1

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.

[3] S. Thoen, L. Van der Perre, B. Gyselinckx and M. Engels, ”Performance Analysis of Combined Transmit-SC/Receive-MRC,” IEEE

Trans. Comm., vol. 49, no. 1, pp. 5-8, January 2001.

[4] Z. Chen, J. Yuan and B. Vucetic, ”Analysis of Transmit Antenna Selection/Maximal-Ratio Combining in Rayleigh Fading Channels,”

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China, September 2006.

18

[6] M. Nakagami, ”The m-distribution: A General Formula of Intensity Distribution of Rapid Fading,” in Statistical Methods in Radio Wave

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[15] A. Annamalai and C. Tellambura, ”A Moment-Generating Function (MGF) Derivative-Based Unified Analysis of Incoherent Diversity

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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.