Analysis of Alamouti Code Transmission over TDMA-Based Cooperative Protocol

Abstract

In this paper, we examine a general time-division multiple-access (TDMA) cooperative protocol. The wireless network consists of a source terminal, a destination terminal and n - number of intermediate relay terminals that operate in amplify-and-forward (AF) mode with a fixed-gain. Starting with the moment generating function (MGF) of the end-to-end output signal-to-noise ratio (SNR), the symbol error rate (SER) performance of M-ary phase shift keying (M-PSK) modulation schemes is theoretically investigated over independent and identical (i.i.d) Nakagami-m fading environments with maximal ratio combining (MRC) scheme at the destination. Subsequently, numerical and simulation results are provided to verify the accuracy of the formulation.

Full text

Page 1

Analysis of Alamouti Code Transmission over
TDMA-based Cooperative Protocol
Saman Atapattu∗and Nandana Rajatheva†
Telecommunications Field of Study, School of Engineering and Technology,
Asian Institute of Technology, Klong Luang, Pathumthani 12120, Thailand
satapattu@ieee.org∗, rajath@ait.ac.th†
Abstract—In this paper, we examine a general time-division
multiple-access (TDMA) cooperative protocol. The wireless net-
work consists of a source terminal, a destination terminal and n
- number of intermediate relay terminals that operate in amplify-
and-forward (AF) mode with a fixed-gain. Starting with the mo-
ment generating function (MGF) of the end-to-end output signal-
to-noise ratio (SNR), the symbol error rate (SER) performance
of M-ary phase shift keying (M-PSK) modulation schemes is
theoretically investigated over independent and identical (i.i.d)
Nakagami-m fading environments with maximal ratio combining
(MRC) scheme at the destination. Subsequently, numerical and
simulation results are provided to verify the accuracy of the
formulation.
I. INTRODUCTION
Peer-to-peer relay communication has been introduced to
wireless communication technology as a relay technique where
the communication signals are relayed through one or more
intermediate relay terminal(s), depending on the fading envi-
ronment and the distance between the end-to-end terminals
[1]. Cooperative communication is a special form of relaying
concept which can create a virtual multi-input multi-output
(MIMO) system in multi-user wireless network using relaying
via mobile terminals [2]. Intermediate relay terminal plays a
major role in both scenarios.
Basically, two forms of relay terminals can be found in the
literature, depending on the signaling method. The decode-
and-forward (DF) or regenerative mode is more complex and
gives rise to demodulated error propagation, whereas the AF
or nonregenerative mode is of low complexity in practice and
leads to noise propagation to the destination [3]. According
to the amplification factor, the AF-relay could be a fixed gain
relay or a variable gain relay.
Three different time-division multiple-access (TDMA)
based cooperative protocols have been introduced in [4] having
different degrees of broadcasting and receive collision. These
schemes are well suited in the case of resource allocation
among the cooperative nodes. Cooperation with space-time
coding (STC) is an interesting concept, known as distributed
space-time codes (DSTC) which has been analyzed in several
recent works [5], [6]. DSTCs are used as a powerful tool of
designing relay protocols in order to maximize the diversity
advantage in a fading environment as well as for having
the coding gain. Therefore, performance benefits could be
gained from the TDMA-based relying network incorporated
with STCs.
In multiple-relay scheme, the transmission between relays
and the destination can be considered as a common channel
signaling or as a orthogonal channel signaling. The latter
method is more advantageous because of the minimum re-
ceiver collision and low complexity combining requirements,
especially for MRC [7], [8].
Nabar et al. [4] introduce three different time-division
multiple-access (TDMA) based transmission protocols with a
single relay for space time signal designs. In this work we use
the protocol I given in [4] with the Alamouti code to analyze
the exact SER of a multiple-relay assisted cooperative network
with Nakagami fading channels. The MGF approach is used to
derive the exact SER using the equivalent single-input single-
output (SISO) model derived here.
The remainder of this paper is organized as follows. Section
II describes the system model under consideration. Analytical
expressions for end-to-end SNR and MGF of SNR are derived
in section III and IV, respectively for both single-relay and
multiple-relay scenarios. The performance analysis of the
system is presented in section V. In particular, exact SER
is analyzed for the M-PSK modulation schemes and also
numerical and simulation results are presented as a means
of verification of the theoretical formulation. Finally, some
concluding remarks are made in section VI.
Notation: The E(·) and | a | denote the expectation operator
and the magnitude of the complex value a, respectively.
The superscriptsT,Hand∗stand for transpose, conjugate
transpose and element-wise conjugate. X → Y signifies the
link between terminals X and Y .
II. SYSTEM MODEL
We consider a multibranch wireless cooperative network
with multiple-relay channels having one source (S), n− num-
ber of cooperative relay terminals (R1,..,Ri,..,Rn) and one
destination (D) as shown in Fig.1.
TABLE I
TRANSMISSION PROTOCOL
Time Slot 1
S → D,Ri
S → D, Ri→ D
Time Slot 2
978-1-4244-1645-5/08/$25.00 ©2008 IEEE1226

Page 2

Time slot 2
X2
Time slot 1
X1
SD
R1
Ri
Rn
hsd
hsr
hsr
hsr
1
i
n
hr d
1
hr d
i
hr d
n
Fig. 1. Illustration of multiple-relay wireless network with the source terminal
S, the destination terminal D and the relay terminals Ri,
i = 1,2,...,n
A. Protocol
The relay terminals (R1,..,Ri,..,Rn) are willing to partic-
ipate for the cooperation between S and D. As described by
Nabar et al. [4] in protocol I, two signals in two consecutive
time slots are transmitted to D as summarized in Table I. In
this paper, we analyze the same protocol for space-time signals
transmitted through both single-relay and multiple-relay sce-
narios. Relay terminals operate in AF mode. In particular it is
used to transmit the Alamouti code [9], C =
as the space-time signal set.
In the first time slot, the source S broadcasts the first
symbol x1 that is received by all n number of relays and
D. In the second time slot, the relay terminals and S terminal
communicate with the destination D. Relays forward a new
signal (amplify and forward version of the first signal) and
S forwards the second symbol x2 to D. Similarly, the two
corresponding symbols −x∗
the third and the fourth time slots, respectively as x1and x2.
Therefore, D receives only one signal within the first and the
third time slots and (n+1) signals within the second and the
fourth time slots.
?
x1
−x∗
x2
x∗
21
?
2and x∗
1are transmitted to D in
B. Channel
The fading channels remain the same over the duration of
four time slots and change independently afterwards. There-
fore, we can consider the channels with slow-flat fading
coefficients. In this model, the relay system is operating over
independent and identical (i.i.d.) Nakagami-m fading channels.
It is assumed that perfect synchronization is available at the
receivers. Similar to [2] and [8], perfect channel state informa-
tion (CSI) are realized at the receivers (i.e. S → Richannel
is known to the relay terminal Ri, while S → D, Ri→ D
and S → Richannels are known to the destination terminal).
hxy represents the fade sample through the X → Y link
and the magnitude of hxy, is assumed to follow Nakagami-m
distribution with probability density function (pdf) is given by
[10]
f(| hxy|) =2mm| hxy|2m−1
Γ(m)
exp?−m | hxy|2?
(1)
where E(| hxy|2) = 1 and Γ(·) is the gamma function. Then,
αxy=| hxy|2follows the gamma distribution pdf is given by
[11]
f(αxy) =mmαm−1
Γ(m)
Furthermore, it is assumed that S → D and Ri→ D,∀ i =
1,...,n, transmit through orthogonal channels (e.g, frequency
division multiple access (FDMA) / code division multiple
access (CDMA)) as in [7], [8]. Therefore D receives two and
(n + 1) number of independent signals through relay paths
and direct path in the cases of single-relay an multiple-relay
respectively.
xy
exp(−mαxy).
(2)
C. Receiver
The independent copies of received signals through the
direct and relay paths are combined using maximal ratio
combiner (MRC). Combining of signals received through relay
paths should be preceded by noise normalization. The reason
is discussed in the following section.
III. SIGNAL TO NOISE RATIO
In this section, we derive an input-output relation as a
SISO model for a single relay network. The expressions for
equivalent end-to-end SNRs are obtained for both single-relay
and multiple-relay scenarios from derived SISO model.
A. Single-relay
D receives two independent signals through relay path and
direct path, because two transmissions between S → D and
R → D are orthogonal to each other.
The received signal at D in the first time slot can be written
as
yd1=
Esdhsdx1+ nd1
where nd1∼ CN(0,N0) denotes the additive white Gaussian
noise (AWGN) at terminal D in the first time slot. Without
loss of generality, we assume that all the AWGN terms can
be represented as a circularly symmetric complex Gaussian
random variable having zero mean and variance N01.
In the first time slot, the received signal at the relay terminal
can be modeled as
?
where nr is the AWGN at R. D receives two independent
signals through the relay path and through the direct path
within second time slot. The received signal through the direct
path can be expressed as
?
where nd2is the AWGN at D. R operates in AF mode with
amplification factor G. Therefore the received signal at D
through the relay path is
?
and d denote the relay path and the direct path respectively), and the second
subscript denotes the time slot. Furthermore, Exy indicates the transmitted
power from X node to Y node through X → Y link.
?
(3)
yr=
Esrhsrx1+ nr
(4)
yd2=
Esdhsdx2+ nd2
(5)
yr2=
EsrhrdGyr+ nr2
(6)
1The first subscript of y and n denotes the path of received signal (e.g., r
1227

Page 3

where nr2is the AWGN at D. Therefore total effective noise
in (6) is
noise terms in (5) and (6) do not have identical variance
(power) because of noise contribution at R. Therefore MRC
at the destination should be preceded by noise normalization
in order to have the same variance N0. Therefore, D receiver
normalizes yr2by ω, where ω2= Erd|hrd|2G2+ 1. Finally,
the signal received at D through relay R can be expressed as
˜ yr2=1
ω
where ˜ nr2 is the normalized noise term with ˜ nr2 | hrd ∼
CN(0,N0).
Similar formulation can be applied for the two symbols −x∗
and x∗
received signals at D during the third and the fourth time slots.
Therefore the input-output relationship for the Alamuoti code
transmission can be written as a single-input single-output
(SISO) model
˜ Y = HX +˜ N
√ErdhrdGnr+ nr2. It is obvious that effective
?
EsrErdhsrhrdGx1+ ˜ nr2
(7)
2
1(the second row of the Alamouti code) to obtain the
(8)
where
˜ YT=(yd1˜ yr2yd2y∗
?
XT=(x1x2)1×2
˜ NT=(nd1˜ nr2nd2n∗
with A =√Esdhsdand B =1
The receiver combines these signals with MRC in order
to separate the two symbols x1 and x2. After MRC at the
destination terminal, the total signal to noise ratio (SNR) for
a particular signal xican be written as
γxi=(2 | A |2+ | B |2)
d3˜ y∗
r4y∗
0
−A∗
d4)1×6
0
−B∗
HT=
A
0
B
0
0
A
A∗
0
?
2×6
d3˜ n∗
r4n∗
√EsrErdGhsrhrd.
d4)1×6
ω
N0
.
(9)
The same scenario with same protocol has been analyzed with
non-orthogonal channels in [12]. It is obvious that the end-to-
end SNR with orthogonal channels is higher than the case
of non-orthogonal channels. Therefore, we are encouraged
to extended same for the case of multiple-relay which is
discussed in the following sub-section.
B. Multiple-relay
As shown in Fig. 1, cooperative network can be generalized
to a multiple-relay network with n number of cooperative
relays. Therefore, there are n-cooperative links and one direct
link between S and D. We assume that the transmission
channels between S → D and Ri → D, ∀ i = 1,2,...,n,
are mutually orthogonal to each other. Similar to the single-
relay case, MRC at the destination should be preceded by noise
normalization.
The signals received at D through the direct path are the
same as previous case (3) and (5). The received signal through
ithrelay (Ri) in the second time slot can be written as
1
ωi
˜ yri2=
?
EsrErdhsrihridGix1+ ˜ nri2
(10)
where ωiis the noise normalization factor for the correspond-
ing path i and Giis the amplification factor at Ri. Since S
broadcasts same signal to all relays, same power (Esr) should
be transmitted towards S → Ri. Eventhough the transmitted
power of different relays could be different to each other,
we take it as the same (Erd) for convenience and following
formulation would be valid for any case.
After MRC at D, the effective SNR of a particular symbol
xican be expressed as
?
γxi=
2 | A |2+
n
?
N0
i=1
| Bi|2
?
(11)
where Bi=
Different choices of amplification factor Gi have been
proposed in the literature. Fixed gain relays are considered
for our subsequent analysis, rather than considering variable
gain relays. Then the amplification factor Gican be given as
[13],
1
?E (|ysri|2)
Therefore, we emphasize that the amplification factors are
same and it can be simply denoted as G.
1
ωi
√EsrErdhsrihridGi.
Gi=
=
1
√Esr+ N0
.
(12)
IV. MOMENT GENERATING FUNCTION
In what follows, we derive the closed form expression for
moment generating function (MGF) of SNR of the above
transmission scheme. Incorporating (11) and (12), the total
equivalent SNR for a particular signal xican be written as
?
+
ω2
i
γxi=2Esd| hsd|2
n
?
i=1
1
EsrErd
(Esr+ N0)| hsri|2| hrid|2
?1
N0.
(13)
Let us consider k1=
Erd
(Esr+N0), | hsd|2= αsd, | hsri|2= αsriand | hrid|2= αrid
notations for convenience.Then (13) can be further written as
?
i=1
We use the approach of moment generation function (MGF)
to evaluate the exact SER. The MGF of γxican be written as
1
N0, k2=
EsrErd
(Esr+N0), k3= Esd, k4=
γxi=2k3αsd+
n
?
k2
k4αrid+ 1αsriαrid
?
k1.
(14)
Mγxi(s) = Eαsd,αsri,αrid
?e−sγxi?.
(15)
Assuming that αsd, αsriand αrid are mutually independent
random variables, ∀ i = 1,2,...,n. When αridis given, (15)
can be written as
?e−2k1k3αsds?
×
i=1
Mγxi|αrid(s) =Eαsd
n
?
Eαsri
?
e
−
k1k2αsriαrid
(k4αrid+1)s
?
.
(16)
1228

Page 4

Since αsriand αsdare gamma distributed random variables
as given in (2), (16) can be averaged with respect to αsriand
αsdto yield
1
?1 +2k1k3
×
i=1
1 +
Mγxi|αrid(s) =
ms?m
?
n
?
1
k1k2αrid
m(k4αrid+1)s
?m.
(17)
After some simple algebraic manipulations, (17) can be written
in the form
Mγxi|αrid(s) =
1
?1 +2k1k3
×
i=1
?
ms?m?
1 +
1 +k1k2
k4ms
G(αrid)
k1k2s+k4m)m
?mn
n
?
?
(αrid+
m
?
(18)
where G(αrid) =
am−1,...,a1,a0are real constants. Following [14, eq. A.36],
the right side of (18) can be decomposed into the partial
fractions as
am−1αm−1
rid
+ ... + a1αrid+ a0
?
with
Mγxi|αrid(s) =
1
?1 +2k1k3
×
i=1
ms?m?
1 +
1 +k1k2
k4ms
?mn
Av
k1k2s+k4m)v
n
?
?
m
?
v=1
(αrid+
m
?
(19)
where Av =
with b =
which follows gamma distribution, we get the unconditional
MGF. Following [15, eq. (3.383.5)], the closed form solution
for the unconditional MGF of γxican be expressed as
1
?1 +2k1k3
×
v=1
1
(m−v)!
m
dm−v
dαm−v
rid
((αrid+ b)mG(αrid)) |αrid=−b
k1k2s+k4m. By averaging (19) with respective to αrid
Mγxi(s) =
ms?m?
1 +
1 +k1k2
k4ms
?mn
?
m
?
Avmmbm−vψ(m,m + 1 − v,mb)
?n
(20)
where ψ(α;β;z) is the confluent hypergeometric function of
the second kind.
V. PERFORMANCE ANALYSIS
SER of the multiple-relay network is analyzed in the first
sub-section and the formulation is justified using simulation
results in the second sub-section.
A. Symbol error rate
The average SER for M-PSK can be written as [9]
?(M−1)π/M
Pe=1
π
0
Mγxi
?gpsk
sin2θ
?
dθ
(21)
where gpsk
ψ(α;β;z) function is tabulated in all major mathematical soft-
ware such as Mathematica and Matlab. Closed form solution
for exact SER in the general case seems difficult to deal with
analytically. Therefore, (20) and (21) can be used to evaluate
exact SER with numerical integration techniques for any M-
PSK modulation scheme.
We consider the channels between the relays and the des-
tination to be AWGN (i.e., hrid= 1) [4], and other links are
effected by Nakagami-m fading. Although this assumption is
conceptual, our analytical approach, to evaluate SER, would be
simplified. Furthermore this corresponds to a scenario where
the destination and relay terminals are stationery and have a
line-of-sight connection, while the source terminal is moving.
Then the MGF of Mγxican be reduced to the form
= sin2(π/M). It should be noted that the
Mγxi(s) =
1
(c1s + 1)m(c2s + 1)mn
and c2 =
(22)
where c1 =
(M=2) can be evaluated as
?π/2
=1
π
0
Following [11, eq. 5A.58], we can present the closed form
solution as
?
21 −c1
−c1
c2
2k1k3
m
k1k2
m(k4+1). The SER for BPSK
Pe=1
π
0
Mγxi
?
?
1
sin2θ
?
?m?
dθ
?π/2
sin2θ
sin2θ + c1
sin2θ
sin2θ + c2
?mn
dθ
(23)
Pe=
c1
c2
?mn−1
c2
?
?m(1+n)−1
?mn−1
k=0
?
m−1
?
?c2
?
c1
− 1
?k
BkIk(c2)
k=0
1 −c1
c2
?k
CkIk(c1)
?
(24)
where
Bk=
Ak
?m(1 + n) − 1
k
?, Ck=
?mn − 1
(mn − 1)!
?
mn−1
?
n=0
?k
n
?
?m(1 + n) − 1
n
?An,
Ak= (−1)mn−1+k
k
?
mn
?
(2n − 1)!!
n!2n(1 + c)n
n=1
n?=k+1
(m(1 + n) − n),
Ik(c) = 1 −
?
c
1 + c
1 +
k
?
n=1
?
.
In the case of QPSK (M=4), a closed form solution for SER
can be found for Rayleigh fading channels (m=1) following
[11, eq.5A.56]. Otherwise, (i.e. m > 1), we can use the
numerical integration techniques.
Although we consider independent and identical channel co-
efficients throughout our analytical analysis, the same formu-
lation is still valid for independent and non-identical channel
coefficient with Nakagami-m fading.
1229

Page 5

1.0E-06
1.0E-05
1.0E-04
1.0E-03
1.0E-02
1.0E-01
0481216
Transmit SNR (dB)
SER
Theoretical
Simulation
n = 1, 2, 3
Fig. 2.
for different number of cooperative nodes (n = 1,2,3) with Rayleigh fading
environment (i.e. m = 1)
SER variation of the simulated results and the theoretical results
B. Numerical and Simulation Results
We provide some numerical results on the analytical SER
for the BPSK modulation scheme with Alamouti code trans-
mission over Nakagami multipath fading. Theoretical and
simulated SER curves for different number of cooperative links
are presented with transmit SNR (defined as SNR=
each link. Without loss of generality, we can consider that the
same power is transmitted from each node.
Fig. 2 and Fig. 3 show the SER performance at different
number of cooperative nodes (n = 1,2,3) in a Rayleigh
fading (i.e., m = 1) environment and Nakagami-2 fading
environment, respectively. It is obvious that the analytical
results match perfectly with their simulation counterparts. We
can also notice that the number of cooperative paths has a
major impact on performance enhancement of the network.
This is due to the diversity advantage from the relay paths.
Performance of a different protocol, which the S communi-
cates only with n-number of relays in the first time-slot, has
been analyzed Fig. 2 and 3 in [16]. The protocol discussed
here shows better performance than the protocol in [16].
Exy
N0) in
VI. CONCLUSION
Performance analysis of the Alamouti transmission with M-
PSK symbols through multiple AF-Relay network over Nak-
agami fading channels has been discussed. We first develop
a SISO model and derive the end-to-end SNR expressions
for both single-relay and multiple-relay for this particular
protocol. The closed form solution for the MGF of SNR is
analytically presented for orthogonal channels network with
fixed gain relays. Subsequently, MGF approach is used to
analyze exact SER when M-PSK modulated Alamouti code is
transmitted. Our formulation is applicable to cooperative links
with any number of relay paths; and remains valid for a large
class of fading models. SER analysis revealed that multibranch
network has the diversity advantage.
1.0E-10
1.0E-09
1.0E-08
1.0E-07
1.0E-06
1.0E-05
1.0E-04
1.0E-03
1.0E-02
1.0E-01
0481216
Transmit SNR (dB)
SER
Theoretical
Simulation
n = 1, 2, 3
Fig. 3.
different number of cooperative nodes (n = 1,2,3) with Nakagami-2 fading
environment
SER variation of the simulated results and the theoretical results for
REFERENCES
[1] R. Pabst et al., “Relay-based deployment concepts for wireless and
mobile broadband radio,” IEEE Communications Magazine, vol. 42,
no. 5, pp. 80-89, Sep. 2004.
[2] A. Sendonaris, E. Erkip, and B. Aazhang, “User cooperation diver-
sity Part I: Implementation aspects and performance analysis,” IEEE
Trans. Commun., vol. 51, no. 11, pp. 1927-1938, Nov. 2003.
[3] J. N. Laneman, D. Tse, and G. W. Wornell, “Cooperative diversity
in wireless networks: Efficient protocols and outage behavior,” IEEE
Trans. Inform. Theory, vol. 50, no. 12, pp. 3062-3080, Dec. 2004.
[4] R. U. Nabar, H. B¨ olckei, and F. W. Kneub¨ uhler, “Fading relay channels
performance limits and space-time signal design,” IEEE J. Select. Areas
Commun., vol. 22, no. 6, pp. 1099-1109, Aug. 2004.
[5] J. N. Laneman and G. W. Wornell, “Distributed space - time - coded
protocols for exploiting cooperative diversity in wireless networks,”
IEEE Trans. Inform. Theory, vol. 49, no. 10, pp. 2415-2425, Oct. 2003.
[6] S. Yiu, R. Schober, and L. Lampe, “Distributed space-time block cod-
ing,” IEEE Trans. Commun., vol. 54, no. 7, pp. 1195-1206, July. 2006.
[7] P. A. Anghel and M. Kaveh, “Multi-user space-time coding in coopera-
tive networks,” in Proc. Int. Conf. Acoustics, Speech, Signal Processing,
Hong-Kong, pp. 73-76, Apr. 2003.
[8] A. Ribeiro, X. Cai, and G. B. Giannakis, “Symbol error probabilities
for general cooperative links,” IEEE Trans. Commun., vol. 4, no. 3,
pp. 1264-1273, Mar. 2005.
[9] S. M. Alamouti, “A simple transmit diversity technique for wireless
communications,” IEEE J. Select. Areas
pp. 1451-1458, Oct. 1998.
[10] M. Nakagami, “The m-distribution-A general formula for intensity
distribution of rapid fading,” in Statistical Methods in Radio Wave
Propagation, W. G. Hoffman, Ed. Oxford, U. K.: Pergamon, 1960.
[11] M. K. Simon and M. -S. Alouini, Digital Communication Over Fading
Channels, 2nd ed. New York: Wiley, 2005.
[12] S. Atapattu and N. Rajatheva, “Performance evaluation of Alamouti STC
through amplify-forward cooperative relay network over Nakagami-
m fading channels,” International Workshop on Cooperative Wireless
Communications and Networking (CoNET-07), Korea, Dec. 2007.
[13] C. S. Patel, G. L. St¨ uber, and T. G. Pratt, “Statistical properties of
amplify and forward relay fading channels,” IEEE Trans. Vehic. Technol.,
vol. 55, no. 1, pp. 1-9, Jan. 2006.
[14] A. V. Oppenheim, A. S. Willsky, and S. H. Nawab, Signal and System,
2nd ed. Prentice-Hall, 1997.
[15] I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, series and
Products, 6th ed. New York: Academic Press, 2000.
[16] S. Atapattu and N. Rajatheva, “Exact SER of Alamouti Code Trans-
mission through Amplify-and-Forward Multiple-Relay Network over
Nakagami-m Fading Channels,” Wireless Personal Multimedia Commu-
nications (WPMC’07), India, Dec. 2007.
Commun., vol. 16, no. 8,
1230