Performance evaluation of generalized selection combiners (GSC) over slow fading channels with...

Wireless Pers Commun (2011) 56:207–236DOI 10.1007/s11277-009-9829-6

Performance Evaluation of Generalized SelectionCombiners Over Slow Fading with Estimation Errors

Fawaz S. Al-Qahtani · Arun K. Gurung ·Salam A. Zummo · Seedahmed S. Mahmoud ·Zahir M. Hussain

Published online: 15 October 2009© Springer Science+Business Media, LLC. 2009

Abstract In this paper we derive closed-form expressions for the single-user adaptivecapacity of generalized selection combining (GSC) system, taking into account the effectof imperfect channel estimation at the receiver. The channel considered is a slowly vary-ing spatially independent flat Rayleigh fading channel. The complex channel estimate andthe actual channel are modelled as jointly Gaussian random variables with a correlationthat depends on the estimation quality. Three adaptive transmission schemes are analyzed:(1) optimal power and rate adaptation; and (2) constant power with optimal rate adaptation,and (3) channel inversion with fixed rate. In addition to deriving an exact expression for thecapacity of the aforementioned adaptive schemes, we analyze the impact of channel estima-tion error on the capacity statistics and the symbol error rate for GSC systems. The capacitystatistics derived in this paper are the moment generating function, complementary cumu-lative distribution function and probability density function for arbitrary number of receiveantennas. Moreover, exact closed-form expressions for M-PAM/PSK/QAM employing GSCare derived. As expected, the channel estimation error has a significant impact on the systemperformance.

F. S. Al-Qahtani (B) · A. K. Gurung · Z. M. HussainSchool of Electrical and Computer Engineering, RMIT University, Melbourne, VIC 3000, Australiae-mail: [email protected]

A. K. Gurunge-mail: [email protected]

Z. M. Hussaine-mail: [email protected]

S. A. ZummoElectrical Engineering Department, KFUPM, Dhahran 31261, Saudi Arabiae-mail: [email protected]

S. S. MahmoudFuture Fibre Technologies Pty Ltd., 10 Hartnett Close, Mulgrave, VIC 3170, Australiae-mail: [email protected]

123

208 F. S. Al-Qahtani et al.

Keywords Adaptive transmission · Channel capacity ·Generalized selection combining (GSC) · Rayleigh fading

1 Introduction

An efficient way to combat multi-path fading and dramatically improve the system per-formance over fading channels is diversity. By applying diversity methods, multiple cop-ies of the information signal can be sent over independent fading channels and combinedby the receiver to maximize the power of received signal. The main principle of diver-sity is to ensure that the same information reaches the receiver on statistically independentchannels. As the channels are statically independent, the probability that antennas are ina fading dip simultaneously is lower than the probability that one antenna is in a fadingdip.

Diversity array combining techniques have attracted a lot of research attention in the lastdecade because they are capable of providing high speed wireless communications over amultipath environment. The receiver combines multiple copies of the desired signal, each ofwhich being affected by an independent channel of the other signals. The most commonlyused diversity combining techniques are equal gain combining (EGC), maximal ratio com-bining (MRC), selection combining (SC), and a hybrid combination of SC and MRC, orcalled generalized selection combining (GSC). GSC is a technique to choose fixed subset ofsize M of a large number of available diversity channels of size L and then combine themusing MRC which combines coherently the individually received branch signals so as to max-imize the instantaneous output signal-to-noise ratio (SNR) [1–3]. It is well known that MRCis the optimal linear combining scheme. However, with receiver MRC, most of the systemcomplexity concentrates at the receiver side. The main advantage of GSC over MRC is thatGSC scheme reduces the number of radio frequency (RF) chains required at the receiver [2].

System performance analysis often considers perfect channel estimates under the assump-tion that the estimation errors are negligible. However, in practical systems, the branch SNRestimates are usually combined with noise which makes it difficult to estimate them perfectly.In practice, a diversity branch SNR estimate can be obtained either from a pilot signal or datasignals (by applying a clairvoyant estimator) [4]. For example, if a pilot signal is inserted toestimate the channel, a Gaussian error may arise in due the large frequency separation or timedispersion. Thus, from a theoretical and practical viewpoints, it is important to quantify howGSC performance degrades due to the estimation error. Previous works on the analysis ofimperfect channel estimation with no diversity can be found in [5–7]. In [4], Gans consideredthe channel estimation error of the MRC per branch SNR as being complex Gaussian andderives the probability density function (PDF) of the output combiner.

The pioneering work of Shannon [10] has established the significance of channel capac-ity as the maximum possible rate at which information can be transmitted over a channel.Spectral efficiency of adaptive transmission techniques has received extensive interest in thelast decade. In [12], the capacity of a single-user flat fading channel with perfect channelinformation at the transmitter and the receiver is derived for various adaptation policies,namely, (1) optimal rate and power adaptation (opra), (2)optimal rate adaptation (ora) andconstant power, and (3) channel inversion with fixed rate (cifr). The first scheme requireschannel information at the transmitter and receiver, whereas the second scheme is morepractical since the transmission power remains constant. The last scheme is a suboptimaltransmission adaptation scheme, in which the channel side information is used to main-tain a constant received power by inverting the channel fading [12]. In [13], the general

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Performance Evaluation of Generalized Selection 209

theory developed in [12] is applied to derive closed-form expressions for the capacity ofRayleigh fading channels under different adaptive transmission and diversity combiningtechniques. Recently, there has been some work dealing with the channel capacity of differ-ent fading channels employing different adaptive schemes such as [14,15], and the referencestherein.

The objective of this paper is to investigate the impact of channel estimation error on theadaptive capacity, capacity statistics and the average symbol error rate (SER) for GSC overslow Rayleigh fading. The channel estimation error is quantified in terms of a correlationcoefficient between the perfect channel and its estimate. This problem has been consideredfrom SER point of view before in [8] and [9]. However, to the authors’ best knowledge,this is the first attempt to derive the adaptive capacity, capacity statistics of GSC (L, M)with estimation errors for an arbitrary number of receive antennas. In this contribution, weinvestigate the adaptive capacity of GSC under different adaptive transmission in Rayleighfading with estimation errors. Also, we derive the capacity statistics of GSC scheme whichare valid for arbitrary number of receive antennas including moment generating function(MGF), cumulative distribution function (CDF) and, probability distribution function (PDF).The contributions of this paper are three-fold. First, we derive the capacity statistics of GSCsubject to Rayleigh fading for an arbitrary number of receive antennas. Secondly, we deriveclosed-form expressions for the channel capacity of GSC in independent and identically dis-tributed (i.i.d.) Rayleigh fading channels with the following adaptive transmission schemes:(1) opra; (2) ora with constant transmit power; and (3) cifr. Finally, we derive exact closedforms of SER for PAM/PSK/QAM with M-ary signaling.

The paper is organized as follows. In Sect. 2, the system model used in this paperis discussed. The capacity statistics are derived in Sect. 4. In Sect. 3, we derive closed-form expressions for the channel capacity under different adaptation schemes. Section 5presents the closed-form expressions for average SER for M-PAM/PSK/QAM of GSC (L,M) in independent and identically distributed (i.i.d.) Rayleigh fading channels. Results arepresented and discussed in Sect. 6. The main outcomes of the paper are summarized inSect. 7.

2 System Model

In this section, we provide a brief description of GSC and the channel estimation model usedthroughout this paper. The channel estimation error used in this paper is characterized by thecorrelation between the true and estimated channel. Whereas the probability density functionof effective SNR under channel estimation error can be expressed in terms of the correlationcoefficient ρ. We consider an L-branch diversity receiver in slow fading channels. Assumingperfect timing and no inter-symbol interference (ISI), the received signal on the lth branchcan be expressed as

rl = gls + nl , l = 1, . . . , L , (1)

where gl is a zero-mean complex Gaussian distributed channel gain, nl is the complex additivewhite Gaussian noise (AWGN) sample with a variance of N0/2, and s is the data symbol takenfrom a normalized unit-energy signal set with an average power Ps . Under the assumption ofchannel estimation error at the receiver which employs GSC by combining the strongest Mdiversity branches out of a total of L diversity branches, the PDF of the received instantaneousSNR γ at the output of the combiner is given by [8]

123


pγ (γ ) =L−M∑

l=0

D∑

k=1

μDl−kl

�kl

[γ (1 − ρ2)

]k−1

×k−1∑

n=0

1

n!(

k − 1n

)[Mρ2γ

�l(1 − ρ2)(l + M)

]n

exp

(−γ�l

), (2)

where

Dl ={

M, for l = 01, for l �= 0

(3)

and the coefficients of partial fractions are given as follows:μ00 =

(LM

), μ0

l = (−1)M+l−1

(M

M+l

) (Ml

)M−1(

LM

)(L − M

l

)for l �= 0, μM−k

0 = ∑L−Mi=l (−1)i+M−k−1

(LM

)

(L − Mi

) (Mi

)M−kif k < M < L , and (−1)M+l−1

(M

M+l

) (Ml

)M−1(

LM

)(L − M

l

)

for l �= 0. The variable �l = γ[(M + l(1 − ρ2))/(l + M)

]and ρ denotes the correlation

between the actual channel gains and their estimates and it can be expressed as

ρ = cov(gl , gl)√var(gl)var(gl)

=√

1 − ε2. (4)

The actual channel gain g is related to the channel estimate g [4] as follows

gl =√

1 − ε2gl + εzl , (5)

where zl is a complex Gaussian random variable independent of g with zero-mean and a unitvariance and ε ∈ [0, 1] is a measure of the accuracy of the channel estimation. The actualchannel gain is scaled to keep the covariance of the estimated channel and the true channelto be the same. For ε = 0, the estimated channel is fully correlated with the true channel(perfect channel estimation ρ = 1). Note that for perfect channel estimation, the pdf of (2)reduces to the well-known PDF at the output of a GSC combiner which is given by

pγ (γ ) =L−M∑

l=0

D∑

k=1

μDl−kl γ k−1(l + M)

M(k − 1)!γ exp

(−(l + M)γ

Mγ

). (6)

3 Adaptive Capacity Policies

In this section, we derive close-form expressions for the capacity of different adaptive schemesemploying GSC over Rayleigh fading channels. In the derivation, we will rely on the mainresults from [13].

3.1 Power and Rate Adaptation

Given an average transmit power constraint, the channel capacity Copra in (bits/s) of a fadingchannel [12,13] is given by

Copra = B

ln 2

∞∫

γ0

ln

(γ

γ0

)pγ (γ )dγ, (7)

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where B (in hertz) is the channel bandwidth and γ0 is the optimum cutoff SNR satisfying thefollowing condition

∞∫

γ0

(1

γ0− 1

γ

)pγ (γ )dγ = 1. (8)

To achieve the capacity in (7), the channel fading level must be tracked at both transmitter andreceiver. The transmitter has to adapt its power and rate accordingly by allocating high powerlevels and transmission rates for good channel conditions (large γ ). Since the transmission issuspended when γ < γ0, this policy suffers from outage, whose probability Pout is definedas the probability of no transmission and is given by

Pout = 1 −∞∫

γ0

pγ (γ )dγ. (9)

Substituting (2) into (8) yields the equality

L−M∑

l=0

D∑

k=1

μDl−kl

�kl

[γ (1 − ρ2)

]k−1k−1∑

n=0

1

n!(

k − 1n

)[Mρ2

(1 − ρ2)(l + M)

]n

×[�l

γ0�

(n + 1,

γ0

�l

)− �

(n,γ0

�l

)]= 1. (10)

In order to obtain the optimal cutoff SNR γ0, in (10), we follow the following procedure. Letx = �l

γ0and define fGSC(x) as

fGSC(x) =L−M∑

l=0

D∑

k=1

μDl−kl

�kl

[γ (1 − ρ2)

]k−1k−1∑

n=0

1

n!(

k − 1n

)

×k−1∑

n=0

[Mρ2

(1 − ρ2)(l + M)

]n [� (n + 1, x)

x− � (n, x)

]. (11)

Now, differentiating the function fGSC(x)with respect to x over the interval (0,+∞) resultsin

f′GSC(x) =

L−M∑

l=0

D∑

k=1

μDl−kl

�kl

k−1∑

n=0

1

n!(

k − 1n

)[Mρ2

(1 − ρ2)(l + M)

]n

×[� (n + 1, x)

x2

](12)

Hence, f′GSC(x) < 0,∀x > 0, meaning that f

′GSC is a strictly decreasing function of x . From

(11) it can be observed that

(1) limx→0

fGSC(x) = +∞

(2) limx→+∞ fGSC(x) =

D∑k=1

μDl −kl

�kl

k−1∑n=0

1n!(

k − 1n

)[Mρ2

(1−ρ2)(l+M)

]n

Note however that fGSC(x) is a continuous function of x , which leads to a unique positiveγ0 such that fGSC(x) = 0. We thereby conclude that for each γ > 0 there is a unique γ0

123


satisfying (11). Numerical results using MATLAB shows that γo ∈ [0, 1] as γ increases, andγ0 → 1 as γ → ∞.

Now, substituting (2) into (7) yields the channel capacity with opra scheme as follows

Copra =L−M∑

l=0

D∑

k=1

μDl−kl

�kl

[γ (1 − ρ2)

]k−1k−1∑

n=0

1

n!

×(

k − 1n

)[Mρ2γ

(1 − ρ2)(l + M)

]n ∫ ∞

γ0

ln

(γ

γ0

)γ n exp

(−γ�l

)dγ

︸︷︷︸I1

. (13)

We evaluate I1 by taking the help of the following identity [13] given by

Js(μ) =∞∫

1

t s−1 ln(t)e−μt dt = �(s)

μs

{E1(μ)+

s−1∑

k=1

1

kPk(μ)

}, (14)

where E1 denotes the Exponential integral of the first order [17] defined as

E1(x) =∞∫

1

exγ

γdγ, x ≥ 0, (15)

and Pk(μ) denotes the Poisson distribution [17] given by

Pk(x) = �(k, x)

�(k)= e−x

k−1∑

i=0

xn

n! . (16)

Upon substituting (14) into (13), the following closed-form expression for capacity Copra perunit bandwidth (in bits/s/Hz) can be obtained as follows

Copra =L−M∑

l=0

D∑

k=1

μDl−kl

�kl

[γ (1 − ρ2)

]k−1k−1∑

n=0

1

n!

×(

k − 1n

)[Mρ2

�l(1 − ρ2)(l + M)

]n{

E1

(γ0

γ

)+

n+1∑

k=1

1

kPk

(γ0

γ

)}. (17)

The above capacity expression allows us to examine the limiting cases for (M = L , andM = 1) more conveniently for MRC and SC, respectively. For M = L , the μ(L−k)

l coeffi-

cient becomes μ00 = 1 and μ(L−k)

0 = 0 for all k = 1, . . . , L − 1. The opra capacity in (17)after some manipulation reduces to the closed-form expression for MRC capacity Cmrc

opra perunit bandwidth (in bits/s/Hz) as follows

Cmrcopra = [

(1 − ρ2)]L−1

L−1∑

n=0

(L − 1

n

)[Mρ2

(1 − ρ2)

]n{

E1

(γ0

γ

)+

n+1∑

k=1

1

kPk

(γ0

γ

)}.

(18)

Letting M = 1 in (17) resulting in μ00 = 1, and μ0

l = (−1)l(

L/(l + 1)

(L − 1

l

))for all

l = 1, . . . , L − 1. Therefore, the capacity Cscopra per unit bandwidth (in bits/s/Hz) after some

123


manipulation can be obtained as follows

Cscopra =

L−1∑

k=0

(−1)k(

Lk + 1

)E1

((1 + k)γ0

γ[k + 1 − kρ2

])

exp

(−γ0(k + 1)

γ[k + 1 − kρ2

]). (19)

3.1.1 Asymptotic Approximation

We can obtain asymptotic approximation FOR Copra using the series representation of Expo-nential integral of first order function [17] expressed as

E1(x) = −E − ln(x)−+∞∑

i=1

(−x)i

i.i ! , (20)

where E = 0.5772156659 is the Euler–Mascheroni constant. Then, the asymptotic approx-imation C∞

opra per unit bandwidth (in bits/s/hertz) can be shown to be

Copra =L−M∑

l=0

D∑

k=1

μDl−kl

�kl

[γ (1 − ρ2)

]k−1k−1∑

n=0

1

n!(

k − 1n

)[Mρ2

�l(1 − ρ2)(l + M)

]n

×{(

−E − ln

(γ0

γ

)+(γ0

γ

))+

n+1∑

k=1

1

kPk

(γ0

γ

)}(21)

3.1.2 Upper Bound

The capacity expression of Copra can be upper bounded by applying Jensen’s inequality to(7) as follows

CUBopra = ln (E[γ ]) . (22)

We evaluate the expression in (22) using the of pdf of γ given in (2) and the following identity[17]

∞∫

0

xne−μx dx = n!μ−n−1, for Re[μ] > 0, (23)

and simplify the resulting expression to obtain the capacity (7) upper bound as follows

CUBopra

B=(

L−M∑

l=0

D∑

k=1

μDl−kl �2−k

l

[γ (1 − ρ2)

]k−1

×k−1∑

n=0

(n + 1)!n!

(k − 1

n

)[Mρ2

(1 − ρ2)(l + M)

]n). (24)

3.2 Constant Transmit Power

By adapting the transmission rate to the channel fading condition with a constant power, thechannel capacity Cora [10,11] is given by

Cora = B

ln 2

∞∫

0

ln (1 + γ ) pγ (γ )dγ. (25)

123


Substituting (2) into (25) results in

Cora

B=

L−M∑

l=0

D∑

k=1

μDl−kl

�kl

[γ (1 − ρ2)

]k−1k−1∑

n=0

1

n!(

k − 1n

)[Mρ2

�l(1 − ρ2)(l + M)

]n

×∫ ∞

0ln (1 + γ ) γ n exp

(− γ

�l

)dγ

︸︷︷︸I3

. (26)

The integral I3 in (26) can be evaluated by taking use of equality [17]

∞∫

0

ln(1 + y)yt−1exydy = (t − 1)!ext∑

i=1

�(−t + i, x)

xi, (27)

where �(., .) is the complementary incomplete Gamma function which can be related to theexponential integral function Et (x) through [17]

Et (x) = xt−1�(1 − l, x). (28)

The expression in (27) can be further evaluated using (28) yielding

∞∫

0

ln (1 + γ ) γ n exp

(−γ�l

)dγ = exp

(1

�l

) n+1∑

j=1

�n+1l En+2− j

(1

�l

). (29)

Inserting (29) into (26) results in a closed-form expression for the capacity Cora per unitbandwidth (in bits/s/hertz) given by

Cora =L−M∑

l=0

D∑

k=1

μDl−kl

�kl

[γ (1 − ρ2)

] k−1∑

n=0

�l

n!

×(

k − 1n

)[Mρ2

(1 − ρ2)(l + M)

]n

exp

(1

�l

) n+1∑

j=1

En+2− j

(1

�l

). (30)

The capacity Cora can be expressed in another form as follows. It has been shown in [13] thatthe integral in (27) has the following form

∞∫

0

ln(1 + y)yt−1exydy = �(t)

xt

[Pt (−x) E1 (x)+

t−1∑

i=1

Pi (x) Pt−i (−x)

i

]. (31)

Substituting (31) into (26) yields another closed-form expression for the capacity Cora perunit bandwidth (in bits/s/hertz)

Cora =L−M∑

l=0

D∑

k=1

μDl−kl �1−k

l

[γ (1 − ρ2)

]k−1k−1∑

n=0

(k − 1

n

)[Mρ2

(1 − ρ2)(l + M)

]n

×⎡

⎣Pn+1

(−1

�l

)E1

(1

�l

)+

n∑

i=1

Pi

(1�l

)Pn+1−i

(−1�l

)

i

⎤

⎦. (32)

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For M = L , it yields the MRC capacity for ora policy per unit bandwidth (in bits/s/hertz)

Cmrcora = [

(1 − ρ2)]L−1

L−1∑

n=0

(L − 1

n

)[ρ2

(1 − ρ2)

]n

×⎡

⎣Pn+1

(−1

γ

)E1

(1

γ

)+

n∑

i=1

Pi

(1γ

)Pn+1−i

(−1γ

)

i

⎤

⎦. (33)

For M = 1, The ora capacity for SC is

Cscora

B=

L−1∑

k=0

(−1)k(

Lk + 1

)exp

((1 + k)

γ[k + 1 − kρ2

])

× E1

((1 + k)

γ[k + 1 − kρ2

]).

(34)

3.2.1 Asymptotic Approximation

Following the same procedure in Sect. 3.1, the asymptotic approximation C∞ora per unit band-

width (in bits/s/hertz) can be computed as

C∞ora =

L−M∑

l=0

D∑

k=1

μDl−kl �1−k

l

[γ (1 − ρ2)

] k−1∑

n=0

(k − 1

n

)[Mρ2

(1 − ρ2)(l + M)

]n

×⎡

⎣Pn+1

(−1

�l

)(−E − ln

(1

�l

)+ 1

�l

)+

n∑

i=1

Pi

(1�l

)Pn+1−i

(−1�l

)

i

⎤

⎦

(35)

3.2.2 Upper Bound

The capacity Cora can be upper bounded by applying Jensen’s inequality to (7) as follows

CUBora

B=(

1 +L−M∑

l=0

D∑

k=1

μDl−kl �2−k

l

[γ (1 − ρ2)

]k−1

×k−1∑

n=0

(n + 1)!n!

(k − 1

n

)[Mρ2

(1 − ρ2)(l + M)

]n). (36)

3.2.3 Higher SNR Region

The Shannon capacity can be approximated at high SNR region using the fact log2(1+γ ) =log2(γ ) as γ → ∞ for x > 0 yields an asymptotically tight bounds for (32) in high SNRregion as follows

ChighSNR =L−M∑

l=0

D∑

k=1

μDl−kl �1−k

l

[γ (1 − ρ2)

] k−1∑

n=0

×(

k − 1n

)[Mρ2γ

�l(1 − ρ2)(l + M)

]n [ψ (n + 1)− ln

(1

�l

)], (37)

123


where ψ(x) = ddx

ln (�(x)). For integer values of x , the function is given by ψ(x) =−E +∑x−1

i1i .

3.2.4 Lower SNR Region

We can approximate Shannon capacity in low SNR region by the square capacity of theargument (γ ) in low SNR region as log2(1 + γ ) ≈ √

γ [16]. Upon using this approximationalong with definition of the incomplete Gamma function yields the approximated Shannoncapacity at low SNR per unit bandwidth (in bits/s/hertz) as

CLowSNR1 ≈L−M∑

l=0

D∑

k=1

μDl−kl

�k−3/2l

[γ (1 − ρ2)

]

×k−1∑

n=0

�(n + 3/2)

n!(

k − 1n

)[Mρ2γ

(1 − ρ2)(l + M)

]n

. (38)

3.2.5 Lower SNR Region II

Another approximation for the Shannon capacity in low SNR region can be found by exploit-ing the fact log2(1+γ ) ≈ 1

ln(2) (γ − 12γ

2), which results the approximated Shannon capacityin low SNR region per unit bandwidth (in bits/s/hertz) as

CLowSNR2 ≈ 1

ln(2)

L−M∑

l=0

D∑

k=1

μDl−kl �2−k

l

n![γ (1 − ρ2)

] k−1∑

n=0

(k − 1

n

)

×[

Mρ2γ

(1 − ρ2)(l + M)

]n [�(n + 2)− 1

2�(n + 3)�l

)]. (39)

3.3 Channel Inversion with Fixed Rate

In this subsection, we consider two schemes; namely, truncated channel inversion with fixedrate referred to as tifr, and channel inversion with fixed rate with no truncation, which weshall refer to as cifr. Channel inversion is an adaptive transmission technique whereby thetransmitter uses the channel information feedback by the receiver in order to invert the fadingchannel. Accordingly, the channel appears to the encoder/decoder as a time-invariant AWGNchannel. As a result, channel inversion suffers a large capacity penalty compared to the pre-vious adaptation techniques (opra and ora), although it is much less complex to implement.The channel inversion technique requires a fixed code design and fixed rate modulation. Inthis case, the channel capacity Ccifr can be derived from the capacity of an AWGN channelwith a received SNR and is given by [12,13]

Ccifr = B ln

⎛

⎜⎜⎜⎝1 + 1∞∫

0

1γ

pγ (γ )dγ

⎞

⎟⎟⎟⎠. (40)

123


The channel capacity for the truncation scheme [13] Ctifr is given by

Ctifr = B ln

⎛

⎜⎜⎜⎝1 + 1∞∫γ0

1γ

pγ (γ )dγ

⎞

⎟⎟⎟⎠ (1 − Pout), (41)

where Pout is the outage probability. Note that, the cutoff level γ0 can be chosen to eithermaximize Ctifr or achieve a specific Pout.

3.3.1 tifr

The integral in (41) can be evaluated using the PDF of the combiner output SNR given in (2)as

∞∫

γ0

1

γp(γ )dγ =

L−M∑

l=0

D∑

k=1

μDl−kl

�kl

[γ (1 − ρ2)

] k−1∑

n=0

1

n!

×(

k − 1n

)[Mρ2γ

(1 − ρ2)(l + M)

]n

�

(n,γ0

�l

). (42)

Furthermore, the outage probability, Pout is derived from the CDF of the combiner outputSNR, and is given by

Pout = Prob (γ ≤ γ0) =γ0∫

0

pγ (γ )dγ = 1 −∞∫

γ0

pγ (γ )dγ

= 1 −L−M∑

l=0

D∑

k=1

μDl−kl

�k−1l

[γ (1 − ρ2)

] k−1∑

n=0

1

n!

×(

k − 1n

)[Mρ2γ

(1 − ρ2)(l + M)

]n

�

(n + 1,

γ0

�l

). (43)

Substituting (42) and (43) in (41) yields a closed-form expression for the tifr capacity perunit bandwidth (in bits/s/Hz) that is given by

Ctifr = B ln

⎛

⎜⎜⎜⎝1+ 1

∑L−Ml=0

∑Dk=1

μDl −kl�k

l

[γ (1−ρ2)

]k−1∑k−1n=0

1n!(

k−1n

)[Mρ2

(1−ρ2)(l+M)

]n�(

n, γ0�l

)

⎞

⎟⎟⎟⎠

×L−M∑

l=0

D∑

k=1

μDl−kl

�k−1l

[γ (1−ρ2)

] k−1∑

n=0

1

n!(

k−1n

)[Mρ2

(1−ρ2)(l+M)

]n

�

(n+1,

γ0

�l

). (44)

The tifr capacity per unit bandwidth (in bits/s/Hz) for the case M = L is

Cmrctifr

B=

⎛

⎜⎜⎝1 + 1

1γ

[(1 − ρ2)

]L−1∑L−1n=0

(L − 1

n

)[ρ2

(1−ρ2)

]n�(

n, γ0γ

)

⎞

⎟⎟⎠

× [(1 − ρ2)]L−1

L−1∑

n=0

(L − 1

n

)[ρ2

(1 − ρ2)

]n

�

(n + 1,

γ0

γ

). (45)

123


The capacity Ccifr per unit bandwidth (in bits/s/hertz) for SC is expressed as:

Csctifr

B= ln

⎛

⎜⎜⎝1 + γ[k + 1 − kρ2

]

∑L−1k=0 (−1)k

(L

k + 1

) [k + 1 − kρ2

]E1

(γ0(k+1)[k+1]

)

⎞

⎟⎟⎠

×L−1∑

k=0

(−1)k(

Lk + 1

)k + 1

γ[k + 1 − kρ2

] exp

(γ0(k + 1)

γ[k + 1 − kρ2

]). (46)

Moreover, the expression in (46) can be used to derive an asymptotic approximation of C∞tifr

as follows

Csc∞tifr

B= ln

⎛

⎜⎜⎝1 + γ[k + 1−kρ2

]

∑L−1k=0 (−1)k

(L

k + 1

) [k + 1 − kρ2

] (−E−log(γ0(k+1)[k+1]

)+(γ0(k+1)[k+1]

))

⎞

⎟⎟⎠

×L−1∑

k=0

(−1)k(

Lk + 1

)k + 1

γ[k + 1 − kρ2

] exp

(γ0(k + 1)

γ[k + 1 − kρ2

]). (47)

3.3.2 cifr

If we set γ = 0 , we get the cifr capacity, where in this case the Pout is equivalent to zero.From (40) and (2), we get

∞∫

0

1

γp(γ )dγ =

L−M∑

l=0

D∑

k=1

μDl−kl

�kl

[γ (1 − ρ2)

] k−1∑

n=0

1

n!(

k − 1n

)[Mρ2γ

(1 − ρ2)(l + M)

]n

.

(48)

By inserting (48) and (2) in (40) yields the cifr capacity per unit bandwidth (in bits/s/Hz) asfollows

Ccifr

B=

⎛

⎜⎜⎜⎝1 + 1

limγ→0∑L−M

l=0∑D

k=1μ

Dl −kl�k

l

[γ (1 − ρ2)

]∑k−1n=0

1n!(

k − 1n

)[Mρ2γ

(1−ρ2)(l+M)

]n

⎞

⎟⎟⎟⎠ .

(49)

For MRC (M = L), the cifr capacity per unit bandwidth (in bits/s/Hz) can be obtained as

Ccifr

B=

⎛

⎜⎜⎝1 + γ

limγ→01γ

[(1 − ρ2)

]M−1∑L−1n=0

(L − 1

n

)[ρ2

(1−ρ2)

]n� (n)

⎞

⎟⎟⎠ . (50)

The cifr capacity for SC is

Ccifr

B= ln

⎛

⎜⎜⎝1 + γ[k + 1 − kρ2

]

limγ→0∑L−1

k=0 (−1)k(

Lk + 1

) [k + 1 − kρ2

]E1

((k+1)[k+1]

)

⎞

⎟⎟⎠ . (51)

123


4 Capacity Statistics

In this section, we focus on deriving the exact analytical expressions for capacity statisticsof GSC over Rayleigh fading channels, assuming perfect channel knowledge at the receiverand no channel knowledge at the transmitter with average input-power constraint. The non-ergodic capacity of GSC system is given in [bit/s/Hz] by

C = log2(1 + γ ). (52)

4.1 Moment Generating Function

The MGF of GSC capacity system in the presence of Gaussian channel estimation errors isgiven by

�C (τ ) = E[eτC

]= E

[(1 + γ )

τln(2)

]. (53)

Expressing the expectation in an integral form over the PDF pγ (γ ) and inserting (2), weobtain

�C (τ ) =∞∫

0

(1 + γ )τ

ln(2) pγ (γ )dγ =L−M∑

l=0

D∑

k=1

μDl−kl

�l k

[γ (1 − ρ2)

]k−1k−1∑

n=0

1

n!

×(

k − 1n

)[Mρ2γ

�l(1 − ρ2)(l + M)

]n ∫ ∞

0(1 + γ )

τln(2) γ n exp

(− γ

�l

)dγ

︸︷︷︸I4

.

(54)

In order to evaluate the integral I4, we make the change-of-variables x = 1 + γ to yield

∞∫

0

(1 + γ )τ

ln(2) γ n exp

(−γ�l

)dγ = exp

(1

�l

) ∞∫

0

xτ

ln(2) (x − 1)k−1 exp

(−x

�l

)dx

= exp

(1

�l

) n∑

i=0

(k − 1

i

)(−1)k−i−1

∞∫

1

x

(τ

ln(2)+i)

exp

(−x

�l

)dx .

= exp

(1

�l

) n∑

i=0

(k − 1

i

)(−1)k−i−1�

(n + 1,

τ

ln(2)+ i

)�

(τ

ln(2)+i+1)

l . (55)

The result in (55) is obtained by the taking the use of binomial expression and the help of(23). Substituting (55) into (54) results in a closed-form expression for the MGF which isexpressed as

�C (τ ) =L−M∑

l=0

D∑

k=1

μDl−kl

�kl

[γ (1 − ρ2)

]k−1k−1∑

n=0

1

n!(

k − 1n

)[Mρ2

�l(1 − ρ2)(l + M)

]n

× exp

(1

�l

) n∑

i=0

(k − 1

i

)(−1)k−i−1�

(n + 1,

τ

ln(2)+ i

)�l

(τ

ln(2)+i)

.

(56)

123


For the case M = L , the MGF can be finally expressed as follows

�mrcC (τ ) = 1

γ

[(1 − ρ2)

]L−1L−1∑

n=0

1

n!(

L − 1n

)[ρ2

γ (1 − ρ2)

]n

× exp

(1

γ

) n∑

i=0

(n − 1

i

)(−1)n−i−1�

(n + 1,

τ

ln(2)+ i

)γ

(τ

ln(2)+i+1)

.

(57)

For M = 1, MGF in (56) reduces to

�scC (τ ) =

L−1∑

k=0

(−1)k(

Lk + 1

)((k + 1)

γ[k + 1 − kρ2

]) τ

ln(2)

exp

((k + 1)

γ[k + 1 − kρ2

])

×�(

τ

ln(2)+ 1,

((k + 1)

γ[k + 1 − kρ2

]))

. (58)

Furthermore, the integral in (54) can be expressed in another form with help of the integralrepresentation of the confluent hypergeometric function (a, b; z) [17]

(a, b; z) = 1

�(a)

∞∫

0

e−zt ta−1(1 + t)b−a−1dt, (59)

which results in the new closed-form expression for the MGF given by

�C (τ ) =L−M∑

l=0

D∑

k=1

μDl−kl

�kl

[γ (1 − ρ2)

]k−1k−1∑

n=0

(k − 1

n

)[Mρ2

�l(1 − ρ2)(l + M)

]n

×(

n + 1,τ

ln(2)+ 2 + n; 1

�l

). (60)

Note that using an alternative notation for(a, b; z) = z−a2 F0(a, 1+a −b; .;−1/z)where

2 F0(., ; . : .) is a generalized hypergeometric serious [17], the MGF of the capacity can besimply written as

�C (τ ) =L−M∑

l=0

D∑

k=1

μDl−kl

�k−1l

[γ (1 − ρ2)

] k−1∑

n=0

(k − 1

n

)[Mρ2

(1 − ρ2)(l + M)

]n

×2 F0

(n + 1,− τ

ln(2);−�l

). (61)

123


4.2 Complementary Cumulative Distribution Function (CCDF)

We evaluate the CDF of (52) in the presence of Gaussian channel estimation errors as follows

FC = Prob(C ≤ C

) =2C −1∫

0

pγ (γ )dγ = 1 −∞∫

2C −1

pγ (γ )dγ

= 1 −L−M∑

l=0

D∑

k=1

μDl−kl

�k−1l

[γ (1 − ρ2)

] k−1∑

n=0

1

n!

×(

k − 1n

)[Mρ2

(1 − ρ2)(l + M)

]n

�

(n + 1,

2C − 1

�l

). (62)

Thus, the complementary CDF can be obtained from (62) as follows

FC =L−M∑

l=0

D∑

k=1

μDl−kl

�k−1l

[γ (1 − ρ2)

]L−1k−1∑

n=0

1

n!(

k − 1n

)[Mρ2

(1 − ρ2)(l + M)

]n

×�(

n + 1,2C − 1

�l

). (63)

For M = L , the CCDF of the capacity simplifies to

FMRCC (C) = [

(1 − ρ2)]L−1

L−1∑

n=0

(L − 1

n

)[ρ2

(1 − ρ2)

]n

�

(n + 1,

2C − 1

γ

). (64)

For M = 1, the CCDF in (63) reduces to

FSCC (C) =

L−1∑

k=0

(−1)k(

Lk + 1

)exp

(2C − 1(k + 1)

γ[k + 1 − kρ2

]). (65)

4.3 Probability Density Function

The PDF of the capacity is defined as the derivative of the CCDF with respect to C , whichcan be written as

pC (C) = 2C ln(2)L−M∑

l=0

D∑

k=1

μDl−kl

�kl

[γ (1 − ρ2)

]k−1k−1∑

n=0

1

n!(

k − 1n

)

×[

Mρ2(2C − 1)

�l(1 − ρ2)(l + M)

]n

exp

(− (2

C − 1)

�l

). (66)

Note that (66) can also be obtained from (2) by performing a transformation of randomvariables γ → C .

1) M = L:

pmrcC (C) = 2C ln(2)

1

γ[(1 − ρ2)]M−1

k−1∑

n=0

1

n!(

k − 1n

)

×[

Mρ2(2C − 1)

γ (1 − ρ2)

]n

exp

(− (2

C − 1)

γ

). (67)

123


2) M = 1:

pscC (C) = 2C ln(2)

L−1∑

k=0

(−1)k(

Lk + 1

)(k + 1

γ[k + 1 − kρ2

])

exp

((2C − 1)(k + 1)

γ[k + 1 − kρ2

])

.(68)

5 Symbol Error Rate

In wireless communications, the symbol error rate (SER) is one of the prime factors in deter-mining the performance of the system over fading channels. In this section, we derive theSER for different modulation schemes employing GSC with channel estimation errors. Fora given instantaneous SNR γ over an AWGN channel, the SER for different modulationschemes with M-ary signaling is given by Pe,PAM(γ ) = aPAM Q(

√gPAMγ ) , Pe,PSK(γ ) =

aPSK Q(√

gPAMγ ) , and Pe,QAM(γ ) = 1 − [1 − aQAM Q(√

gQAMγ )]2, where aPAM =2(

1 − 1Md

), gPAM = 6

M2d −1

, aPSK = 2, gPSK = 2 sin2(π

Md

), aQAM = 2

(1 − 1√

Md

),

and gQAM = 3Md−1 . In order to obtain the SER in fading channels, the conditional SER is

averaged over the PDF of γ . Thus, we obtain the following expression for SER of GSC usingMPAM modulation

SER =L−M∑

l=0

D∑

k=1

μDl−kl

�kl

[γ (1 − ρ2)

]k−1

×k−1∑

n=0

1

n! ×(

k − 1n

)[Mρ2γ

�l(1 − ρ2)(l + M)

]n ∞∫

0

Q(√

gPAMγ )γn exp

(γ

�l

)dγ

︸︷︷︸I5

.

(69)

In order to compute I5 in (69) we use the following formula

I (p, q,m) = �(m)

qm

∞∫

0

Q(√

px)

e−qx xm−1dx . (70)

By applying Craig’s formula given by Q(x) = 1π

∫ π/20 exp(− x2

2 sin2 θ)dθ , and the equality∫∞

0 xv−1e−μx dx , the integral in (70) can be evaluated. Upon using the Gaussian hypergeo-

metric function 2 F1(a, b; c; z) = ∑∞k=0

(a)k (b)k(c)k

zk

k! , the integral in (70) can be simplifiedto

I (p, q,m) = 1

2√π(1 + s)m

�(m + 12 )

�(m + 1)× 2 F1

(m,

1

2; m + 1; 1

s + 1

), (71)

where s = p2q . Using the fact that 2 F1(a, b; c; z) = (1 − z)c−a−b

2 F1(c − a, c − b; c; z),(71) can be expressed as

I (p, q,m) = 1

2√π(1 + s)m

�(m + 12 )

�(m + 1)

(s

s + 1

) 12 × 2 F1

(1,m,+1

2; m + 1; 1

s + 1

).

(72)

123


Substituting p = gPAM, q = 1�l

, and m = n +1 in (72), the following expression is obtained

I (gPAM, b, k) = 1

2√π (2 +�l gPAM)

n+1

�(n + 32 )

�(n + 2)

(�l gPAM

�l gPAM + 2

) 12

× 2 F1

(1, n + 1,+1

2; n + 2; 1

gPAM2b + 1

). (73)

In order to get the SER for MPAM modulation, (73) is substituted in (69), and the finalexpression for the SER of MPAM using GSC is given by

SERPAM = aPAM

L−M∑

l=0

D∑

k=1

μDl−kl

�kl

[γ (1 − ρ2)

]k−1k−1∑

n=0

1

n!(

k − 1n

)[Mρ2

�l(1 − ρ2)(l + M)

]n

×(

�l gPAM�l gPAM+2

) 12

2√π (2 +�l gPAM)

n+1

�(n + 32 )

�(n + 2)2 F1

(1, n + 1,+1

2; n + 2; 1

�l gPAM + 2

).

(74)

Similarly, the SER for MPSK modulation can be expressed as

SERPSK = aPSK

L−M∑

l=0

D∑

k=1

μDl−kl

�kl

[γ (1 − ρ2)

]k−1k−1∑

n=0

1

n!(

k − 1n

)[Mρ2

�l(1 − ρ2)(l + M)

]n

×(

�l gPSK�l gPSK+2

) 12

2√π (2 +�l gPSK)

n+1

�(n + 32 )

�(n + 2)2 F1

(1, n + 1,+1

2; n + 2; 1

�l gPSK + 2

).

(75)

Assuming matched filter detection and perfect channel estimation for M-QAM system, theaverage SER for M-QAM with GSC diversity reception in Rayleigh fading environment canbe obtained by averaging its conditional SER over the PDF of γ at the output of GSC diversityreceiver resulting in

SERQAM = 1 −

⎧⎪⎪⎨

⎪⎪⎩1 − aQAM

L−M∑

l=0

D∑

k=1

μDl−kl

�kl

[γ (1 − ρ2)

]

×k−1∑

n=0

1

n!(

k − 1n

)[Mρ2

�l(1 − ρ2)(l + M)

]n

×(

�l gQAM�l gQAM+2

) 12

2√π(2 +�l gQAM

)n+1

�(n + 3

2

)

�(n + 2)2 F1

(1, n + 1,+1

2; n + 2; 1

�l gQAM + 2

)⎫⎪⎪⎬

⎪⎪⎭

2

.

(76)

123


Equation (72) can be further simplified when m is restricted to a positive integer value in(71) to give

I (p, q,m) = 1

2

⎡

⎣1 − μ

m−1∑

j=0

(2 jj

)(1 − μ2

4

) j⎤

⎦, (77)

where μ =√

pp+2q . Therefore, the integral I5 in (69) can be substituted by (77) to yield

SERPSK = aPSK

2

L−M∑

l=0

D∑

k=1

μDl−kl

�kl

[γ (1 − ρ2)

]k−1k−1∑

n=0

1

n!(

k − 1n

)[Mρ2

�l(1 − ρ2)(l + M)

]n

×⎡

⎣1 −√

�l gPSK

�l gPSK + 1

n∑

j=0

(2 jj

)(1

4(�l gPSK + 1)

) j⎤

⎦. (78)

The SER for MQAM is given by

SERQAM = 1 −⎧⎨

⎩1 − aQAM

2

L−M∑

l=0

D∑

k=1

μDl−kl

�kl

[γ (1 − ρ2)

]k−1k−1∑

n=0

1

n!(

k − 1n

)

[Mρ2

�l(1 − ρ2)(l + M)

]n

⎡

⎣1 −√

�l gQAM

�l gQAM + 1

n∑

j=0

(2 jj

)(1

4(�l gQAM + 1)

) j⎤

⎦

⎫⎬

⎭

2

. (79)

We remark that asγ → ∞, 2 F1(.) → 1. For example, the SERPSK asymptotically approachesthe following expression

SERPSK = aPSK

L−M∑

l=0

D∑

k=1

μDl−kl

�kl

[γ (1 − ρ2)

]k−1k−1∑

n=0

1

n!(

k − 1n

)[Mρ2

�l(1 − ρ2)(l + M)

]n

×(

�l gPSK�l gPSK+2

) 12

2√π (2 +�l gPSK)

n+1

�(n + 3

2

)

�(n + 2). (80)

For M = L , the SER for MSPK in (75) reduces to

SERPSK = aPSK

[1

γ(1 − ρ2)

] k−1∑

n=0

1

n!(

k − 1n

)[ρ2

γ (1 − ρ2)

]n

×(

γ gPSKγ gPSK+2

) 12

2√π (2 + γ gPSK)

n+1

�(n + 3

2

)

�(n + 2)2 F1

(1, n + 1,+1

2; n + 2; 1

γ gPSK + 2

)

= aPSK

2

[1

γ(1 − ρ2)

] k−1∑

n=0

1

n!(

k − 1n

)[ρ2

γ (1 − ρ2)

]n

×⎡

⎣1 −√

γ gPSK

γ gPSK + 1

n∑

j=0

(2 jj

)(1

4(γ gPSK + 1)

) j⎤

⎦. (81)

123


0 5 10 15 20 25 300

2

4

6

8

10

12

SNR dB

C [B

its/S

ec/H

z]

opraoratifr

Fig. 1 Capacity per unit bandwidth for a Rayleigh fading with GSC diversity (L = 4,M = 3) for differentadaptation. Schemes with perfect estimation ρ2 = 1

For MQAM, it can be represented as

SERQAM = 1 −

⎧⎪⎪⎨

⎪⎪⎩1 − aQAM

[1

γ(1 − ρ2)

] k−1∑

n=0

1

n!(

k − 1n

)[ρ2

γ (1 − ρ2)

]n

×(

γ gQAMγ gQAM+2

) 12

2√π(2 +�l gQAM

)n+1

�(n + 3

2

)

�(n + 2)2 F1

(1, n + 1,+1

2; n + 2; 1

γ gQAM + 2

)⎫⎪⎪⎬

⎪⎪⎭

2

= 1 −{

1 − aQAM

2

[1

γ(1 − ρ2)

] k−1∑

n=0

1

n!(

k − 1n

)[ρ2

γ (1 − ρ2)

]n

×⎡

⎣1 −√

γ gQAM

γ gQAM + 1

n∑

j=0

(2 jj

)(1

4(γ gQAM + 1)

) j⎤

⎦

⎫⎬

⎭

2

. (82)

For SC, i.e., when M = 1, the SER for MPSK

SERPSK = aPSK

M−1∑

k=0

(−1)k(

Mk + 1

)⎡

⎢⎣1 −√√√√ 1

1 + (k+1)gPSKγ (k+1−kρ2)

⎤

⎥⎦, (83)

and the SER for MQAM,

SERQAM = 1 −

⎧⎪⎨

⎪⎩1 − aQAM

M−1∑

k=0

(−1)k(

Mk + 1

)⎡

⎢⎣1 −√√√√ 1

1 + (k+1)gQAMγ (k+1−kρ2)

⎤

⎥⎦

⎫⎪⎬

⎪⎭

2

.

(84)

123


0 5 10 15 20 25 300

2

4

6

8

10

12

SNRdB

C[b

its/s

ec/H

z]

ρ2=0.9

UpperBound(ρ2=0.9)

approx−ρ2=0.9

ρ2=0.5

approx−ρ2=0.5

ρ2=0.1

approx−ρ2=0.1

Fig. 2 Capacity per unit bandwidth for a Rayleigh fading with GSC diversity (L = 4,M = 3) and variousvalues of different ρ2 under power and rate adaptation

0 5 10 15 20 25 300

2

4

6

8

10

12

SNRdB

C[b

its/s

ec/H

z]

ρ2=0.9

(UpperBoundρ2=0.9)

approx ρ2=0.9

ρ2=0.5

ρ2=0.1

approx ρ2=0.9

Fig. 3 Capacity per unit bandwidth for a Rayleigh fading channel with SCD (L = 4,M = 3) and variousvalues of different ρ2 under rate adaptation and constant power

6 Numerical Results

In this section, we provide some numerical results that illustrate the mathematical derivationof the channel capacity per unit bandwidth as a function of average receiver SNR γ in dBfor different adaptation policies with GSC over slow Rayleigh fading with weight estimationerrors. All curves provided are obtained using the closed-form expressions.

Figure 1 shows the comparison of the capacity per unit bandwidth for opra, ora and tifrpolicies for GSC (L = 4,M = 3). The result indicates how the opra policy achieves thehighest capacity for any average receive SNR, γ . From the same figure, it can be noticed

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0 5 10 15 20 25 300

2

4

6

8

10

12

Average γ per branch (dB)

Cap

acity

per

Uni

t Ban

dwid

th [B

its/S

ec/H

z] ρ=0.9

upper bound ρ=0.9

Aprro−ρ=0.9

ρ=0.5

Appro−ρ=0.5

ρ=0.1

Appro−ρ=0.1

Fig. 4 Capacity per unit bandwidth for a Rayleigh fading channel with SCD (L = 4,M = 3) and variousvalues of different ρ2 under rate adaptation and constant power

−20 −15 −10 −5 0 5 10 15 20−1

0

1

2

3

4

5

6

7

8

SNRdB

C[b

its/s

ec/H

z]

\ExactUpperBoundLowSNR1Low2SNR2approxHighSNR

Fig. 5 Exact and approximated capacity per unit bandwidth for a Rayleigh fading with GSC diversity for(L = 4,M = 3) under imperfect channel estimation ρ2 = 0.9

that ora achieves less capacity than opra. However, both opra and ora achieve the same resultwhen there is no power adaptation implemented at the transmitter as in opra. As expected,Fig. 1 shows that the tifr scheme achieves less capacity compared to the other adaptationpolicies. The results in Fig. 1 is plotted for the case of fully estimated channel (ρ2 = 1).Figure 2 compares Copra for different values of correlation between the channel and its esti-mate; namely, ρ2 = 0.1, ρ2 = 0.5, and ρ2 = 0.9. It can be noticed that the highest Copra thatcan be achieved is when ρ2 = 1 as shown in Fig. 1 with perfect estimation. Furthermore,Copra decreases when the value of ρ2 decreases where in this case the weight error increases.It can be observed from Fig. 2 that there is almost a 3 dB difference in Copra between ρ2 = 0.9

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−10 −5 0 5 10 15 20 25 300

2

4

6

8

10

12

Average γ per branch (dB)

Cap

acity

per

Uni

t Ban

dwid

th [B

its/S

ec/H

z] ExactapproxUpperboundHigh−SNRLow−SNR1Low−SNR2

Fig. 6 Exact and approximated capacity per unit bandwidth for a Rayleigh fading with diversity for (L =4,M = 1) under imperfect channel estimation ρ2 = 0.9

0 5 10 15 20 25 301

2

3

4

5

6

7

8

9

10

11

12

SNRdB

C[b

its/s

ec/H

z]

ρ2=0.9

ρ2=0.5

ρ2=0.1

Fig. 7 Capacity per unit bandwidth for a Rayleigh fading with GCD diversity (L = 4,M = 3) and variousvalues of ρ2 under truncated channel inversion

and ρ2 = 0.1. In Fig. 2, we showed the asymptotic capacity approximation as expressed in(21) and the upper bound in (24). We observe that the asymptotic capacity provide a goodmatch to the exact capacity values which can be taken as a tight measure in case that theexact capacity can not be obtained analytically. Figure 3 shows the plot of Cora as well asits asymptotic approximation and upper bound as a function of the average received SNRγ for (L = 4,M = 3). As can be seen from the same figure that the ora policy is moresensitive to the estimation error than the opra policy by 1 dB difference between ρ2 = 0.9and ρ2 = 0.1. Figure 4 considers a special case of GSC with (L = 4,M = 1) in whichthe capacity shown in Fig. 3 is collapsed into the well known SC scheme. Figure 5 depicts

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−5 0 5 10 15 20 25 300

1

2

3

4

5

6

7

8

9

10

Cut−off γ0 SNR dB

Cap

acity

per

Uni

t Ban

dwid

th [B

its/S

ec/H

z] ρ2=0.1

ρ2=0.5

ρ2=0.9

Fig. 8 Capacity per unit bandwidth for a Rayleigh fading with GSC diversity (L = 4,M = 3) and variousvalues of ρ2 versus optimal cutoff SNR γ0 with truncated channel inversion with γ = 15d B

−5 0 5 10 15 20 25 300

1

2

3

4

5

6

7

8

9

10

Cut−off γ0 SNR dB

Cap

acity

per

Uni

t Ban

dwid

th [B

its/S

ec/H

z] ρ2=0.1

appox−ρ2=0.1

ρ2=0.5

approx−ρ2=0.5

ρ2=0.9

approx−ρ2=0.9

Fig. 9 Capacity per unit bandwidth for a Rayleigh fading with GSC diversity (L = 4,M = 1) and variousvalues of ρ2 versus optimal cutoff SNR γ0 with truncated channel inversion with γ = 15 dB

the exact closed-form expression capacity of (32) for (L = 4,M = 3) and ρ = 0.9 as wellas the corresponding approximations given in (35–39). It can be observed that the upperbound for SC and the high-SNR approximation for the SC scheme match on each other forSNR ≥ 5 dB which show a tight approximation of the exact average capacity. Furthermore,the approximations for low SNR region for the SC scheme are twofold: (1) the low-SNRapproximation becomes tight for SNR values less than −5 dB, whereas; (2) the expressionthe low-SNR approximation II becomes tight to exact capacity between 0 and 10 dB asshown in Fig. 6. Figure 7 illustrates the capacity of the tifr policy for different values ofρ2. The tifr policy performs the worst compared to the other adaptation policies. However,

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0 10 20 30 40 5010

−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Average SNR per branch(dB)

Pou

t

ρ2=0.1

ρ2=0.5

ρ2=0.9

ρ2=0.99

ρ2=1.0

Fig. 10 Probability of outage for a Rayleigh fading with with GCD diversity (L = 4,M = 3) and variousvalues of ρ2 and γ0 = 0.7

−4 −2 0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

C[bits/sec/Hz]

Pro

babi

lity

Den

sity

Fun

ctio

n

ρ2=0.1

ρ2=0.5

ρ2=0.9

ρ2=1.0

Fig. 11 Probability density function PC (C) for a GSC with (L = 4,M = 3) at γ = 15 dB for differentvalues of ρ2

the loss due to the estimation errors is very high, which is almost 15 dB difference betweenρ2 = 0.9 and ρ2 = 0.1. Figure 8 depicts the behavior of Ctifr against the optimal cutoffSNR, γ0 for different values of ρ2 and GSC (L = 4,M = 3), and Fig. 9 shows the samebehavior for the case GSC (L = 4,M = 1), which illustrates the Ctifr performance for SCscheme as expressed in (46). As can be seen that Ctifr becomes almost flat when the value ofρ2 decreases which experiences a large capacity loss. Figure 10 shows the behavior of theoutage probability for different values of ρ2. It can be observed that when there is no datato be transmitted because of the outage event, the tifr policy suffers an outage probabilitywhich is larger than the outage probability suffered by the opra policy. In addition, we note

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0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

C[bits/sec/Hz]

Com

plem

enta

ry C

umul

ativ

e D

ensi

ty F

unct

ion

ρ2=0.1

ρ2=0.5

ρ2=0.9

ρ2=1.0

Fig. 12 Cumulative distribution function FC (C) for a GSC with (L = 4,M = 3) at γ = 15 dB for differentvalues of ρ2

0 5 10 15 20 25 30 35 4010

−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

SNRdB

Sym

bol E

rror

Rat

e

ρ2=0.1

asymptotic ρ2=0.1

ρ2=0.8

asymptotic ρ2=0.8

ρ2=0.9

asymptotic ρ2=0.9

ρ2=0.99

asymptotic ρ2=0.99

ρ2=1

Fig. 13 Symbol error probability performance for QPSK modulation with GSC with (L = 4,M = 3) andvarious values of ρ2

that the correlation ρ2 has a greater influence on tifr than it has on opra and the decrease inoutage probability with increase in γ is faster for opra than tifr.

Figure 11 depicts the PDF curves for different values of the correlation coefficient squaredρ2, considering an average SNR per branch of γ = 15 dB and (L = 4,M = 1). This fig-ure shows that the capacity distribution has a Gaussian-like shape even in the presence ofchannel estimation errors. As expected, the distribution of C shift towards the left indicatinga decreasing value of its mean as the value of ρ2 decreases. Figure 12 considers the samesetting in Fig. 11 and depicts the CCDF curves for different values of ρ2, with very similarobservations. The expressions in (74–76 , 78), and (79) generalize the M-PSK and the squareM-QAM average symbol error rates results with GSC (L ,M) over slow Rayleigh fading

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0 5 10 15 20 25 30 35 4010

−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

SNRdB

Sym

bol E

rror

Rat

e 4P

SK

mrc

ρ=0.1

asymptotic ρ=0.1

ρ=0.8

asymptotic ρ=0.8

ρ=0.9

asymptotic ρ=0.9

ρ=0.99

asymptotic ρ=0.99

ρ=1

Fig. 14 Symbol error probability performance for QPSK modulation under various values of ρ2 for a GSCwith (L = 4,M = 1) given by (75)

0 5 10 15 20 25 30 35 4010

−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

SNRdB

Sym

bol E

rror

Rat

e

ρ2=0.1

asymptotic ρ2=0.1

ρ2=0.8

asymptotic ρ2=0.8

ρ2=0.9

asymptotic ρ2=0.9

ρ2=0.99

asymptotic ρ2=0.99

ρ2=1

Fig. 15 Symbol error probability performance for 16-QAM modulation under various values of ρ2 for GSCwith (L = 4,M = 3)

with weight estimation errors. These expressions are collapsed into M-PSK and the squareM-QAM average SER results for MRC and conventional SC. For instance, for particularcase of M = L (i.e. MRC), we showed the average SER for M-PSK and the square M-QAMSER over Rayleigh fading with estimation error in (81) and (82), respectively. Similarly, for(M = 1) (i.e. SC) the average SER for M-PSK and the square M-QAM are expressed in (81)and (82), respectively.

123


0 5 10 15 20 25 30 35 4010

−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

SNRdB

Sym

bol E

rror

Rat

e

ρ2=0.1

asymptotic ρ2=0.1

ρ2=0.8

asymptotic ρ2=0.8

ρ2=0.9

asymptotic ρ2=0.9

ρ2=0.99

asymptotic ρ2=0.99

ρ2=1

Fig. 16 Symbol error probability performance for 16-QAM modulation under various values of ρ2 for aspecial case of GSC with (L = 4,M = 4) given by (82)

7 Conclusions

The closed-form expressions for the channel capacity per unit bandwidth for three differentadaptation policies including their approximations and upper bounds over a slow Rayleighfading channel for GSC (L ,M) with estimation error is derived. Furthermore, we presentedan upper bound as well as asymptotically tight approximations for ora policy for the highand low SNR regions. The result showed that Copra outperforms Cora and Ctifr , and is lesssensitive to the estimation error compared to other policies. However, Ctifr performs theworst among the other policies because it suffers a large capacity penalty due to the estima-tion error whereas it is less complex to implement. It is worth to mention that the derivedcapacity expressions represent general formulas for GSC (L ,M) with estimation error overslow Rayleigh fading channel from which those spacial limited cases of GSC (i.e, M = L ,MRC; M = 1, SC; ρ2 = 1, perfect estimation; ρ2 = 0 no diversity) can be derived. Also,The second part of this paper analyzed the impact of channel estimation error on the capacitystatistics and SER performance of GSC over slow Rayleigh fading channels. Starting withPDF of instantaneous branch SNR, we derived exact closed-forms for capacity statistics:moment generating function, probability density function and cumulative distribution func-tion. In addition, the exact analytical expressions of SER for M-PAM/PSK/QAM modulationwere obtained. The derived expressions are valid for arbitrary number of receivers.

Acknowledgment The third author acknowledges the support of King Fahd University of Petroleum andMinerals under grant no. SB090006.

References

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6. Dong, X., & Beaulieu, N. C. (2003). SER of two-dimensional signaling in Rayleigh fading with channelestimation error. In Proceedings of international conference on communication (ICC), Anchorage,AK, pp. 2763–2767.

7. Annamalali, A. Jr., (2001). The effect of Gaussian error in selection diveristy combiners. WirelessCommunications and Mobile Computing, 1, 419–439.

8. Annamalali, A., & Tellambura, C. (2002). Analysis of hybrid selection/maximal-ratio diversity withGaussian errors. EEE Transactions on Communications, 1(3), 498–511.

9. Ma, Y., Zhang, D., & Schober, R. (2005). Effect of channel estimation errors on M-QAM with GSCdiversity in fading channels. In Proceddings of IEEE International Conference on Communications.Seoul, Korea, pp. 2190–2194.

10. Shannon, C. E. (1948). A mathematical theory of communication. Bell System Technical Journal,27, 397–423.

11. Fochini, G., & Gans, M. (1998). On limits of wireless communications in a fading environment whenusing multiple antennas. Wireless Personal Communications, 6–3, 311–335.

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Author Biographies

Fawaz S. Al-Qahtani received the B.Sc. in Electrical Engineeringfrom King Fahad University of Petroleum and Minerals (KFUPM),Saudi Arabia in 2000 and his M.Sc. in Digital Communication Sys-tems from Monash University, Australia in 2005. He was a GraduateAssistance at Dammam Technical Collage, Saudi Arabia. He workedas an Executive Engineer at Ericsson Telecommunication Company inSaudi Arabia in 2001. He joined the Department of Electrical and Com-puter Engineering (SECE), RMIT University, Melbourne, Australia in2005. He worked as well at International House College as a residen-tial tutor for 2 years (2005–2007), Melbourne University. During hisPh.D. candidacy at the School of Electrical and Computer Engineering,RMIT University, he also delivered signal processing and communica-tions tutorials and laboratory classes at both undergraduate and post-graduate levels and wrote 22 technical papers. His research interests aredigital communications, channel modelling, applied signal processing,and space-time coding for OFDM systems, MIMO capacity and coop-

erative networks. He is a student member of IEEE and served as a reviewer in many conferences and inter-national journals.

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Arun K. Gurung finished B.E. and M.E. at IOE Pulchowk, Lalitpur,Nepal and BUPT, Beijing, China in 2002 and 2007, respectively. Hewas a Lecturer at KEC, Lalitpur, Nepal during 2002–2003. Currently,he is Ph.D. candidate at RMIT University, Melbourne, Australia. Hiscurrent research areas include wireless communications topics OFDM,MIMO, Relay/Cooperative techniques.

Salam A. Zummo was born in 1976 in Saudi Arabia. He receivedthe B.Sc. and M.Sc. degrees in Electrical Engineering from King FahdUniversity of Petroleum and Minerals (KFUPM), Dhahran, Saudi Ara-bia, in 1998 and 1999, respectively. He received his Ph.D. degree fromthe University of Michigan at Ann Arbor, USA, in June 2003. He iscurrently an Associate Professor in the Electrical Engineering Depart-ment and the Dean of Graduate Studies at KFUPM. He is also theCoordinator of the Communications, Electronics and Optics NationalScience and Technology Program at KFUPM. Dr. Zummo was awardedPrince Bandar Bin Sultan Award for early Ph.D. completion in 2003and the British Council/BAE Research Fellowship Award in 2004 and2006. He has more that 50 publications in international journals andconference proceedings in the many areas in wireless communicationsincluding error control coding, diversity, MIMO, iterative receivers,multiuser diversity, multihop networks, user cooperation, interferencemodeling and networking issues for wireless communication systems.

Dr. Zummo is a Senior Member of the IEEE.

Seedahmed S. Mahmoud received the B.Sc. (honour, first class)in Electronic Engineering from University of Gezira, Sudan, (1996)and his M.Sc. in Electronic System Engineering from UniversityPutra Malaysia, UPM (1998). He received his Ph.D. degree in Elec-trical Engineering (digital communication) from RMIT University,Australia in 2004. He was a former Teaching Assistant in Sudan,a Research Assistant at UPM and a Research Fellow at the Schoolof Electrical and Computer Engineering, RMIT University where hesupervised and co-supervised postgraduate researches and taught somecourses for undergraduate level. In September 2006 he joined the R&Dlab of the Future Fibre Technologies Pty Ltd as a senior signal pro-cessing engineer. He wrote over 35 technical papers in refereed inter-national journals and conferences in various areas of electrical engi-neering. He is also serving as a reviewer for IEEE Transactions onVehicular Technology and IEEE Wireless Communication Magazine.His research interests are signal processing for communications and

biomedical engineering, perimeter intrusion detection and classification, and time-frequency analysis.

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Zahir M. Hussain took the first rank in Iraq in the General Bacca-laureate Examinations 1979 with an average of 99%. He received theB.Sc. and M.Sc. degrees in Electrical Engineering from the Univer-sity of Baghdad in 1983 and 1989, respectively, and the Ph.D. degreein Electrical Engineering from Queensland University of Technology,Brisbane, Australia, in 2002. From 1989 to 1998 he researched and lec-tured on electrical engineering and mathematics. In 2001 he joined theSchool of Electrical and Computer Engineering (SECE), RMIT Uni-versity, Melbourne, Australia, as a researcher then lecturer of signalprocessing and communications. He was the academic leader of a 3Gcommercial communications project at RMIT 2001–2002. In 2002 hewas promoted to Senior Lecturer. Since 2001 he has been the seniorsupervisor for 15 Ph.D. candidates at RMIT (with nine completionsbetween 2001 and 2009); also the second supervisor for another sevenPh.D. candidates. Dr. Hussain has over 160 internationally refereedtechnical publications on signal processing, communications, and elec-

tronics. His work on single-bit processing has led to an Australian Research Council (ARC) Discovery Grant(2005–2007). In 2005 he was promoted to Associate Professor of Signal Processing. He got the RMIT 2005and 2006 Publication Awards (also shared the 2004 Publication Award with Professor Richard Harris, nowChair of Telecommunications at Massey University, NZ). In 2006, he was the Head of the CommunicationEngineering Discipline at SECE, RMIT. In 2007, he got the RMIT Teaching Award. Dr. Hussain is a mem-ber of Engineers Australia (formerly IEAust), IET (formerly IEE), and a senior member of IEEE and theAustralian Computer Society (ACS). He worked on the technical program committees of many conferencesand served as a reviewer for leading IEEE and Elsevier journals.

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