AmplifyandForward Relay Transmission with EndtoEnd Antenna Selection.

George K. Karagiannidis
Aristotle University of Thessaloniki, Greece, and Khalifa University, UAE
ABSTRACT In this paper, the performance of a dualhop AmplifyandForward (AF) multiantenna relay network, with endtoend (e2e) best antenna selection, is investigated. To investigate the performance of this system, we first derive the exact outage probability in closedform. It is then used to obtain expressions for e2e SNR moments and the average symbol/bit error rate (SER/BER) valid for a large class of practical modulation schemes. A simple and accurate BER approximation is also derived to quantify the performance at high SNR. Our analytical results, show that the e2e antenna pair selection scheme achieves the same diversity order as for the case where all antennas are used. To further confirm the validity of our analysis, Monte Carlo simulation results are also presented.
 Journal of Electronics (China) 28(3).

Conference Paper: Outage probability and SER of twohop MIMO relaying systems with a fixedgain relay and antenna selection
Communications and Networking in China (CHINACOM), 2012 7th International ICST Conference on; 01/2012  Sarnoff Symposium (SARNOFF), 2012 35th IEEE; 01/2012
Page 1
AmplifyandForward Relay Transmission with
EndtoEnd Antenna Selection
Himal A. Suraweera∗, George K. Karagiannidis†, Yonghui Li‡, Hari K. Garg∗, A. Nallanathan§and Branka Vucetic‡
∗Department of Electrical and Computer Engineering, National University of Singapore, Singapore
†Department of Electrical and Computer Engineering, Aristotle University of Thessaloniki, Greece
‡School of Electrical and Information Engineering, University of Sydney, Australia
§Division of Engineering, Kings College London, United Kingdom
Email: {elesaha, eleghk}@nus.edu.sg, geokarag@auth.gr, {lyh,branka}@ee.usyd.edu.au, arumugam.nallanathan@kcl.ac.uk
Abstract—In this paper, the performance of a dualhop
AmplifyandForward (AF) multiantenna relay network, with
endtoend (e2e) best antenna selection, is investigated. To in
vestigate the performance of this system, we first derive the
exact outage probability in closedform. It is then used to obtain
expressions for e2e SNR moments and the average symbol/bit
error rate (SER/BER) valid for a large class of practical modu
lation schemes. A simple and accurate BER approximation is also
derived to quantify the performance at high SNR. Our analytical
results, show that the e2e antenna pair selection scheme achieves
the same diversity order as for the case where all antennas are
used. To further confirm the validity of our analysis, Monte Carlo
simulation results are also presented.
I. INTRODUCTION
Future wireless networks will support relay based com
munications since they offer significant performance benefits,
including increased spatial diversity, capacity and extended
coverage [1], [2]. One of the most commonly used relaying
protocols is AmplifyandForward (AF). Some AF relays use
the source to relay channel knowledge for amplifying the
received signal. These kind of relays are in general known
as channel state information (CSI)assisted relays.
In the available literature, various performance aspects of
dualhop relay networks have been investigated. While most of
such performance studies assume single antennas at each node,
several recently published papers consider multiple antenna
deployments at the source/relay and the destination (see, for
e.g. [3]–[10]). In [3], the impact of receive diversity on the
symbol error rate (SER) performance of a fixed gain and
CSIassisted relay network has been investigated. In [4], the
network capacity of several signalling and routing methods
for multipleinput multipleoutput (MIMO) relay systems has
been compared. In [5], the impact of multiple antennas on
the performance of a decodeandforward (DF) distributed
cooperative network has been examined. In [6], Louie et
al. have analyzed a dualhop CSIassisted AF relay network
with beamforming at the source and maximal ratio combining
(MRC) at the destination providing exact and asymptotic error
performance results. Assuming fixedgain AF relays, the per
formance of the same network has been studied by Costa and
A¨ ıssa in [7]. More recently, antenna selection techniques in
This research work was supported by the National University of Singapore
under Research Grant Numbers R263000421112 & R263000436112.
multiantenna relay systems has also come under consideration
[8]–[10]. In [8], optimal SNRbased transmit antenna selection
rules at the source/relay nodes for the AF halfduplex MIMO
relay channel, have been derived. In [9], the performance of a
dualhop AF relay system with transmit antenna selection at
the source and MRC at the destination has been investigated.
In this paper, a dualhop AF MIMO relay network with the
best antenna selected at each end is considered. This system
has a reduced performance compared to the transmit beam
forming/MRC system studied in [6]. However, this simple
antenna selection method requires much less feedback over
head [8]. In [10], assuming an ideal relay gain, the probability
density function (pdf) of the system’s e2e SNR has been
derived. However, in [10] other important performance metrics
of the system has not been presented. This paper fills this
gap. We have derived closedform expressions for the exact
outage probability, moments of the e2e SNR and the error
performance including the average bit error rate (BER) for
various modulations and the average SER of MQAM. To gain
further valuable insights into the system performance at high
SNR, we also present asymptotic SER/BER approximations
which have also not been investigated in [10].
The rest of the paper is organized as follows. In Section II,
the dualhop AF system model with best antenna selection is
described. Closedform expressions for the statistics of the e2e
SNR are presented in Section III and in Section IV, the average
error performance, diversity and array gains are investigated.
We then proceed by verifying the new results in Section V.
Finally, conclusions are drawn in Section VI.
II. SYSTEM AND CHANNEL MODEL
We consider a wireless communications system where a
source, S, equipped with NS antennas communicates with
the destination, D, equipped with ND antennas, through a
relay, R, equipped with NR antennas. It is assumed that S
does not have a direct link to D. Hence, in this network, the
diversity gain expected from the use of multiple antennas is
of significant interest since cooperative diversity can no longer
be achieved.
The system operation is as follows: the source and the relay
transmit on orthogonal channels. We consider a time multi
plexing scheme, for which communications from S to D takes
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the WCNC 2010 proceedings.
9781424463985/10/$26.00 ©2010 IEEE
Page 2
places in two time slots. An NS× NRmatrix, HSR, is used
to denote the channel paths between the source and the relay
and an ND×NRmatrix, HRD, is used to denote the channel
paths between the relay and the destination. Their entries are
modeled as i.i.d complex Gaussian random variables (RVs)
corresponding to Rayleigh fading. Prior to sending data, with
the help of pilot symbols, at each hop, a single transmit/receive
antenna pair that maximizes the received SNR is selected. The
same relay system model has been considered in [10] and is
an extension of the pointtopoint system first studied in [11].
We assume that the destination has perfect channel knowledge
of the selected S − R and R − D antenna paths for coherent
decoding of the signals.
The received signal at the relay during the first time slot
can be written as
?
where h1is the complex channel between the selected source
transmit antenna to the relay receive antenna, P1is the transmit
power, x is the transmitted scalar symbol with zero mean
and unit variance, nR is the additive white Gaussian noise
(AWGN) satisfying E[nR2] = σ2
at the relay is then multiplied by a gain G, and transmitted
to the destination during the second time slot. The received
signal at the destination can be written as
√P2h2G
yR=
P1h1x + nR,
(1)
1. The received scalar signal
yD=
?√P1h1x + nR
?
+ nD,
(2)
where P2 is the relay transmit power, h2 is the complex
Rayleigh channel between the selected relay transmit antenna
to the destination receive antenna and nD is the AWGN
satisfying E[nD2] = σ2
assisted AF relaying, the gain G is chosen as [2]
2. According to the principles of CSI
G2=
1
P1h12+ σ2
1
(3)
and the e2e SNR is of the form:
γeq1=
γ1γ2
γ1+ γ2+ 1,
(4)
where γ1 = h12¯ γ1, γ2 = h22¯ γ2, and ¯ γi =
1,2. However, the e2e SNR in (4) is not easily tractable [2].
Fortunately, (4) can be tightly upper bounded by
Pi
σ2
i
for i =
γeq1< γeq2=
γ1γ2
γ1+ γ2.
(5)
The form of γeq2 in (5) has the advantage of mathematical
tractability over that in (4) and it is a tight upper bound for
γeq1, specially at medium to high average SNR.
III. OUTAGE PROBABILITY AND SNR MOMENTS
In this section, we derive important performance measures
for the dualhop relay system under investigation. This in
cludes the outage probability, asymptotic outage probability
and the e2e SNR moments.
To study the outage probability of (4) and (5), it is necessary
to obtain the distribution of the e2e SNR, of the RV
γ1γ2
γ1+ γ2+ c,
where c ≥ 0 is a constant.
A. Outage Probability
Z =
(6)
The outage probability, Po, defined as the probability that
γeq1 drops below an acceptable SNR threshold γT, is an
important quality of service (QoS) measure. Mathematically,
Po= Pr(γeq1< γT) = Fγeq1(γT).
(7)
To derive the outage probability of γeq1, we first present a
theorem for the cumulative distribution function (cdf) and the
pdf of the RV Z, in (6).
Theorem 1: The cdf and pdf of Z are given respectively by
?
¯ γ1¯ γ2
(−1)k?Nr−1
?
FZ(z) = 1 − 2NtNr
z2+ cz
Nt−1
?
l=0
(−1)l?Nt−1
l
?
√1 + l
(8)
× e−(1+l)z
¯ γ1
Nr−1
?
?
k=0
(1 + l)(1 + k)(z2+ cz)
¯ γ1¯ γ2
k
?
√1 + k
e−(1+k)z
¯ γ2
× K1
2
?
,
and
pZ(z) = 2NtNr
?
(−1)k?Nr−1
z2+ cz
¯ γ1¯ γ2
Nt−1
?
?
?
?
l=0
(−1)l?Nt−1
l
?
√1 + l
(9)
× e−(1+l)z
?
?1 + l
where Nt = NSNR, Nr = NRND and Kν(x) is the νth
order modified Bessel function of the second kind [12, Eq.
(8.432.6)].
Proof: The proof is given in the Appendix.
We now present the following corollary for the outage
probability and the e2e SNR pdf of the system.
Corollary 1: The outage probability and the e2e SNR pdf
of the system can now be obtained by first substituting z = γT
and z = γ into (8) and (9) respectively. The outage probability
and SNR pdf of the system follow by substituting c = 1 into
the resultant expressions.
¯ γ1
Nr−1
?
k=0
k
√1 + k
?
?
e−(1+k)z
¯ γ2
?
(2z + c)
×
(1 + l)(1 + k)
(z2+ cz)¯ γ1¯ γ2K0
2
(1 + l)(1 + k)(z2+ cz)
¯ γ1¯ γ2
?
¯ γ1¯ γ2
?
+
¯ γ1
+1 + k
¯ γ2
K1
2
(1 + l)(1 + k)(z2+ cz)
??
,
B. Outage Probability at High SNR
We now analyze the system’s e2e asymptotic outage prob
ability.
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the WCNC 2010 proceedings.
Page 3
Theorem 2: The outage probability in the large SNR regime
is given by
⎧
⎪
P∞
o
?
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
⎨
?
γT
¯ γ1
1 +
?Nt+ O
1
μN
?
?
γT
¯ γ1
?N
?Nt
+ O
?
Nt< Nr
?
μNr
??
γT
¯ γ1
?
γT
¯ γ1
?N
Nt= Nr= N
1
γT
¯ γ1
?Nr+ O
γT
¯ γ1
?Nr
Nt> Nr
(10)
where μ =
O(x) if limx→0
Proof: The proof is given in the Appendix.
Note, that the presented expression in (10) is much simpler
compared to (8) and allows us to study important factors such
as the diversity gain of the system.
¯ γ2
¯ γ1and a function of x, f(x), is represented as
f(x)
x
= 0.
C. Moments of the e2e SNR
We now characterize the moments of γeq2given in (5). The
moments of the e2e SNR are important performance metrics
directly related to the system’s performance [6], [7]. It can be
showed that the moments of γeq2are given by
?∞
where Fγeq2(·) denote the cdf of γeq2and E(·) is the expec
tation operator. After substituting (8) with c = 0 into (11)
and using [12, Eq. (6.621.3)], the moments of the e2e can be
evaluated in closedform. Therefore, E[γn
E[γn
eq2] = n
0
γn−1(1 − Fγeq2(γ))dγ,
(11)
eq2] is expressed as
?Nt− 1
E[γn
eq2] =8√πNtNrn!(n + 1)!
¯ γ1¯ γ2
Nt−1
?
l=0
(−1)l
l
?
(12)
×
Nr−1
?
k=0
(−1)k
?
?Nr− 1
k
?
×
2F1
n + 2,3
2;n +3
2;
1+l
¯ γ1+1+k
1+l
¯ γ1+1+k
¯ γ2−2
¯ γ2+2
?
?
(1+l)(1+k)
¯ γ1¯ γ2
(1+l)(1+k)
¯ γ1¯ γ2
?
?
Γ?n +3
2
??
1+l
¯ γ1+1+k
¯ γ2+ 2
(1+l)(1+k)
¯ γ1¯ γ2
?n+2,
where Γ(x) is the gamma function and2F1(a,b;c;z) is the
Gauss hypergeometric function [12].
IV. AVERAGE ERROR PERFORMANCE
In this section, we derive closedform expressions for the
e2e average BER of the system under consideration for var
ious modulation formats. Although in [10], an average BER
expression for MPSK modulation has been derived, it is in
the form of an integral expression. We consider γeq2instead
of (4) for mathematical tractability purposes and because it
provides a tight upper bound for CSIassisted AF relaying in
the medium to high SNR regime. Our results apply for all
modulation formats that have a BER expression of the form:
Pb= a Eγ[Q(
?
bγ)],
(13)
0510 15 2025
10
−6
10
−5
10
−4
10
−3
10
−2
10
−1
10
0
Outage Probability
S−R link average SNR in dB
Analytical
Asymptotic (High SNR)
Simulations
Fig. 1.
NS= ND= 2,3,4 and NR= 2. γT= 5 dB and ¯ γ2= ¯ γ1.
Outage probability against ¯ γ1 for various antenna configurations.
0510
S−R link average SNR in dB
152025 30
10
−6
10
−5
10
−4
10
−3
10
−2
10
−1
10
0
Outage Probability
Analytical
Asymptotic (High SNR)
Simulations
Fig. 2.
NS= 4, NR= 2 and ND= 1,2,3. γT= 5 dB and ¯ γ2= 1.5¯ γ1.
Outage probability against ¯ γ1 for various antenna configurations.
where a,b > 0 and Q(x) =
Qfunction.
Consider the following integral defined as
1
√2π
?∞
xe−y2
2 dy is the Gaussian
T(a,b,s) =
a
√2π
?∞
0
Fγeq2
?t2
b
?
e−st2dt.
(14)
Substituting (8) into (14), T(a,b,s) can be expressed as
?∞
Nt−1
?
?∞
With the help of
Eq. (6.621.3)], T(a,b,s) has a closedform solution given by
(16) at the top of the next page.
T(a,b,s) =
a
√2π
0
e−st2dt −aNtNr
Nr−1
?
?
b
?
2
π¯ γ1¯ γ2
(15)
×
l=0
(−1)l?Nt−1
?
l
?
√1 + l
k=0
(−1)k?Nr−1
?
k
?
√1 + k
×
0
t2e−
1+l
b¯ γ1+1+k
b¯ γ2+st2K1
2t2
b
?π
?
(1 + l)(1 + k)
¯ γ1¯ γ2
?
dt.
?∞
0e−qx2dx =
1
2q, and employing [12,
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the WCNC 2010 proceedings.
Page 4
T(a,b,s) =
a
2√2s−3πaNtNr
√2b2
Nt−1
?
l=0
(−1)l?Nt−1
l
?
¯ γ1
Nr−1
?
k=0
(−1)k?Nr−1
k
?
¯ γ2
2F1
?
5
2,3
2;2;
1+l
b¯ γ1+1+k
1+l
b¯ γ1+1+k
b¯ γ2+s−2
b¯ γ2+s+2
?
b
?
(1+l)(1+k)
b¯ γ1¯ γ2
(1+l)(1+k)
¯ γ1¯ γ2
b
?
?
?
1+l
b¯ γ1+1+k
b¯ γ2+ s +2
b
(1+l)(1+k)
¯ γ1¯ γ2
?2.5.
(16)
Now, using integration by parts, it can be shown that the
average BER in (13) can be rewritten as
?
A. Average SER of MQAM
The classical form of the average SER for Mary quadrature
amplitude modulation (MQAM) is given by
?
?
order to derive the average SER we need to evaluate
I(a,b) = a Eγ[Q2(
Pb= Ta,b,1
2
?
.
(17)
Ps= a Eγ
Q
??
1 − 1/√M
bγ
??
− a
?
1 −
1
√M
?
Eγ
?
Q2??
bγ
??
(18)
,
where a = 4
?
and b = 3/(M −1). Therefore, in
?
?u2
bγ)].
(19)
After integration by parts (19) can be rewritten as
√2a
√π
0
We are unaware of a closedform solution to this integral.
Nevertheless, using a Qfunction tight approximation given
by [15, Eq. (14)]
1
12e−x2
I(a,b) can be efficiently evaluated. Substituting (21) into (20)
we get
?∞
+
2√2π
0
Therefore, Pscan be written as
?
×
3
B. Diversity and Array Gain at High SNR
At sufficiently high SNRs, diversity gain defines the slope
of the average BER against average SNR in a loglog scale
while Gain dB determines the shift of the curve with respect
to the average BER curve of ¯ γ−Gd
1
SNRs can be closely approximated as [13, Prop. 1]
I(a,b) =
?∞
Q(u)Fγeq2
b
?
e−u2
2du.
(20)
Q(x) ?
2 +1
4e−2x2
3 ,
(21)
I(a,b) =
a
6√2π
a
0
Fγeq2
?u2
?u2
b
?
?
e−u2du
(22)
?∞
Fγeq2
b
e−7
6u2du.
Ps? Ta,b,1
2
?
−1
2
?
?
1 −
a,b,7
1
√M
??
?
.
(23)
?T (a,b,1)
+ T
6
. The average BER at high
P∞
b
=2qaξ Γ(q +3
2)
√π
(b¯ γ1)−(q+1)+ O(¯ γ−(q+1)
1
),
(24)
where q = min(Nt,Nr) − 1 and
⎧
⎪
Eq. (24) implies that the array gain Gaand the diversity gain
Gdof the system can be written as
?2qaξ Γ(q +3
and
ξ =
⎪
⎩
⎨
Nt
?
μNr
Nt< Nr
Nt= Nr= N
Nt> Nr
1 +
1
μN
?
N
Nr
(25)
Ga= b
2)
√π
?−
1
q+1
,
(26)
Gd= min(Nt,Nr).
(27)
From (27) we can infer that selecting the best antenna pair at
each end can achieve the maximum possible diversity order
of this system.
We now present an approximation for the average SER of
MQAM modulation at high SNR. Using (10) and after some
manipulations I(a,b) at high SNR can be expressed as
2qaξ
√π(q + 1)
0
× (b¯ γ1)−(q+1)+ O(¯ γ−(q+1)
where erfc(·) is the complementary error function. With the
aid of [14, Eq. (2.8.5.7)] the integral in (28) can be evaluated
in closedform as follows
2qq!aξ
π(2q + 3)
× (b¯ γ1)−(q+1)+ O(¯ γ−(q+1)
Finally, using (24) and (29) the average SER for MQAM in
the high SNR regime can be expressed as
?
?
V. NUMERICAL AND SIMULATION RESULTS
We have confirmed the correctness of the derived analytical
results in Section III and IV, through comparison with Monte
Carlo simulations. Figs. 1 and 2 show the e2e SNR outage
for various numbers of antennas at S and D. We see that the
outage probability is significantly improved as the number of
deployed antennas increase. In order to verify our analysis at
high SNRs, we have also plotted the outage curves obtained
I∞(a,b) =
?∞
u2(q+1)erfc(u)e−udu
(28)
1
),
I∞(a,b) =
2F1
?
q +3
2,q + 2;q +5
).
2;−1
?
(29)
1
P∞
S =2qaξ
√π
Γ(q +3
q + 1
2)
−
?
1 −
1
√M
?
q!
√π(2q + 3)
(30)
×2F1
q +3
2,q + 2;q +5
2;−1
??
(b¯ γ1)−(q+1).
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the WCNC 2010 proceedings.
Page 5
0510152025
10
−6
10
−5
10
−4
10
−3
10
−2
10
−1
10
0
Average Bit Error Rate
S−R link average SNR in dB
Analytical
Asymptotic (High SNR)
Simulations
Fig. 3.
the relay is equipped with a single antenna.
Average BER with BPSK and ¯ γ2= 3¯ γ1. NS= ND= 2,3,4 and
0510
S−R link average SNR in dB
152025 30
10
−6
10
−5
10
−4
10
−3
10
−2
10
−1
10
0
Average Symbol Error Rate
Analytical
Asymptotic (High SNR)
Simulations
Fig. 4.
and the relay is equipped with a single antenna.
Average SER with 16QAM and ¯ γ2= 3¯ γ1. NS= ND= 2,3,4,5
from (10). It is apparent that these results are able to accurately
predict system’s outage at medium to high SNR region.
Due to space limitations, results of the system’s e2e SNR
moments are not presented. However, extensive simulations
again confirmed the correctness of (12). In Figs. 3 and 4 we
have evaluated the average BER and average SER versus SNR
performance for BPSK and 16QAM modulations, respec
tively. These figures clearly demonstrate the accuracy of the
error performance analysis using (17) and (23), in comparison
with the results obtained from simulations. We have also
verified the accuracy of the high SNR approximations. In the
case of BPSK, the exact and the asymptotic average BER
results shown in Fig. 3 are almost indistinguishable when
¯ γ1> 15 dB. In all cases, increasing the number of antennas
has a positive impact on the error performance. However, as
seen from Fig. 4 increasing the number of antennas higher than
five, may not produce further significant performance gains.
We note that increasing the number of deployed antennas also
contribute to high infrastructure costs.
VI. CONCLUSIONS
Antenna selection methods provide a good cost/performance
tradeoff. The performance of a dualhop AF relay system with
e2e best antenna selection over Rayleigh fading channels was
investigated. A range of closedform results has been derived
to evaluate the outage probability, the e2e SNR moments
and the average BER/SER of the system. In order to gain
further insights, asymptotic results were also derived including
the array and diversity gains at high SNRs. The asymptotic
analysis confirms that the system achieves the maximum
possible diversity order, which is equal to the minimum of the
product of the number of antennas available at the source/relay
and relay/destination, respectively.
APPENDIX
A. Proof of Theorem I
The cdf of Z can be written as
?∞
After applying some algebraic manipulations to (31), we
obtain
?∞
where Pγ1(x) = 1 − Fγ1(x) denotes the complementary cdf
of γ1and pγ2(x) is the pdf of γ2.
To obtain the cdf of Z, we need the cdf of γ1and the pdf
of γ2. Note that the cdf of γ1is given by
?
and the pdf of γ2is given by
?
Substituting (33) and (34) into (32), solving the resultant
integral using [12], and after some algebraic manipulations,
we arrive at the desired result. By taking the derivative of
(8) with respect to z, taking into account that
−K0(x) −K1(x)
B. Proof of Theorem II
To obtain an expansion of Fγeq1(γT), we first note that
lim
FZ(z) =
0
Pr
?
xγ1
γ1+ x + c< z
?
pγ2(x)dx
(31)
FZ(z) = 1 −
0
Pγ1
?
z +z2+ cz
w
?
pγ2(z + w)dw, (32)
Fγ1(x) = 1 − Nt
Nt−1
l=0
(−1)l
1 + l
?Nt− 1
l
?
e−(1+l)x
¯ γ1 ,
(33)
pγ2(x) =Nr
¯ γ2
Nr−1
k=0
(−1)k
?Nr− 1
k
?
e−(1+k)x
¯ γ2
.
(34)
dK1(x)
dx
=
x
, we obtain the pdf of Z.
¯ γ1,¯ γ2→∞Fγeq1(γT) =
We substitute ¯ γ2 = μ¯ γ1 and c = 0 into (8) and write the
outage probability as
lim
¯ γ1,¯ γ2→∞Fγeq2(γT)
(35)
Fγeq2(γT) = 1 −
2γT
√μ¯ γ1NtNr
Nt−1
?
(−1)k?Nr−1
(1 + l)(1 + k)
μ
l=0
(−1)l?Nt−1
?
?
l
?
√1 + l
(36)
× e−(1+l)γT
¯ γ1
Nr−1
?
?
k=0
k
√1 + k
e−(1+k)γT
μ¯ γ1
× K1
?
2γT
¯ γ1
.
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the WCNC 2010 proceedings.
Page 6
Now by substituting x =γT
¯ γ1we can rewrite (36) as
Fγeq2(x) = 1 −2NtNr
√μ
x
Nt−1
?
(−1)k?Nr−1
(1 + l)(1 + k)
μ
l=0
(−1)l?Nt−1
l
?
√1 + l
(37)
× e−(1+l)x
Nr−1
?
?
k=0
k
?
√1 + k
e−(1+k)x
μ
× K1
?
2x
?
.
Next, employing an approximation for small arguments of x,
0 < x <<√α + 1, given by
Kα(x) ?2α−1Γ(α)
xα
,
(38)
Fγeq2(x) can be simplified as
Fγeq2(x) ? 1 − NtNr
Nt−1
?
l=0
(−1)l?Nt−1
?
l
?
1 + l
(39)
×
Nr−1
?
k=0
(−1)k?Nr−1
k
1 + k
e−(1+l+1+k
μ )x.
Using the McLaurin series representation for the exponential
function in (39) yields
Fγeq2(x) ? −NtNr
Nt−1
?
l=0
(−1)l?Nt−1
?
l
?
1 + l
(40)
×
Nr−1
?
k=0
(−1)k?Nr−1
k
1 + k
∞
?
κ=1
?
−
?
1 + l +1+k
μ
?
x
?κ
κ!
.
Now, applying the Binomial expansion to
realizing that the first or the second summation vanishes for
κ < min(Nt,Nr), (40) simplifies to
?
1 + l +1+k
μ
?κ
and
Fγeq2(x) ? −NtNr(−x)min(Nt,Nr)
Nt−1
?
l=0
(−1)l?Nt−1
?
l
?
1 + l
(41)
×
Nr−1
?
k=0
(−1)k?Nr−1
k
?
1 + k
(1 + l)min(Nt,Nr)+
1+k
μ
?min(Nt,Nr)
min(Nt,Nr)!
.
If Nt< Nr, we get
Fγeq2(x) ? −Nr
Nr−1
?
k=0
(−1)k?Nr−1
Nt−1
?
k
?
1 + k
(42)
×
? xNt+ O?xNt?.
(−x)Nt
(Nt− 1)!
l=0
(−1)l
?Nt− 1
l
?
(1 + l)Nt−1
If Nr< Nt, similarly we get
Fγeq2(x) ? −Nt
Nt−1
?
(−x)Nr
μNr(Nr− 1)!
? μ−NrxNr+ O?xNr?.
Finally, if Nt= Nr= N, we rewrite (41) as
k=0
(−1)k?Nt−1
Nr−1
?
k
?
1 + k
(43)
×
l=0
(−1)l
?Nr− 1
l
?
(1 + k)Nr−1
Fγeq2(x) ? −N2(−x)N
Nt−1
?
l=0
(−1)l?N−1
l
?
?
1 + l
(44)
×
N−1
?
?
¯ γ1into (42)(44) gives the result in (10) and
the proof is completed.
REFERENCES
k=0
(−1)k?N−1
1
μN
k
?
1 + k
?
(1 + l)N+
1+k
μ
?N
N!
?
1 +
xN+ O?xN?.
Substituting x =γT
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