Pairwise Error Probability of Distributed Space–Time Coding Employing Alamouti Scheme in Wireless...

Wireless Pers Commun (2009) 51:231–244DOI 10.1007/s11277-008-9640-9

Pairwise Error Probability of Distributed Space–TimeCoding Employing Alamouti Scheme in Wireless RelaysNetworks

Trung Q. Duong · Ngoc-Tien Nguyen · Trang Hoang ·Viet-Kinh Nguyen

Published online: 13 November 2008© Springer Science+Business Media, LLC. 2008

Abstract In this paper, we analyze the pairwise error probability (PEP) of distributedspace–time codes, in which the source and the relay generate Alamouti space–time code in adistributed fashion. We restrict our attention to the space–time code construction for Proto-col III in Nabar et al. (IEEE Journal on Selected Areas Communications 22(6): 1099–1109,2004). In particular, we derive two closed-form approximations for PEP when the relay iseither close to the destination or source and an upper bound for any position of the relay.Using the alternative definition of Q-function, we can express these PEPs in terms of finiteintegral whose integrand is composed of trigonometric functions. We further show that withonly one relay assisted source-destination link, system still achieves diversity order of two,assuming single-antenna terminals. We also perform Monte-Carlo simulations to verify theanalysis.

Keywords Distributed space–time codes · Relay channels · Pairwise error probability(PEP)

This paper was presented in part at the 7th International Symposium on Communications and InformationTechnologies, Sydney, Australia, Oct., 2007.

T. Q. Duong (B)Blekinge Institute of Technology, Ronneby 372 25, Swedene-mail: [email protected]

N.-T. NguyenMinistry of Posts and Telematics (MPT), Hanoi, Vietname-mail: [email protected]

T. HoangHeterogeneous Silicon Integration Department, CEA/LETI, Grenoble, Francee-mail: [email protected]

V.-K. NguyenUniversity of Technology-Hanoi National University, Hanoi, Vietname-mail: [email protected]

123

232 T. Q. Duong et al.

1 Introduction

It is well-known that the multiple-input multiple-output (MIMO) systems can remarkablyimprove the capacity and reliability of wireless communications over fading channels usingmultiplexing scheme and space–time coding [2–5]. However, the demand for low-cost andsmall-size portable devices has prohibited practical implementation of MIMO system.Recently, with the increasing interests in ad-hoc networks, cooperative diversity has beenproposed to exploit MIMO benefits in a distributed fashion [6–8].

Distributed space–time coding (DSTC) has been considered to achieve cooperativediversity in wireless relay networks, in which different relays work as co-located transmitantennas and construct a space–time code to realize spatial diversity gain [1,9–11]. Recently,a closed form expression of bit-error-rate (BER) has been presented for DSTC in [11]. Also,a simple DSTC for two relays has been investigated in [12,13]. More specifically, Alamoutischeme has been applied into relay systems where two single-antenna1 relays simultaneouslyreceive a noisy signal from the source and generate Alamouti space–time codes in a distrib-uted fashion before relaying the signals to the destination. This model has been demonstratedto obtain a full diversity order, i.e., the degree of diversity is order of two [14]. In practice, suchcode design is difficult due to distributed and ad-hoc nature, as opposed to codes designed forco-located (MIMO) systems. Furthermore, coherent reception of multiple relays transmissionobliges synchronization (at the symbol and carrier level) among multiple transmit-receivepairs, which enhances the receiver complexity. The problem we are interested in is whetherwe still achieve a full diversity order with only one relay.

The considered relay network model is similar to the Protocol III in [1]. Unlike [1], inwhich the channel of relay-destination link was assumed static, in this paper we analyze thePEP taking into account random fading channels for all links. The source communicates withthe destination during the first hop. In the second hop, both relay and source communicatewith the destination. We also assume the channel information is only available at the destina-tion. A key feature of our work is that the relay simply amplifies the signal and transmits tothe destination without any sort of signal regeneration, called non-regenerative or amplify-and-forward protocol. This simplification of relay operations avoid imposing bottlenecks onthe rate of the relays.

Our contribution in this work can be briefly described as follows. We analyze the PEPof DSTC, in which the relay and the source construct a distributed-Alamouti space-timecode (each terminal transmits one row of Alamouti code [15] to the destination). We thensuccessfully derive two closed-form expressions for PEP as the relay approaches both endsand an upper bound for any position of relay. Assessing two tight approximations of PEPin the high signal-to-noise ratio (SNR) regime, we further show that the considered distrib-uted-Alamouti system can achieve a full diversity gain with only one relay assisted the directcommunication. We also perform Monte-Carlo simulations to validate our analysis.

The rest of this paper is organized as follows. In Sect. 2, we briefly review the cooperativesystem of one non-regenerative relay based on Alamouti scheme. Two closed-form approxi-mations for PEP when the relay is either near to the destination or source and an upper boundfor any position of relay are derived in Sect. 3. Using the asymptotic (high SNR) PEP formulaswe show that the distributed system employing Alamouti codes achieves a diversity gain oforder two in Sect. 4. Numerical results are given in Sect. 5. Finally, Sect. 6 concludes the paper.

Notation: Throughout the paper, we shall use the following notations. Vector is written asbold lower case letter and matrix is written as bold upper case letter. The superscripts ∗ and †

1 In this paper, we only consider the single-antenna terminal.

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Pairwise Error Probability of Distributed Space–Time Coding 233

Fig. 1 Schematic of relay channel

stand for the complex conjugate and transpose conjugate, respectively. III n represents the n×nidentity matrix. ‖AAA‖F denotes Frobenius norm of the matrix AAA and |x | indicates the envelopeof x . Ex {.} is the expectation operator over the random variable x . A complex Gaussian dis-tribution with mean µ and variance σ 2 is denoted by CN (µ, σ 2). Let us denote Nm (mmm,��)

as a complex Gaussian random vector with mean vector mmm and covariance matrix ��. logis the natural logarithm. � (a, x) is the incomplete gamma function defined as � (a, x) =∫ ∞

x ta−1e−t dt and Kn(.) is the nth-order modified Bessel function of the second kind.

2 System Models

We consider a wireless relay network with three terminals as shown in Fig. 1. Every terminalhas a single antenna, which can not transmit and receive simultaneously. The relay terminalassists in communication by simply amplifying and forwarding received signals. We alsoassume all terminals are synchronized in the symbol level and channel remains constant fora coherence time (at least two symbol-intervals) and changes independently to a new valuefor each coherence time.

The source transmits the first row of Alamouti code to the destination during the first hopwith average transmit power per symbol Ps. For the first hop transmission, the receive signalat the relay is given by

yyyR = hSRsss1 + nnnR (1)

where hSR ∼ CN (0,�SR) is Rayleigh-fading channel coefficient for the source-relay link,sss1 = [ s1 s2 ] is the first row of Alamouti code, si (i = 1, 2) is selected from signal constel-lation S, and nnnR ∼ CN (0, N0) is complex additive white Gaussian noise (AWGN) at therelay.

In the second hop, the source sends the second row of Alamouti code to the destination,whereas the relay retransmits a scaled version of yyyR to the destination with the same powerconstraint as in the first hop. The relaying gain is determined only to satisfy the averagepower constraint with statistical channel state information (CSI) on hSR. For the second hoptransmission, the receive signal at the destination is readily written as

yyyD = hRDωyyyR + hSDsss2 + nnnD (2)

where ω is the scalar relaying gain, hRD ∼ CN (0,�RD) and hSD ∼ CN (0,�SD) areRayleigh-fading channel coefficients for the relay-destination and source-destination links,

123


respectively, sss2 = [−s∗2 s∗

1 ] is the second row of Alamouti code, and nnnD ∼ CN (0, N0) is theAWGN at the destination. Note that all the random quantities hA and nnnB, A ∈ {SD, SR, RD},B ∈ {R, D}, are statistically independent and the variations in �A capture the effect ofdistance-related path loss in each link. To constrain transmit power at the relay, we have

E

{∥∥ωyyyR

∥∥2

F

}= E

{‖sssi‖2F

}(3)

yielding

ω2 =(

�SR + 1

SNR

)−1

(4)

where SNR = PsN0

is the common SNR of each link without fading [8]. The receive signal atthe destination in (2) is now formed as follows

yyyD = hhhSSS + nnn (5)

where hhh = [hRDhSRω hSD

], SSS =

[s1 s2

−s∗2 s∗

1

]

is Alamouti space–time code, and nnn =hRDωnnnR + nnnD ∼ N2

(0, N0

(1 + |hRD|2 ω2

)III 2

).

3 Pairwise Error Probability

As mentioned above, assuming the destination knows channel information for all links, itis easy to see that yyyD|hA is a Gaussian random vector with mean vector hhhSSS and covariancematrix N0

(1 + |hRD|2 ω2

)III 2. Hence, the maximum-likelihood (ML) decoding of the system

can be readily seen to be

SSS = arg minS

∥∥yyyD − hhhSSS

∥∥2

F (6)

where the minimization is performed over all possible codeword matrices SSS. With the MLdecoding in (6), the PEP, given the channel coefficients hA, of mistaking SSS by EEE is obtained as

P(SSS → EEE | hA) = Q

(√‖hhh(SSS − EEE)‖2

F

2N0(1 + |hRD|2 ω2

)

)

(7)

It is important to note that SSS and EEE (SSS �= EEE) are the two possible codewords of Alamoutispace–time code, hence, we have

(SSS − EEE) (SSS − EEE)† = Psd2E III 2 (8)

where

d2E = (|s1 − e1|2 + |s2 − e2|2

)> 0 (9)

Applying the alternative definition of Q-function [16]: Q(x) = 1π

∫ π/20 exp

(− x2

2 sin2 θ

)dθ

into (7) and integrating over all channel realizations, we obtain the unconditional PEP asfollows:

P(SSS → EEE) = Eγ {P(SSS → EEE |γ )} = 1

π

π/2∫

0

φγ

(SNRd2

E

4 sin2 θ

)

dθ (10)

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where

γ = ‖hhh‖2F

1 + |hRD|2 ω2= |hRD|2 |hSR|2 ω2 + |hSD|2

1 + |hRD|2 ω2, (11)

and φγ (ν) � Eγ {exp (−νγ )} is the moment-generating function (MGF) of the random var-iable γ . To evaluate the integration in (10), next we discuss two specific cases: (i) when therelay is close to the destination (ii) when the relay is close to the source.

3.1 The Relay is Close to the Destination

If the relays are much closer to the destination than the source, then we may have 1 +|hRD|2 ω2

high SNR≈ |hRD|2 ω2. In this case (11) can be approximated as

γ ≈ |hSR|2 + |hSD|2|hRD|2 ω2

(12)

Recalling that hA, A ∈ {SD, SR, RD}, are assumed to be statistically independent, theMGF of γ , φγ (ν), can be determined by

φγ (ν) = φ|hSR|2(ν) φy

( ν

ω2

)(13)

where y = |hSD|2|hRD|2 . Since hA ∼ CN (0,�A), it is obvious that |hA|2 obeys an exponential

distribution with hazard rate 1/�A. The probability density function (p.d.f.) of |hA|2 can bewritten as

p|hA|2(x) = 1

�Aexp(−x/�A), (14)

yielding

φ|hSR|2(ν) = E|hSR|2{exp

(− |hSR|2 ν)} = (1 + ν�SR)−1 (15)

Following Lemma 1 in the Appendix 6, p.d.f. of y is given by

pY (y) = �SD�RD(�SD + �RD y)−2 , (16)

yielding

φy

( ν

ω2

)=

∞∫

0

exp(− νy

ω2

)�SD�RD(�SD + �RD y)−2 dy

= ν�SD

ω2�RDexp

(ν�SD

ω2�RD

)

�

(

−1,ν�SD

ω2�RD

)

(17)

where (17) follows immediately from the change of variable u = �SD + �RD y and [17,Eq. (3.381.1)]. Substituting (13), (15), and (17) into (10), we obtain

P (SSS → EEE) = 1

π

π/2∫

0

(1 + ξ�SR

)−1[ξ exp

(ξ)�

(−1, ξ)]

dθ (18)

123


where

ξ = SNRd2E

4 sin2 θ(19)

and

ξ = �SD

�RDω2 ξ (20)

3.2 The Relay is Close to the Source

On the other hand, if the relays are much close to the source, the following approximation

may be hold 1 + |hRD|2 ω2high SNR≈ 1. In this case, (11) can be readily written as

γ ≈ |hRD|2 |hSR|2 ω2 + |hSD|2 (21)

yielding

φγ (ν) = φz(ω2ν

)φ|hSD|2(ν) (22)

with

φ|hSD|2(ν) = E|hSD|2{exp

(−ν |hSD|2)} = (1 + �SDν)−1 (23)

and φz(ω2ν

), where z = |hRD|2 |hSR|2, is given by

φz(ω2ν

) = Ez{exp

(−ω2νz)}

=∞∫

0

exp(−ω2νz

) 2

�SR�RDK0

(

2

√z

�SR�RD

)

dz (24)

= λ exp(λ) �(0, λ) (25)

where λ = (ω2�SR�RDν

)−1, (24) follows immediately from [14, Theorem 3], and (25) can

be obtained from the change of variable t = ω2 · ν · z along with [17, Eq. (8.352.4)].Combining (22), (23), (25), and (10), the PEP becomes

P (SSS → EEE) = 1

π

π/2∫

0

(1 + ξ�SD

)−1(

ζ exp (ζ ) � (0, ζ )

)

dθ (26)

where ξ is given in (19) and ζ = (ω2�SR�RDξ

)−1.

We can clearly see that the PEPs are given in closed-form expressions when the relay iseither close to the destination or source, shown in (18) and (26), respectively. These results canbe readily calculated by common mathematical software packages such as MATHEMATICAor MAPLE.

3.3 An Upper Bound of Pairwise Error Probability

So far, the closed-form approximations of PEP have been considered for the case when therelay is close to both ends. In this subsection, we will derive the upper bound of PEP for anyposition of relay. From (11), the value of γ can be shown as

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γ >|hRD|2 |hSR|2

1ω2 + |hRD|2 = γ0. (27)

Applying (27) into (10), we can upper bound the PEP as

P (SSS → EEE) <1

π

π/2∫

0

φγ0

(SNRd2

E

4 sin2 θ

)

dθ

= 1

π

π/2∫

0

1

1 + ξ�SR

[

1 + ρξ�SR exp (ρ) � (0, ρ)

]

dθ (28)

where (28) follows immediately from Lemma 2 in the Appendix, ξ is given in (19), and

ρ =[

ω2�RD(1 + ξ�SR

)]−1

4 Diversity Order

Noting that, in [1] the diversity order was obtained from the Chernoff bound (not a tight bound)with an assumption that the relay-to-destination link is static (non-fading) and deduced fromthe upper bound of PEP (assuming fading relay-destination link) when the relay is close tothe source. In this section, we quantify the effect of our relay protocol on the PEP curve in thehigh-SNR regime when the relay approaches both ends. Using the two tight approximationsof PEP derived Sects. 3.1 and 3.2, we can assess the diversity order of distributed-Alamoutisystems by the following theorem.

Theorem 1 (Achievable Diversity Order) The non-regenerative cooperation of our schemeprovides maximum diversity order, i.e., D = 2, when the relay is near both ends.

Proof The diversity has been defined as the absolute values of the slopes of the error proba-bility (e.g., PEP) curve plotted on a log–log scale in high SNR regime [16], i.e.,

D � limSNR→∞

− log P (SSS → EEE)

log (SNR)(29)

As can be seen from (10), the PEP is expressed in a form of finite integral whose integrandis the MGF of random variable γ . Therefore, the asymptotic behavior of the MGF φγ (ν) atlarge SNR reveals a high-SNR slope of the PEP curve, we have

D = limSNR→∞

− log φγ (ν) |θ=π/2

log (SNR). (30)

• The relay is close to the destinationSubstituting (15) and (17) into (13), the diversity order D in (30) is now shown as

D = limx→∞ log(1+αx)log x − limx→∞ log(βx exp(βx)�(−1,βx))

log x

= 1 − (−1) = 2 (31)

where α = d2E �SR

4 , β = d2E �SD

4ω2�RD, x = SNR, and (31) follows immediately by applying

l’Hospital rule.• The relay is close to the source

Substituting (23) and (25) into (22), the diversity order D is now given by

123


Fig. 2 Collinear topology with an exponential-decay path loss model where �SD ∝ d−α , �SR = ε−α�SD,and �RD = (1 − ε)−α �SD with α = 4

D = limx→∞ log(1+ηx)log x − limx→∞

log( ϕx exp( ϕ

x )�(0,ϕx ))

log x

= 1 − (−1) = 2 (32)

where η = d2E �SD

4 , ϕ = 4(d2

Eω2�SR�RD)−1

, and (32) follows immediately froml’Hospital rule.

This completes the proof.2 �

5 Numerical Results

In this section, we validate our analysis by comparing with simulation. In the followingnumerical examples, we consider the non-regenerative relay protocol employing Alamouticode as in Sect. 2. We assume collinear geometry for locations of three communicating termi-nals, as shown in Fig. 2. The path loss of each link follows an exponential-decay model: if thedistance between the source and destination is equal to d , then�SD ∝ d−α where the exponentα = 4 corresponding to a typical non line-of-sight propagation. Then, �SR = ε−α�SD and�RD = (1 − ε)−α �SD. For Alamouti code transmission, the source send symbols selectedfrom binary phase-shift keying (BPSK). The reason we use BPSK modulation is to simplifythe calculation of PEP as described in the following. For the normalization, the BPSK con-stellation points are −1 and 1. When only one of the corresponding symbols in SSS and EEE aredifferent, for example s1 = e1 and s2 �= e2, we have d2

E = 4. On the other hand, when bothcorresponding symbols in the transmit and receive codewords are different from each other,it holds d2

E = 8.Figures 3 and 4 draw the PEP versus SNR when the relay approaches the destination for

ε = 0.6, 0.7, 0.8 with d2E = 4 and the source for ε = 0.2, 0.3, 0.4 with d2

E = 8. As canbe clearly seen from both figures, analytical and simulated PEP curves match exactly whenthe relay is located near by both ends. Observe that the PEP slops for ε = 0.2, ε = 0.3,and ε = 0.4 are identical at the high SNR regime, as speculated in Theorem 1. Similarobservation can be made for ε = 0.6, ε = 0.7, and ε = 0.8.

However when the relay moves far away from both ends, analytical and simulated curvesdo not completely agree together, for example ε = 0.4 and ε = 0.6, since the approximationsin (12) and (21) only satisfy when the relay is closely located to the destination and source,respectively.

In Fig. 5, the PEP at SNR=20 dB with d2E = 4 is depicted as a function of the fraction

ε. It is clear to see that the performance is decreased when the relay approaches both end

2 It has been confirmed by using mathematical software package MATHEMATICA.

123


Fig. 3 Pairwise error probabilityof BPSK versus SNR innon-regenerative relay channelsemploying Alamouti schemewhen ε = 0.6 , 0.7 , 0.8 withd2

E = 4 (the relay is close to the

destination). �SD = 116

Fig. 4 Pairwise error probability of BPSK versus SNR in non-regenerative relay channels employingAlamouti scheme when ε = 0.2 , 0.3 , 0.4 with d2

E = 8 (the relay is close to the source). �SD = 116

and the symmetric geometry (ε = 0.5) shows an optimal performance. For comparison, wealso plot the upper bound of PEP. The upper bound closely matches the simulated curvewhen the relay approaches the destination. It can be explained in the following: At the rela-tively high SNR regime, we have ω2 ≈ 1

�SRwhich makes the numerator of (11) becomes

|hRD|2|hSR|2�SR

+ |hSD|2. In addition, when the relay is close to the destination, �SR is relativelysmall leading to γ ≈ γ0, given in (27).

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Fig. 5 Pairwise error probability of BPSK as a function of the fraction ε with d2E = 4 and SNR = 15 dB in

non-regenerative relay channels employing Alamouti scheme. We also plot the upper bound of PEP

6 Conclusion

In this paper, we have analyzed the PEP of cooperative system, in which the source and therelay generate Alamouti space–time code in a distributed fashion to exploit the benefit ofMIMO system in relay fading channels. Specifically, two tight approximations of PEP as therelay approaches both ends and an upper bound of PEP for any position of relay have beenderived in closed-form expressions. We have quantified the effect of PEP in the high SNRregime and shown that the full diversity order can be achieved.

Appendix 1: Auxiliary Results

The following lemmas will be helpful in the paper.

Lemma 1 Let X and Y be statistically independent and not necessarily identically dis-tributed (i.n.i.d.) exponential random variables with hazard rate �x and �y, respectively.Suppose that the ratio of Z takes the form

Z = X

Y(33)

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

pZ (z) = �x�y(�x + �yz

)−2 (34)

Proof Note that

pZ (z) =∞∫

0

y · pXY (yz, y) dy (35)

123


=∞∫

0

y

�x�yexp

[

−(

z

�x+ 1

�y

)

y

]

dy (36)

= �x�y(�x + �yz

)−2 (37)

where (36) follows immediately from (14) and the statistical independence of X and Y . Thiscompletes the proof. �Lemma 2 Let X and Y be statistically independent and not necessarily identically dis-tributed (i.n.i.d.) exponential random variables with hazard rate �x and �y, respectively.Suppose that the ratio of Z takes the form

Z = XY

a + Y, a > 0. (38)

Then, we obtain the MGF of random variable Z as

φZ (ν) = 1

1 + ν�x

[

1 + τν�x exp (τ ) � (0, τ )

]

(39)

with

τ = a

�y (1 + ν�x )(40)

Proof The cumulative distribution function (c.d.f.) of Z is given by

FZ (z) = Pr{ XY

a + Y≤ z}

= EY{

FZ |Y (z)}

= EY

{

1 − exp

[

− z (a + Y )

�x Y

]}

= 1 − 1

�y

∞∫

0

exp

[

− z (a + y)

�x y− y

�y

]

dy (41)

The p.d.f. of Z is obtained by differentiating (41) with respect to z as follows

pZ (z) = 1

�x�y

∞∫

0

(a + y

y

)

exp

[

− z (a + y)

�x y− y

�y

]

dy

= 1

�x�y

∞∫

0

exp

[

− z (a + y)

�x y− y

�y

]

dy

+ 1

�x�y

∞∫

0

a

yexp

[

− z (a + y)

�x y− y

�y

]

dy (42)

= 2

�x�yexp

(

− z

�x

) √az�y

�xK1

(

2√

az

�x�y

)

+ 2a

�x�yexp

(

− z

�x

)

K0

(

2√

az

�x�y

)

. (43)

123


where (43) follows immediately from [17, eq. (3.471.9)]. The MGF of Z can be expressedby

φZ (ν) = EZ {exp (−νz)}

= 2

�x�y

√a�y

�x

∞∫

0

z1/2 exp

[

−(

1

�x+ ν

)

z

]

K1

(

2√

az

�x�y

)

dz

+ 2a

�x�y

∞∫

0

exp

[

−(

1

�x+ ν

)

z

]

K0

(

2√

az

�x�y

)

dz (44)

= 1

1 + ν�x

[

1 + τν�x exp (τ ) � (0, τ )

]

(45)

where (44) is obtained from (43), (45) follows immediately from [17, Eq. (8.353.4)] andsome simple manipulations, and τ is given in (40). This completes the proof. �

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Author Biographies

Trung Q. Duong was born in Hoi An Town, Quang Nam Province, Vietnam, in 1979. He received the B.S.degree in electrical-electronics engineering from Ho Chi Minh City University of Technology, Vietnam, in2002, and the M.S. degree in computer engineering from Kyung Hee University, South Korea, in 2005. In April2004, he joined the Faculty of Electrical Engineering, Faculty of Ho Chi Minh City University of Transport,Vietnam. He was a recipient of the Korean Government IT Scholarship Program for International GraduateStudents. In December 2006, he was awarded the Best Paper Award of IEEE Student Paper Contest—IEEESeoul Section. He finished a two-year Ph.D. course in radio communications engineering from Kyung HeeUniversity, South Korea, in 2007. In December 2007, he joined the staff of the School of Engineering, BlekingeInstitute of Technology (BTH), Sweden. His current research interests include wireless and mobile commu-nications.

Ngoc-Tien Nguyen was born in Hanoi, Vietnam, on February 26, 1964.He received the B.S. degree in electronics and communications fromMilitary Technical Academy, Vinh Yen, Vietnam, in 1985, and the M.S.degree in electronics and telecommunications from Hanoi National Uni-versity. Currently, he is a Ph.D. candidate at the Vietnam Posts andTelecommunications Institute of Technology (PTIT). Since 1993, he hasbeen working at the Ministry of Posts and Telematics (MPT) of Viet-nam. His research interests include multi-user detection for DS-CDMAsystems, OFDM, MC-CDMA, and MIMO.

Trang Hoang received the B.S. and M.S. degree in electronics-telecommunications engnieering from Ho Chi Minh City University ofTechnology, Viet Nam in 2002 and 2004, respectively. Currently, he is aresearch staff in CEA/LETI-Minatec, Heterogeneous Integartion on Sil-icon Division, Grenoble working towards his Ph.D. degree in the fieldof MEMS at University Joseph Fourier, France. His research interest isin the domain of integration of passive components, SAW devices, ICdesign, SoC and MEMS fabrication.

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Viet-Kinh Nguyen received the Ph.D. degree in electrical engineeringfrom Poland Academy of Sciences, in 1977. Nowadays, he is a professorteaching at University of Technology, Hanoi National University, Viet-nam. His current research interests are CDMA, MC-CDMA, OFDM andMIMO.

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