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Performance of Multichannel Reception withTransmit Antenna Selection in ArbitrarilyDistributed Nagakami Fading Channels
Juan M. Romero-Jerez, Member, IEEE, and Andrea J. Goldsmith, Fellow, IEEE
Abstract
We present exact expressions for the average bit error rate (BER) and symbol error rate (SER) of different
modulation techniques of a wireless system with multiple transmit and receive antennas. The receive antennas
are assumed to use maximal ratio combining (MRC) or post-detection equal gain combining (EGC), whereas the
transmit antenna that maximize the output signal-to-noise ratio (SNR) is selected. Exact expressions of the moment
generating function (MGF) of the output SNR and all its derivatives are also derived. These expressions are used
to obtain high order statistics and the performance of the proposed scheme for different scenarios. We consider an
independent but non-identically distributed (i.n.d.) Nakagami-m fading channel where the average SNR and fading
parameters from the different transmit antennas are arbitrary and may be different from each other. For the case
when the Nakagami fading parameter m has an integer value in every channel, results are given in closed-form
as a finite sum of simple terms. For the case when fading parameters take any real value, our results are given in
terms of the multivariate Lauricella hypergeometric function F(n)A . Numerical results for the error rates of different
modulation techniques are presented. The effect of unbalance average SNR on performance is also investigated.
Index Terms
Transmit antenna selection (TAS), Lauricella´s hypergeometric functions, Nakagami-m fading.
J. M. Romero-Jerez is with the Departamento de Tecnologıa Electronica, E.T.S.I. Telecomunicacion, University of Malaga, 29071 Malaga,Spain (e-mail: romero@dte.uma.es).
A. J. Goldsmith is with the Department of Electrical Engineering, Stanford University, Stanford, CA 94305 USA (e-mail: an-drea@ee.stanford.edu).
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I. INTRODUCTION
MULTIPLE transmit and receive antenna systems offer substantial performance improvement
in wireless systems by increasing their spectral efficiency and/or by reducing the effects of
the channel impairments [1]. In systems with base stations or access points communicating with small
low power terminals, the terminals may be limited to a single transmit chain due to power complexity
constrains. In this case, transmit antenna selection (TAS) has been proposed as a means to provide transmit
diversity gain and reduced implementation complexity [2]-[5]. In these systems, the antenna at the transmit
end that provides the highest instantaneous post-processing signal-to noise ratio (SNR) is selected, while
at the received end maximal ratio combining (MRC) is typically proposed, though other diversity schemes
at the receiver are also possible.
In this paper we study the performance of a TAS system where at the receive end MRC or post-detection
equal gain combining (EGC) is employed under Nakagami fading. The Nakagami distribution is widely
used as a model of wireless fading channels due to its good fit to experimental results in a variety of
fading scenarios as well as its analytical versatility. Nakagami fading models cover both severe and weak
fading conditions via the fading parameter m, and includes Rayleigh fading as a special case. Additionally,
Nakagami fading can be used to closely approximate Rice and Hoyt fading models [6].
Previous work on TAS systems obtained closed-form results for the average bit error rate (BER) in
Rayleigh fading [3], [4]. For Nakagami fading, only approximate results have been reported for the average
BER when the fading parameter of the Nakagami distribution takes any real value [5], [7]. For integer
values of this parameter, an exact formula of the BER in terms of an infinite series is provided in [5], but
only for two transmit antennas. For an arbitrary number of transmit antennas, a closed-form expression of
the BER is derived in [7] for integer fading parameter. All of these works consider MRC at the receive end
and that the fading between each transmit and receive antenna is identically distributed. Additionally, these
works provide average BER results only for binary modulation schemes, such as binary phase-shift keying
(BPSK). Higher-oder modulation techniques are common in todays digital communications systems such
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as Wi-Fi (IEEE 802.11), WiMax (IEEE 802.16) and 3G cellular networks. Closed-form expressions for
the average SER for these modulations are presented in [8] for a TAS system with generalized selection
diversity (GSC) at the receiver, but the results are restricted to independent and identically distributed
(i.i.d.) Rayleigh fading.
In this paper we derive exact expressions for the average error rate of TAS systems for different binary
and M-ary modulations. Our system model is more general than in previous work in that we consider an
independent but non-identically distributed (i.n.d.) Nakagami fading channel where the average received
SNR and fading parameters from the different transmit antennas are arbitrary and may be different from
each other. For arbitrary values of the fading parameters our error rate expressions are given in terms
of the Lauricella’s hypergeometric function. When the fading parameters take integer values the derived
expressions are given as a finite sum of simple terms. We also derive exact expression of the MGF of
the output SNR as well as all its derivatives, from which the moments of the output SNR can be readily
computed, which provides a useful characterization of wireless systems [10].
The remainder of this paper is organized as follows. In Section II we describe the system model and
present results for the MGF of the output SNR and all its derivatives. In Section III, exact expressions
of the average BER or SER are derived for different modulation schemes. In Section IV some numerical
results for the error rates of TAS systems with different modulations are given. Finally, concluding remarks
are presented in Section V.
II. SYSTEM MODEL AND SNR STATISTICS
We consider a wireless link with L transmit and N receive antennas. At the receive end, MRC or
post-detection EGC is used, whereas at the transmit end, the antenna that maximizes the instantaneous
output SNR after receive combining is selected for transmission. No channel knowledge is required at
the transmitter and the feedback from the receiver to the transmitter for antenna selection is considered
to be error-free. Let hij denote the channel gain between transmit antenna j and receive antenna i where
all channel gains are assumed to be independent and constant during a symbol interval. We assume
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that |hij | follows a Nakagami distribution with fading parameter mj and squared mean α2j = E
[|hij |2
],
i.e., for a given transmit antenna j the channels at the different receive antennas are assumed to be
identically distributed. However, we allow the channels from different transmit antennas to have different
fading parameters and squared means. This model is quite general and allows the performance analysis of
macrodiversity systems where the transmit antennas are located at different cell-sites and will typically be
at different distances to a multichannel receiver. This is, for example, the case of downlink transmissions
in CDMA networks for users in soft hand-off, where the multiple receive channels are provided by the
branches of the RAKE receiver of the mobile unit. Note also that the problem stated here is isomorphic to
a system with N transmit antennas employing maximal ratio transmission (MRT) and L receive antennas
with selection combining (SC), where the average powers and fading parameters at any of the receive
antennas are arbitrary.
For MRC and post-detection EGC combining it is known that, for transmit antenna j selected for
transmission, the instantaneous output SNR per symbol can be written as
γj = (Esj/No) · ΣL
i=1 |hij |2 , (1)
where Esjdenotes the the average symbol energy before transmission from antenna j and No denotes the
single-sided power spectral density of the Gaussian noise, which is considered here to be equal at every
receive antenna. Therefore, the output SNR, conditioned on antenna j being selected for transmission,
will follow a Gamma distribution, with pdf
fj(x) =
(mj
γj
)mjN1
Γ(mjN)xmjN−1e−xmj/γj , (2)
where mj is the fading parameter corresponding to the j-th transmit antenna and can take any real value
greater than or equal to 0.5, Γ(α) =∫∞0 zα−1e−zdz denotes the Gamma function and γj = α2
j · (Esj/No)
is the average SNR at every receive antenna from transmit antenna j . The corresponding cdf will be
Fj(x) =γ(mjN, xmj/γj)
Γ(mjN), (3)
where γ(α, x) =∫ x0 zα−1e−zdz denotes the incomplete Gamma function.
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The unconditional instantaneous output SNR per symbol will be γs = maxjγj, and its cdf can be
calculated as
Fo(x) =L∏
j=1
Fj(x), (4)
while the pdf will be
fo(x) =L∑
j=1
fj(x) ·L∏
k=1,k =j
Fk(x). (5)
A. SNR Statistics for Arbitrary Fading Parameters
We now calculate the MGF of the output SNR and all its derivatives when the fading parameters mj
can take any real value. Let Ψ(n)(s) be the MGF n-th order derivative of the output SNR, which can be
written as
Ψ(n)(s) =∫ ∞
0xnesxfo(x)dx. (6)
Combining (2)-(6) we can write
Ψ(n)(s) =1∏L
j=1 Γ(mjN)
L∑j=1
(mj
γj
)mjN ∫ ∞
0xn+mjN−1
× e−x(mj/γj−s)L∏
k=1,k =j
γ(mkN, xmk/γk) dx.
(7)
The integrand in (7) can be expressed in terms of the confluent hypergeometric function of the first kind
1F1(a; b; x) =∞∑
n=0
(a)n
(b)n
xn
n!(8)
where (a)n = Γ(a+n)/Γ(a) denotes the Pochhammer symbol [11, eq. 6.1.22]. By considering the relation
γ(a, x) = (1/a)e−xxa1F1(1; 1 + a, x), [12, eq. (8.351.2)], we can write
Ψ(n)(s) = N
⎛⎝ L∏j=1
(mj/γj)mjN
Γ(mjN + 1)
⎞⎠ L∑j=1
mj
×∫ ∞
0xΛ(n)−1e−x(
∑L
k=1mk/γk−s)
×L∏
k=1,k =j
1F1(1; 1 + mkN ; xmk/γk) dx,
(9)
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where Λ(n) = n + N∑L
k=1 mk. With the help of [13, eq. (2.4.2)] we finally obtain
Ψ(n)(s) = N
⎛⎝ L∏j=1
(mj/γj)mjN
Γ(mjN + 1)
⎞⎠ Γ(Λ(n))
(∑L
k=1 mk/γk − s)Λ(n)
×L∑
j=1
mjF(L−1)A
⎛⎜⎝(Λ(n), 1, . . . , 1︸ ︷︷ ︸L−1 terms
;
1 + m1N, . . . , 1 + mLN︸ ︷︷ ︸L−1 terms;
; X1(s), . . . , XL(s)︸ ︷︷ ︸L−1 terms
⎞⎟⎠ ,
(10)
where Xk(s) = (mk/γk)/(∑L
l=1 ml/γl −s) and the symbol over the terms 1+mkN and Xk(s) indicates
that the terms corresponding to k = j (the index of the outer summation) are not included in the arguments
of the function F(n)A . The function F
(n)A is the Lauricella’s hypergeometric function of n variables, defined
as [13]
F(n)A (a, b1, . . . , bn; c1, . . . , cn; x1, . . . , xn)
=∞∑
m1=0
· · ·∞∑
mn=0
(a)m1+...mn(b1)m1 · · · (bn)mnxm11 · · ·xmn
n
(c1)m1 · · · (cn)mnm1! · · ·mn!
for |x1| + · · · + |xn| < 1,
(11)
To calculate F(n)A numerically, it may be useful to write (11) in a more compact form as
F(n)A (a, b1, . . . , bn; c1, . . . , cn; x1, . . . , xn)
=∞∑
q=0
(a)q
∑Ω(q,n)
n∏k=1
(bk)qk
(ck)qk
xqkk
qk!,
(12)
where Ω(q, n) is the set of n-tuples such that Ω(q, n) = (q1, . . . , qn) : qk ∈ 0, 1, . . . , q,∑nk=1 qk = q.
Note that the MGF of the output SNR is simply Ψ(0)(s). Also, the moments of the output SNR are
calculated as
E[γns ] = Ψ(n)(0). (13)
Using (10) and (13), the mean and variance of the output SNR distribution can be readily evaluated, as
well as high order metrics such as the skewness and the Kurtosis [10]. The amount of fading (AoF) can
also be calculated from our result. The AoF was introduced in [14] as a measure of the severity of a
fading channel. It is generally independent of the fading power and is defined as
AoF =V ar(γs)
E2[γs]=
E[γ2s ]
E2[γs]− 1. (14)
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Although we have been able to provide a compact exact expression for the MGF of the output SNR
and all its derivatives in very general conditions, the computational complexity of our expressions is high.
We now simplify (10) for several important cases. If the average symbol energy Esjis the same for every
transmit antenna j, and the channel gains from every transmit to every receive antenna are identically
distributed, (10) simplifies to
Ψ(n)(s) = NL(m/γ)mNL m
(Lm/γ − s)n+mNL
× Γ(n + mNL)
ΓL(mN + 1)F
(L−1)A
⎛⎜⎝n + mNL, 1, . . . , 1︸ ︷︷ ︸L−1 terms
;
1 + mN, . . . , 1 + mN︸ ︷︷ ︸L−1 terms;
;m/γ
Lm/γ − s, . . . ,
m/γ
Lm/γ − s︸ ︷︷ ︸L−1 terms
⎞⎟⎟⎟⎟⎠ ,
(15)
where in this case a Lauricella function must be evaluated once, as opposed to the L − 1 evaluations
needed in (10) In this case (12) can be expressed in a somewhat simpler form as
F(n)A (a, 1, . . . , 1; c, . . . , c; x, . . . , x)
=∞∑
q=0
(a)qxq∑
Ω(q,n)
1
(c)q1 · · · (c)qn
,(16)
On the other hand, for the case of 2 transmit antennas, (10) reduces to
Ψ(n)(s) =Γ(n + N(m1 + m2))
NΓ(m1N)Γ(m2N)
× (m1/γ1)m1N (m2/γ2)
m2N
(m1/γ1 + m2/γ2 − s)n+NmT
×⎡⎣ 1
m12F1
⎛⎝n + NmT , 1; 1 + m!N ;m1/γ1
m1
γ1+ m2
γ2− s
⎞⎠+
1
m22F1
⎛⎝n + NmT , 1; 1 + m2N ;m2/γ2
m2
γ1+ m2
γ2− s
⎞⎠⎤⎦ ,
(17)
with mT = m1 + m2, where we have used the fact that 2F1 = F(1)A is the Gaussian hypergeometric
function.
B. SNR Statistics for Integer Fading Parameters
In this section we calculate the SNR statistics when the fading parameters mj are constrained to take
integer values. In this case, from (3) and (4) and with the help of [12, eq. (8.352.1)], the cdf of the output
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SNR can be written as
Fo(x) =L∏
k=1
⎛⎝1 − e−xmk/γk
mkN−1∑l=0
1
l!
(xmk
γk
)l⎞⎠ . (18)
This expression is very easy to evaluate, however, for subsequent derivations a more convenient expression
of (18) will be given as a summation of terms of the type xaebx, with a and b being constants. This can
be accomplished with the help of the following two lemmas.
Lemma 1: Let z1, z2, . . . , zL be a set of L variables. Then, the following equality holds:
L∏k=1
(1 − zk) =L∑
i=0
(−1)i∑
τ(i,L)
L∏k=1
zikk , (19)
where τ(i, L) is the set of L-tuples such that τ(i, L) =(i1, . . . , iL) : ik ∈ 0, 1,∑L
k=1 ik = i
.
Proof: It is clear that we can write∏L
k=1(1 − zk) = 1 − (z1 + z2 + . . . + zL) + (z1z2 + z1z3 . . . +
zL−1zL)− (z1z2z3 + . . .+ zL−2zL−1zL) + . . .+ z3z3 · · · zL. This summation can be written as expressed in
(19) by noting that the i-th term, i = 0..L, is given by the sum of the
⎛⎜⎜⎜⎝ L
i
⎞⎟⎟⎟⎠ combinations of products
of i variables zk, and this number of combinations of i terms is also the number of elements in the set
τ(i, L). Note also that when all zk variables are equal, (19) collapses to Newton’s binomial.
Lemma 2: Let us consider a set of L polynomials, where the order of the k-th polynomial is Lk and
the coefficient of xl is denoted as bk,l. The product of the L polynomials satisfies the following equality
L∏k=1
⎛⎝ Lk∑l=0
bk,lxl
⎞⎠ =U∑
l=0
⎛⎝ ∑ω(l,L)
L∏k=1
bk,lk
⎞⎠xl, (20)
with
U =L∑
k=1
Lk,
and where ω(l, L) is the set of L-tuples such that ω(l, L) =(l1, . . . , lL) : lk ∈ 0, 1, · · · , Lk,∑L
k=1 lk = l
.
Proof: It is well known that the product of a set of polynomials is another polynomial whose degree is
the sum of the degrees of the polynomials in the set and the coefficient of xl in the resulting polynomial
is the sum of terms of the form∏L
k=1 bk,lk such that∑L
k=1 lk = l. Thus, equality (20) follows in a
straightforward manner.
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Obviously, (18) is in the form of (19), therefore, after some manipulation we can write
Fo(x) =M∑i=0
(−1)i∑
τ(i,L)
e−x∑M
k=1ikmk/γk
×L∏
k=1
⎡⎣ik(mkN−1)∑l=0
1
l!
(xmk
γk
)l⎤⎦ ,
(21)
and applying now equality (20) to (21) we finally obtain
Fo(x) =L∑
i=0
(−1)i∑
τ(i,L)
e−x∑L
k=1ikmk/γk
U∑l=0
Cl,Lxl, (22)
with
Lk = ik(mkN − 1)
Cl,L =∑
ω(l,L)
L∏k=1
1
lk!
(mk
γk
)lk
.
On the other hand, the pdf of the output SNR can be calculated from (5) as
fo(x) =L∑
j=1
(mj
γj
)mjN1
(mjN − 1)!xmjN−1e−xmj/γj
×L∏
k=1,k =j
⎛⎝1 − e−xmk/γk
mkN−1∑l=0
1
l!
(xmk
γk
)l⎞⎠ .
(23)
Following the same approach as the one used for the cdf, we can rewrite (23) in the form
fo(x) =L∑
j=1
(mj
γj
)mjN1
(mjN − 1)!xmjN−1e−xmj/γj
×L−1∑i=0
(−1)i∑
τ(i,L−1)
e−x∑L
k=1,k =jikmk/γk
U∑l=0
Cl,L−1xl,
(24)
where the terms are defined as in (22) and the symbol over a given term or expression indicates that
the argument for k = j (the index of the outer summation) is not included in the definition of the term.
Finally, the MGF of the output SNR and its derivatives is calculated substituting (24) into (6), yielding
Ψ(n)(s) =L∑
j=1
(mj
γj
)mjN1
(mjN − 1)!
×L−1∑i=0
(−1)i∑
τ(i,L−1)
U∑l=0
Cl,L−1
×∫ ∞
0xn+mjN+l−1e
−x
(mjγj
+∑L
k=1,k =j
ikmkγk
−s
)dx.
(25)
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By performing the integration we obtain
Ψ(n)(s) =L∑
j=1
(mj
γj
)mjN1
(mjN − 1)!
L−1∑i=0
(−1)i
× ∑τ(i,L−1)
U∑l=0
Cl,L−1(n + mjN + l − 1)!(mj
γj+∑L
k=1,k =jikmk
γk− s
)n+mjN+l .
(26)
III. DERIVATION OF THE AVERAGE ERROR RATES
In this section we present exact expressions for the average BER and SER of different modulation
techniques. The average error rates can be calculated promediating the conditional error probability (CEP),
i.e., the error rate under AWGN, over the output SNR, that is:
Pe =∫ ∞
0PE(x)fo(x)dx, (27)
where PE(x) denotes the CEP. When the cdf of the output SNR has a more compact form, which is our
case, it may be preferable to compute the average error rate in terms of the cdf. This can be done by
integrating (27) by parts, yielding:
Pe = −∫ ∞
0P
′E(x)Fo(x)dx, (28)
where P′E(γ) is the first order derivative of the CEP.
A. MRC Diversity
1) Binary Modulation: For binary phase-shift keying (BPSK) and coherent frequency-shift keying
(BFSK) the CEP is given by:
PE(γ) = Q(√
bγ)
(29)
with b = 2 for BPSK and b = 1 for BFSK, and where Q(x) = (√
2π)−1∫∞x e−z2/2dz is the Gaussian
Q-function. For non-coherent BFSK (NCBFSK) and differential BPSK (DBPSK) the CEP is given by.
PE(γ) = 0.5e−bx, (30)
with b = 1 for DBPSK and b = 0.5 for NCBFSK. It can be easily observed that if we substitute (30)
into (27), the average bit error rate is given in terms of the MGF of the output SNR, which has been
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calculated in the previous section. Specifically, for DBPSK and NCFSK, the average BER will be given
by:
Pe = 0.5 · Ψ(0)(−b) (31)
However, for the sake of compactness, to compute the BER of the considered binary modulations we will
use the following expression for the CEP [9, eq. (8.100)]:
PE(γ) =Γ(b, aγ)
2Γ(b). (32)
It is easy to check that (32) includes the cases of both (29) and (30), and therefore the error rates of the
different binary modulations considered here can be evaluated using the same final expressions by just
changing the values of the specified constants. The value of the constants in (32) are: (a, b) = (1, 0.5) for
BPSK, (a, b) = (0.5, 0.5) for BFSK, (a, b) = (1, 1) for DBPSK and (a, b) = (0.5, 1) for NCBFSK.
The average BER can be calculated by substituting (3)-(4) into (28) which, by recognizing that P′E(γ) =
−abγb−1e−aγ/2Γ(b) and with the help of [12, eq. (8.351.2)], yields
Pe =ab
2Γ(b)
⎛⎝ L∏j=1
(mj/γj)mjN
Γ(mjN + 1)
⎞⎠ I1(a, b), (33)
where
I1(a, b) =∫ ∞
0xΛ(b)−1e−x(
∑L
k=1mk/γk+a)
×L∏
k=1
1F1(1; 1 + mkN ; xmk/γk) dx,
(34)
and where Λ(·) is defined as in (9). With the help of [13, eq. (2.4.2)] we can solve (34) as
I1(a, b) =Γ(Λ(b))
(∑L
k=1 mk/γk + a)Λ(b)F
(L)A (Λ(b), 1, . . . , 1︸ ︷︷ ︸
L terms
;
1 + m1N, . . . , 1 + mLN ; X1(−a), . . . , XL(−a))
(35)
where Xk(·) is defined as in (10).
When the fading parameters mk are integer numbers a closed form expression of the average BER, as
a function of a finite sum of simple terms, can by found by substituting (22) into (28), resulting in
Pe =ab
2Γ(b)
L∑i=0
(−1)i∑
τ(i,L)
U∑l=0
Cl,LH1(a, b), (36)
12
where
H1(a, b) =∫ ∞
0xl+b−1e−x(
∑L
k=1ikmk/γk+a)dx
=Γ(l + b)(∑L
k=1 ikmk/γk + a)l+b ,
(37)
2) M-ary Modulation: QPSK and M-QAM with rectangular constellation have a CEP given by [9, eq.
(8.10)]:
PE(γ) = aQ(√
bγ)− cQ2
(√bγ)
, (38)
where (a, b, c) = (2, 1, 1) for QPSK and (a, b, c) = ((4(√
M − 1)/√
M, 3/(M − 1), 4(√
M − 1)2/M) for
M-QAM with rectangular constellation. The derivative of (38) is given by
P′E(γ) = (c − a)
√b
8πγe−bγ/2 − cb
2πe−bγ
1F1
(1;
3
2;bγ
2
), (39)
where we have used the relation [11, eq. (7.1.21)]
Q(x) =1
2
[1 − 2x√
2π1F1
(1;
3
2;x2
2
)]. (40)
Substituting (39) into (28) and following an approach very similar to the one used to obtain (33)-(35),
we obtain the average SER
Pe =
⎛⎝ L∏j=1
(mj/γj)mjN
Γ(mjN + 1)
⎞⎠×⎛⎝(a − c)
√b
8πI1(b/2, 1/2) +
cb
2πI2(b)
⎞⎠ (41)
where
I2(b) =∫ ∞
0xΛ(1)−1e−x(
∑L
k=1mk/γk+b)
× 1F1
(1;
3
2;bx
2
)L∏
k=1
1F1(1; 1 + mkN ; xmk/γk) dx,
(42)
which can be solved as
I2(b) =Γ(Λ(1))
(2Y (b)/b)−Λ(1)F
(L+1)A (Λ(1), 1, . . . , 1;︸ ︷︷ ︸
L+1 terms
3/2,
1 + m1N, . . . , 1 + mLN ; Y (b), X1(−b), . . . , XL(−b)) ,
(43)
where Y (b) = (b/2)/(∑L
l=1 ml/γl + b).
13
When the fading parameters are integer numbers, the average SER can be obtained as a closed-form
expression by substituting (22) and (39) into (28), yielding
Pe =L∑
i=0
(−1)i∑
τ(i,L)
U∑l=0
Cl,L⎛⎝(a − c)
√b
8πH1(b/2, 1/2) +
cb
2πH2(b)
⎞⎠ ,
(44)
where
H2(b) =∫ ∞
0xle−x(
∑L
k=1ikmk/γk+b)
1F1
(1;
3
2;bx
2
)dx
=l!(∑L
k=1 ikmk/γk + b)l+1
× 2F1
(l + 1, 1;
3
2;
b/2∑Lk=1 ikmk/γk + b
).
(45)
B. Post-detection EGC with NCMFSK
In post-detection EGC with non-coherent M-ary frequency-shift keying (NCMFSK) signaling, square-
law detectors at every receive branch are used to obtain the M decision variables, and the CEP is given
as [15], [16]
PE(γ) =1
(N − 1)!
M−1∑r=1
(−1)r+1
⎛⎜⎜⎜⎝ M − 1
r
⎞⎟⎟⎟⎠×
r(N−1)∑q=0
βq,r(q + N − 1)!
(r + 1)q+Ne−γ
1F1
(q + N ; N ;
γ
r + 1
)
=M−1∑r=1
(−1)r+1
⎛⎜⎜⎜⎝ M − 1
r
⎞⎟⎟⎟⎠r(N−1)∑
q=0
βq,r exp( −γr
r + 1
)
×q∑
n=0
⎛⎜⎜⎜⎝ q + N − 1
q − n
⎞⎟⎟⎟⎠ q!
n!(r + 1)q+N+nγn,
(46)
where βq,r is the coefficients of xq in the expansion⎛⎝N−1∑q=0
xq
q!
⎞⎠r
=r(N−1)∑
q=0
βq,rxq. (47)
14
An iterative algorithm is used in [15] and [16] to calculate these coefficients. However, they can be
expressed in closed-form by noting that (47) is a special case of (20), and thus we can write
βq,r =∑
ω(q,r)
r∏k=1
1
lk!, (48)
where ω(q, r) is the set of r-tuples such that ω(q, r) = (l1, . . . , lr) : lk ∈ 0, 1, · · · , N − 1,∑rk=1 lk = q.
It is shown in [15] that in this case the average SER can be written as a function of the n-th order
derivative of the MGF, and is obtained by substituting (46) into (27), which with the help of (6) yields
Pe =M−1∑r=1
(−1)r+1
⎛⎜⎜⎜⎝ M − 1
r
⎞⎟⎟⎟⎠r(N−1)∑
q=0
βq,r
×q∑
n=0
⎛⎜⎜⎜⎝ q + N − 1
q − n
⎞⎟⎟⎟⎠ q!
n!(r + 1)q+N+nΨ(n)
( −r
r + 1
).
(49)
Using the results for Ψ(n)(s) obtained in Section II we thus obtain the exact results of average SER for
arbitrary or integer values of the fading parameter in Nakagami fading.
IV. COMPUTATIONAL METHODS AND NUMERICAL RESULTS
In this section we provide numerical results for various system configurations and modulation tech-
niques. The effect of different average receive power from the different transmit antennas as well as
different fading severity as measured by the AoF in (14) is also investigated. In practical downlink wireless
systems such as Wi-Fi, WiMax or cellular systems, the number of antennas at the portable handset terminal
is typically low. Therefore in this section all the presented numerical results assume that the number of
receive antennas N is set to 2. We provide results for a TAS systems with multichannel reception when
the number of transmit antennas L is set to 2 and 4. All the results shown here have been analytically
obtained by the direct evaluation of the expressions developed in this paper: either (10), (33), (35), (41)
and (43) for non-integer values of m, or (26), (36), (37), (44) and (45) for integer values. It can also be
easily demonstrated analytically that all the expressions presented in this paper involving the Lauricella
function are convergent, as the convergence condition given in (11) is always fulfilled.
15
In the next subsection we describe the computational methods used to obtain the numerical results
presented here for computations involving the Lauricella function.
A. Computational Methods
The Lauricella function F(n)A is typically computed with a finite summation approximating the infinite
summation given in (12) as
F(n)A (a, b1, . . . , bn; c1, . . . , cn; x1, . . . , xn)
=qmax∑q=0
(a)q
∑Ω(q,n)
n∏k=1
(bk)qk
(ck)qk
xqkk
qk!,
(50)
where the minimum value of qmax for a desired approximation depends on the values of the system
parameters and the type of modulation. For example, for L = 4, N = 2, mk = 3 and γk = 10 dB for
k = 1 . . . L, the evaluation of BER for BPSK using (33) yields a relative error with respect to the exact
value (which can be obained using (36), as we are considering integer values for the fading parameters)
of only 0.5% by setting qmax = 12. For QSPK, to obtain the same precision using (41) to approximate
(44), we must set qmax = 32. Note that as qmax increases the number of terms of the inner summation
in (12) increases exponentially. If a large qmax is required to obtain the desired accuracy, then (50) may
have a high computational complexity. In this case an alternate method with lower complexity can be
obtained from [13, eq. (2.4.2)]:
F(n)A (a, b1, . . . , bn; c1, . . . , cn; x1, . . . , xn)
≈Np∑k=0
wkta−1k
n∏i=1
1F1(bi; ci; xitk)
(51)
where tk and wk are, respectively, the k-th zero and weight of the Laguerre polynomial of order Np
[11, eq. (25.4.45)]. Numerical results show that for Np = 30 this approximation provides an excellent
agreement with the exact results for almost all cases.
B. Numerical Results
In Fig. 1 we show numerical results for the BER of BPSK with MRC reception as a function of the
average SNR per receive antenna for different values of the Nakagami fading parameter m and number
16
of transmit antennas. We consider here that the fading from the different transmit antennas are equally
distributed. As expected, as the fading parameter increases, the diversity gain increases too, resulting in a
higher slope of the curves, and with diminishing returns as m increases. The same behavior is observed in
Figs. 2 and 3, where the SER for QPSK and 16-QAM, respectively, with MRC reception, is plotted under
the same conditions of Fig. 1. Fig. 4 shows numerical results when EGC is employed at the receiver with
NCMFSK modulation for M = 4. A loss of around 2 to 3 dB is appreciated in this case with respect to
QPSK, due to the non-coherent reception.
We now explore the effect of unbalanced average received SNR from the different transmit antennas.
In the following figures we assume that the average received SNR resulting from one of the transmit
antennas, that we denote as antenna 1, is C times higher then the average SNR resulting from the rest of
the transmit antennas. This higher average SNR resulting from one of the transmit antennas may be due
to a Line Of Sight (LOS) signal component between that transmit antenna and the receive array. Such a
situation may occur in a cell-site diversity scheme. Fig. 5 shows the average SER of 16-QAM for different
values of C as a function of the average received SNR per receive antenna, which is defined here as
SNR =1
L
L∑k=1
γk.
The fading parameter is assumed to be m = 1 (Rayleigh fading) for all the channels. It is clear from the
figure that when all the fading parameters are equal, the unbalanced average SNR has a detrimental effect
on performance, as part of the benefit of the diversity is lost. Fig. 6 shows again results for the average
SER for 16-QAM for different values of C but in this case m1 = 5 and mk = 1 for k = 1. In this case,
transmit antenna 1 is dominant due to the presence of a strong specular component to antenna 1. We can
see that, in this case, better performance is obtained as C increases, because the signals received from
the dominant transmit antenna have a more favorable channel than for the rest of transmit antennas. Fig.
7 shows the effect of the SNR unbalance factor C on the amount of fading for different values of the
fading parameter, which is assumed to be the same for all transmit antennas. Note that, as C increases,
more power is received from the dominant antenna and therefore the AoF increases, and the benefit of
17
diversity decreases. In all cases, for a C higher than around 10 dB, the AoF is nearly the same as when
there is only one transmit antenna and therefore the benefits of transmit selection diversity will be greatly
reduced, and this reduction is more dramatic as the number of transmit antennas increases or the fading
parameter decreases.
V. CONCLUSIONS
We have presented exact expressions of the SNR statistics and average error rates for a system
employing TAS and a multichannel receiver using MRC or post-detection EGC in Nakagami fading. We
considered several modulation techniques including M-QAM with rectangular constellation and QPSK,
as well as a very general channel model where the fading from every transmit antenna are not necessarily
identically distributed. When the fading parameters of the channel between every transmit antenna and
the multichannel receiver are integers our results are given in terms of a finite sum of simple terms.
When the fading parameters are arbitrary real numbers our results are given in terms of the Lauricellas
hypergeometric function F(n)A . Numerical results for the error rates of different modulations has been
presented for a variety of scenarios and system configurations, including the case when there is a dominant
signal from one of the transmit antennas to the receiver.
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2004.
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19
0 5 10 15 20 25 3010
−8
10−7
10−6
10−5
10−4
10−3
10−2
10−1
100
Average SNR
Ave
rage
BE
RL=2L=4
m=0.5, 1, 1.7, 3.2
Fig. 1. Average BER vs. average SNR for BPSK for different values of the fading parameter and number of transmit antennas. N=2.
0 5 10 15 20 25 3010
−8
10−7
10−6
10−5
10−4
10−3
10−2
10−1
100
Average SNR
Ave
rage
SE
R
L=2L=4
m=0.5, 1, 1.7, 3.2
Fig. 2. Average SER vs. average SNR for QPSK for different values of the fading parameter and number of transmit antennas. N=2.
20
0 5 10 15 20 25 3010
−8
10−7
10−6
10−5
10−4
10−3
10−2
10−1
100
Average SNR
Ave
rage
SE
RL=2L=4
m=0.5, 1, 1.7, 3.2
Fig. 3. Average SER vs. average SNR for 16-QAM for different values of the fading parameter and number of transmit antennas. N=2.
0 5 10 15 20 25 3010
−8
10−7
10−6
10−5
10−4
10−3
10−2
10−1
100
Average SNR
Ave
rage
SE
R
L=2L=4
m=0.5, 1, 1.7, 3.2
Fig. 4. Average SER vs. average SNR for NCMFSC (M=4) with post-detection EGC for different values of the fading parameter and
number of transmit antennas. N=2.
21
0 5 10 15 20 25 3010
−8
10−7
10−6
10−5
10−4
10−3
10−2
10−1
100
Average SNR
Ave
rage
SE
RL=2L=4
C=0, 5, 10, 15, 20 dB
Fig. 5. Average SER vs. average SNR for 16-QAM with unbalanced received SNR. m=1. N=2.
0 5 10 15 20 25 3010
−8
10−7
10−6
10−5
10−4
10−3
10−2
10−1
100
Average SNR
Ave
rage
SE
R
L=2L=4
C=0, 5, 10, 15, 20 dB
Fig. 6. Average SER vs. average SNR for 16-QAM for unbalanced SNR and fading parameter. m1=5. mk=1, k = 1. N=2.
22
0 5 10 150
0.5
1
1.5
C(dB)
Am
ount
of F
adin
g
L=2L=4
m=0.5, 1, 1.5, 2
Fig. 7. Effect of unbalanced SNR on AoF. N=2.
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