Symbol Error Probability of Distributed-AlamoutiScheme in Wireless Relay Networks

Trung Q. DuongRadio Communications Group,

Blekinge Institute of Technology, SE-372 25, Ronneby, Sweden,e-mail: quang.trung.duong@bth.se

Dac-Binh Ha, Hoai-An Tran, Nguyen-Son VoFaculty of Electrical and Electronics Engineering,

Ho Chi Minh City University of Transport, Vietnam

Abstract—In this paper, we analyze the maximum likelihooddecoding performance of non-regenerative cooperation employ-ing Alamouti scheme. Specifically, we derive two closed-formexpressions for average symbol error probability (SEP) whenthe relays are located near by the source or destination. Theanalytical results are obtained as a single integral with finite limitsand an integrand composed solely of trigonometric functions.Assessing the asymptotic (high signal-to-noise ratio) behaviorof SEP formulas, we show that the distributed-Alamouti codesachieves a full diversity order. We also perform Monte-Carlosimulations to validate the analysis.

Index Terms—Alamouti, distributed space-time codes (DSTCs),non-regenerative relays, symbol error probability (SEP).

I. INTRODUCTION

A. Distributed-Alamouti Space–Time Coding Scheme

Alamouti scheme [1] has been considered as the onlyorthogonal space-time block code that can provide full rateand full diversity over complex constellation with symbol-wise maximum likelihood (ML) decoding complexity. Re-cently, there exists an extension of Alamouti scheme intocooperative/relay systems where relays simultaneously receivea noisy signal from the source and construct Alamouti space-time codes in a distributed fashion before relaying the signalsto the destination [2]–[6].

B. Related Work

Distributed-Alamouti scheme dated back to the work in[2], [3], where two single-antenna relays assisted the source-destination communication by forming the Alamouti space–time block code in relay networks. The authors conjectureda diversity order of distributed-Alamouti scheme around onefrom their simulation. In [5], the average bit error rate (BER)was shown to be proportional to log (SNR) /SNR2.

In [4], distributed-Alamouti system was created by usingonly one single-antenna relay in Protocol III, i.e., the relay andsource construct a distributed-Alamouti space-time code andeach terminal transmits each row of Alamouti code. Recently,exact closed-form expressions for pairwise error probability ofthis scheme has been analyzed in [6] where it has been shownthat a full diversity order is achieved.

0This research was supported in part by Consultant Application of ScienceTechnology of Transport Company (CAST) - Vietnam.

More recently, we generalized the construction of dis-tributed space–time codes using amicable orthogonal designin relay networks [7]. It has been showed that our scheme canachieve full diversity order and single-symbol ML decodingcomplexity with the sacrifice of bandwidth efficiency.

C. Motivation and Results

Unlike most of previous works [2], [3], [5], [8], whichfocused on the average bit error rate (BER) of distributed-Alamouti space–time code for binary phase-shift keying(BPSK) modulation, we analyze the average symbol errorprobability (SEP) for M -ary phase-shift keying (M -PSK) byusing the moment generating function (MGF) method [9],[10]. More specifically, we derive two closed-form expressionsof average SEP for M -PSK modulation when the relays areclose to either the destination or source. Since the asymptoticbehavior of the MGF at large signal-to-noise (SNR) reveals ahigh-SNR slope of the SEP curve [9], [11], we show thatdistributed-Alamouti scheme achieves full diversity, i.e., adiversity order of two.

D. Organization and Notations

The paper is organized as follows. In the following sec-tion, we briefly review the cooperative system of two non-regenerative relays based on Alamouti scheme. We then de-rive two exact closed-form expressions of average SEP anddiversity orders when two relays are closely located to bothends in Section III and Section IV, respectively. We show thattwo average SEP curves agree exactly with the Monte-Carlosimulation in Section V. Section VI provides the conclusionand future work.

Notation: Throughout the paper, we shall use the followingnotations. Vector is written as bold lower case letter and matrixis written as bold upper case letter. The superscripts ∗ and †stand for the complex conjugate and transpose conjugate. IIInrepresents the n × n identity matrix. ‖AAA‖F denotes Frobe-nius norm of the matrix AAA and |x| indicates the envelopeof x. Ex . is the expectation operator over the randomvariable x. A complex Gaussian distribution with mean µand variance σ2 is denoted by CN (µ, σ2). log is the naturallogarithm. Γ (a, x) is the incomplete gamma function definedas Γ (a, x) =

∫∞xta−1e−tdt and K0 (.) is the zeroth-order

modified Bessel function of the second kind.

978-1-4244-1645-5/08/$25.00 ©2008 IEEE 648

S

R

R

D

Destination

second hop

h f

Source

first hop

Relays

Fig. 1. Dual-hop non-regenerative relay channels.

II. SYSTEM MODEL

We consider a dual-hop relay channel, as shown in Fig. 1,where the channel remains constant for Tcoh coherence time(an integer multiple of the dual-hop interval) and changesindependently to a new value for each coherence time. Allterminals are in half-duplex mode and, hence, transmissionoccurs over two time slots, each with the interval of twosymbol periods.

In the first time slot, the source transmits two symbols sss =[ s1 s2 ], selected from M -PSK signal constellation S, withaverage transmit power per symbol Ps. During the first hop thereceived signals yyyi = [ yi (1) yi (2) ] at ith relay is givenby

yyyi = hisss+nnnRi (1)

where yi (j) is the jth symbol received at the ith relay, hi ∼CN (0,Ωh) is the Rayleigh-fading channel coefficient for thesource-ith relay link, nnnRi is complex additive white Gaussiannoise (AWGN) of variance N0, and i = 1, 2.

During the second time slot, the two relays constructAlamouti space–time scheme from the two received signals,and then retransmit a scaled version to the destination with thesame power constraint as in the first hop, whereas the sourceremains silent. To simplify relaying operation, the relayinggain is determined only to satisfy the average power constraintwith statistical channel state information (CSI) on hhh (not itsrealizations) at the relay. With this semi-blind relaying, theoutput signal of the two relays are defined as

XXX =[x1 (1) x2 (1)x1 (2) x2 (2)

]= G

[y1 (1) −y∗2 (2)y1 (2) y∗2 (1)

](2)

where xi (j) is the jth symbol transmitted from the ith relayand G is the scalar relaying gain defined in the following. Thereceived signal at the destination can be described as

r (j) =2∑i=1

fixi (j) + nD (j) (3)

where fi ∼ CN (0,Ωf) is the Rayleigh-fading channel coeffi-cient for the ith relay-destination link and nD (j) is complexadditive white Gaussian noise (AWGN) of variance N0. Notethat all the random variable hi and fi, i = 1, 2 are statistically

independent and the variations in ΩA, A ∈ h, f capture theeffect of distance-related path loss in each link. To constraintransmit power at the relay, we have

E

‖XXX‖2

F

= 4G2 (ΩhPs +N0) = E

‖sss‖2

F

(4)

yielding

G2 =12

(Ωh +

1SNR

)−1

(5)

where SNR = PsN0

is the common SNR of each link withoutfading [12]. The receive signals at the destination in (3) nowcan be equivalently described in the matrix form as

rrr =HHHsss+nnn (6)

where

rrr =[r (1)r∗ (2)

],

HHH = G

[h1f1 −h∗2f2h2f

∗2 h∗1f

∗1

],

which is complex orthogonal, i.e.,

HHH†HHH = G22∑i=1

|hi|2 |fi|2III2,

and

nnn = G

[f1nR1 (1) −f2n∗R2

(2)f∗1n

∗R1

(2) f∗2nR2 (1)

]+

[nD (1)n∗D (2)

]which is zero-mean AWGN of the variance

N0

(G2

2∑i=1

|fi|2 + 1)

. It is easy to see that the ML

decoding of the symbol vector sss turns out very simple astwo symbols in sss are independently decomposed from oneanother. Therefore, the instantaneous receive SNR is readilywritten as

γ = SNR

G22∑i=1

|hi|2 |fi|2

G22∑i=1

|fi|2 + 1(7)

To examine the statistical characteristic of γ given in (7) weconsider two special cases. If the relays are much closer to the

destination than the source, then we may have G22∑i=1

|fi|2 1. In this special case, we express γ in the following

γ = SNR

2∑i=1

|hi|2 |fi|2

2∑i=1

|fi|2(8)

On the other hand, if the relays are near the source, then the

following expression may hold G22∑i=1

|fi|2 1. Therefore,

the instantaneous receive SNR γ is as follows

γ = SNRG22∑i=1

|hi|2 |fi|2 (9)

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III. CLOSED-FORM EXPRESSION OF THE AVERAGE SEPAND DIVERSITY ORDER

In this section, on account of the statistical behavior of theinstantaneous receive SNR for two special cases as shownin previous section, we now derive the close-form expressionof the average SEP and then deduce the diversity order ofdistributed-Alamouti scheme in such cases.

A. When the relays are close to the destination

With the instantaneous receive SNR in (8), the MGF of γ,averaged over channel coefficients, is given by

φγ (ν) Eγ exp (−νγ) = Ehi,fiexp (−νγ) (10)

Since Ai ∼ CN (0,ΩA), A ∈ h, f and i = 1, 2, it is obviousthat |Ai|2 obeys an exponential distribution with hazard rate1/ΩA. The probability density function (p.d.f.) of |Ai|2 can bewritten as

p|Ai|2 (x) =1

ΩAexp (−x/ΩA) (11)

Since hi and fi are statistically independent with each other,the MGF of γ in (10) can be written as

φγ (ν) = EfiEhi

exp (−νγ)

= Efi

2∏i=1

(1 + SNRν

|fi|2|f1|2 + |f2|2

)−1

= Ez

(1 + ξ

z

z + 1

)−1 (1 + ξ

1z + 1

)−1

(a)=

∫ ∞

0

[(1 + (1 + ξ) z) (1 + ξ + z)

]−1dz

=2 log (1 + ξ)ξ (2 + ξ)

(12)

where z = |f1|2|f2|2 , ξ = ΩhSNRν, and (a) follows immediately

from Theorem 2 in Appendix. Using the well-known MGFapproach [10], [13] along with (12), we obtain the averageSEP of the distributed-Alamouti scheme with M -PSK in relaychannels as1

Pe =1π

∫ π− πM

0

φγ

(gMPSK

sin2 θ

)dθ

=1π

∫ π− πM

0

2 sin4 θ log(1 + ψ

sin2 θ

)ψ(2 sin2 θ + ψ

) dθ (13)

where ψ = ΩhSNRgMPSK and gMPSK = sin2 (π/M).

B. When the relays are close to the source

In this section, we will consider the case when the tworelays are close to the source. Following the same steps as in

1The result can be applied to other binary and M -ary signals in astraightforward way (see, e.g., [13]).

Section III-A and from (9), the MGF of γ can be describedas

φγ (ν) = Ehi,fi

2∏i=1

exp(−G2SNRν |hi|2 |fi|2

)(14)

=[∫ ∞

0

exp(−G2SNRνt

)pT (t) dt

]2

(15)

=

[∫ ∞

0

2Ω

exp(−G2SNRνt

)K0

(2

√t

Ω

)dt

]2

(16)

=[λ exp (λ) Γ (0, λ)

]2

(17)

where T = |hi|2 |fi|2, Ω = ΩhΩf, λ =(G2SNRνΩ

)−1, (16)

follows immediately from Theorem 3 in Appendix, and (17)can be obtained from the change of variable v = G2SNRνtalong with [14, eq. (8.353.4)]. Therefore, similarly as in theprevious subsection, the SEP is given by

Pe =1π

∫ π− πM

0

[τ sin2 θ exp

(τ sin2 θ

)Γ(0, τ sin2 θ

)]2

dθ

(18)

where τ =(G2SNRΩgMPSK

)−1.

C. High-SNR Characteristic: Diversity Order

We now assess the effect of cooperative diversity on theSEP behavior in a high-SNR regime. The diversity impact ofnon-regenerative cooperation on a high-SNR slope of the SEPcurve can be quantified by the following theorem.

Theorem 1 (Achievable Diversity Order): The non-regenerative cooperation of distributed-Alamouti schemeprovides maximum diversity order, i.e.,

D limSNR→∞

− logPe

log (SNR)= 2. (19)

Proof: See Appendix C

IV. NUMERICAL RESULTS

In this section, we validate our analysis by comparing withMonte-Carlo simulation. In the following numerical examples,we consider the non-regenerative relay protocol employingAlamouti code as in Section II. We assume collinear geometryfor locations of three communicating terminals, as shown inFig. 2. The path-loss of each link follows an exponential-decaymodel: if the distance between the source and destinationis equal to d, then Ω0 ∝ d−α where the exponent α = 4corresponding to a typical non line-of-sight propagation. Then,Ωh = ε−αΩ0 and Ωf = (1 − ε)−α Ω0.

Fig. 3 and Fig. 4 show the SEP of QPSK versus SNR whenthe relay approaches the destination with ε = 0.7 and ε = 0.8and the source with ε = 0.2 and ε = 0.3, respectively. Asseen from both figures, analytical and simulated SEP curvesmatch exactly. Observe that the PEP slops for ε = 0.7 andε = 0.8 are identical at the high SNR regime, as speculated inTheorem 1. The same observation can be obtained for ε = 0.2and ε = 0.3. For comparison, two examples demonstrate the

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performance is decreased when the relay approaches both end,e.g., the SEP for ε = 0.3 is slightly less than that for ε = 0.2and a 3dB-gain in SEP performance can be obtained withε = 0.7 compared to the case with ε = 0.8.

APPENDIX

A. Auxiliary Results

The following lemmas will be useful in the paperLemma 1: Let a > 0 be a finite constant and

f (x) =2 log (1 + ax)ax (2 + ax)

, x > 0 (20)

We have

fx↑ limx→∞

− log f (x)log x

= 2 (21)

Proof: It follows immediately from (20) and (21) that

fx↑ = limx→∞

− log log (1 + ax)log x︸ ︷︷ ︸

(b)=0

+ limx→∞

log (ax)log x︸ ︷︷ ︸

(c)=1

+ limx→∞

log(1 + ax

2

)log x︸ ︷︷ ︸

(d)= 1

= 2 (22)

where (b), (c), and (d) follow immediately by applyingl’Hospital rule.

Lemma 2: Let

g (ζ) =[ζ exp (ζ) Γ (0, ζ)

]2, ζ > 0 (23)

Let ζ = αβx+1x2 and α, β > 0 be finite constants. We have

gx↑ limx→∞

− log g (x)log x

= 2 (24)

Proof: Substituting ζ = αβx+1x2 into (23), it follows

immediately from (24) that

gx↑ = −2[

limx→∞

− log(βx+1x2

)log x

+ limx→∞

βx+ 1x2 log x

+ limx→∞

Γ(0, αβx+1

x2

)log x

]= −2

[−2 + lim

x→∞log (1 + βx)

log x︸ ︷︷ ︸(e)=1

+ limx→∞

βx+ 1x2 log x︸ ︷︷ ︸(f)= 0

+ limx→∞

Γ(0, αβx+1

x2

)log x︸ ︷︷ ︸

(g)= 0

]= 2 (25)

where (e), (f) follow immediately by l’Hospital rule and (g)follows from the Laguerre2 polynomial series representationof incomplete gamma function [14] together with l’Hospitalrule .

2The incomplete gamma function can be described as Γ (0, x) =

e−x∞∑

n=0

Ln(x)n+1

where Ln (x) =n∑

m=0(−1)m (n!xm) / (m! (n − m)!m!)

is the Laguerre polynomial of order n.

B. A Ratio and Product Distribution

Theorem 2 (Ratio Distribution): Let

X ∼ Υ (1/Ω)Y ∼ Υ (1/Ω)

be statistically independent and identically distributed (i.i.d.)exponential r.v.’s. Suppose the ratio Z of the form

Z =X

Y(26)

Then, we obtain the p.d.f. of random variable Z as

pZ (z) = (z + 1)−2 (27)

Proof: Note that

pZ (z) =∫ ∞

0

ypXY (yz, y) dy

=∫ ∞

0

y

Ω2exp

(−y z + 1

Ω

)dy

= (z + 1)−2 (28)

Theorem 3 (Product Distribution): Let

X ∼ Υ (1/Ωx)Y ∼ Υ (1/Ωy)

be statistically independent and not necessarily identicallydistributed (i.n.i.d.) exponential r.v.’s. Suppose the product Tof the form

T = XY (29)

Then, we have

pT (t) =2

ΩxΩyK0

(2

√t

ΩxΩy

)(30)

where K0 (.) is the zeroth-order modified Bessel function ofthe second kind.

Proof: Note that

FT (t) = PrXY ≤ t= EX

FT |X (t)

= EX

1 − exp

[− t

xΩy

]

= 1 − 1Ωx

∫ ∞

0

exp[− t

xΩy− x

Ωx

]dx (31)

The p.d.f. of T follows immediately from differentiating (31)with respect to t.

pT (t) =1

ΩxΩy

∫ ∞

0

1x

exp[− t

xΩy− x

Ωx

]dx

=2

ΩxΩyK0

(2

√t

ΩxΩy

)(32)

where the last equality follows from the change of variableu = x

Ωxalong with [14, eq. (8.432.6)] as desired.

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C. Proof of Theorem 1

Since the asymptotic behavior of the MGF φγ (ν) at largeSNR reveals a high-SNR slope of the SEP curve, we have [11]

D = limSNR→∞

− log φγ (gMPSK)log (SNR)

. (33)

Hence, it follow immediately from (12) and Lemma 1 witha = ΩhgMPSK that D = 2 when the relays are close to thedestination. Also, from (17) and Lemma 2 with α = 2

ΩhΩfgMPSK

and β = Ωh, we can obtain D = 2 as in (19) when the relaysare much close to the source.

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S DR

d

dε ( )1 dε−

0Ω

hΩ fΩ

Fig. 2. Collinear topology with an exponential-decay path loss model whereΩ0 ∝ d−α, Ωh = ε−αΩ0, and Ωf = (1 − ε)−α Ω0 with α = 4.

Fig. 3. Symbol error probability of QPSK versus SNR in non-regenerativerelay channels employing Alamouti scheme when ε = 0.7 and ε = 0.8 (closethe destination). Ω0 = 1.

Fig. 4. Symbol error probability of QPSK versus SNR in non-regenerativerelay channels employing Alamouti scheme when ε = 0.2 and ε = 0.3 (closethe source). Ω0 = 1.

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