# Performance Analysis of Dual Hop Relaying over Non-Identical Weibull Fading Channels.

**ABSTRACT** In this paper, we present closed form expressions for tight lower bounds of the performance of dual-hop non- regenerative relaying over independent non-identical Weibull fading channels. Since it is hard to find a closed form expression for the probability density function (PDF) of the signal-to-noise ratio (SNR) for the Weibull fading distribution, we use an approximate value instead. Novel closed form expressions for the PDF, outage probability and the moments of the approximate value of the SNR at the destination are derived. Also, the average SNR and amount of fading are determined. Moreover, closed form expressions (in terms of the tabulated Meijer's G-function) are found for the average symbol error probability (for several modulations schemes) as well as the Shannon capacity. It should be noted that the Meijer's G-function is widely available in many scientific software packages, such as MATHEMATICAL and MAPLEreg. Finally, simulations results are also shown to verify the analytical results.

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- The Twelfth COTA International Conference of Transportation Professionals; 07/2012
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**ABSTRACT:**In this paper, we consider a dual-hop wireless communication system with non-regenerative relay node and we study its performance over the α–μ fading channels. Specifically, we derive a closed-form expression for the moment generating function (MGF) of the harmonic mean of end-to-end signal-to-noise ratio (SNR) assuming the α–μ fading models. We also derive closed-form expressions for the end-to-end outage probability and average bit error rate of coherent modulation techniques. The obtained expressions can be reduced to study the performance of dual-hop communication systems over other fading channel models by using the proper values for the α and μ parameters. Numerical results are provided and conclusion remarks are drawn.International Journal of Electronics 01/2014; 101(6). · 0.75 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**In this letter, outage probability of dual-hop decode-and-forward (DF) relaying scheme is analyzed over mixed Rayleigh and generalized Gamma fading channels. Cooperation model considered in this work consists of a source, a relay and a destination. It is assumed that source-relay and relay-destination channels experience Rayleigh fading and generalized Gamma fading, respectively. Exact outage probability expression is derived and outage performance is illustrated for both direct transmission and DF relaying scheme.Wireless Personal Communications 07/2013; 71(2). · 0.98 Impact Factor

Page 1

Performance Analysis of Dual Hop Relaying over

Non-Identical Weibull Fading Channels

Salama S. Ikki, Student member IEEE and Mohamed H. Ahmed, Senior Member IEEE

Faculty of Engineering and Applied Sciences

Memorial University of Newfoundland

St. John’s, NL, Canada A1B 3X5

e-mail: s.s.ikki@mun.ca, mhahmed@mun.ca

Abstract—In this paper, we present closed form expressions

for tight lower bounds of the performance of dual-hop non-

regenerative relaying over independent non-identical Weibull

fading channels. Since it is hard to find a closed form expression

for the probability density function (PDF) of the signal-to-noise

ratio (SNR) for the Weibull fading distribution, we use an

approximate value instead. Novel closed form expressions for

the PDF, outage probability and the moments of the approximate

value of the SNR at the destination are derived. Also, the average

SNR and amount of fading are determined. Moreover, closed

form expressions (in terms of the tabulated Meijer’s G-function)

are found for the average symbol error probability (for several

modulations schemes) as well as the Shannon capacity. It should

be noted that the Meijer’s G-function is widely available in many

scientific software packages, such as MATHEMATICA R ? and

MAPLE R ?. Finally, simulations results are also shown to verify

the analytical results.

Index Terms—Error rate, outage probability, channel capacity,

dual-hop relaying, amplify-and-forward, Weibull fading channel.

I. INTRODUCTION

Dual-hop relaying communication has a number of advan-

tages over direct-link transmission in terms of connectivity,

power saving, and channel capacity. Relaying techniques en-

able connectivity when traditional direct transmission is not

practical due to large path-loss and/or power constraints. The

relaying concept can be applied to cellular, wireless local area

network (WLAN) and hybrid networks.

The performance analysis of dual-hop wireless communi-

cation systems operating over fading channels has been an

important area of research in the past few years. The authors in

[1]–[3] have presented a useful semi-analytical framework for

the evaluation of the end-to-end outage probability and average

error probability of dual-hop wireless systems with non-

regenerative relaying over Rayleigh and Nakagami-m fading

channels. In [4]–[6] the authors have studied the performance

bounds for multi-hop relayed transmissions with fixed-gain re-

lays over Nakagami-(Rice), Nakagami-(Hoyt), and Nakagami-

m fading channels using the moments based approach. Re-

cently, the authors in [7] have proposed and characterized four

channel models for multi-hop wireless communications.

However, another well known fading channel model, namely

the Weibull model, has not yet received as much attention

as other fading distributions, despite the fact that it exhibits

an excellent fit to experimental fading channel measurements

for both indoor as well as outdoor environments [8], [9].

Considering related work on diversity combining in Weibull

fading, the authors in [10] have presented an analysis for the

evaluation of the generalized selection combining technique

performance over independent Weibull fading channels. Re-

cently, some other contributions dealing with switched and

selection diversity, as well as second-order statistics over

Weibull fading channels have been presented in [11]–[13],

where useful performance criteria including the average output

SNR, outage probability, and the bit-error rate performance

have been studied.

In this paper, we focus on non-regenerative (amplify-and-

forward) dual-hop systems and study their end-to-end perfor-

mance over independent non-identical, Weibull fading chan-

nels. The main contribution of this paper is to derive closed

form expressions for the probability density function (PDF)

and the cumulative distribution function (CDF) of the approx-

imate value of the end-to-end SNR of this dual hop system.

Moreover, the average symbol error probability (ASEP) for

several modulations schemes and the average channel capacity

are expressed in terms of the tabulated Meijer’s G-function1

[14, eq. (9.301)].

The rest of this paper is organized as follows. In Section II,

the system and channel models are described. In Sections III,

the probability density function and cumulative distribution

function have been obtained . In Section IV and V Error

probability and channel capacity have been introduced. Results

obtained from numerical analysis and Monte-Carlo simulation

are compared in Section VI. Finally, conclusions are presented

in Section VII.

II. SYSTEM AND CHANNEL MODELS

A. System Model

As shown in Fig. 1, a source node (S) communicates

with the destination node (D) through a relay node (R). The

channel coefficients between S and R (h) and between R and

1Note that G[.] can be expressed in terms of more familiar generalized

hypergeometric functionspFq(.;.;.) [14, eq. (9.14/1)] using the transforma-

tion presented in [14, eq. (9.303)], with p and q being positive integers. In

addition, both G[.] andpFq(.;.;.) are included as built-in functions in most of

the popular mathematical software packages such as Maple or Mathematica.

978-1-4244-2517-4/09/$20.00 ©2009 IEEE

Page 2

D (g) are flat Weibull fading coefficients. In addition, h and g

are mutually-independent and non-identical. We also assume

here without loss of generality that all additive white Gaussian

noise (AWGN) terms of the two links have zero mean and

equal variance N0.

Given that the relaying gain equals

[3] where Es is the transmitted signal energy; it is straight

forward to show that the end-to-end SNR at the destination

node can be written as

?Es/(h2+ N0) [1]–

Fig. 1. Illustration of a multi branch cooperative diversity network.

γout=

γhγg

γh+ γg+ 1

(1)

where γh= h2Es/N0is the instantaneous SNR of the source

signal at the relay and γg = g2Es/N0 is the instantaneous

SNR of the relayed signal by the relay at the destination. In

order to simplify the performance analysis calculations, (1)

should be expressed in a more mathematically tractable form.

To achieve it, we proposed in [15] a tight upper bound for

γoutgiven by

γout≤ min(γh,γg) = γub

(2)

Our subsequent analysis exclusively relies on γub, as this

upper bound has been shown to be quite accurate [15],

[16]. Moreover, we provide extensive simulation results to

complement the bounds. A lower bound can be formulated

as in [16] where γlb= 0.5min(γh,γg). As the lower bound

is different from the upper bound only by a factor of half, the

following analysis can easily be extended to the lower bound,

but is omitted here for brevity.

B. Channel Model

As mentioned above, we assume that the fading channel

coefficients (h and g) are Weibull random variables. Hence,

the PDF of h can be written as

fh(x) =2υh

Ωhx2υh−1exp

?

−x2υh

Ωh

?

(3)

where Ωhis the power scaling parameter given by Ω1/υh

E?h2?/Γ(1 + 1/υh), Γ(•) is the Gamma function [14, eq.

of fading and E(•) is the statistical average operator. For

special case υh= 1, the Rayleigh model may be considered.

h

=

(8.310.1)], υhis the fading parameter expressing the severity

The PDF of g is the same as in (3) after replacing the

subscript h with g. The PDF of the instantaneous signal-to-

noise ratio (SNR) of the S → R link, γh= h2Es/N0, can be

expressed as

?βh

where ¯ γh = E?h2?Es/N0 is the average value of γh, and

link, γg, is the same as in (4) after replacing the subscript h

with g.

Finally, to capture the effect of the path-loss on the per-

formance metrics we use the following model, which is

widely accepted in the literature: E?h2?= (dS,D/dS,R)α, and

terminal i and j, and α is the power exponent path.

fγh(γ) =

¯ γh

?υh

υhγυh−1exp

?

−

?βhγ

¯ γh

?υh?

(4)

βh= Γ(1 + 1/υh). Also, the PDF of the SNR of the R → D

E?g2?

= (dS,D/dR,D)αwhere di,j is the distance between

III. STATISTICS OF THE END-TO-END SNR

A. Outage Probability

In noise limited systems, the probability of outage is defined

as the probability that the instantaneous SNR falls below a

specified threshold γth. This threshold is a minimum value of

the SNR, below which the quality of service is unsatisfactory.

Consequently, the outage probability is given by

Pout= Fγub(γth) = Pr(γub≤ γth)

where Fγub(γth) is the cumulative distribution function (CDF)

of γub, which can be calculated as

(5)

Pout=1 − Pr(γh> γth)Pr(γg> γth)

=1 − exp

?

−

?βhγth

¯ γh

?υh?

exp

?

−

?βgγth

¯ γg

?υg?

(6)

B. PDF and the Moments

The PDF of γubcan be found by taking the derivative of

the CDF with respect to γth, yielding

?

×

¯ γh

fγub(γ)=exp

−

?υh

?βhγ

υhγυh−1+

¯ γh

?υh?

exp

?βg

?

−

?υg

?βgγ

υgγυg−1

¯ γg

?υg?

??βh

¯ γg

?

(7)

In this paper and in order to get a closed form expression

of the nthmoments (μn= E(γn

υh= υg= υ and this is a valid assumption in most practical

applications [9], [11]. This condition is required to facilitate

the mathematical solution2. Hence, the nthmoment of γub,

with the help of [14, eq. (6.455.1)], can be written as

ub)) of γub, we assume that

μn=

Γ(1 + n/υ)

?υ

??

βh

¯ γh

+

?βg

¯ γg

?υ?n/υ

(8)

2if υh ?= υg the performance metrics can be upper bounded and lower

bounded as υ = max(υh,υg) and υ = min(υh,υg), respectively.

Page 3

If we have identical channel (i.e. ¯ γh = ¯ γg = ¯ γ) it can be

shown that (8) greatly simplifies to

μn=Γ(1 + n/υ)

2n/υ?

By setting n = 1 in (8), the average total SNR (¯ γub) can be

obtained. Furthermore, the first two moments of γub can be

used in order to evaluate the amount of fading (AF) at the

destination [17, Chapter 1]. The AF is defined as the ratio of

the variance to the square mean of ¯ γub

β

¯ γ

?n

(9)

?AF = μ2/¯ γ2

ub− 1?.

IV. ERROR PERFORMANCE ANALYSIS

The most straightforward approach to obtain the average

error probability, P (e), is to average the conditional symbol

error probability, P (e|γ), over the PDF of the end-to-end SNR

as follows

?∞

where P (e|γ) can be found as follows [17]

1) For coherent modulation (like M-ary phase shift keying

(M-PSK)), the error is in the form of A erfc?√Bγ?,

eq. (8.250/4)].

2) For non-coherent modulation (like M-ary differen-

tial PSK (DBPSK)), the error is in the form of

Aexp(−Bγ).

The particular values of A and B depend on the considered

modulation schemes and can be found in [17]. P (e) is

obtained in closed-form expressions for each one of the above

two cases.

1) P(e) for Coherent Modulation:

The integral in (10) can be evaluated in a closed form

as follows. Firstly, by expressing the exponential and

erfc(•) functions in terms of the Meijer’s G-function as

exp(−g (x)) = G1,0

and

?√Bx

Then, P(e) with the help of [18, eq. (24)] can be written

in a closed form as

??βh

? ??βh

P (e) =

0

P (e|γ )fγub(γ)dγ

(10)

where erfc(•) is the complementary error function [14,

0,1

?

g (x)|−

0

?

(11)

erfc

?

=

1

√πG2,0

1,2

?

Bx|1

0, 1/2

?

(12)

P(e) =

¯ γh

?υ

?υ

(B

+

?βg

?υ?k

¯ γg

?υ?

????

υAlυ−1?k/π

Bυ?√2π?l+k−2×

Λ(k,0), Λ(l,−υ)

Gk,2l

2l,k+l

¯ γh

+

?βg

l

kk

¯ γg

l)

Λ(l,1−υ), Λ(l,1/2−υ)

?

(13)

where Λ(•,•) is defined as Λ(p,q)

1)/p,··· .,(q + p − 1)/p with p being positive integer

and q real constant and G[•] is the Meijer’s G-function,

while l and k are the smallest positive integers such that

=

q/p,(q +

l

k= υ

(14)

For example, if υ = 1.25 we have to choose k = 4 and

l = 5.

If ¯ γh = ¯ γg = ¯ γ it can be shown that (13) greatly

simplifies to the compact form

P(e) = 2(β/¯ γ)υ υAlυ−1?k/π

?

Bυ?√2π?l+k−2×

Λ(k,0), Λ(l,−υ)

Gk,2l

2l,k+l

2k(β/¯ γ)l

(B

l)

l

kk

????

Λ(l,1−υ), Λ(l,1/2−υ)

?

(15)

2) P(e) for Non-Coherent modulation:

By expressing the exponential function as a Meijer’s G-

function as in (11), and with the help of [18, eq. (24)],

P(e) for non-coherent modulations can be written in a

closed form as

??βh

⎡

⎣

P(e)=

¯ γh

??

?υ

+

?βg

¯ γg

?βg

?υ?

?υ?k

υAlυ−1/2√k

Bυ?√2π?l+k−2×

Λ(l,1 − υ)

Λ(k,0)

Gk,l

l,k

⎢

βh

¯ γh

?υ

(B/l)lkk

+

¯ γg

???????

⎤

⎦

(16)

Integers l and k must be chosen so that (14) holds. If

¯ γh= ¯ γg= ¯ γ it can be shown that (16) simplifies to the

compact form

P(e)=2(β/¯ γ)υ

υAlυ−1/2√k

Bυ?√2π?l+k−2×

2k(β/¯ γ)l

(B/l)lkk

Gk,l

l,k

?

?????

Λ(l,1 − υ)

Λ(k,0)

?

(17)

V. AVERAGE CHANNEL CAPACITY

The channel capacity, in the Shannon’s sense, is an im-

portant performance measure since it provides the maximum

achievable transmission rate under which the errors are recov-

erable. The average channel capacity can be expressed as

?∞

where BW is the transmitted signal bandwidth. The reason of

the

time slots for transmitting the data. By substituting (7) into

(18) the average channel capacity can be obtained in closed

form as

??βh

Gk+2l,l

2l,k+2l

kk

¯C =BW

2

0

log2(1 + γ)fγup(γ)dγ

(18)

1

2factor is that we need two orthogonal channels or two

¯C =BW

2¯ γh

?υ

?υ

+

?βg

?βg

¯ γg

?υ?k

?υ?

????

υ√k/l

ln(2)(2π)2l+k−3/2×

? ??βh

¯ γh

+

¯ γg

Λ(l,−υ),Λ(l,1−υ)

Λ(k,0),Λ(l,−υ),Λ(l,−υ)

?

(19)

Page 4

VI. NUMERICAL AND SIMULATION RESULTS

An asymmetric network geometry is examined where R

is located on a straight line between S and D. Direct path

length S → D is normalized to be 1. In all presented results,

the path-loss exponent α is chosen to be equal to 3. In this

Section, we show numerical results of the analytical bit error

rate (BER) for binary phase shift keying (BPSK) modulation,

outage probability and the average channel capacity. We plot

the performance curves in terms of average BER, outage

probability and Shannon capacity versus the SNR of the

transmitted signal (Es/N0 dB). We also show the results of

the computer simulations for verification.

Fig. 2 shows the error performance for arbitrary values

of fading parameters for dS,R = 0.6. It can be seen form

Fig. 2 that our lower bound is tight enough especially for

medium and high SNR regime. For example, for SNR value

as low as 10 dB the difference between the exact BER and

the analytical BER is less than 7% and it is more accurate

for higher values of SNR. This shows that the analytical BER

expressions are almost exact. This trend (the tightness of our

bound) is valid at any different values of fading parameters and

distances. Also, Fig. 2 demonstrates that the relaying technique

improves the BER performance in comparison with the direct

transmission. These results are obvious because the relaying

technique benefits from the path-loss reduction.

05 1015 20 25 30

10

−4

10

−3

10

−2

10

−1

10

0

Es/N0dB

BER

E(h2) = (1/0.6)3, E(g2) = (1/0.4)3, v = 1.8.

Exact (Simulation)

Analytical Lower Bound

Direct Transmission

Fig. 2.

systems over Weibull fading channels.

Error performance for dual-hop and single-hop communication

Fig. 3 illustrates the outage performance determined from

(6) for dS,R = 0.6. It is clear again that the difference

between the exact (from simulation) and analytical results

(lower bound) for Pout is very small for medium and high

values of Es/N0. Again here, the relaying technique improves

the outage at the destination due to the path loss reduction.

Finally, Fig. 4 also shows that our lower bound is tight

for the normalized average channel capacity in medium and

high SNR regime. Here, it should be mentioned that the

amount of resources required for the relaying technique is

05 1015202530

10

−4

10

−3

10

−2

10

−1

10

0

Es/N0dB

Pout

E(h2) = (1/0.6)3, E(g2) = (1/0.4)3, v = 1.8.

Exact (Simulation)

Analytical Lower Bound

Direct Treansmission

Fig. 3. probability for dual-hop and single-hop communication systems over

Weibull fading channels.

twice that for direct transmission since the relay does not

simultaneously transmit and receive. This is the reason why

the direct link capacity outperforms the channel capacity

for relaying technique. However, it is evident that at low

SNR region, the capacity of the dual-hop relaying slightly

outperforms that of the direct transmission.

It should be noted that for Figs. 2, 3 and 4 that the tightness

of the performance improves as Es/N0 increases; however,

the proposed lower bound slightly loses its tightness at low

Es/N0. This is due to the fact that the accuracy of end-to-end

SNR approximation (in (2)) improves as Es/N0increases.

05 1015 2025 30 35 40

10

0

10

1

10

2

E(h2) = (1/0.6)3, E(g2) = (1/0.4)3, v = 1.8.

Es/N0dB

C/BW

Exact (Simulation)

Analytical Upper Bound

Direct Transmission

Fig. 4.

systems over Weibull fading channels.

Channel Capacity for dual-hop and single-hop communication

VII. CONCLUSION

We have analyzed the performance of dual-hop commu-

nication systems, operating over independent, but not neces-

Page 5

sarily identically distributed, Weibull fading channels. Using

the PDF formula of the end-to-end SNR, novel closed-form

expressions for the average output SNR, AF, error probability

(for a broad class of digital modulations) as well as the

Shannon average capacity were obtained. Computer simulation

results verified the accuracy and the correctness of the pro-

posed analysis. We also compared the performance of the dual-

hop relaying systems with that of the traditional single hop

transmission. Extending this work to analyze the performance

of (decode-and-forward and amplify-and-forward) cooperative

diversity networks over independent non-identical Weibull

channels is currently underway.

REFERENCES

[1] M. O. Hasna and M.-S. Alouini, ”End-to-End Performance of Trans-

mission Systems With Relays Over Rayleigh-Fading Channels,” IEEE

Trans. on Wirless Commun., vol. 2, pp.1126-1131, Nov. 2003.

[2] M. O. Hasna and M.-S. Alouini, ”Harmonic mean and end-to-end

performance of transmission systems with relays,” IEEE Trans. on

Commun., vol. 52, pp.130-135, Jan. 2004.

[3] M. O. Hasna and M.-S. Alouini, ”Outage probability of multihop

transmission over Nakagami fading channels,” IEEE Commun. Letter,

vol. 7, pp.216-218, May. 2003.

[4] G. K. Karagiannidis, ”Performance bounds of multihop wireless com-

munications with blind relays over generalized fading channels,” IEEE

Trans. on Wirless Commun., vol. 5, pp.498-503, March. 2006.

[5] G. K. Karagiannidis, ”Moments-based approach to the performance

analysis of equal-gain diversity in Nakagami-m fading,” IEEE Trans.

on Commun., vol. 52, pp.685-690, May. 2004.

[6] G. K. Karagiannidis, T. A. Tsiftsis, and R.K. Mallik, ”Bounds for Mul-

tihop Relayed Communications in Nakagami-m Fading,” IEEE Trans.

on Commun., vol. 54, pp.18-22, Jan. 2006.

[7] J. Boyer, D. D. Falconer, and H. Yanikomeroglu, ”Multihop diversity in

wireless relaying channels,” IEEE Trans. on Commun., vol. 52, pp.1820-

1830, Oct. 2004.

[8] F. Babich and G. Lombardi, ”Statistical analysis and characterization

of the indoor propagation channel,” IEEE Trans. on Commun., vol. 48,

pp.455-464, Mar. 2000.

[9] N.C. Sagias and G. K. Karagiannidis, ”Gaussian class multivariate

Weibull distributions: Theory and applications in fading channels,” IEEE

Trans. on Information Theory., vol. 51, pp.3608-3619 , Oct. 2005.

[10] M.-S. Alouini and M. K. Simon, ”Performance of generalized selection

combining over Weibull fading channels,” in Proc. Vehicular Technology

Conf., vol. 3, Atlantic City, NJ, pp. 1735-1739, 2001.

[11] N. C. Sagias, D. A. Zogas, G. K. Karagiannidis, and G. S. Tombras,

”Performance analysis of switched diversity receivers in Weibull fading,”

IEE Electron. Letter, vol. 39, pp.1472-1474 , Oct 2003.

[12] N. C. Sagias, P. T. Mathiopoulos, and G. S. Tombras, ”Selection diversity

receivers in Weibull fading: Outage probability and average signal-

tonoise ratio,” IEE Electron. Letter, vol. 39, pp.1859-1860 , Dec 2003.

[13] N. C. Sagias, G. K. Karagiannidis, D. A. Zogas, P. T. Mathiopoulos,

and G. S. Tombras, ”Performance analysis of dual selection diversity in

correlated Weibull fading channels,” IEEE Trans. on Commun., vol. 52,

pp. 1063-1067, Jul. 2004.

[14] I. S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series and

Products, Academic Press, San Diego, Calif, USA, 5th edition, 1994.

[15] S. Ikki and M. H. Ahmed, ”Performance Analysis of Cooperative

Diversity Wireless Networks over Nakagami-m Fading Channel,” IEEE

Commun. letter, vol. 11, pp. 334-336, April, 2007.

[16] P. A. Anghel and M. Kaveh, ”Exact symbol error probability of a

cooperative network in a Rayleigh-fading environment,” IEEE Trans.

on Wirless Commun., vol. 3, pp. 1416-1421, Sept. 2004.

[17] M. K. Simon and M.-S. Alouini, Digital Communication over Fading

Channels, John Wiley and Sons, New York, NY, USA, 2000.

[18] V. S. Adamchik and O. I. Marichev, ”The algorithm for calculating in-

tegrals of hypergeometric type functions and its realization in REDUCE

system,” in Proc. Int. Conf. Symbolic and Algebraic Computation,

Tokyo, Japan, 1990, pp. 212-224.

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