Performance analysis of space shift keying (SSK) modulation with multiple cooperative relays
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Mesleh et al. EURASIP Journal on Advances in Signal Processing 2012, 2012:201
http://asp.eurasipjournals.com/content/2012/1/201
RESEARCH Open Access
Performance analysis of space shift
keying (SSK) modulation with multiple
cooperative relays
Raed Mesleh1* , Salama S Ikki2 , El-Hadi M Aggoune1 and Ali Mansour3
Abstract
In this article, space shift keying (SSK) modulation is used to study a wireless communication system when multiple
relays are placed between the transmitter and the receiver. In SSK, the indices of the transmit antennas form the
constellation symbols and no other data symbol are transmitted. The transmitter and the receiver communicate
through a direct link and the existing relays. In this study, two types of relays are considered. Conventional amplify and
forward relays in which all relays amplify their received signal and forward it to the destination in a round-robin
fashion. In addition, decode and forward relays in which the relays that correctly detect the source signal will forward
the corresponding fading gain to the destination in pre-determined orthogonal time slots are studied. The optimum
decoder for both communication systems is derived and performance analysis are conducted. The exact average bit
error probability (ABEP) over Rayleigh fading channels is obtained in closed-form for a source equipped with two
transmit antennas and arbitrary number of relays. Furthermore, simple and general asymptotic expression for the
ABEP is derived and analyzed. Numerical results are also provided, sustained by simulations which corroborate the
exactness of the theoretical analysis. It is shown that both schemes perform nearly the same and the advantages and
disadvantages of each are discussed.
Keywords: SSK, Amplify and forward, Decode and forward, Cooperative communication, Performance analysis, MIMO
Introduction Multiple-input multiple-output (MIMO) technique is
Cooperative communication creates collaboration also one of the major contributions to the progress
through distributed transmission/processing by allowing in wireless communications in recent years and has
different nodes in a wireless network to share resources. been considered in many recent standards such as LTE,
The information for each user is sent out not only by the WiMAX, WINNER [5], and others. Cooperative MIMO
user, but also by other collaborating users. This includes techniques promise a significant enhancement in spectral
a family of configurations in which the information can efficiency and network coverage for future wireless com-
be shared among transmitters and relayed to reach final munication systems ([6], and references therein). The use
destination in order to improve the system’s overall of multiple antennas at the transmitter and the receiver in
capacity and coverage [1,2]. Recently, cooperative tech- a MIMO system may not be feasible in all applications due
nologies have also made their way toward next generation to size, cost, and hardware considerations [7]. Therefore,
wireless standards, such as IEEE 802.16 (WiMAX) [3] or multiple relays can be used as a virtual antenna array to
LTE [4], and have been incorporated into many modern emulate MIMO communications.
wireless applications, such as cognitive radio and secret Space shift keying (SSK) is a MIMO technique which
communications. activates a single transmit-antenna during each time
instant and uses the activated antenna index to implicitly
*Correspondence: rmesleh.sncs@ut.edu.sa convey information [8]. The fundamental idea of SSK is
1 Electrical Engineering Department and Sensor Networks and Cellular Systems originally proposed in [9], which was further developed
(SNCS) Research Center, University of Tabuk, 71491 Tabuk, Saudi Arabia into spatial modulation (SM) in [10,11]. Activating single
Full list of author information is available at the end of the article
© 2012 Mesleh et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons
Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction
in any medium, provided the original work is properly cited.Mesleh et al. EURASIP Journal on Advances in Signal Processing 2012, 2012:201 Page 2 of 10
http://asp.eurasipjournals.com/content/2012/1/201
transmit-antenna at a time eliminates inter-channel inter- and accurate expressions for the average error probabil-
ference, avoids the need for inter-antenna synchroniza- ity are also obtained to illustrate the impact of fading
tion, and creates a robust system to channel estimation parameters on the systems under study.
errors since the probability of error is determined by the The remainder of this article is organized as follows: AF
differences between channels associated with the different and DF systems with optimum receiver detector are dis-
transmit antennas rather than the actual channel realiza- cussed in “System model and optimum receiver design”
tion. Thereby, SSK is shown to have lower complexity section. Performance analysis for conventional AF relay-
and enhanced error performance with moderate number ing is given in “Performance analysis of conventional AF
of transmit antennas as compared to other conventional relaying system” section and for DF system in “DF system
MIMO techniques such as space–time coding [12] and performance analysis” section. Numerical and analytical
vertical Bell laboratories layered space–time [13]. How- results are discussed in “Numerical analysis and discus-
ever, the diversity potential of MIMO systems is not fully sion” section and a conclusion at the last section.
exploited in conventional SSK where only receive diversity
gain through the multiple receive antennas is achieved but System model and optimum receiver design
no transmit diversity. Therefore, several recent attempts A MIMO system consisting of Nt transmit antennas, sin-
were made to develop systems based on the SSK concept gle receive antenna, Nr = 1, and M DF relays is depicted
that achieves both transmit and receive diversity [14-18]. in Figure 1.
In this article, a source and a destination in a wireless The transmission is conducted in two phases. In the first
communication system adopting SSK modulation com- phase, each log2 (Nt ) bits are mapped into the index of
municates through a direct link and through a set of one of the transmitting antennas. At each time instant,
multiple relays. Conventional amplify and forward (AF) only one transmit antenna () is active and it transmits an
relays as well as decode and forward (DF) relays are con- energy Es . The other transmit antenna remains silent dur-
sidered. In conventional AF system, all existing relays ing this instant. The transmitted information bits at this
amplify their received signals from the source and for- particular time instance are incorporated in the location
ward them to the destination in a round-robin fashion. of the active transmit antenna and no other data symbol
While in DF system, only the relays that decode the source is transmitted. The received signal at the mth relay input
signals correctly participate in the retransmission process over the MIMO channel can be written as
in a predetermined orthogonal time slots. The receiver,
in turn, assumes full channel knowledge and estimates ys−rm (t) = Es hm, x (t) + ns−rm (t) ,
the activated transmit antenna to retrieve the transmitted (1)
= 1, 2, . . . , Nt and m = 1, . . . , M
information bits.
However, and though important, the use of SSK in
where x(t) is a unit energy deterministic signal, ns−rm (t)
cooperative MIMO is very limited. Recently, the appli-
is the additive white gaussian noise (AWGN) at the mth
cation of SM in a dual-hop non-cooperative scenario is
relay input with both real and imaginary parts having a
proposed in [19] and significant performance gains are
double-sided
power spectral density equal to N0 /2, and
reported as compared to non-cooperative DF system.
hm, = hm, ejφm, ∼ CN (0, h ) is the channel com-
Also, performance analyses of SSK with single AF relay are
plex path
gain
between transmit antenna and the relay m
reported in [20]. In [15], a coherent versus non-coherent
with hm, and ejφm, being the amplitude and the phase of
DF space–time shift keying system is proposed where a
the said channel, respectively. Similarly, the received sig-
matrix dispersion approach is used to activate one of the
nal through the direct link at the receiver can be written
relays similar to activating transmit antennas in SSK. In
as
[21], a space–time SSK aided AF relaying is employed to
avoid the need for a large number of transmit antennas
and mitigate the effects of deep fading. Also, based on ys−d (t) = Es f x (t) + ns−d (t) , (2)
the concept of SSK, an information-guided transmission
scheme is proposed in [22] for multi-relay channel and the where f = f ejθ ∼ CN 0, f is the channel com-
achievable data rate is analyzed. plex path gain between
transmit antenna and the receive
With respect to current literature, our contributions are antenna with f and ejθ being the amplitude and the
threefold: (i) the optimum receiver ML detectors for the phase of the channel; and ns−d is the AWGN at the
signal received via single or multiple relays and through a receiver input with similar characteristics as ns−rm (t).
direct link in AF and DF systems are derived, (ii) the end- In the second transmission phase, the relays participate
to-end average error probability for the systems under in retransmitting the source message to the destination.
study are computed in closed-form without resorting to Based on the relays type, two systems are discussed in
Monte Carlo numerical simulations, and (iii) approximate what follows.Mesleh et al. EURASIP Journal on Advances in Signal Processing 2012, 2012:201 Page 3 of 10
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Figure 1 SSK system model with multiple DF relays. The system considers a transmitter with two transmit antennas, a receiver with single
receive antenna, and M relays. The communication is conducted through the relays and through a direct link.
Conventional AF relaying where D is the decision metric defined as [23]
In conventional AF relaying, all the relays participate in ⎧ ⎫
re-sending the source signal to the destination in pre- ⎪
⎨ ⎪
⎬
determined time slots. Therefore, M + 1 time slots are ∗
D = Re ys−d (t) × f (t) dt
⎪
⎩ ⎪
⎭
needed for each symbol transmission. The received signal Ts
at the destination can be written as
1
− f (t) × f ∗ (t) dt
2
yrm −d (t) = Am gm ejφgm Es hm, ejφm, x (t) Ts
⎧ ⎫ (6)
⎪
⎨ ⎪
⎬
Signal part
(3) M
∗
+ Am gm ejφgm ns−rm (t) + nrm −d (t), + Re ydm (t) × sm, (t) dt
⎪
⎩ ⎪
⎭
m=1 Ts
Noise part
1
− sm, (t) × s∗m, (t) dt
2
where gm = gm ejφgm ∼ CN 0, gm denotes the Ts
channel complex path gain between the relay m and the
receiver, Am = 1 where Re(·) denotes the real part of complex number, Ts
Es h +N0 is the amplification factor at ∗
is the symbol
√ time,
(·)jφ is the complex conjugate, and
the relay m, and nrm −d (t) is the AWGN with both real
sm, (t) = Gm hm, e m, x (t).
and imaginary parts having a double-sided power spectral
density equal to N0 /2.
DF relaying
It is assumed that the receiver has full channel state
In the DF relaying system, only the relays that correctly
information (CSI). Therefore, the received signal can be
detect the active transmit antenna index will forward the
simplified to
channel path gain multiplied by the unit energy deter-
ministic signal to the destination. To simplify the analy-
yrm −d (t) = Gm hm, ejφm, x (t) + n̂m (t) , (4) sis, a genie-aided receiver at each relay is assumed. This
receiver is able to determine exactly which symbols in
A2m Es |gm |
2 the transmitted data frame are erroneously detected at
where Gm = . the relay. At each symbol position, only those relays that
A2m |gm | +1
2
The optimum ML detector, assuming Nt transmit correctly detect the symbol are allowed to forward a
antennas and perfect time synchronization, is then given message in the second phase. In other words, with this
by [23] genie-aided system, the decoding set C, i.e., the set of
forwarding relays, actually changes from symbol to sym-
bol. This is different from a practical DF system involving
u = arg max {D } (5)
=1,2,...,Nt an error-detecting code, where the decoding set is fixedMesleh et al. EURASIP Journal on Advances in Signal Processing 2012, 2012:201 Page 4 of 10
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and comprises only those relays that correctly decode After a few algebraic manipulations, the instantaneous
the entire data frame. Nonetheless, this assumption of probability of error, given that transmit antenna one was
the relays’ knowledge of erroneous symbol after detec- active, is reduced to
tion facilitates the error probability derivations. Such an
approach is commonly used in the literature (see [24-27]). Es 2 M
G 2
f2 − f1 + hm,2 − hm,1 < ñ
m
Pe |=1 = Pr
Furthermore, this can be used as bench mark for all prac- 2 2
m=1
tical systems. Hence, the received C signals at d can be
(10)
rewritten as
√ M √
yrm −d (t) = Er hm, gm x (t) + nrm −d (t). (7) where ñ = Es f2 ñ2 − f1 ñ1 + Gm hm,2
m=1
Again, the receiver is assumed to have full CSI. There- ñm,2 − hm,1 ñm,1 , which when conditioned upon
fore, the optimum ML detector, assuming perfect time the fading channels is a random variable with
2
synchronization, is similar to Equations (5) and (6) except
zero-mean and a variance of (N0 /2) Es f2 − f1
that the summation considers only the relays that belong
M √ 2
to set C. +
Gm hm,2 − hm,1 .
m=1
Performance analysis of conventional AF relaying Accordingly, Pe can readily be computed in closed form
system as follows [23,28,29]
Conditional error probability ⎛# ⎞
$ 2 2
A closed-form expressions are derived in what follows for $ M
⎜$ Es f2 − f1 + Gm hm,2 − hm,1 ⎟
the case of Nt = 2 transmit antennas. A generalization ⎜% ⎟
Pe |=1 = Q ⎜ ⎟
m
to any number of transmit antennas can be obtained by ⎜ 2N0 ⎟
⎝ ⎠
using the union bounding technique as in ([20], Section
III-B). Let us assume that at a particular time instant the
active antenna index is ν. Then, the decision metrics can (11)
be rewritten as Using similar analytical steps, Pe |=2 can be obtained
Es 2 and is equivalent to (11). Substituting Pe |=1 and Pe |=2
D |ν= = fν + Es fν ñ1 in (9), the conditional error probability can be written as
2
M
2 M ⎛# ⎞
Gm
$ 2 2
+ hm,ν + Gm hm,ν ñm,1 ⎜
$P
2 Prm gm PS hm,2 − hm,1 ⎟
M
Pe = Q ⎝$ % f2 − f1 +
s
m=1
2
m=1 ) * ⎠
2 gm Prm + Cm
2 2
Es 2 m=1
D |=ν = Es Re fν f∗ − fν + Es fν ñ2
2 (12)
M
E Er m
+ Gm Re hm,ν h∗m, where Prm = Nrm0 , Ps = NEs0 , and Cm = A2m N0
with Erm
m=1 being the mth relay output energy.
M
Gm 2 M
− hm, + Gm hm, ñm,2 Average error probability using moment generation
2
m=1 m=1 function-based approach
(8) In what follows, the average error probability will be
computed by exploiting the moment-generation function
where ñm,1 = Re Ts n̂m (t) e(−jφm, ) x∗ (t) dt , ñm,2 =
(MGF)-based approach for performance analysis of digital
Re Ts n̂m (t) e(−jφm,ν ) x∗ (t) dt , ñ1 = Re Ts ns−d (t) × communication systems over fading channels.
Let us define γs−d = P2s |f2 − f1 |2 and γrm =
e−jθ x∗ (t) dt , and ñ2 = Re Ts n̂s−d (t) e−jθ x∗ (t) dt . Prm |gm |2 Ps |Hm |2
Prm |gm |2 +Cm
, γs = Ps |hm,2 − hm,1 |2 /2 with |Hm |2 =
The instantaneous
probability of error, Pe f1 , f2 , hm,1 ,
|hm,2 −hm,1 |2
hm,2 , gm conditioned upon the channel impulse re- 2 = Prm |gm |2 . Note that γr and γs
, and γr
sponses (f1 , f2 , hm,1 , hm,2 , and gm ), can explicitly be written are random variables following) exponential
* distribution
1
as follows given by fγr (x) = g Pr exp − g Pr x
and fγs (x) =
) * m m m m
1 1 1
Ps exp − Ps , respectively. The MGF of γs−d is [23]
x
Pe = Pr D 1|=1 < D 2|=1 + Pr D 2|=2 < D1|=2
2 2 h h
Pe (f1 ,f2 ,hm,1 ,hm,2 ,gm )|=1 Pe (f1 ,f2 ,hm,1 ,hm,2 ,gm )|=2 1
Mγs−d (s) = (13)
(9) 1 + sPs fMesleh et al. EURASIP Journal on Advances in Signal Processing 2012, 2012:201 Page 5 of 10
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The cumulative distribution function of γrm is computed To avoid numerical integration, this integral can be
as follows [30,31] approximated as
+ , + ,1 + ,
γr γs M
Fγrm (x) = PrMesleh et al. EURASIP Journal on Advances in Signal Processing 2012, 2012:201 Page 6 of 10
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- + ,
substituted in (22) to obtain the error probability for an |F|2 |f2 −f1 |2 Es |F|2
with = 2 and the term E Q N0 can be
arbitrary number of transmit antennas.
computed as
DF system performance analysis ⎡ ⎛. ⎞⎤ / . 0
Conditional error probability E |F|2 1 Ps f /2
⎣
E Q ⎝ s ⎠ ⎦ = 1− . (28)
Let us assume that antenna number i is used to send N0 2 1 + Ps f /2
the bit at a particular time instance. Then, the decision
metrics, D , = 1, 2 can be rewritten as
In the second scenario, m out of M relays detect the
Er
D |i= = hm,i 2 gm 2 + Es fi 2 signal correctly and in that case the destination will com-
2 2 bine the direct link with the m indirect links to esti-
m∈C
mate the transmitted signal. The probability that this
+ ,
+ Er hm,i gm ñm,1 + Es fi ñ1
M M
m∈C scenario occurs is (Poff )M−m (1 − Poff )m , where
m=1 m
Er Re gm hm,i hm, ejφm,i e−jφm, the summation from m = 1 to M is to consider all possible
2
D |i= =
m∈C values of m. The average error probability for the second
Er 2 2 scenario is then given by
+ gm hm, + Er hm, gm ñm,2
2
M + ,
Es 2 M M−m
+ Es Re fi f ejθi e−jθ − f + Es f ñ2 Pre2 = Poff (1 − Poff )m E
2 m
m=1
(23) ⎡ ⎛# ⎞⎤ (29)
$m
$
Following similar analytical steps as discussed in pre- ⎣Q ⎝% γs−rm −d + γs−d ⎠⎦
vious section, the conditional error probability can be k=1
written as
⎛#$ ⎞ Er |gm | |Hm |2
2
|hm,2 −hm,1 |2
$ E g 2 P h − h 2 + E f − f 2 with (γs−rm −d = N0 , where |Hm |2 = 2
⎜$ r m S m,2 m,1 s 2 1 ⎟
and γs−d = EsN|F|
2
⎜% m∈C ⎟ ).
Pe = Q ⎜ ⎟ 0
⎝ 2N0 ⎠ The exact equation for (29) is calculated in what follows.
Let X1 and X2 be two exponential )distributed * random
1 x
(24) variables with PDFs fX1 (x) = b1 exp − b1 and fX2 (x) =
) *
1
b2 exp − b2 . The PDF of X = X1 X2 is then given by [32]
x
Average error probability
In DF system, the transmitted message is received via a + 6 ,
direct link and through all relays that were able to detect 2 x
fX (x) = K0 2 (30)
the transmitted signal correctly. The average probability b1 b2 b1 b2
that the relay detects the signal incorrectly Poff is given by
⎡ ⎛. ⎞⎤ where Kκ (·) denotes the modified Bessel function of the
Es |Hm | ⎠⎦
2 second kind of order κ. The MGF of X is then written as
Poff = E ⎣Q ⎝ (25) + , + ,
N0 1 −1 1
MX (s) = exp 0, (31)
sb1 b2 sb1 b2 sb1 b2
where |Hm |2 is an exponential
) * random variable with PDF
f|Hm |2 (x) = 1h exp − xh . Hence, Poff can be written as where (0, ·) is the incomplete Gamma function. There-
/ . 0 fore, the MGF of γs−rm −d can be written as
1 Ps h /2
Poff = 1− , (26) ) * + ,
2 1 + Ps h /2 1 −1 1
Mγs−rm −d (s) = e sPr g h
0,
where Ps = Es /N0 . sPr g h sPr g h
The probability that all relays will be off and only direct (32)
M
link communication exist is Poff and the average error
probability in this case can be written as with Pr = Er /N0 . Using similar steps, the MGF of γs−d is
⎡ ⎛. ⎞⎤ written as
E |F| 2
Pre1 = (Poff )M × E ⎣Q ⎝ ⎠⎦ ,
s 1
(27) Mγs−d (s) = (33)
N0 1 + sPs fMesleh et al. EURASIP Journal on Advances in Signal Processing 2012, 2012:201 Page 7 of 10
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0
10
Simulation
−1 Analytical
10
Asymptotic
−2
10
−3
10
BER
−4
10
10
−5 L=2
−6 L=3
10 L=4
−7
10 L=5
−8
10
Figure 2 Simulation, analytical, and asymptotic results for SSK system with Nt =2, arbitrary number of AF relays, and Nr =1. The results
show close match for wide range of SNR values.
Collecting
/ . all formulas,0 the term α1 = The above expression can be approximated as (by sub-
m stituting θ = π/2)
E Q γs−rm −d + γs−d is given by
⎡ ⎛# ⎞⎤
k=1 $m
$
E ⎣Q ⎝% γs−rm −d + γs−d ⎠⎦
π/2 + ,1
m + ,
1 1 1 k=1
(35)
α1 = Mγs−d Mγs−rm −d dθ
π sin2 (θ) sin2 (θ) 1 1 m
0 k=1 ≤ Mγs−d (1) Mγs−rm −d (1)
(34) π
k=1
0
10
Simulation (Exact)
Analytical
−1
10 Asymptotic (High SNR)
−2
10
−3
10
BER
−4
10
−5
10
−6
10
−7
10
Figure 3 Simulation, analytical, and asymptotic results for SSK system with Nt =2, arbitrary number of DF relays, and Nr =1. The results
show close match for wide range of SNR values.Mesleh et al. EURASIP Journal on Advances in Signal Processing 2012, 2012:201 Page 8 of 10
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0
10
SSK 451 Conventional Sim.
SSK 451 Conventional Ana.
−1
10 SSK 441 Conventional Sim.
SSK 441 Conventional Ana.
SSK 831 Conventional Sim.
−2
SSK 831 Conventional Ana.
10
BER
−3
10
−4
10
−5
10
−6
10
0 5 10 15 20 25 30
Figure 4 Simulation and analytical results for conventional relaying SSK with an arbitrary number of transmit antennas and different
number of AF relays.
By collecting all terms, the exact expression for the The initial value theorem [33] states that fγs−rm −d (0) =
average error probability can be obtained and given by lim sMγs−rm −d (s). Therefore, fγs−rm −d (0) can be written as
s→∞
) * + ,
/ . 0 1 1 1
M 1 Ps f /2 fγs−rm −d (0) = lim s e sPr g h 0,
P̄e = (Poff ) × 1− + s→∞ sPr g h sPr g h
2 1+s f /2 + + ,,
1 1
M +
, = ψ (1) − log
M P r g h P r g h
(Poff )M−m (1 − Poff )m
m (38)
m=1
π/2 + ,1
m + , where ψ (1) denotes the digamma function.
1 1 1
Mγs−d Mγs−rm −d dθ. Using the theorem in [34], the PDF of λ is written as
π sin2 (θ) k=1
sin2 (θ)
0
1 1
(36) fλ (x) = xM
(M + 1) Ps f
+ + + ,,,M
Asymptotic analysis: high SNR approximation 1 1
× ψ (1) − log
Although the expression for the average error probability P r g h P r g h
in (36) enables numerical evaluation of the system perfor- (39)
mance and may not be computationally intensive, it does
not offer insight into the effect of the system parameters. Finally, the asymptotic error probability is written as
We now aim at expressing P̄e in a simpler form to ease the
2M (M + 1.5) 1
analysis of the optimization problems. At high SNR, all P̄e = √
relays will be on and the error probability can be written π (M + 1) ! Ps f
+ + + ,,,M (40)
as 1 1
× ψ (1) − log
P r g h P r g h
M +
,
M
P̄e ≈ (Poff )M−m (1 − Poff )m Numerical analysis and discussion
m
m=1 Simulation and analytical results along with asymptotic
∞ (37)
√ results for SSK system with two transmit antennas, dif-
= E [Q (λ)] = fλ (x) Q x dx ferent number of relays, and single receive antenna are
0 shown in Figure 2 using AF conventional relays and inMesleh et al. EURASIP Journal on Advances in Signal Processing 2012, 2012:201 Page 9 of 10
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0
10
−1
10
−2
10
−3
10
BER
−4
10
−5
10
−6
10
−7
10
0 5 10 15 20 25 30
Figure 5 Simulation results for SSK system with Nt =2 and Nt =4, arbitrary number of DF relays, and Nr =1.
Figure 3 using DF relays. Results are plotted as a function only the relays that correctly detect the active transmit
of Et /N0 , where Et = Es + Er . Numerical and analytical antenna participate in the retransmission process. It is
results demonstrate an identical match for a wide range of shown that the performance with four transmit antennas
SNR values. While, asymptotic results show close perfor- degrades the performance by about 1 dB as compared to
mance for pragmatic SNR values. The achieved diversity two transmit antennas. However, it is significant to men-
gain increases with increasing the number of relays and tion that the spectral efficiency with four transmit anten-
this is obvious in the figure. The performances of both sys- nas is double the spectral efficiency with two transmit
tems are nearly the same. However, the spectral efficiency antennas.
of the conventional AF relays is less than that of DF relays
since all relays participate in the retransmission process. Conclusion
While, system complexity of DF relays is higher than that We have introduced an accurate analysis of the perfor-
of AF relays since the relays decode the received signal, mance of SSK modulation over Rayleigh fading channels
use error detection techniques, and then cooperate in the with arbitrary number of relays. Conventional AF relays
retransmission process. as well as DF relays are considered in the study. A sim-
Simulation results for three systems with four and eight ple asymptotic expression for the error probability has
transmit antennas and different number of relays are been derived as well. Numerical results have validated
shown in Figure 4 and compared to analytical results the accuracy of the proposed analytical derivations. Also,
using the bound in (22). The bound demonstrates good it is shown that the complexity and the spectral effi-
matching with the simulation results for Et /N0 values ciency of the two proposed schemes can be traded off
greater than 10 dB. However, for DF system, the analy- while maintaining almost identical performance. Opti-
sis with an arbitrary number of transmit antennas is not mizing the transmitted power and the relays positions as
straightforward. In fact, the selection of the optimum well as comparing to other cooperative MIMO techniques
relay when the source is equipped with more than two will be considered in future works.
transmit antennas is a complicated process. The selection
criteria should be designed such that the selected relays Competing interests
The authors declare that they have no competing interests.
maximizes the euclidian distances between the channel
paths form all transmit antennas to the selected relay. Acknowledgements
This is different than conventional systems where the The authors gratefully acknowledge the support for this study from SNCS
Research center at University of Tabuk under the grant from the Ministry of
relay that maximizes the SNR is the best relay. The anal- Higher Education in Saudi Arabia.
ysis of SSK with more than two transmit antennas and
DF relaying is left for future investigations. Nevertheless, Author details
1 Electrical Engineering Department and Sensor Networks and Cellular Systems
simulation results for Nt = 4 SSK system with 2 and (SNCS) Research Center, University of Tabuk, 71491 Tabuk, Saudi Arabia. 2 INRS,
4 DF relays are shown in Figure 5. In the simulation, Montreal, QC, Canada. 3 Lab STICC, ENSTA-Bretagne, Brest, France.Mesleh et al. EURASIP Journal on Advances in Signal Processing 2012, 2012:201 Page 10 of 10
http://asp.eurasipjournals.com/content/2012/1/201
Received: 17 February 2012 Accepted: 28 August 2012 25. SS Ikki, MH Ahmed, Performance analysis of adaptive
Published: 18 September 2012 decode-and-forward cooperative diversity networks with best-relay
selection. IEEE Trans. Commun. 58(1), 68–72 (2010)
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