, the ergodic sum-rate overRayleigh fading channels [24] is analyzed. This result hasshown that the two-way relaying network with the proposedAlamouti coding scheme outperforms the traditional one-wayrelay system in terms of spectral efficiency. In addition, exactpairwise error probability over Nakagami-m fading channelswas also derived for the system [25]. However, to the bestknowledge of the authors, the exact SEP analysis of the MLdetector for this scheme is still unavailable. Therefore, all theaforementioned factors significantly motivate us to analyzethe asymptotic behavior of the exact SEP for the distributedAlamouti coded two-way relaying network with the fixed-gainAF and the variable-gain AF [24], [25] when the square QAMconstellation is transmitted and the ML receiver is employed.

Notation: Column vectors and matrices are boldface lower-case and uppercase letters, respectively. Notation‖r‖ denotesa 2-norm ofr; (·)T , (·)∗ and(·)H denote the matrix transpose,conjugate, and conjugate transpose, respectively;E[·] denotesthe expected value of the expression in brackets;IN denotesthe N ×N identity matrix.

Fig. 1. Two-way relaying networks proposed in [24]

0000–0000/00$00.00c© 201X IEEE




TABLE ITWO-WAY RELAYING NETWORK PROPOSED IN[24]

Phase Islot1 Islot2 IIslot1 IIslot2 III slot1 III slot2

T1 T1x1→ R T1

x∗2→ R T1−x2→ T2 T1

x∗1→ T2 t11 t12

T2 T2y1→ R T2

y∗2→ R t21 t22 T2−y2→ T1 T2

y∗1→ T1

R r1 r2 Rr1→ T2 R

r2→ T2 Rr1→ T1 R

r2→ T1

II. T WO-WAY RELAYING NETWORKS

Let us first begin with briefly reviewing the two-way re-laying network with a distributed Alamouti space-time blockcode (DASTC) proposed in [24], in which the transmissionscheme, for discussion clarity, is shown in Fig. 1 as well asillustrated in Table I. In this scheme, two sources, denoted byT1 andT2, exchange information through the assistance of anAF relay R. The channel link betweenTi for i = 1, 2 andRis denoted byhi, whereas the channel link betweenT1 andT2 is denoted byh0. Each terminal is equipped with a singleantenna and operates in half-duplex mode. Throughout thispaper, we assume that all link channelsh0, h1, and h2 areindependent and follow quasi-static Rayleigh fading havingzero mean and varianceΩ0, Ω1 and Ω2 respectively. Thereare three communication phases. In Phase I, bothT1 andT2

spend two time slots on transmitting the respectivex and yto R, with each symbol having the same transmission powerP . Thus, the received signal vectorr = [r1, r2]T at the relayR is represented by

r = h1x + h2y + n, (1)

where n is a two by one circularly symmetrical complexGaussian noise vector with zero mean and covarianceσ2I2,x = [x1, x

∗2]

T and y = [y1, y∗2 ]T , with xi andyi for i = 1, 2

being randomly, independently and equally likely chosen froma standardM -ary square QAM constellation. In Phase II, therelay R amplifies the received signalr with amplifying gainG and then, forwards what it has received toT2 while T1

sends two symbols−x2 andx∗1 to T2 within two consecutivetime slots. Similarly, in Phase III, the relayR broadcasts thesignalGr, whereasT2 sends two symbols−y2 andy∗1 to T1

during two time slots. Then, the received signals at the twoterminalsT1 andT2 can be written, respectively, as

t1 = Gh1r + h0y + n1, (2a)

t2 = Gh2r + h0x + n2, (2b)

wherex = [−x2, x∗1]

T and y = [−y2, y∗1 ]T , eachni is a two

by one noise vector, all elements of which are independent cir-cularly symmetrical complex Gaussian with each having zeromean and varianceσ2. In this paper, we consider both fixed-gain AF and variable-gain AF, i.e.,G2 =

(Ω1+Ω2+σ2

)−1for

fixed-gain AF andG2 =(|h1|2 + |h2|2 + σ2

)−1 ≈ (|h1|2 +|h2|2

)−1for variable-gain AF. Substituting (1) into (2) and

removing the self-interferences caused by their own signals,

we arrive at

t1 =[y1 −y2

y∗2 y∗1

] [Gh1h2

h0

]+ ζ1, (3a)

t2 =[x1 −x2

x∗2 x∗1

] [Gh1h2

h0

]+ ζ2, (3b)

where ζi = Ghin + ni are the combined Gaussian noisevectors, with each having zero mean and covariance matrixσ2(1 + G2|hi|2)I2. For purpose of detection, (3) can beequivalently rewritten as

t1 = Hy + ζ1 =[Gh1h2 −h0

h∗0 Gh∗1h∗2

] [y1

y2

]+ ζ1, (4a)

t2 = Hx + ζ2 =[Gh1h2 −h0

h∗0 Gh∗1h∗2

] [x1

x2

]+ ζ2, (4b)

where ti = [ti1, t∗i2]T , and ζi = [ζi1, ζ

∗i2]

T are still whiteGaussian noise vectors with the same covariance matrices asζi. Therefore, when channel state information is perfectlyavailable at the source terminals, the optimal detector forestimation of the transmitted signals is the ML detector, whichis to solve the following optimization problems:

y = arg miny‖t1 −Hy‖2, (5a)

x = arg minx‖t2 −Hx‖2. (5b)

We would like to make the following two comments on thechannel model (4).

1) The average sum-rate analysis on given in [24] hasdemonstrated that the system (4) has a significant spec-tral efficiency gain over the one-way relaying network.

2) Despite the fact thath1 × h2 has the same form asthat in [26], the system (4) can not be modelled as akeyhole fading channel. The major difference betweenthe channel model (4) and the one discussed in [26]is that the noise in [26] is independent of the channelfading, whereas the equivalent noise in (4) depends onthe channel fading.

III. A SYMPTOTIC ANALYSIS ON SEP

The primary purpose of this section is to derive asymptoticSEP formulae under high SNR for the distributed Alamouticoded two-way relaying networks with the fixed-gain AF andthe variable-gain AF [24]. To that end, let us recall that thanksto the Alamouti coding scheme, the channel matrixH in (4) isunitary up to a scale, i.e.,HHH = (G2|h1|2|h2|2 + |h0|2)I2,whereas each noise vectorζi is white Gaussian for the givenhi, with the covariance matrix given byσ2(1 + G2|hi|2)I2.Therefore, the optimal ML detection for (5) is equivalently




reduced to a symbol by symbol detection and its arithmeticaverage SEP for the terminalTi can be expressed by [27]

Pei|h0,h1,h2 =

4(1− 1√

M

)Q

(√di

)− 4(1− 1√

M

)2Q2

(√di

), (6)

wheredi are determined by

di =

3(G2|h1|2|h2|2+|h0|2)ρ

(M−1)(1+G2|hi|2) for the fixed-gain AF3(|h1|2|h2|2+(|h1|2+|h2|2)|h0|2)ρ

(M−1)(|h1|2+|h2|2+|hi|2) for the variable-gain AF

with ρ = Pσ2 being SNR per symbol. Using two alternative

formulae for theQ(ξ) and Q2(ξ) functions [27]: Q(ξ) =1π

∫ π2

0exp

(− ξ2

2 sin2 θ

)dθ andQ2(ξ) = 1

π

∫ π4

0exp

(− ξ2

2 sin2 θ

)dθ

for ξ ≥ 0, equation (6) can be represented by

Pei|h0,h1,h2 =4(1− 1√

M

) 1π

∫ π2

0

exp(− di

2 sin2 θ

)dθ

− 4(1− 1√

M

)2 1π

∫ π4

0

exp(− di

2 sin2 θ

)dθ.

By averaging over all the channel realizations, an arithmeticaverage SEP per terminal is defined asPe = 1

2 (Pe,1 + Pe,2),where

Pe,i = E[Pei|h0,h1,h2

]

= 4(

1− 1√M

)1π

∫ π2

0

Eh0,h1,h2

[exp

(− di

2 sin2 θ

)]dθ

− 4(

1− 1√M

)2 1π

∫ π4

0

Eh0,h1,h2

[exp

(− di

2 sin2 θ

)]dθ.

(7)

For notation simplicity, let4i(θ) be defined as

4i(θ) = Eh0,h1,h2

[exp

(− di

2 sin2 θ

)]. (8)

In order to analyze an asymptotic behavior onPe, we need tofirst establish the following lemma.

Lemma 1:Let 4i(θ) be defined by (8) andτ(θ) =3ρ

2(M−1) sin2 θ. Then, the following two statements are true:

1) For the fixed-gain AF, we have

4i(θ) =2G2Ωi + G4Ω2

i − γ + ln(Ω1Ω2G2)

G2Ω0Ω1Ω2τ2(θ)

+ln τ(θ)

G2Ω0Ω1Ω2τ2(θ)+ O

(ln ρ

ρ3

), (9)

whereγ is the Euler’s constant.2) For the variable-gain AF, we have

4i(θ) =Ωj + 4Ωi

Ω0Ω1Ω2τ2(θ)+ O

(ln ρ

ρ3

). (10)

wherej = 3− i.The proof of Lemma 1 is given in Appendix. Now, our mainresult of this paper can be stated as Theorem 1 below.

Theorem 1:If signal to noise ratioρ is large, the SEP ofthe ML detector has the following two asymptotic formulae.

1) For the fixed-gain AF,

Pe = C1ρ−2 ln ρ + C2ρ

−2 + O

(ln ρ

ρ3

), (11)

where

C1 =(√

M − 1)(M − 1)2(3π√

M + 8√

M + 3π − 8)18Ω0Ω1Ω2πM

and

C2 =(√

M − 1)(M − 1)2

3G2Ω0Ω1Ω2

√M

ln

6G2Ω1Ω2

M − 1+ G2(Ω1 + Ω2)

+Ω2

1 + Ω22

2G−4− 7

6− γ − (

√M − 1)

12π√

M

((G2(Ω1 + Ω2)

+Ω2

1 + Ω22

2G−4− γ + ln

3G2Ω1Ω2

2(M − 1))(6π − 16)

− (7− 12 ln 2)π − 12− 16 ln 2 + 24β

),

with β being the Catalan’s constant.2) For the variable-gain AF,

Pe = Cρ−2 + O

(ln ρ

ρ3

), (12)

where

C =5(Ω1 + Ω2)(

√M − 1)(M − 1)2(3π

√M + 8

√M + 3π − 8)

36Ω0Ω1Ω2πM.

Proof: We consider the following cases.Case 1: fixed-gain AF. Substituting (9) into (7) results in

Pe,i = P(1)e,i − P(2)

e,i ,

where

P(1)e,i =

4π

(1− 1√

M

)∫ π2

0

[2G2Ωi + G4Ω2

i − γ + ln(Ω1Ω2G2)

G2Ω0Ω1Ω2τ2(θ)

+ln τ(θ)

G2Ω0Ω1Ω2τ2(θ)

]dθ + O

( ln ρ

ρ3

)

= Ci,11ρ−2 ln ρ + Ci,21ρ

−2 + O(ln ρ

ρ3),

in which

Ci,11 =(√

M − 1)(M − 1)2

3G2Ω0Ω1Ω2

√M

,

Ci,21 =(√

M − 1)(M − 1)2

3G2Ω0Ω1Ω2

√M(

ln6G2Ω1Ω2

M − 1+ 2G2Ωi + G4Ω2

i −76− γ

),

and

P(2)e,i =

4π

(1− 1√

M

)2∫ π

4

0

[2G2Ωi + G4Ω2

i−γ+ln(Ω1Ω2G2)

G2Ω0Ω1Ω2τ2(θ)

+ln τ(θ)

G2Ω0Ω1Ω2τ2(θ)

]dθ + O

( ln ρ

ρ3

)

= Ci,12ρ−2 ln ρ + Ci,22ρ

−2 + O( ln ρ

ρ3

),

in which

Ci,12 =(3π − 8)(

√M − 1)2(M − 1)2

18G2Ω0Ω1Ω2πM




and

Ci,22 =(√

M − 1)2(M − 1)2

36G2Ω0Ω1Ω2πM

((6π−16

)(2G2Ωi + G4Ω2

i−γ

+ ln3G2Ω1Ω2

2(M − 1))−(7−12 ln 2)π −12−16 ln 2 + 24β

),

with β being the Catalan’s constant. Thus,Pe,i can be writtenas

Pe,i = Ci,1ρ−2 ln ρ + Ci,2ρ

−2 + O

(ln ρ

ρ3

), (13)

where

Ci,1 =(√

M − 1)(M − 1)2(3π√

M + 8√

M + 3π − 8)18Ω0Ω1Ω2πM

and

Ci,2 =(√

M − 1)(M − 1)2

3G2Ω0Ω1Ω2

√M

ln

6G2Ω1Ω2

M − 1+ 2G2Ωi + G4Ω2

i

− 76−γ− (

√M−1)

12π√

M

[(2G2Ωi+G4Ω2

i−γ+ln3G2Ω1Ω2

2(M − 1))

(6π − 16)− (7− 12 ln 2)π − 12− 16 ln 2 + 24β

].

By usingPe = 12 (Pe,1 +Pe,2), we attain (11). This completes

the proof of Statement 1.Case 2: variable-gain AF. Similarly, applying Lemma 1

to (7) yields

Pe,i = Ciρ−2 + O

(ln ρ

ρ3

), (14)

where

Ci =(Ωj +4Ωi)(

√M−1)(M−1)2(3π

√M+8

√M+3π−8)

18πΩ0Ω1Ω2M.

Thus, in view ofPe = 12 (Pe,1 + Pe,2), we obtain (12). This

completes the proof of Theorem 1. ¤Theorem 1 shows that the diversity gain function for the

fixed-gain AF is proportional toln ρ/ρ2, whereas the diversitygain for the variable-gain AF is 2. This benefit from thedisappearance ofln ρ is due to the fact that the relay nodeknows channel state information in the variable-gain AFprotocol.

IV. SIMULATIONS

In this section, Monte-Carlo simulations are performed toverify the accuracy of our analytical results. Both the fixed-gain AF and variable-gain AF are investigated.

Fig. 2(a) and Fig. 2(b) show the simulated symbol error rate(SER) and theoretical SEP curves of the distributed Alamouticoded two-way relaying networks, where the 4QAM, 16QAM,and 64QAM constellation are used. Fig. 2(a) is obtainedunder symmetric channels withΩ0 = Ω1 = Ω2 = 1 andFig. 2(b) is given under asymmetric channels withΩ0 = 2,Ω1 = 5, and Ω2 = 3. SNR is defined asρ = P

σ2 , which isassumed to be same for all considered Rayleigh fading channellinks. It can be seen from Fig. 2(a) and Fig. 2(b) that whenSNR is relatively high, the asymptotic curves from the SEPexpressions given by (11) and (12) match almost perfectly

to the simulated SER curves. In addition, for the same AFprotocol, the SER curves for different QAM constellations areparallel in high SNR region, as derived in Section III. We canalso see that when SNR is low, the SER curves for the variable-gain AF are very close to those for the fixed-gain AF, and thatwhen SNR is high, the slopes for the variable-gain AF alwaysoutperform their corresponding competitors for the fixed-gainAF. All the above observations further affirm the statementsmade in Theorem 1.

5 10 15 20 25 30 35 4010

−6

10−5

10−4

10−3

10−2

10−1

100

SNR (dB)

SE

R

4QAM 16QAM

64QAM

Simulation − fixed−gain AFAsymptotic SER − fixed−gain AFSimulation − variable−gain AFAsymptotic SER − variable−gain AF

(a) Ω0 = Ω1 = Ω2 = 1

0 5 10 15 20 25 30 35 4010

−6

10−5

10−4

10−3

10−2

10−1

100

SNR (dB)

SE

R

4QAM 16QAM

64QAM

Simulation − fixed−gain AFAsymptotic SER − fixed−gain AFSimulation − variable−gain AFAsymptotic SER − variable−gain AF

(b) Ω0 = 2, Ω1 = 5, Ω2 = 3

Fig. 2. Comparison of theoretical SEP and simulated SER for two-wayrelaying networks.

V. CONCLUSION

In this paper, we have analyzed error performance ofthe ML receiver for the two-way relaying network with thedistributed Alamouti space-time block code proposed in [24].The asymptotic SEP formulae have been derived. The resulthas shown that when SNR is large, the SEP for the fixed-gainAF is proportional toln SNR/SNR2, whereas the SEP for thevariable-gain AF is proportional to1/SNR2.




APPENDIX

A. Proof of Lemma 1

We prove Lemma 1 by considering the AF with the fixedand variable gains individually.

1) Fixed-gain AF: In this case, the amplifying gainG isa constant independent ofh` for ` = 0, 1, 2. Since h` areindependent and all of them follow Rayleigh distribution, byfirst taking expectation overh0, we obtain

4i(θ) = Eh0,h1,h2

[exp

(− di

2 sin2 θ

)]

= Ehj ,hi

[exp

(− 3G2ρ|hj |2|hi|2

2(M − 1) sin2 θ(1 + G2|hi|2)

)

(1 +

3Ω0ρ

2(M − 1) sin2 θ(1 + G2|hi|2)

)−1]. (15)

where j = 3 − i. After taking expectation overhj , equa-tion (15) can be further written as

4i(θ) = Ehi

[(1 +

3Ω0ρ

2(M − 1) sin2 θ(1 + G2|hi|2)

)−1

(1 +

3ΩjG2ρ|hi|2

2(M − 1) sin2 θ(1 + G2|hi|2)

)−1]

=1

1 + Ωjτ(θ)Ev

[(v + G−2)2

(v + p)(v + q)

]

=1

1 + Ωjτ(θ)

1 + Ev

[A

v + p

]+ Ev

[B

v + q

], (16)

where v = |hi|2, p = 1+Ω0τ(θ)G2 , q =

1G2(1+Ωjτ(θ)) , A = − G−2Ω2

0τ2(θ)1+Ω0τ(θ)−(1+Ωjτ(θ))−1 and

B = Ω2jτ2(θ)

G2(1+Ωjτ(θ))2(1+Ω0τ(θ)−(1+Ωjτ(θ))−1) . By takingadvantage of the results in [28, Eqs. 3.352, 8.357, and 8.359],the termJ1 in (16) becomes

Ev

[A

v + p

]= − A

Ωiexp

(1 + Ω0τ(θ)

ΩiG2

)Ei

(− 1 + Ω0τ(θ)

ΩiG2

)

= − Ω20τ

2(θ)1 + Ω0τ(θ)− (1 + Ωjτ(θ))−1

· 11 + Ω0τ(θ)(

1− G2Ωi

1 + Ω0τ(θ)+ O

(1ρ2

)), (17)

where Ei(z) is the exponential integral func-tion and Γ(z) is the gamma function. Since

11+Ω0τ(θ) = 1

Ω0τ(θ)

(1− 1

Ω0τ(θ) + O(

1ρ2

))and

11+Ω0τ(θ)−(1+Ωjτ(θ))−1 = 1

Ω0τ(θ)

(1− 1

Ω0τ(θ) + O(

1ρ2

)), it

becomes

Ev

[A

v + p

]= −

(1− 2

Ω0τ(θ)+ O

(1ρ2

))

(1− ΩiG

2

1 + Ω0τ(θ)+ O

(1ρ2

))

= −1 +2 + ΩiG

2

Ω0τ(θ)+ O

(1ρ2

). (18)

Similarly, applying Eqs. (3.352) and (8.214) in [28] to (16)yields

Ev

[B

v + q

]= − B

Ωiexp

(Ω−1

i G−2

1 + Ωjτ(θ)

)

(γ + ln

Ω−1i G−2

1 + Ωjτ(θ)+

∞∑

k=1

( −Ω−1i G−2

1 + Ωjτ(θ)

)k 1kk!

). (19)

By usingexp(

Ω−1i G−2

1+Ωjτ(θ)

)= 1 + O

(1ρ

), we attain

Ev

[B

v + q

]=

−G−2Ω−1i Ω2

jτ2(θ)

(1 + Ωiτ(θ))2(1 + Ω0τ(θ)− (1 + Ωjτ(θ))−1)(γ + ln

Ω−1i G−2

1 + Ωjτ(θ)+ O

(1ρ

)). (20)

Furthermore, with 1(1+Ωjτ(θ))2 = 1

Ω2jτ2(θ)

(1 + O

(1ρ

))and

11+Ω0τ(θ)−(1+Ωjτ(θ))−1 = 1

Ω0τ(θ)

(1 + O

(1ρ

)), we have

Ev

[B

v + q

]

=−Ω−1

i G−2

Ω0τ(θ)

(1 + O

(1ρ

))(γ + ln

Ω−1i G−2

1 + Ωjτ(θ)+ O

(1ρ

))

= − G−2

Ω0Ωiτ(θ)

(γ + ln

G−2Ω−1i

1 + Ωjτ(θ)+ O

(1ρ

)). (21)

Now, substituting (18) and (21) into (16) completes the proofof Statement 1 in Lemma 1.

2) Variable-gain AF:Similar to the discussion on the fixed-gain AF, taking an expectation with respect to the randomvariableh0 first yields

4i(θ) =1

Ω0Ehj ,hi

[exp

(− τ(θ)|hj |2|hi|2|hj |2 + 2|hi|2

)

(1

Ω0+

τ(θ)(|hj |2 + |hi|2)|hj |2 + 2|hi|2

)−1]. (22)

In the following, we develop a novel strategy to particularlydeal with the Gaussian integrals. If we letu = |hj |2 andv = |hi|2, then, bothu andv follow the exponential distribu-tion with unit mean and thus, (22) can be further written as(23), which is given on the top of the next page. By performingtransformsv = ut and u = vt into the two terms in (23)respectively and interchanging the order of integration, wehave (24).

In order to quickly and simply extract the dominant termfrom I1, we notice that

I1 =2Ω0Ω2

i

Ω0τ(θ) + 2

∫ 1

0

(t +

12

)3(t +

Ω0τ(θ) + 1Ω0τ(θ) + 2

)−1

(t2 +

ΩjΩiτ(θ) + Ωj + 2Ωi

2Ωjt +

Ωi

2Ωj

)−2

dt

=2Ω0Ω2

i

Ω0τ(θ) + 2

∫ 1

0

18 + 3

4 t

(t + 1)(t + a)2(t + b)2dt + O

(1ρ3

),

(25)

where a = ΩjΩiτ(θ)+Ωj+2Ωi

4Ωj−√

(ΩjΩiτ(θ)+Ωj+2Ωi)2−8ΩjΩi

4Ωj

and b = ΩjΩiτ(θ)+Ωj+2Ωi

4Ωj+√

(ΩjΩiτ(θ)+Ωj+2Ωi)2−8ΩjΩi

4Ωj.




4i(θ) =1

Ω0Eu,v

[exp

(− τ(θ)uv

u + 2v

)(1

Ω0+

τ(θ)(u + v)u + 2v

)−1]

=1

Ω0ΩjΩi

∫ ∞

0

du

∫ u

0

exp(− τ(θ)uv

u + 2v

)(1

Ω0+

τ(θ)(u + v)u + 2v

)−1

e−( u

Ωj+ v

Ωi)dv

+1

Ω0ΩjΩi

∫ ∞

0

dv

∫ v

0

exp(− τ(θ)uv

u + 2v

)(1

Ω0+

τ(θ)(u + v)u + 2v

)−1

e−( u

Ωj+ v

Ωi)du. (23)

4i(θ) =1

Ω0ΩjΩi

∫ 1

0

(1

Ω0+

τ(θ)(1 + t)1 + 2t

)−1( ΩjΩi(1 + 2t)ΩjΩiτ(θ)t + (Ωi + Ωjt)(1 + 2t)

)2

dt

︸︷︷︸I1

+

1Ω0ΩjΩi

∫ 1

0

(1

Ω0+

τ(θ)(1 + t)2 + t

)−1( ΩjΩi(2 + t)ΩjΩiτ(θ)t + (Ωj + Ωit)(2 + t)

)2

dt

︸︷︷︸I2

. (24)

Since 11+ t

b

= (1− tb ) + O

(1ρ2

), I1 can be rewritten as

I1 =2Ω0Ω2

i

b2(Ω0τ(θ) + 2)

∫ 1

0

(18 + 3

4 t)(1− tb )

2

(t + 1)(t + a)2dt + O

( 1ρ3

)

=Ω0Ω2

i

4b2(Ω0τ(θ) + 2)

∫ 1

0

1 + (6− 2b )t

(t + 1)(t + a)2dt + O

( 1ρ3

).

Because of the fact that∣∣ − 2

b t

(t+1)(t+a)2

∣∣ <2b t

(t+1)2ta = 1ab(t+1) =

2Ωi(t+1) , I1 can be further reduced to

I1 =Ω0Ω2

i

4b2(Ω0τ(θ) + 2)

∫ 1

0

1 + 6t

(t + 1)(t + a)2dt + O

( 1ρ3

).

Now, the integral inI1 can be simply calculated such that

I1 =Ω0Ω2

i

4b2(Ω0τ(θ) + 2)

(1− 6a

a2 − 1︸︷︷︸I11

+6a− 1

a(a− 1)︸︷︷︸I12

−

5 ln 2(a− 1)2︸︷︷︸

I13

+5 ln( 1

a + 1)(a− 1)2︸︷︷︸

I14

)+ O

(1ρ3

). (26)

Once when realize the following asymptotic formulae forsufficiently large SNR:1a = 2Ωjb

Ωi= Ωjτ(θ)+2+ Ωj

Ωi+O

(1ρ

),

ln a−1 = ln τ(θ) + O(1), we can immediately haveI11 =O(1), I12 = Ωjτ(θ) + O(1), I13 = O(1), I14 = O(ln τ(θ)).Therefore,I1 has the following asymptotic formula:

I1=Ω0ΩjΩ2

i τ(θ)4b2(Ω0τ(θ) + 2)

+ O

(ln ρ

ρ3

)=

Ωj

τ2(θ)+ O

(ln ρ

ρ3

). (27)

Using the same strategies as those from (25) to (26), we canattain

I2 =Ω0Ω2

j

Ω0τ(θ) + 1

∫ 1

0

(t + 2)3(

t +Ω0τ(θ) + 2Ω0τ(θ) + 1

)−1

(t2 +

ΩjΩiτ(θ) + Ωj + 2Ωi

Ωit +

2Ωj

Ωi

)−2

dt

=Ω0Ω2

j

Ω0τ(θ) + 1

∫ 1

0

8 + 12t

(t + 1)(t + 2Ωj

Ωia)2(t + 2Ωj

Ωib)2

dt+O( 1ρ3

),

(28)

and furthermore

I2 =2Ω0Ω2

i

b2(Ω0τ(θ) + 1)

[

(Ωi − 3Ωja)Ωi

(2Ωja− Ωi)(2Ωja + Ωi)︸︷︷︸I21

+(3Ωja− Ωi)Ωi

2Ωja(2Ωja− Ωi)︸︷︷︸I22

− Ω2i ln 2

2(2Ωja− Ωi)2︸︷︷︸I23

+Ω2

i ln( Ωi

2Ωja + 1)

2(2Ωja− Ωi)2︸︷︷︸I24

]+ O

(1ρ3

). (29)

Since I21 = O(1), I22 = Ωi

2Ωja + O(1), I23 = O(1), andI24 = O(ln ρ), I2 is expressed by

I2 =2Ω0Ω2

i

b(Ω0τ(θ) + 1)+ O

(ln ρ

ρ3

)=

4Ωi

τ2(θ)+ O

(ln ρ

ρ3

). (30)

Finally, substituting (27) and (30) into (24) finishes the proofof Statement 2 and thus, of Lemma 1. ¤

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