This paper analyzes the performance of composite multipath/shadowing fading channels, specifically focusing on Weibull-lognormal models. The authors present closed-form equations for the probability density function (pdf), amount of fading, outage probability, and channel capacity, utilizing Holtzman's approximation for simplification. Graphical results are provided to illustrate the findings, which are significant for communication systems operating over such composite channels.

1. 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 Composite
Multipath/Shadowing (Weibull-Log Normal)
Fading Channels
1
Rupender Singh, 2
S.K. Soni, 3
Rajesh Birok
1,2,3
Department of Electronics & Communication Engineering
Delhi Technological University (Formerly Delhi College of Engineering), Delhi
rupendersingh04cs39@gmail.com
Abstract: Composite multipath/shadowing fading environments are frequently encountered in different mobile realistic
scenarios. These channels are generally modeled different
Composite multipath/shadowing fading. In this paper we
present the performance analysis of composite (Weibull-
Lognormal shadowed) fading. We adopt efficient tool
proposed by Holtzman to approximate composite (Weibull-
Lognormal shadowed) fading. The performance measures of
fading communication systems such as Probability density
function (PDF) of Signal to Noise ratio (SNR), Amount of
fading (AF), Outage probability (Pout) and Channel
Capacity(C/B) will be calculated. Graphical results will be
presented for different signals and fading parameters. The
different expressions that will be provided are of great
importance in assessing the performance of communication
systems in composite channels.
Keywords: Weibull-Lognormal Shadowed fadingfading (WL),
Probability density function(PDF), Amount of fading(AF),
Outage probability(Pout), Channel Capacity(C/B)
Introduction
Wireless communication channels are impaired by
detrimental effects such as Multipath Fading and
Shadowing [1]. Based on various indoor and outdoor
empirical measurements, there is general consensus that
shadowing be modeled using Log-normal distribution [12-
14]. Fading causes difficulties in signal recovery. When a
received signal experiences fading during transmission, its
envelope and phase both fluctuate over time.
A composite multipath/shadowed fading environment
modeled either as Rayleigh-lognormal, Rician-lognormal
or Nakagami-lognormal are considered in [3-5]. Up to
now, composite multipath/shadowed fading environment
modeled as Weibull-lognormal (WL), has been considered
only in several papers [7, 8]. The Weibull distribution
plays an important role in several scientific fields, but it
has become recently the topic of wireless communications
theory [9], particularly with mobile radio systems
operating in the 800/900 MHz frequency range. The
Weibull model exhibits an excellent fit to experimental
fading channel measurements, for both indoor [10] and
outdoor [11] environments.
In this paper, a simple accurate closed-form using
Holtzmanin [18] approximation for the expectation of the
function of a normal variant is also employed. Then,
simple analytical approximations for the PDF of
Composite/Shadowed (WL) are derived.
System and Channel Models
Here we are taking two uncorrelated channels in presence
of weibull and log normal fading. So PDF of SNR can be
obtained by averaging the PDF of weibull over log normal
fading. Weibull and conditional log normal distributions
given below[1,17]
( / )
=
2
1 +
2
− 1
+
2
≥ 0
Where c is shape parameter for weibull distribution.
( )
=
√2
exp −
(10 − μ)
2
Where µ, σ are mean and variance respectively
of 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] to
simplify Eg. (2.1). Taking Eg. (5-7) in [14], we have
Using 10 = in (2.1)
( ) = ( ).
σ √2
( )
Then finally we have PDF of WL fading
( ) ≈ (μ) + μ + √3 + μ − √3 2.2
Where

2. 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 +
2.3
Using 2.2 and 2.3 we have CDF of WL fading
( ) = {1 − (μ)} + {1 − μ + √3 } + {1 −
μ − √3 } 2.4
Where
( )= −
( )
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 = 0.25
Amount of Fading (AF) is defined as
=
[ ]
( [ ])
− 1 3.1
AF of WL fading can be calculated after some
manipulations 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 c
increases.
Table 3.1 AF for different shape parameter
c AF
0.5 69
1 5
2 1
3 0.460998
4 0.27324
5 0.183105
6 0.132093
7 0.100146
8 0.0787052
9 0.0635701
10 0.0524652
Outage Probability
The outage probability is standard performance criterion of
diversity systems operating over fading channels and it is
defined as the probability that the instantaneous error rate
exceeds a specified value, or equivalently, that combined
SNR 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 +
2
exp (μ)
=
1 +
2
exp (μ + √3 )
=
1 +
2
exp (μ − √3 )
CDF of WL in 2.4 is same as ( ) in 4.2
Fig 4.1 Simulated CDF for Composite/Shadowed (WL) fading for
μ = 1 = 0.25
Channel Capacity
For WL fading, Channel capacity [12] is defined as
= ∫ (1 + ) ( ) 5.1
Using 2.2 and 2.3 in 5.1
=
2 ln(2) ,
, −
2
, 1 −
2
0, − , −
Where = and & =
0 2 4 6 8 10 12 14 16 18 20
0
0.2
0.4
0.6
0.8
1
1.2
1.4
SNR y
PDFofWeibull-LogNormalFading
PDF of Composite/Shadowed (WL)
c=0.5
c=1
c=3
0 2 4 6 8 10 12 14 16 18 20
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
SNR y
CDFofWeibull-LogNormalfading
CDF of Composite/Shadowed(WL)
c=0.5
c=1
c=3
0 2 4 6 8 10 12 14 16 18 20
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Thresold Yth(dB)
OutageProbabilityPout
Outage Probability ofWL shadowed fading
c=0.5
c=1
c=3

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 in
closed form using MeijerG function. The values of C/B
has been calculated and shown in Table 5.1. As we can
conclude from table 5.1 that channel capacity decrease
with increasing shape parameter.
Table 5.1 Channel Capacity (C/B) for different shape
paramter
c C/B
0.5 2.37928
1 1.34641
2 0.904291
3 0.654006
4 0.507786
5 0.413617
6 0.348352
7 0.300611
8 0.264241 +2.02467×10-16
i
9 0.235643
10 0.212584
Conclusion
This paper has established a process for estimating the
distribution of Composite/Shadowed (WL) fading. The
procedure uses the Holtzmanian approximations to
estimate the closed form of composite PDF of WL
fading.Successfully we have achieved closed form
equations for PDF. We calculated amount of fading (AF)
and channel capacity (C/B) in closed form. We have
evaluated outage probability in closed form.
Graphical results have been given for PDF of received
SNR, CDF of received SNR and outage probability .
References
[1] Marvin K. Simon, Mohamad Slim Alouni“Digital Communication
over Fading Channels”, Wiley InterScience Publication.
[2] A. Goldsmith, Wireless Communications, Cambridge University
Press, 2005.
[3] F. Hansen and F.I. Mano, “Mobile Fading- Rayleigh and Lognormal
Superimposed”, IEEETrans. Vehic. Tech., vol. 26, pp. 332–335,
1977.
[4] E. K.Al-Hussaini, A.M. Al-Bassiouni, H. Mourad and H. Al-
Shennawy, “Composite Macroscopic and Microscopic Diversity of
SectorizedMacrocellular and Microcellular Mobile Radio Systems
Employing RAKE Receiver over Nakagami Fading plus Lognormal
Shadowing Channel”, Wireless Personal Communications, vol. 21,
pp. 309–328, 2002.
[5] W.C. Jakes, Microwave Mobile Communication, 2nd ed. Piscataway,
NJ: IEEE Press, 1994
[6] P. M. Shankar, “Error rates in generalized Shadowed Fading
Channels”, Wireless Personal Communications, vol. 28, pp. 233-238,
2004.
[7] Mitic, M. Jakovljevic, "Second-Order Statistics in Weibull-Lognormal
Fading Channels", Conference Proceedings of TELSIKS 2007, , Nis,
Serbia, September 26-28, 2007.
[8] M. Stefanović, A. Mitić, D. Pavlović,
"Poređenjestatističkihkarakteristikasignala u
kanalusaVejbulovimfedingomzarazličitetehnikekombinovanja"
,Infoteh, Jahorina, Conference Proceedings, 2007. (inserbian)
[9] Nikos C. Sagias, George K. Karagiannidis, Petros S. Bithas, P.
TakisMathiopoulos, “On the Correlated Weibull Fading Model and
Its Applications”, Proc. Vehicular Techology Conference IEEE, vol.
4, pp. 2149–2153, Sept. 2005.
[10] H. Hasemi, “The indoor radio propagation channel,” Proc. IEEE,
vol. 81, pp. 943–968, July 1993.
[11] N. S. Adawi, et al., “Coverage prediction for mobile radio systems
operating in the 800/900 MHz frequency range,” IEEE Trans. Veh.
Technol., vol. 37, pp. 3–72, Feb. 1988.
[12] H. Suzuki, “A Statistical Model for Urban Radio Propagation”, IEEE
Trans. Comm., Vol. 25, pp. 673–680,1977.
[13] H. Hashemi, “Impulse response modeling of indoor radio
propagation channels,” IEEE J. Select. Areas Commun., vol. SAC-11,
September 1993, pp. 967–978.
[14] F. Hansen and F.I. Mano, “Mobile Fading-Rayleigh and Lognormal
Superimposed”, IEEE Trans. Vehic.Tech., Vol. 26, pp. 332–335,
1977.
[15] E. Lutz, D. Cygan, M. Dippold, F. Dolainsky, and W. Papke, “The
land mobile satellite communication channel: recording, statistics,
and channel model,” IEEETrans. Veh. Technol., vol. VT-40, May
1991, pp. 375–386.
[16] R. M. Barts and W. L. Stutzman, “Modeling and simulation of
mobile satellite propagation,” IEEE Trans. Antennas Propagat., vol.
AP-40, April 1992, pp. 375–382.
[17] J. Lieblein, “On moments of order statistics form the Weibull
distribution,” Annals Math. Stat., vol. 26, no. 2, June 1955, pp. 330–
333.
[18] J. M.Holtzman, "A simple, accurate method to calculate spread
multipleaccess error probabilities," IEEE Trans.Commu., vol. 40, no.
3, pp. 461- 464, Mar. 1992.