DSpace at VNU: Pairwise Error Probability of Distributed Space-Time Coding Employing Alamouti Scheme in Wireless Relays Networks

DSpace at VNU: Pairwise Error Probability of Distributed Space-Time Coding Employing Alamouti Scheme in Wireless Relays...

DOI 10.1007/s11277-008-9640-9

Pairwise Error Probability of Distributed Space–Time

Coding Employing Alamouti Scheme in Wireless Relays

Networks

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 distributed space–time codes, in which the source and the relay generate Alamouti space–time code in a distributed 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 is either 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 finite

integral whose integrand is composed of trigonometric functions We further show that with only 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 the analysis

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 Information Technologies, Sydney, Australia, Oct., 2007.

T Q Duong (B)

Blekinge Institute of Technology, Ronneby 372 25, Sweden

e-mail: quang.trung.duong@bth.se

N.-T Nguyen

Ministry of Posts and Telematics (MPT), Hanoi, Vietnam

e-mail: nntien@mpt.gov.vn

T Hoang

Heterogeneous Silicon Integration Department, CEA/LETI, Grenoble, France

e-mail: trang.hoang@cea.fr

V.-K Nguyen

University of Technology-Hanoi National University, Hanoi, Vietnam

e-mail: kinhnv@vnu.edu.vn

1 Introduction

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

Distributed space–time coding (DSTC) has been considered to achieve cooperative diversity in wireless relay networks, in which different relays work as co-located transmit antennas 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, Alamouti scheme has been applied into relay systems where two single-antenna1relays simultaneously receive 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 demonstrated

to obtain a full diversity order, i.e., the degree of diversity is order of two [14] In practice, such

code design is difficult due to distributed and ad-hoc nature, as opposed to codes designed for co-located (MIMO) systems Furthermore, coherent reception of multiple relays transmission

obliges synchronization (at the symbol and carrier level) among multiple transmit-receive pairs, which enhances the receiver complexity The problem we are interested in is whether

we 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], in which the channel of relay-destination link was assumed static, in this paper we analyze the PEP taking into account random fading channels for all links The source communicates with the destination during the first hop In the second hop, both relay and source communicate with 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 to the destination without any sort of signal regeneration, called non-regenerative or amplify-and-forward protocol This simplification of relay operations avoid imposing bottlenecks on the rate of the relays

Our contribution in this work can be briefly described as follows We analyze the PEP

of DSTC, in which the relay and the source construct a distributed-Alamouti space-time code (each terminal transmits one row of Alamouti code [15] to the destination) We then successfully derive two closed-form expressions for PEP as the relay approaches both ends and an upper bound for any position of relay Assessing two tight approximations of PEP

in 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 direct communication 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 cooperative system 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 bound for any position of relay are derived in Sect.3 Using the asymptotic (high SNR) PEP formulas

we show that the distributed system employing Alamouti codes achieves a diversity gain of order two in Sect.4 Numerical results are given in Sect.5 Finally, Sect.6concludes the paper

Notation: Throughout the paper, we shall use the following notations Vector is written as

bold 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.

Fig 1 Schematic of relay channel

stand for the complex conjugate and transpose conjugate, respectively III n represents the n ×n

identity matrix.AAAFdenotes Frobenius norm of the matrix A A A and |x| indicates the envelope

of x.Ex{.} is the expectation operator over the random variable x A complex Gaussian

dis-tribution with meanµ and variance σ2is denoted byCN (µ, σ2) Let us denote ˜ N m (m m ,  )

as a complex Gaussian random vector with mean vector m m m and covariance matrix   log

is the natural logarithm. (a, x) is the incomplete gamma function defined as  (a, x) =

∞

x t a−1e −t dt and K n (.) 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 terminal has a single antenna, which can not transmit and receive simultaneously The relay terminal assists in communication by simply amplifying and forwarding received signals We also assume all terminals are synchronized in the symbol level and channel remains constant for

a coherence time (at least two symbol-intervals) and changes independently to a new value for each coherence time

The source transmits the first row of Alamouti code to the destination during the first hop with average transmit power per symbolPs For the first hop transmission, the receive signal

at the relay is given by

where hSR∼CN (0, SR) is Rayleigh-fading channel coefficient for the source-relay link, sss1 = [ s1s2] is the first row of Alamouti code, s i (i = 1, 2) is selected from signal

constel-lationS, and n nR ∼CN (0, N0) is complex additive white Gaussian noise (AWGN) at the

relay

In the second hop, the source sends the second row of Alamouti code to the destination,

whereas the relay retransmits a scaled version of yyyRto the destination with the same power constraint as in the first hop The relaying gain is determined only to satisfy the average

power constraint with statistical channel state information (CSI) on hSR For the second hop transmission, the receive signal at the destination is readily written as

yyyD= hRDωyyyR+ hSDsss2+ nnnD (2) whereω is the scalar relaying gain, hRD ∼ CN (0, RD) and hSD ∼ CN (0, SD) are

Rayleigh-fading channel coefficients for the relay-destination and source-destination links,

respectively, sss2= [ −s2∗s∗

1] is the second row of Alamouti code, and nnnD∼CN (0, N0) is the

AWGN at the destination Note that all the random quantities hAand n nB,A∈ {SD,SR,RD},

B∈ {R,D}, are statistically independent and the variations in A capture the effect of distance-related path loss in each link To constrain transmit power at the relay, we have

E

ωyyyR2 F



=E

sss i2 F



(3) yielding

ω2=



SR+ 1 SNR

−1

(4) whereSNR= Ps

N0 is the common SNR of each link without fading [8] The receive signal at the destination in (2) is now formed as follows

where h h = hRDhSRω hSD , S S =

s1 s2

−s∗

2 s∗ 1

is Alamouti space–time code, and n n =

hRDωnnnR+ nnnD∼ ˜N2 0, N0 1+ |hRD|2ω2

III2

3 Pairwise Error Probability

As mentioned above, assuming the destination knows channel information for all links, it

is easy to see that yyyD|hAis a Gaussian random vector with mean vector h h hS S S and covariance

matrix N0 1+ |hRD|2ω2

III2 Hence, the maximum-likelihood (ML) decoding of the system can be readily seen to be

ˆSˆSˆS = arg min S

yyyD− hhhSSS2

where the minimization is performed over all possible codeword matrices S S S With the ML

decoding in (6), the PEP, given the channel coefficients hA, of mistaking S S S by E E E is obtained as

P(SSS → EEE| hA) = Q



hhh(SSS − EEE)2

F

2N0 1+ |hRD|2ω2



(7)

It is important to note that S S S and E E E (S S = EEE) are the two possible codewords of Alamouti

space–time code, hence, we have

(SSS − EEE) (SSS − EEE)†=Psd2E III2 (8) where

d2E= |s1− e1|2+ |s2− e2|2

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

π

π/2



− x2

2 sin2θ



d θ

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

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

π

π/2



0

φ γ

 SNRd E2

4 sin2θ



γ = hhh2F

1+ |hRD|2ω2 = |hRD|2|hSR|2ω2+ |hSD|2

andφ γ (ν)
Eγ {exp (−νγ )} is the moment-generating function (MGF) of the random

var-iableγ To evaluate the integration in (10), next we discuss two specific cases: (i) when the relay 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|2ω2high SNR≈ |hRD|2ω2 In this case (11) can be approximated as

γ ≈ |hSR|2+ |hSD|2

Recalling that hA,A ∈ {SD,SR,RD}, are assumed to be statistically independent, the MGF ofγ , φ γ (ν), can be determined by

φ γ (ν) = φ |hSR|2(ν) φ y  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|2can be written as

p |hA | 2(x) = 1

A

yielding

φ |hSR|2(ν) =E|hSR|2

 exp − |hSR|2ν= (1 + νSR)−1 (15) Following Lemma1in the Appendix6, p.d.f of y is given by

yielding

φ y  ν

ω2



=

∞



0

exp



−νy

ω2



SDRD(SD+ RDy )−2d y

= νSD

ω2RD

exp



νSD

ω2RD







−1, νSD

ω2RD



(17)

where (17) follows immediately from the change of variable u = SD+ RDy and [17,

Eq (3.381.1)] Substituting (13), (15), and (17) into (10), we obtain

P (SSS → EEE) = 1

π

π/2



0

1+ ξSR

−1

ˆξ exp 1, ˆξdθ (18)

ξ =SNRd2E

and

ˆξ = SD

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|2ω2high SNR≈ 1 In this case, (11) can be readily written as

yielding

with

φ |hSD|2(ν) =E|hSD|2

 exp −ν |hSD|2

= (1 + SDν)−1 (23)

andφ z ω2ν, where z = |hRD|2|hSR|2, is given by

φ z ω2ν=Ez

exp −ω2νz

=

∞



0

exp −ω2νz 2

SRRDK0

 2



z

SRRD



whereλ = ω2SRRDν−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+ ξSD

−1

ζ exp (ζ)  (0, ζ)



whereξ is given in (19) andζ = ω2SRRDξ−1

We can clearly see that the PEPs are given in closed-form expressions when the relay is either close to the destination or source, shown in (18) and (26), respectively These results can

be readily calculated by common mathematical software packages such as MATHEMATICA

or 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 the relay is close to both ends In this subsection, we will derive the upper bound of PEP for any position of relay From (11), the value ofγ can be shown as

γ > |hRD|2|hSR|2 1

ω2 + |hRD|2 = γ0. (27) Applying (27) into (10), we can upper bound the PEP as

P (SSS → EEE) < 1

π

π/2



0

φ γ0

 SNRd2

E

4 sin2θ



dθ

= 1

π

π/2



0

1

1+ ξSR

1+ ρξSRexp(ρ)  (0, ρ)

where (28) follows immediately from Lemma2in the Appendix,ξ is given in (19), and

ρ =

ω2RD 1+ ξSR

−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 from the upper bound of PEP (assuming fading relay-destination link) when the relay is close to the source In this section, we quantify the effect of our relay protocol on the PEP curve in the high-SNR regime when the relay approaches both ends Using the two tight approximations

of PEP derived Sects.3.1and3.2, we can assess the diversity order of distributed-Alamouti systems by the following theorem

Theorem 1 (Achievable Diversity Order) The non-regenerative cooperation of our scheme

provides 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.,

SNR →∞

− log P (SSS → EEE)

As can be seen from (10), the PEP is expressed in a form of finite integral whose integrand

is the MGF of random variableγ Therefore, the asymptotic behavior of the MGF φ γ (ν) at

largeSNRreveals a high-SNR slope of the PEP curve, we have

D= lim

SNR →∞

− log φ γ (ν) | θ=π/2

• The relay is close to the destination

Substituting (15) and (17) into (13), the diversity order D in (30) is now shown as

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

whereα = d2

E SR

4 ,β = d2

E SD

4ω2RD, 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

Fig 2 Collinear topology with an exponential-decay path loss model whereSD∝ d −α,SR= ε −α SD , andRD= (1 − ε) −α SD withα = 4

D = limx→∞loglog x (1+ηx)− limx→∞log( ϕ

xexp( ϕ

x )  (0, ϕ x )) log x

where η = d2E SD

4 , ϕ = 4 d2

E ω2SRRD

−1 , and (32) follows immediately from l’Hospital rule

This completes the proof.2

5 Numerical Results

In this section, we validate our analysis by comparing with simulation In the following numerical examples, we consider the non-regenerative relay protocol employing Alamouti code 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 the

distance between the source and destination is equal to d, then SD∝ d −αwhere the exponent

α = 4 corresponding to a typical non line-of-sight propagation Then, SR =  −α SDand

RD= (1 − ) −α SD For Alamouti code transmission, the source send symbols selected from binary phase-shift keying (BPSK) The reason we use BPSK modulation is to simplify the 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 are different, for example s1= e1and s2= e2, we have d2E= 4 On the other hand, when both corresponding symbols in the transmit and receive codewords are different from each other,

it holds d E2 = 8

Figures3and4draw the PEP versusSNRwhen the relay approaches the destination for

 = 0.6, 0.7, 0.8 with d2

E = 4 and the source for  = 0.2, 0.3, 0.4 with d2

E = 8 As can

be clearly seen from both figures, analytical and simulated PEP curves match exactly when the relay is located near by both ends Observe that the PEP slops for = 0.2,  = 0.3,

and = 0.4 are identical at the high SNR regime, as speculated in Theorem 1 Similar observation can be made for = 0.6,  = 0.7, and  = 0.8.

However when the relay moves far away from both ends, analytical and simulated curves

do not completely agree together, for example = 0.4 and  = 0.6, since the approximations

in (12) and (21) only satisfy when the relay is closely located to the destination and source, respectively

In Fig.5, the PEP atSNR=20 dB with d2

E = 4 is depicted as a function of the fraction

 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.

Fig 3 Pairwise error probability

of BPSK versus SNR in

non-regenerative relay channels

employing Alamouti scheme

when = 0.6 , 0.7 , 0.8 with

d E2 = 4 (the relay is close to the

destination).SD = 1

16

Fig 4 Pairwise error probability of BPSK versusSNR in non-regenerative relay channels employing Alamouti scheme when = 0.2 , 0.3 , 0.4 with d2

E = 8 (the relay is close to the source) SD = 1

16

and the symmetric geometry ( = 0.5) shows an optimal performance For comparison, we

also plot the upper bound of PEP The upper bound closely matches the simulated curve when the relay approaches the destination It can be explained in the following: At the rela-tively highSNRregime, we haveω2 ≈ 1

SR which makes the numerator of (11) becomes

|hRD | 2|hSR | 2

SR + |hSD|2 In addition, when the relay is close to the destination,SRis relatively small leading toγ ≈ γ0, given in (27)

Fig 5 Pairwise error probability of BPSK as a function of the fraction with d2

E= 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 the relay generate Alamouti space–time code in a distributed fashion to exploit the benefit of MIMO system in relay fading channels Specifically, two tight approximations of PEP as the relay approaches both ends and an upper bound of PEP for any position of relay have been derived in closed-form expressions We have quantified the effect of PEP in the high SNR regime 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

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

p Z (z) =  x  y  x +  y z−2

(34)

Proof Note that

p Z (z) =

∞



0

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... the following numerical examples, we consider the non-regenerative relay protocol employing Alamouti code as in Sect.2 We assume collinear geometry for locations of three communicating termi-nals,...

III2

3 Pairwise Error Probability< /b>

As mentioned above, assuming the destination knows channel information for all links, it

is easy to see that yyyD|hAis