Int. Journal of Electrical & Electronics Engg.
Vol. 2, Spl. Issue 1 (2015)
e-ISSN: 1694-2310 | p-ISSN: 1694-2426
On the Performance Analysis of Composite Multipath/Shadowing (Weibull-Log Normal) Fading Channels 1
Rupender Singh, 2S.K. Soni, 3Rajesh 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 fading. Weibull and conditional log normal distributions Composite multipath/shadowing fading. In this paper we given below[1,17] present the performance analysis of composite (Weibull( / ) Lognormal shadowed) fading. We adopt efficient tool 2 proposed by Holtzman to approximate composite (Weibull1+ Lognormal shadowed) fading. The performance measures of = − 1 fading communication systems such as Probability density 2 function (PDF) of Signal to Noise ratio (SNR), Amount of fading (AF), Outage probability (Pout) and Channel 2 Capacity(C/B) will be calculated. Graphical results will be + ≥0 presented for different signals and fading parameters. The different expressions that will be provided are of great Where c is shape parameter for weibull distribution. 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 [1214]. 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 NITTTR, Chandigarh
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( )
=
√2
exp −
(10
2
− μ)
Where µ, σ are mean and variance respectively of RV w and ξ=10/ln10. PDF of Instantaneous SNR γ is ( )=
( )=∫ √
Where
exp −
( / ) ( )
(
)
−
1+ 2.1
= 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) ( )=
( ).
(μ) +
μ + √3
σ √2
(
)
Then finally we have PDF of WL fading ( )≈ Where
+
μ − √3
2.2
108
Int. Journal of Electrical & Electronics Engg.
( )=
−
( )
Vol. 2, Spl. Issue 1 (2015)
2 3 4 5 6 7 8 9 10
1+
( )
2.3 Using 2.2 and 2.3 we have CDF of WL fading ( )=
μ − √3 Where ( )=
{1 − (μ)} + {1 −
μ + √3
} 2.4
−
1+
( )
} + {1 −
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 .
c=0.5 c=1 c=3 1.2
PDF of Weibull-Log Normal Fading
1
0.8
0.6
0.4
0
(
)=
(
)=
[ ≤
]=∫
Using 2.2 and 2.3 in 4.1
0.2
0
2
4
6
8
10 SNR y
12
14
16
18
20
Fig 2.1 Simulated PDF for Composite/Shadowed (WL) fading μ = 1 = 0.25 CDF of Composite/Shadowed(WL)
1
c=0.5 c=1 c=3
0.9
exp Where =
=
0.8
0.7 CDF of Weibull-Log Normal fading
1 0.460998 0.27324 0.183105 0.132093 0.100146 0.0787052 0.0635701 0.0524652
2.5
PDF of Composite/Shadowed (WL)
1.4
e-ISSN: 1694-2310 | p-ISSN: 1694-2426
=
0.6
1+
1 − exp
2
exp (μ)
1+
+
0.4
1−
4.2
2
exp (μ + √3 ) 2 1+
0.3
(
Outage Probability of WL shadowed fading
1
0.2
4.1
+
1 − exp
exp (μ − √3 ) CDF of WL in 2.4 is same as
0.5
( )
) in 4.2
0.9
c=0.5 c=1 c=3
0.1
0.8 0
2
4
6
8
10 SNR y
12
14
16
18
20
0.7 Outage Probability Pout
0
Fig 2.2 Simulated CDF for Composite/Shadowed (WL) fading forμ = 1 = 0.25 [ [
]
])
−1
−1
0
2
4
6
8
10 Thresold Yth(dB)
12
14
16
18
20
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
3.2
Table 3.1 AF for different shape parameter
109
0.1 0
In Table 3.1, AF is given for different shape parameter c. We can conclude that AF decreases as shape parameter c increases.
c 0.5 1
0.4
0.2
3.1
AF of WL fading can be calculated after some manipulations using 2.2 and 2.3 approximations =
0.5
0.3
Amount of Fading (AF) is defined as =(
0.6
AF 69 5
= ∫ Using 2.2 and 2.3 in 5.1
Where
=
2 ln(2)
= and
&
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(1 + ) ( ) , ,
=
5.1 − ,1 − 2 2 0, − , − EDIT-2015
Int. Journal of Electrical & Electronics Engg.
Vol. 2, Spl. Issue 1 (2015)
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 0.5 1 2 3 4 5 6 7 8 9 10
C/B 2.37928 1.34641 0.904291 0.654006 0.507786 0.413617 0.348352 0.300611 0.264241 +2.02467×10-16i 0.235643 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.
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e-ISSN: 1694-2310 | p-ISSN: 1694-2426
[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. AlShennawy, “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.
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