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Noncoherent Decode-and-Forward Cooperative Systems with Maximum Energy Selection Ha X.. While finding the optimal thresholds requires information on the average signal-to-noise ratios SNR

Noncoherent Decode-and-Forward Cooperative Systems with Maximum Energy Selection

Ha X Nguyen1, Cuu V Ho2, Chan Dai Truyen Thai3, Danh T Nguyen4

1ha.nguyen@ttu.edu.vn, School of Engineering, Tan Tao University

Tan Duc Ecity, Duc Hoa, Long An Province, Vietnam

2cuu.hv@cb.sgu.edu.vn, Department of Electronics and Telecommunications, Saigon University

273 An Duong Vuong, District 5, Ho Chi Minh City, Vietnam

3chan.thai@ifsttar.fr, Univ Lille Nord de France

F-59000, Lille, IFSTTAR, LEOST, F-59650, Villeneuve d’Ascq, France

4nthanhdanh0410@gmail.com, Faculty of Electronics and Telecommunications, University of Science

Vietnam National University, Ho Chi Minh City, Vietnam

Abstract—This paper investigates the performance of a

max-imum energy selection receiver of an adaptive

decode-and-forward (DF) relaying scheme for a cooperative wireless system.

In particular, a close-form expression for the bit-error-rate (BER)

is analytically derived when the system is deployed with binary

frequency-shift keying (BFSK) modulation The thresholds used

at the relays to address the issue of error propagation are

opti-mized to minimize the BER While finding the optimal thresholds

requires information on the average signal-to-noise ratios (SNRs)

of all the transmission links in the system, the approximate

threshold at each relay that requires only information on the

average SNR of the source-corresponding relay is investigated.

It is also shown that the system achieves a full diversity order

with the approximate thresholds Both analytical and simulation

results are provided to validate our theoretical analysis.

I INTRODUCTION

Frequency shift keying (FSK) is a popular modulation

scheme in noncoherent communications in which the receiver

does not require any channel state information (CSI) to decode

the transmitted signals [1] Consequently, using FSK signals

in cooperative systems has been focused recently since there

is a complexity advantage in decoding [2]–[7] It is due to

the fact that there are many wireless fading channels involved

in the systems [8], [9], which makes the task of channel

estimation more difficult With the decode-and-forward (DF)

protocol employing FSK in cooperative systems, reference [3]

proposed maximum likelihood (ML) and suboptimal piecewise

linear (PL) schemes to decode the signals at the destination

However, it was shown that the system could not achieve a

full diversity order due to the error forwarding at the relays

References [6], [7] proposed to use a threshold at the relays

to address the issue of error propagation for binary

frequency-shift keying (BFSK) modulation While the destination in [6]

combines all the signals from the retransmitting relays, the

destination in [7] selects only one signal with the largest

magnitude of the energy difference to decode Unfortunately,

designing the optimal thresholds to minimize the average

bit-error-rate (BER) of the system relies on the MATLAB

Op-timization Toolbox and a theoretical analysis of the diversity

order is not available

This paper studies the maximum energy selection (MES)

receiver, i.e., selecting the maximum output from the square-law detectors of all branches to perform a detection, for a threshold-based (i.e., adaptive) DF cooperative system While the destination in [7] relies on the maximum magnitude of the energy difference, the destination in this paper employs the maximum energy from the square-law detectors to detect the transmitted signal The approximate thresholds that achieve full diversity are provided in this paper Note that the direct link between the source and destination is considered in this work while the work in [7] assumes that there is no such a link

II SYSTEMMODEL

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Fig 1 System description of the proposed scheme.

Fig 1 illustrates the signal transmission from the source (node 0) to destination (node K + 1) with the assistance

of K half-duplex relays (node i, i = 1, , K) The relays retransmit signals to the destination in orthogonal channels

In this paper, we assume that the fading channel coefficient between transmit nodei and receive node j, denoted by hi,j, and the noise component at receive node j, denoted by ni,j, are modeled as zero-mean complex Gaussian random variables with variances σ2

i,j and N0, respectively The instantaneous signal-to-noise ratio (SNR) of the channel between node i and node j, which is denoted by γi,j, is given as γi,j =

Ei|hi,j|2/N0 where Ei is the average transmitted energy of

nodei The corresponding average SNR is γi,j=Eiσ2

i,j/N0.

In the first phase, the source broadcasts the signal xmand

the received signals at node i, i = 1, , K + 1, are written

as

y0,i=

E0h0,ixm+n0,i, i = 1, 2, , K + 1 (1)

wherexmis themth symbol of an BFSK constellation

Without loss of generality, assume that the first symbol

from the signal constellation is transmitted The outputs of

the square-law detector for the first and second symbols at

node i, i = 1, , K + 1 are written, respectively, as

y0,i,1 = E0h0,i+n0,i,12, (2)

As in [6], the difference of the outputs of the

square-law detector, namely θ0,i = y0,i,1 − y0,i,2, is considered

as a reliability measure of the detection at node i

There-fore, node i decodes and retransmits a BFSK signal only if

θ0,i > θth

r .When nodei transmits a correct bit in the second

phase, the outputs of the square-law detector for the first and

second symbols at the destination are

yi,K+1,1 = Eih0,i+ni,K+1,12, (4)

yi,K+1,2 = ni,K+1,22 (5)

Meanwhile, the outputs of the square-law detector for the first

and second symbols at the destination can be written as follows

if nodei transmits an incorrect bit:

yi,K+1,1 = ni,K+1,12, (6)

yi,K+1,2 = Eih0,i+ni,K+1,22, (7)

Ifθ0,i< θth

r , nodei remains silent in the second phase and

the outputs of the square-law detector for the first and second

symbols at the destination are

yi,K+1,1 = ni,K+1,1|2, (8)

yi,K+1,2 = ni,K+1,2|2 (9) Finally, the destination compares and chooses the maximum

output from all the outputs of the square-law detectors, i.e.,

employs the maximum energy selection, to detect the

trans-mitted information In other words, the decision rule is of the

following form:



ˆi, ˆm = arg max

i=0, ,K m=1,2

yi,K+1,m (10) III BER COMPUTATIONS ANDTHRESHOLDS

In this section, the BER analysis for MES scheme is first

carried out for a network with arbitrary qualities of

source-relay and source-relay-destination links Then, the optimal thresholds

are chosen to minimize the average BER are discussed Finally,

the approximate thresholds are proposed to achieve a full

diversity order

A BER Computations

The law of total probability is employed to compute the average BER of the system First, denoteΩ1,Ω2, andΩ3 as the sets of the relays that forward a correct bit, an incorrect bit, and remain silent, respectively It is clear thatK = |Ω1|+

|Ω2| + |Ω3| where |Ω| denotes the cardinality of set Ω The probability of occurrence for the specific set{Ω1, Ω2, Ω3} is [7]:

P (Ω1, Ω2, Ω3) = 

i∈(Ω 1 ∪Ω 2 )



1− I1(θth

r , γ0,i) 

i∈Ω 3

I1(θth

r , γ0,i)

i∈Ω 1



1− I2(θth

r , γ0,i) 

i∈Ω 2

I2(θth

r , γ0,i) (11)

whereA∪B denotes the union of sets A and B The function

I1(θth

r , γ0,i) is the probability of the event θ0,i< θth

r and is

computed as [7]:

I1(θth

r , γ0,i) = 1 +γ0,i

2 +γ0,i



1− e−θ th

r /(1+γ0,i)

2 +γ0,i



1− e−θ th r

 (12)

On the other hand, I2(θth

r , γ0,i) is the probability of error

at node i, i = 1, , K, given the event θ0,i > θth

r and is

determined by [7]

I2(θth

r , γ0,i) = 1

2 +γ0,i

1

1− I1(θth

r , γ0,i)e

−θ th

Now letWw,m(w ∈ {Ω1∪{0}}), Vv,m (v ∈ Ω2) andRr,m (r ∈ Ω3) denote the outputs of the square-law detector for the mth symbol, m = 1, 2, measured at the destination With the assumption that the first symbol from the signal constellation is transmitted, the probability density functions (pdfs) ofWw,m,

Vv,m andRr,m are given, respectively, by

fW w,m(x) =

fw,1(x), m = 1

fV v,m(x) =

fv,2(x), m = 1

fRr,m(x) = fr,2(x), m = 1 or m = 2 (16) wherefk,1(x) = 1

N 0 (1+γk,K+1)e−x/(N0 (1+γk,K+1)),x ≥ 0 and

fk,2(x) = 1

N 0e−x/N0,x ≥ 0

An error occurs at the destination if among the 2(K + 1) statisticsWw,m,Vv,m andRr,m,w ∈ {Ω1∪{0}}, n ∈ Ω2, r ∈

Ω3, m = 1, 2, the one with the largest value is 1) Case 1 (Θ = 1) one ofWw,1, 2) Case 2 (Θ = 2) one ofVv,1, and 3) Case 3 (Θ = 3) one ofRr,1 Thus, given the set{Ω1, Ω2, Ω3}, the BER can be computed as

PΩ 1 ,Ω 2 ,Ω 3(ε) = 3

i=1

PΩ 1 ,Ω 2 ,Ω 3(ε, Θ = i)

=

w∈Ω 1 ∪{0}

PWw,2− Ww,2< 0+

v∈Ω 2

PVv,2− Vv,2< 0

+

r∈Ω 3

PRr,2− Rr,2< 0 (17)

where Ww,2 = max i=w

m=1,2(Wi,m, Ww,1, Vv,m, Rr,m),

m=1,2(Ww,m, Vi,m, Vv,1, Rr,m), and

Rr,2= max i=r

m=1,2(Ww,m, Vv,m, Ri,m, Rr,1)

The conditional BER PΩ1,Ω2,Ω3(ε, Θ = i), i = 1, 2, 3, can

be computed1 as (18), (19), and (20) on the top of this page,

where(G1∪ G2) = Ω means thatG1andG2are two disjoint

subsets of Ω and the union of those disjoint subsets is Ω

Obviously, the average BER with a given threshold θth

r can

be expressed as

θth

r

=

Ω 1 ∈P(S) Ω 2 ∈P(S\Ω 1 ) 3

i=1

PΩ 1 ,Ω 2 ,Ω 3(ε, Θ = i)P (Ω1, Ω2, Ω3) (21)

where P(Ω) denotes the power set of Ω The set S =

{1, , K}

B Optimal and Approximate Thresholds

Given the closed-form expression of the average BER in

(21), one can choose the thresholdθth

r to minimize the average

BER of the system by using the MATLAB Optimization

Toolbox The optimization problem can be set up as follows:

th

r = arg min

θ th r BER(θth

It is clear from (21) that the system need to collect

in-formation on the average SNRs of all the transmission links

to find the optimal thresholds Unfortunately, an close-form

solution for optimal threshold values is very difficult, if not

impossible, to find Therefore, to further reduce the complexity

of the system2, in what follows, we propose approximate

thresholds and prove that by using those thresholds, the system

can achieve the maximum diversity order

Lemma 1: If the relays use the threshold θth

r = Q log cγ whereγ = E0/N0, the system achieves a full diversity order

ofK + 1 for any Q ≥ K and a positive constant c

Proof:

To simplify our derivation, we consider the i.i.d case, i.e.,

γ0,i = γi,K+1 = γ0,K+1 = γ0, i = 1, , K where γ0 =

E0σ2/N0 andN0= 1 Since θth

r =Q log cγ and

lim

γ0→∞

1−1 cγ

1+γ0

γ0

it follows from (11) that3

P (Ω1, Ω2, Ω3)≤I1(θth

r , γ0,i)|Ω3|

I2(θth

r , γ0,i)|Ω2|

 (log(γ)/γ)|Ω3 |

1/γQ+1|Ω2|

(24)

1 The pdfs of W w,2 , V v,2 and R r,2 are given in Appendix A.

2 By using the approximate thresholds, besides the information collection,

the system does not need to find the optimal thresholds centrally and send to

the relays, hence, reducing the complexity and implementation costs of the

system.

3 With two positive real functions f (x) and g(x), we say f(x)  g(x) if

lim supx→∞f (x)g(x) = d where d < ∞ is a positive constant.

On the other hand, the conditional BER PΩ 1 ,Ω 2 ,Ω 3(ε) =

3

i=1PΩ1,Ω2,Ω3(ε, Θ = i) can be evaluated from the large SNR behavior by considering the value of the first non-zero order derivative of the PDF at the origin [10] According to [11], one can verify that

PΩ 1 ,Ω 2 ,Ω 3(ε, Θ = 1) =

0 fWw,1(x)e−x/N 0dx

 (1/γ0)|Ω1|+|Ω2|+1, (25)

PΩ 1 ,Ω 2 ,Ω 3(ε, Θ = 2) =

0 fVv,1(x)e−x/N 0 (1+γ0)dx



(1/γ0)|Ω1 |+|Ω 2 |, if |Ω1| + |Ω2| > 1 (26)

PΩ 1 ,Ω 2 ,Ω 3(ε, Θ = 3) =

0 fRr,1(x)e−x/N 0dx

 (1/γ0)|Ω1 |+|Ω 2 |+1 (27) Thus, one has

PΩ1,Ω2,Ω3(ε)



⎧

⎪

⎪

(1/γ0)K+1, if |Ω2| = 0 (1/γ0)Q+1+|Ω3 |, if |Ω1| = 0 and |Ω2| = 1 (1/γ0)K+|Ω2 |(Q+1), if |Ω1| > 0 and |Ω2| > 0

(28)

Therefore, for sufficiently large values of SNR and θth

Q log cγ where Q ≥ K, it follows from (21) that

BER

θth r

So Lemma 1 is proved

IV SIMULATIONRESULTS

This section presents analytical and simulation results for the BER performance of different noncoherent DF cooperative systems In conducting the simulations, it is assumed that the noise components at the receivers, i.e., relays and destination are modeled as i.i.d.CN (0, 1) random variables Fig 2 plots the average BERs of the proposed scheme, PL scheme and the scheme in [6] in a two-relay system when the variances of Rayleigh fading channels are set to be2σ2

0,i = 0.1σ2

i,K+1 = 5σ2

0,K+1 = 1, i = 1, 2 From the figure, both the analytical (shown as marker symbols) and simulation (shown in line with marker symbols) results are identical, hence verifying our analysis in Section III The figure also shows that the BER of the proposed scheme is significantly better than the BER of the PL scheme It is institutively clear since the

PL scheme suffers from the error propagation The scheme

in [6] outperforms the other two schemes due to the fact that the destination in [6] combines all the signals from the retransmitting relays besides dealing with the problem of error propagation However, the proposed scheme does not require any statistical information of the fading channels to perform a

PΩ1,Ω2,Ω3(ε, Θ = 1) = (|Ω1| + 1)

K+|Ω 3 | l=0



K + |Ω3| l

 ⎡

⎣

i∈(Ω 1 ∪Ω 2 ∪{0}) (G 1 ∪G 2 )=((Ω 1 ∪Ω 2 ∪{0})\{i})



(−1)K+|Ω 3 |+|G 2 |−l 1

N0

1 +γi,K+1

 ⎛

N 0 (1+γ t,K+1 )+N0(1+γ1

i,K+1 )+K+|ΩN30|−l+1

⎞

⎠

⎤

⎦

+(|Ω1| + 1) (K + |Ω3|)

N0

K+|Ω 3 |−1 l=0



K + |Ω3| − 1 l

 ⎡

⎣

(G 1 ∪G 2 )=(Ω 1 ∪Ω 2 ∪{0})

 (−1)K+|Ω 3 |+|G 2 |−l−1⎛

t∈G 2

1

N 0

⎞

⎠

⎤

⎦ (18)

PΩ1,Ω2,Ω3(ε, Θ = 2) =

K+|Ω 3 |+1 l=0



K + |Ω3| + 1

2

⎡

⎣

i∈(Ω 1 ∪Ω 2 ∪{0})\{v} (G 1 ∪G 2 )=((Ω 1 ∪Ω 2 ∪{0})\{i,v})



(−1)K+|G 2 |+|Ω 3 |−l+1 1

N0

1 +γi,K+1

 ⎛

t∈G 2

1

N 0

⎞

⎠

⎤

⎦

N0

K+|Ω 3 | l=0



K + |Ω3|

2

⎡

⎣

(G 1 ∪G 2 )=((Ω 1 ∪Ω 2 ∪{0})\{v})

 (−1)K+|G 2 |+|Ω 3 |−l⎛

t∈G 2

1

N 0 (1+γt,K+1)+(N0(1+γ1

N 0

⎞

⎠

⎤

⎦ (19)

PΩ1,Ω2,Ω3(ε, Θ = 3) = |Ω3|

K+|Ω 3 | l=0



K + |Ω3| l

 ⎡

⎣

i∈((Ω 1 ∪Ω 2 ∪{0})) (G 1 ∪G 2 )=((Ω 1 ∪Ω 2 ∪{0})\{i})



(−1)K+|G 2 |+|Ω 3 |−l 1

N01 +γi,K+1

 ⎛

t∈G 2

1

N 0

⎞

⎠

⎤

⎦

+|Ω3| (K + |Ω3|)

N0

K+|Ω 3 |−1 l=0



K + |Ω3| − 1 l

 ⎡

⎣

(G 1 ∪G 2 )=(Ω 1 ∪Ω 2 )

 (−1)K+|G 2 |+|Ω 3 |−l−1⎛

N 0

⎞

⎠

⎤

⎦ (20)

detection Such the information is required for the PL scheme

and the scheme in [6]

Fig 3 presents the average BERs obtained by simulation and

analysis for two different schemes in a three-relay cooperative

system Here σ2

0,i =σ2

i,K+1 =σ2

0,K+1 = 1, i = 1, 2, 3 The figure again confirms the analysis performed in Section III At

sufficient large values of SNR, the proposed scheme yields a

superior performance compared to the PL scheme

V CONCLUSION

This paper studies the maximum energy selection receiver

for an adaptive decode-and-forward (DF) relaying system with

BFSK signals A closed-form BER expression is obtained and used to choose the optimal thresholds to minimize the average BER Approximate thresholds are proposed and the diversity order is verified Performance comparison reveals that the proposed scheme outperforms the other two schemes with a lower complexity

This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 102.04-2012.33

0 5 10 15 20 25 30

10−6

10−5

10−4

10−3

10−2

10−1

100

Average Power per Node (dB)

PL

Two−threshold [6]

MES (Opt threshold − Simulation)

MES (Opt threshold − Analysis)

MES (Approx threshold − Simulation)

MES (Approx threshold − Analysis)

Fig 2 BERs of a two-relay network with different schemes when

i,K+1 = 5σ 2

0,K+1 = 1.

10−8

10−6

10−4

10−2

100

Average Power per Node (dB)

PL

MES (Opt threshold − Simulation)

MES (Opt threshold − Analysis)

MES (Approx threshold − Simulation)

MES (Approx threshold − Analysis)

Fig 3 BERs of a three-relay network with different schemes when

0,i = σ 2

i,K+1 = σ 2

0,K+1 = 1.

APPENDIXA

PDFS OFWw,1,Vv,1,ANDRr,1RANDOM VARIABLES

The pdf ofWw,1,Vv,1, andRr,1can be found, respectively,

as follows:

fWw,2(x) = dxd P (Ww,2< x) =

i∈(Ω 1 ∪Ω 2 ∪{0})

fi,1(x)×



j∈((Ω 1 ∪Ω 2 ∪{0})\{i})

Fj,1(x) (F1,2(x))K+|Ω3 |+K + |Ω3|

e−x/N0(F1,2(x))K+|Ω3 |−1 

j∈(Ω 1 ∪Ω 2 ∪{0})

Fj,1(x) (30)

fVv,2(x) = d

dxP (Vv,2< x) =i∈(Ω

1 ∪Ω 2 ∪{0})\{v}

fi,1(x)×



j∈((Ω 1 ∪Ω 2 ∪{0})\{i,v})

Fj,1(x) (F1,2(x))K+|Ω3 |+1+(K + |Ω3| + 1)

N0

× e−x/N 0(F1,2(x))K+|Ω3 | 

j∈(Ω 1 ∪Ω 2 ∪{0})\{v}

Fj,1(x) (31)

fRr,2(x) = d

dxP (Rr,2< x) =i∈(Ω

1 ∪Ω 2 ∪{0})

fi,1(x)×



j∈((Ω 1 ∪Ω 2 ∪{0})\{i})

Fj,1(x) (F1,2(x))K+|Ω3 |+(K + |Ω3|)

N0

× e−x/N 0(F1,2(x))K+|Ω3 |−1 

j∈(Ω 1 ∪Ω 2 ∪{0})

Fj,1(x) (32)

whereFk,1(x) = 1 − e−x/(N 0 (1+γ k,K+1 )) andFk,2(x) = 1 −

e−x/N0

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PΩ...

of the system2, in what follows, we propose approximate

thresholds and prove that by using those thresholds, the system

can achieve the maximum diversity order

Lemma