Int. Journal of Electrical & Electronics Engg. Vol. 2, Spl. Issue 1 (2015) e-ISSN: 1694-2310 | p-ISSN: 1694-2426

NITTTR, Chandigarh EDIT -2015 108

On the Performance Analysis of CompositeMultipath/Shadowing (Weibull-Log Normal)

Fading Channels1Rupender Singh, 2S.K. Soni, 3Rajesh Birok

1,2,3Department of Electronics & Communication EngineeringDelhi Technological University (Formerly Delhi College of Engineering), Delhi

rupendersingh04cs39@gmail.com

Abstract: Composite multipath/shadowing fading environments are frequently encountered in different mobile realisticscenarios. These channels are generally modeled differentComposite multipath/shadowing fading. In this paper wepresent the performance analysis of composite (Weibull-Lognormal shadowed) fading. We adopt efficient toolproposed by Holtzman to approximate composite (Weibull-Lognormal shadowed) fading. The performance measures offading communication systems such as Probability densityfunction (PDF) of Signal to Noise ratio (SNR), Amount offading (AF), Outage probability (Pout) and ChannelCapacity(C/B) will be calculated. Graphical results will bepresented for different signals and fading parameters. Thedifferent expressions that will be provided are of greatimportance in assessing the performance of communicationsystems in composite channels.

Keywords: Weibull-Lognormal Shadowed fadingfading (WL),Probability density function(PDF), Amount of fading(AF),Outage probability(Pout), Channel Capacity(C/B)

IntroductionWireless communication channels are impaired bydetrimental effects such as Multipath Fading andShadowing [1]. Based on various indoor and outdoorempirical measurements, there is general consensus thatshadowing be modeled using Log-normal distribution [12-14]. Fading causes difficulties in signal recovery. When areceived signal experiences fading during transmission, itsenvelope and phase both fluctuate over time.A composite multipath/shadowed fading environmentmodeled either as Rayleigh-lognormal, Rician-lognormalor Nakagami-lognormal are considered in [3-5]. Up tonow, composite multipath/shadowed fading environmentmodeled as Weibull-lognormal (WL), has been consideredonly in several papers [7, 8]. The Weibull distributionplays an important role in several scientific fields, but ithas become recently the topic of wireless communicationstheory [9], particularly with mobile radio systemsoperating in the 800/900 MHz frequency range. TheWeibull model exhibits an excellent fit to experimentalfading channel measurements, for both indoor [10] andoutdoor [11] environments.In this paper, a simple accurate closed-form usingHoltzmanin [18] approximation for the expectation of thefunction of a normal variant is also employed. Then,simple analytical approximations for the PDF ofComposite/Shadowed (WL) are derived.

System and Channel ModelsHere we are taking two uncorrelated channels in presenceof weibull and log normal fading. So PDF of SNR can beobtained by averaging the PDF of weibull over log normal

fading. Weibull and conditional log normal distributionsgiven below[1,17] ( / )= 2 1 + 2 − 1+ 2 ≥ 0Where c is shape parameter for weibull distribution.( )= √2 exp − (10 − μ)2Where µ, σ are mean and variance respectivelyof RV w and ξ=10/ln10.PDF of Instantaneous SNR γ is

( ) = ( / ) ( )( ) = ∫ − 1 +

√ exp − ( )2.1

Where = 10/ 10=4.3429

It is difficult to calculate the results directly, in this work,we adopt the efficient tool proposed by Holtzmanin[9] tosimplify Eg. (2.1). Taking Eg. (5-7) in [14], we haveUsing 10 = in (2.1)

( ) = ( ). σ √2 ( )Then finally we have PDF of WL fading( ) ≈ (μ) + μ + √3 + μ − √3 2.2

Where

Int. Journal of Electrical & Electronics Engg. Vol. 2, Spl. Issue 1 (2015) e-ISSN: 1694-2310 | p-ISSN: 1694-2426

109 NITTTR, Chandigarh EDIT-2015

( ) = ( ) − ( ) 1 +2.3Using 2.2 and 2.3 we have CDF of WL fading( ) = {1 − (μ)} + {1 − μ + √3 } + {1 −μ − √3 } 2.4Where( )= − ( ) 1 + 2.5

Fig 2.1 Simulated PDF for Composite/Shadowed (WL) fading μ =1 = 0.25

Fig 2.2 Simulated CDF for Composite/Shadowed (WL) fading forμ =1 = 0.25Amount of Fading (AF) is defined as= [ ]( [ ]) − 1 3.1

AF of WL fading can be calculated after somemanipulations using 2.2 and 2.3 approximations

= − 1 3.2

In Table 3.1, AF is given for different shape parameter c.We can conclude that AF decreases as shape parameter cincreases.

Table 3.1 AF for different shape parameter

c AF0.5 691 5

2 13 0.4609984 0.273245 0.1831056 0.1320937 0.1001468 0.07870529 0.0635701

10 0.0524652

Outage ProbabilityThe outage probability is standard performance criterion ofdiversity systems operating over fading channels and it isdefined as the probability that the instantaneous error rateexceeds a specified value, or equivalently, that combinedSNR of MRC falls below a predetermined threshold .( ) = [ ≤ ] = ∫ ( ) 4.1

Using 2.2 and 2.3 in 4.1( ) = 1 − exp + 1 −exp + 1 − exp 4.2

Where= 1 + 2exp (μ)= 1 + 2exp (μ + √3 )= 1 + 2exp (μ − √3 )CDF of WL in 2.4 is same as ( ) in 4.2

Fig 4.1 Simulated CDF for Composite/Shadowed (WL) fading forμ = 1 = 0.25Channel Capacity

For WL fading, Channel capacity [12] is defined as= ∫ (1 + ) ( ) 5.1Using 2.2 and 2.3 in 5.1= 2 ln(2) ,, − 2 , 1 − 20, − , −Where = and & =

0 2 4 6 8 10 12 14 16 18 200

0.2

0.4

0.6

0.8

1

1.2

1.4

SNR y

PD

F o

f W

eib

ull-L

og N

orm

al F

adin

g

PDF of Composite/Shadowed (WL)

c=0.5c=1c=3

0 2 4 6 8 10 12 14 16 18 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SNR y

CD

F o

f W

eib

ull-L

og N

orm

al fa

din

g

CDF of Composite/Shadowed(WL)

c=0.5c=1c=3

0 2 4 6 8 10 12 14 16 18 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Thresold Yth(dB)

Outa

ge P

robability P

out

Outage Probability of WL shadowed fading

c=0.5c=1c=3

Int. Journal of Electrical & Electronics Engg. Vol. 2, Spl. Issue 1 (2015) e-ISSN: 1694-2310 | p-ISSN: 1694-2426

NITTTR, Chandigarh EDIT -2015 110

Where B is the bandwidth. Equation can be represented inclosed form using MeijerG function. The values of C/Bhas been calculated and shown in Table 5.1. As we canconclude from table 5.1 that channel capacity decreasewith increasing shape parameter.

Table 5.1 Channel Capacity (C/B) for different shapeparamter

c C/B0.5 2.379281 1.346412 0.9042913 0.6540064 0.5077865 0.4136176 0.3483527 0.3006118 0.264241 +2.02467×10-16i9 0.235643

10 0.212584

ConclusionThis paper has established a process for estimating thedistribution of Composite/Shadowed (WL) fading. Theprocedure uses the Holtzmanian approximations toestimate the closed form of composite PDF of WLfading.Successfully we have achieved closed formequations for PDF. We calculated amount of fading (AF)and channel capacity (C/B) in closed form. We haveevaluated outage probability in closed form.Graphical results have been given for PDF of receivedSNR, CDF of received SNR and outage probability .

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