fading

Wireless Pers CommunDOI 10.1007/s11277-010-9938-2

Statistical Models for Fading and Shadowed FadingChannels in Wireless Systems: A Pedagogical Perspective

P. M. Shankar

© Springer Science+Business Media, LLC. 2010

Abstract A unified analysis of statistical models for describing fading, shadowing, andshadowed fading channels is presented from a pedagogical viewpoint. The different proba-bility density functions such the Rayleigh, Nakagami, gamma, generalized gamma, Weibull,lognormal, Nakagami-lognormal, K distribution, generalized K distribution, and Nakagamiinverse Gaussian distribution are presented and the relationships among them are detailed.These density functions are compared in terms of two quantitative measures, namely theamount of fading and outage probability. A general approach to fading and shadowed fadingchannels using a cluster based approach is also presented to link several of the distributions. Itis expected that this overview will be very helpful to students and educators who are engagedin the study of wireless systems and the adverse impact of fading and shadowing in wirelessdata transmission.

Keywords Wireless channels · Fading · Shadowing · Shadowed fading · Outageprobabilities · Amount of fading · Rayleigh · Nakagami · Rician · Double Rayleigh ·Double Nakagami · Generalized K · Nakagami inverse Gaussian · Generalized gamma ·Weibull distribution

1 Introduction

Courses in communications and wireless communications are offered at the undergraduateand graduate level programs in electrical engineering at universities and colleges aroundthe world. Of the many topics in wireless systems, fading is one of the important conceptsrequired for a good understanding of problems in signal reception and ways to improve over-all system performance [1]. This is also a topic that is often described in different formsmaking it a little difficult for the students and instructors to fully comprehend its impact.While several publications and books cover the topic of fading, a unified approach to explain

P. M. Shankar (B)Department of Electrical and Computer Engineering, Drexel University, Philadelphia, PA 19104, USAe-mail: [email protected]

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P. M. Shankar

Fig. 1 The multipath transmission in wireless systems is shown with three distinct paths between the trans-mitter and receiver

the concept of fading in a pedagogical context is not available [1–4]. This manuscript presentsa concise unified approach to explain the concepts of fading in wireless systems.

The brief background section introduces the terminology of fading, shadowing, and shad-owed fading channels. This is followed by sections on fading models, shadowing models, andshadowed fading models. A simple general model is then described which can encompassall three. The last section contains a discussion of the main aspects of the manuscript andconclusions.

2 Background

In wireless communications, the transmitted signals often do not reach the receiver directly[1,2]. As the power is lost due to attenuation and absorption, the signals reach the receiverafter undergoing scattering, diffraction, reflection, etc. from the buildings, trees, and otherstructures in the medium (channel) between the transmitter and the receiver. Thus, thereexist multiple paths for the signal to reach the receiver and the signals arriving throughthese paths add in phase. Since the amplitude and phase of the signal from each of thesepaths can be treated as random variables, the received power will also be random. Thisrandom fluctuation of power is identified as ‘fading’ in wireless systems. These power fluc-tuations have a very short period and hence, the fading is referred to as ‘short term fading’.The transmission and reception of signals is sketched in Fig. 1. It shows three differentpaths between the transmitter and the receiver and that these paths are independent. It isalso possible that instead of the simple scenario drawn in Fig. 1, signals might be reach-ing the receiver after multiple scattering in the channel such that signal in a path encoun-ters more than one object in its path. This phenomenon causes fluctuations in the receivedsignal that have a period longer than those associated with short term fading. These fluc-tuations are identified as ‘long term fading’ or ‘shadowing’. Figure 2 shows the typicalreceived signal profiles. The thick line represents the attenuated signal reaching the receiver.The heavy dotted line with a slow variation is the shadowing or long term fading and the‘short term fluctuations’ seen riding on the heavy dotted line represent the short term fading.Thus, the realistic case of the received signal consists of the transmitted signal that reachesthe receiver as it undergoes attenuation and passes through the ‘shadowed fading channel’resulting in randomly varying and attenuated signals.

It must be noted that the short term fading might be accompanied by frequency dependenteffects which limit the bandwidth capability of the channel [1,2]. Our discussion in this workdoes not address such ‘frequency selective fading’ channels and only examines channels

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Statistical Models for Fading and Shadowed Fading Channels

Fig. 2 Profiles of the receivedpower are shown. The heavy darkline represents the attenuatedsignal reaching the receiver.The heavy dotted line with slowvariations is the shadowing orlong term fading. The ‘short termfluctuations’ are seen riding onthe heavy dotted line representthe short term fading

that are considered as ‘flat’ implying that the short term fading does not alter the frequencycharacteristics or bandwidth capability of the channel. We also assume that neither the trans-mitter nor the receiver is in motion so that we exclude the effects of any frequency modulationcaused by the Doppler effect, commonly labeled as ‘Doppler fading’ [1,2]. In other words,we will only be considering channels that are considered as ‘slow’ (relative speed of thetransmitter/receiver negligible) instead of the ‘fast’ (relative speed of the transmitter/receiverhigh) channels. Thus, the analysis below deals with wireless channels where fading can bedescribed as ‘slow’ and ‘flat’.

We will now explore different ways to describe the statistical fluctuations of the receivedsignal arising from fading and shadowing occurring separately as well as concurrently.

3 Models for Short Term Fading

Short term fading in wireless channels has been described using several models. We will nowlook at these models starting with the Rayleigh fading model followed by others.

3.1 Rayleigh Fading

To understand fading, we have to examine the manner in which the signals from the trans-mitter reach the receiver. The simplest way to visualize this situation is through the use of themultipath phenomenon. A typical multipath scenario is sketched in Fig. 1 where the transmit-ter sends a simple sinusoidal signal at a carrier frequency of f0. Use of a sinusoidal signal isa reasonable approach since we are only dealing with a ‘flat’ channel that does not introduceany frequency dependent changes. The received signal er (t) arising from the propagation ofthe signal via multiple paths in the channel can be expressed as

er (t) =N∑

i=1

ai cos (2π f0t + φi ). (1)

The number of multiple paths is N which can be treated as equivalent to the number ofscattering/reflecting/diffracting centers or objects in the channel. The i th multipath signalcomponent has an amplitude ai and a phase φi . The Eq. 1 can be rewritten in terms ofinphase and quadrature notation as

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P. M. Shankar

er (t) = cos (2π f0t)N∑

i=1

ai cos (φi )− sin (2π f0t)N∑

i=1

ai sin (φi ) (2)

where the first summation (associated with the cosine term) is identified as the inphase termand the second summation (associated with the sine term) is identified as the quadratureterm. If the locations of the structures are completely random, one can safely assume thatthe phase φ’s will be uniformly distributed in the range {0, 2π}. Under conditions of largeN, the amplitude of the received signal can then be expressed as

er (t) = X cos (2π f0t)− Y sin (2π f0t) (3)

where

X =N∑

i=1

ai cos (φi ) ; Y =N∑

i=1

ai sin (φi ) . (4)

X and Y will be independent identically distributed (i.i.d) Gaussian random variables of zeromean by virtue of the central limit theorem [5]. This Gaussianity of X and Y also leads to theenvelope of the received signal, A, given by (X2 + Y 2)1/2 to be Rayleigh distributed [5]. Theprobability density function (pdf) of the received signal envelope, fR(a), will be given by

fR (a) = a

σ 2 exp

(− a2

2σ 2

)U (a) . (5)

In Eq. 5, σ 2 is the variance of the random variables X (or Y) and U(.) is the Unit step function.The subscript (R) of the pdf in Eq. 5 and subscripts in all the other pdfs later in this manuscriptmerely indicate the nature of the statistics associated with fading, i.e., in this case, Rayleigh.Note that if the envelope of the signal is Rayleigh distributed, the power, P = A2, will havean exponential probability density function, given by

fR (p) = 1

P0exp

(− P

P0

)U (p) . (6)

Once again, the subscript R relates to the nature of the statistics which in this case is classifiedas Rayleigh. In Eq. 6, 2σ 2 has been replaced by the average power P0 of the received signal.

Rayleigh density function is not the only pdf that can be used to model the statistics ofshort term fading. It has limited application in a broader context because of its inability tomodel fading conditions that result in significant degradation in performance of wireless sys-tems. To understand this point, it is necessary to quantify the level of fading. The parameterused to measure levels of fading is the amount of fading (AF) defined as [1]

AF =⟨P2

⟩ − 〈P〉2

〈P〉2 (7)

Note that in Eq. 7, 〈.〉 represents the statistical average. Making use of the moments of theexponential pdf in Eq. 6, given by Papoulis and Pillai [5]

⟨Pk

⟩

R= Pm

0� (k + 1)

� (k), kth moment (8)

where �(.) is the gamma function, Eq. 7 becomes

AF R = 1 (9)

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Thus, the amount of fading in a channel that has Rayleigh pdf for the envelope is equal tounity. If the amount of fading is larger than unity, we have severe fading conditions, we willclassify that channel as a pre-Rayleigh and if the fading conditions are better than Rayleigh,AF will be less than one, and, we identify such a channel as post-Rayleigh. Thus, it is obviousthat Rayleigh pdf is inadequate to model all fading conditions that exist in wireless channels,and, other models need to be explored. One of such models is based on the Nakagami-m pdf[1,6]. For the remainder of this paper, this pdf will be simply identified as the Nakagami pdf.But, before we look at the Nakagami model, let us go back to the multipath model describedearlier and make minor modifications to it by considering a direct path or a line-of-sight(LOS) between the transmitter and receiver. Such a multipath scenario results in the Ricianfading channel as described below.

3.2 Rician Fading

By including a direct path between the transmitter and receiver, represented by a0 cos(2π f0t)where a0 is a constant, Eq. 3 becomes

eRice (t) = X cos (2π f0t)− Y sin (2π f0t)+ a0 cos (2π f0t)

= (X + a0) cos (2π f0t)− Y sin (2π f0t) (10)

The received power will now be given by

P = (X + a0)2 + Y 2 = X ′2 + Y 2 (11)

where X ′ is a Gaussian random variable with a non-zero mean equal to a0. The pdf of thepower will be given by

fRi (p) = 1

2σ 2 exp

(− p + a2

0

2σ 2

)I0

( a0

2σ 2

√p)

U (p) (12)

where I0(.) is the modified Bessel function of the first kind. Equation 12 is the pdf of thereceived signal power in a Rician fading channel which differs from the Rayleigh channelbecause of the existence of a LOS path in addition to multiple indirect paths. Because ofthe existence of the direct path, the amount of fading will be less than what is observed inRayleigh fading. The Rician factor K0 is defined as

K0 = a20

2σ 2 (13)

The quantity K0 is a measure of the strength of the LOS component and when K0 → 0, wehave Rayleigh fading and as K0 increases, the fading in the channel declines. If the averagereceived power is PRi , it can be expressed as

PRi = 〈p〉Ri = 2σ 2 + a20 . (14)

We now have,

2σ 2 = 1

K0 + 1PRi (15)

and

a20 = K0

K0 + 1PRi (16)

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P. M. Shankar

Using Eqs. 14–16, the pdf of the received power in Rician fading becomes

fRi (p) = K0 + 1

PRiexp

[−K0 − (K0 + 1)

p

PRi

]I0

(2

√K0 (K0 + 1)

PRip

)U (p) (17)

Note that when K0 = 0, Eq. 19 becomes Eq. 6. The amount of fading in a Rician channelcan be obtained from the moments of the pdf in Eq. 17 and can be expressed as

AF Ri = 1 + 2K0

(1 + K0)2 (18)

It can be easily seen that as K0 increases, the amount of fading decreases and as K0 → ∞,the amount of fading becomes zero. In other words, the existence of the direct path can reducethe levels of fading in wireless channels. When K0 becomes unity, Eq. 18 becomes zero, theamount of fading in a Rayleigh channel.

We will now look at the most commonly used model to describe short term fading inwireless channels, namely, the Nakagami distribution.

3.3 Nakagami Fading

Based on the original work by Nakagami, the pdf of the received signal power in short termfading can be expressed as [1,6]

fN (p) = mm pm−1

Pm0 � (m)

exp

(−m

p

P0

)U (p) , m ≥ 1

2(19)

where m is called the Nakagami parameter and�(.) is the gamma function. The average poweris P0. Equation 19 is the pdf of the received signal power in a Nakagami fading channel and iscommonly identified as the Nakagami-m type pdf. While other forms of the Nakagami pdfsexist, since we will be only discussing the pdf in Eq. 19, it will be identified as the Nakagamipdf.

Using the moments of the Nakagami pdf in Eq. 19 given as⟨Pk

⟩

N= � (m + k)

� (m)mkPk

0 . (20)

the amount of fading in a Nakagami channel becomes,

AF N = 1

m. (21)

Equation 21 also provides a rationale for calling ‘m’ the fading parameter since the amountof fading depends entirely on it.

Even though the Nakagami pdf has shown to be good fit based on experimental observa-tion, we can also provide a simple semi-analytical means to justify its use. The approach isbased on the concept of ‘clustering’ or ‘bunching’ of scatterers [7–10]. Consider the case ofa wireless channel which contains a large number of scattering centers (buildings, trees, etc.which reflect, scatter, diffract, refract) as described earlier. However, instead of all of thembeing located in a purely random way in the channel, they are now clustered together. Let usassume that there are n such clusters of scatterers within the channel, with each cluster havingsufficiently large number of scatterers. We will also assume that these clusters are locatedrandomly within the confines of the channel, with the received signal coming from theseclusters or groups. Since we assumed that each of these groups has sufficiently large number

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of scatterers, without any loss of generality, the inphase X and quadrature Y componentsfrom them would be Gaussian distributed. Let us define Z as

Z =n∑

i=1

X2i , (22)

where n is an even number ≥ 2. This means that for n = 2, i = 1 represents X and i = 2represents Y and no information is lost in expressing Z in terms of X alone. Note also thatZ represents the power. We will now extend Eq. 22 to include the cases of n being an oddnumber as well since Eq. 22 is simply the sum of the squares of Gaussian random numbers.The probability density function of Z will be a chi-square density function given by Papoulisand Pillai [5]

f (z) = z(n2 )−1

z( n

2 )0 �

( n2

) exp

(− z

z0

)U (z) (23)

where z0 is a constant. If we replace (n/2) by m, and z by p, the density function will beidentical to the Nakagami pdf for the power in Eq. 19 with the condition that m> 1/2 sincethe smallest value of n = 1 (except for scaling factors). Rayleigh fading (exponential pdf forthe power) is still possible with m = 1, so that n = 2 and Z will be equal to

Z =2∑

i=1

X2i = X2

1 + X22 = X2

1 + Y 21 (24)

since X1 and Y1 are identically distributed. The pdf can now be generalized to have m takenon-integer values in which case the Eq. 23 will be identical to Eq. 19 [7–10]. Please notethat treating the collection of scatterers as individual ones or treating them as ‘clusters’ isnot in conflict since we can have a case of a single cluster with a large number of scatterersgiving rise to the Rayleigh case described in connection with Eq. 6.

Two important aspects of the Nakagami fading channel can be easily seen. First, whenm = 1, Eq. 19 becomes identical to the exponential pdf in Eq. 6. Second, the choice of dif-ferent values of m permits the amount of fading to vary from zero (m = ∞) to 2 (m = 1/2).This means that at very high values of m (m → ∞), the fading vanishes and the channelbecomes a pure Gaussian channel suggesting that the channel suffers only from additivewhite Gaussian noise (and possibly other forms of noise such as impulsive noise as well ifsuch noises exist in the channel). Still, the Nakagami pdf in Eq. 19 does not allow fadingconditions that might lead to values of AF > 2. Note that it is possible to know the amount offading present in the channel (if so required), by conducting measurements over a long periodand evaluating AF in Eq. 7 from the observed signal powers and estimating the moments.Thus, it is necessary to have models which allow modeling of fading channels which exhibitfading levels far severe than what are observed in a Nakagami channels.

A wide range of fading conditions can be modeled using other probability density func-tions and we will now examine the use of gamma and Weibull distributions in this context.

3.4 Gamma, Generalized Gamma, and Weibull Fading

There are several probability density functions that have been used to model such channelswhere fading is worse than in a Nakagami channel [1]. The first and simplest approach toundertake such a modeling is to rename Eq. 19. Note that Eq. 19 is a gamma probabilitydensity function [1]. In this case, values of m less than 1/2 are allowed. Instead of using the

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P. M. Shankar

pdf for power in Eq. 10, the gamma model for short term fading can be expressed as in termsof the pdf as

fG (p) = pm−1

Pmg � (m)

exp

(− p

Pg

)U (p) , m > 0 (25)

where Pg is related to the average power. Equations 19 and (25) have identical forms exceptfor two factors, namely, m is allowed to take any positive values and that the average powerscorresponding to the pdf in Eqs. 19 and 25 are different. Thus, use of Eq. 25 permits toinclude conditions where the fading could be worse than what is observed in a Nakagamichannel. The average power can be obtained from the moments of the gamma pdf as

〈P〉G = m Pg. (26)

Even removing the restriction on m and consequently changing the Nakagami fading togamma fading might be insufficient to model the fading observed in wireless channels. Toaccommodate such channels, a generalized Nakagami or generalized gamma channel [4],[11–13] can be defined by scaling the power by (1/s) where s is a positive number. Let usdefine Pgg as

Pgg = P1s , s > 0 (27)

The probability density function of the power in Eq. 27 can be obtained and is expressed as[5]

f(

pgg) = s pms−1

gg

� (m) Pmg

exp

(− ps

gg

Pg

)U

(pgg

), 0 < s < ∞ (28)

Without any loss of generality, we can replace Pgg by p and we have the expression for thepdf of the received power in a generalized gamma fading channel as

fGG (p) = s pms−1

� (m) Pmg

exp

(− ps

Pg

)U (p) , 0 < s < ∞ (29)

Equation 29 will be generalized Nakagami pdf if m is restricted to values larger than 1/2. Themoments of Eq. 29 are given by

⟨Pk

⟩

GG= �

(m + k

s

)

� (m)P

(ks

)

g (30)

resulting in

〈P〉GG =[�

(m + 1

s

)

� (m)

]P

(1s

)

g (31)

Equation 31 reduces to Eq. 26 when s = 1. Thus, the parameter s permits an additional levelof flexibility in modeling fading channels. Using the moments of the generalized gamma pdfgiven in Eq. 30, the amount of fading in a generalized gamma fading channel becomes

AFGG = �(m + 2

s

)� (m)

[�

(m + 1

s

)]2 − 1 (32)

Note that when s becomes unity,

AFGG = AFG = AFN (33)

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It is also possible to redefine the generalized gamma distribution so that instead of the depen-dence of fading on two parameters, namely m and s, a new fading parameter mw can be used.This leads to the Weibull fading model [4,14,15]. The simplest way to generate the Weibullfading conditions is to start from the Rayleigh fading or the associated exponential pdf of thepower in Eq. 6. Let us define a new variable W

W = P1

mw (34)

where P is exponentially distributed with a pdf of the form in Eq. 6. Equation 34 suggeststhat the power observed in this fading channel is best described in terms of a scaled versionof the power received in a typical Rayleigh channel. Using the concept of transformation ofthe variables, the probability density function of W can be obtained as [5]

f (w) = mw

Pgwmw−1 exp

(−w

mw

Pg

)U (w) (35)

Without any loss of generality, we can replace w with the variable p and hence, the pdf of thepower in a Weibull fading channel becomes

fW (p) = mw

Pgpmw−1 exp

(− pmw

Pg

)U (p) (36)

Certainly, Eq. 36 is much simpler than Eq. 29 for the generalized gamma fading channeland more complicated than the Nakagami (or gamma) channels. The fading parameter isidentified as mw . It must be noted that Eq. 36 can also be obtained as a special case of thegeneralized gamma pdf by putting m = 1 and s = mw pointing to Weibull fading beingsimpler than the generalized gamma fading and justifying the scaling employed in Eq. 34.

The moments of the pdf in Eq. 36 can be expressed as

⟨Pk

⟩= �

(1 + k

mw

)P

(k

mw

)

g (37)

The amount of fading now becomes

AFW =�

(1 + 2

mw

)

[�

(1 + 1

mw

)]2 − 1 (38)

One can now compare the amount of fading existing in the channels by comparing Eqs. 6,25 and 29. Note that m in Eqs. 19 and 25 is identical.

The Weibull channel, gamma channel, and, the generalized gamma channels are com-pared in terms of the amount of fading and the results are shown in Fig. 3. It is clear that thegeneralized gamma channel offers a means to model fading channels with widely varyinglevels of fading. For values of s > 1, the generalized fading channel has less fading thanthe gamma channel and for values of s< 1, the generalized fading channel has higher levelsof fading. Comparing the gamma channel and Weibull channel, it is clear that each of themoffers a different way of looking at the fading taking place in the channel. Note that simplicityis offered by the Nakagami pdf with m taking values equal to and beyond 1/2 (or gamma pdfwith m taking values less than 1/2).

Now that we have looked at different fading models, we will now examine the models todescribe shadowing in wireless systems.

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P. M. Shankar

Fig. 3 Amount of Fading (AF) is plotted for the Rayleigh (R), gamma (G), generalized gamma (GG) andWeibull (W) channels. For values of m > 0.5, the gamma channel is also the Nakagami channel. For the GGchannel, two values of s are shown. For s>1, the GG channel has lower values of amount of fading than theNakagami channel and for s<1, the GG channel has higher levels of fading. If the amount of fading>1 (abovethe horizontal line corresponding to the Rayleigh channel), we have a pre-Rayleigh channel. Otherwise, wehave a post-Rayleigh channel with AF <1 which lies below the value of R = 1

4 Models for Shadowing

As mentioned earlier, in wireless systems, it is often observed that local average power variesrandomly from location to location within a given geographical region as shown in Fig. 2[1]. This has been attributed to the existence of shadowing terrain, buildings, structures etc.Measurements have suggested that the density function of the average power can be mod-eled in terms of a lognormal probability density function or a Gaussian probability densityfunction if the power is expressed in decibel (dB) units [1,16,17]. The simplest way to arguefor the case of a Gaussian pdf for shadowing (expressed in dB) is to invoke the central limittheorem for products [5].

Shadowing can be described in terms of multiple scattering and the received signal powercan be expressed as the product of powers. The received power Z can be expressed as

Z =J∏

i=1

Pi (39)

where J is the number of multiple scattering elements and Pi is the fraction of the powerscattered at each instance. Converting Eq. 39 into decibels, we have

10 log10 (Z) = ZdBm =J∑

i=1

10 log10 (Pi ) (40)

If J is sufficiently large, the probability density function of the power on the left hand sideof Eq. 40 will be Gaussian and it can be written as

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f (zdBm) = 1√2πσ 2

dB

exp

[− (zdBm − µ)2

2σ 2dB

](41)

where µ is the average power in dBm and σdB is is the standard deviation of shadowing.Converting back to power units in Watts or milliWatts, the lognormal pdf of shadowingbecomes

fL (z) = A0

σdB y√

2πexp

[−

(10 log10 z − µ

)2

2σ 2dB

]U (z) (42)

where

A0 = 10

loge (10)(43)

Using the moments of the pdf of Eq. 42 given as

⟨Zk

dB

⟩

L= exp

[k

A0µ+ 1

2

(k

A0

)2

σ 2dB

](44)

the amount of fading in a shadowing channel can be obtained as

AFL = exp

(σ 2

dB

A20

)− 1 (45)

Lognormal pdf in Eq. 42 is not the only pdf that has been proposed for modeling the shad-owing seen in wireless systems. Based on the analysis of terrestrial data, it was argued thata simple gamma distribution can also be used model shadowing [16,18]. The probabilitydensity function of the shadowing power Z can be expressed as

fG (z) = zc−1

yc0� (c)

exp

(− z

y0

)U (z) , c > 0. (46)

Since the severity of shadowing is expressed in terms of the standard deviation of shadowingσdB, it is necessary to establish the relationship between (µ, σdB) and (c and y0). This can bedone by comparing the moments of the lognormal pdf in Eq. 42 and moments of the pdf inEq. 46 after conversion into decibel units. These parameters are related as [16,18]

σ 2dB = A2

0ψ′ (c) (47)

µ = A0[loge (y0)+ ψ (c)

](48)

where ψ(.) and ψ ′(.) are the digamma and trigamma functions [19, 8-360].Even though short term fading and shadowing are two different effects, often, the wireless

signal is subject to both at the same time as seen in Fig. 2. We will now look at ways ofmodeling the statistical characteristics of signals in such shadowed fading channels.

5 Shadowed Fading Channels

When evaluating the performance of wireless systems, it is necessary to consider the simul-taneous effect of fading and shadowing on the received signal [1,16]. Such co-existence of

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P. M. Shankar

fading and shadowing also points to higher values of the amount of fading which is a simplemeasure of fluctuations in the channel.

As mentioned earlier, the consequence of shadowing is the loss of deterministic nature ofthe mean power of the short term faded signal. Indeed, the average power becomes random,and, for the case of a Nakagami faded signal, and Eq. 19 needs to be rewritten as

f (p |z ) = mm pm−1

zm� (m)exp

(−m

p

z

)U (p) . (49)

The average power P0 in Eq. 19 has been replaced by a random variable z. The pdf in Eq. 49is conditioned on z. Taking fading and shadowing simultaneously, the pdf of the receivedsignal power can now be expressed as

f (p) =∞∫

0

f (p |z ) f (z) dz (50)

where f(z) is the probability density function of the mean power. If we treat f(z) to be log-normal as discussed earlier, the Nakagami-lognormal pdf for the received power becomes[1,16]

fNLN (p) =∞∫

0

mm pm−1

zm� (m)exp

(−m

p

z

)A0√

2πσ 2dB z2

exp

[−

(10 log10 z − µ

)2

2σ 2dB

]dz. (51)

The Nakagami-lognormal pdf in Eq. 51 is in integral form and no closed solution exists forthe pdf on the left hand side of Eq. 51. Therefore, the evaluation of performance of wirelesssystems in shadowed fading channels (shadowing and fading concurrently present) usingEq. 51 is very cumbersome. Since it was argued that a gamma shadowing is an excellentmatch to the lognormal shadowing seen in wireless systems, Eq. 51 can be rewritten usingthe gamma pdf in Eq. 46 for z as

fNG (p) =∞∫

0

mm pm−1

zm� (m)exp

(−m

p

z

)zc−1

yc0� (c)

exp

(− z

y0

)dz. (52)

The subscript (N G) on the left hand side identifies the pdf as Nakagami-gamma. An analyti-cal solution exists for Eq. 52 and the resulting pdf is known as the generalized K distributiongiven as [21–23]

fGK (p) = 2

� (m) � (c)

(b

2

)c+m

p(c+m

2 )−1 Kc−m(b√

p)

U (p) (53)

where b is a parameter related to the average power and Kc−m(.) is the modified Besselfunction of the second kind of order (c-m).Using the moments of the pdf in Eq. 53 [19]

⟨pk

⟩

G K=

(2

b

)2k� (m + k) � (c + k)

� (m) � (c)(54)

we have

〈P〉G K = 〈P |Z 〉Z = cy0 = mc

(2

b

)2

. (55)

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Note that if m=1, Eq. 51 is the pdf for the Rayleigh-lognormal channel and Eq. 53 becomesthe K distribution (or K -fading) as [20]

fK (p) = 2

� (c)

(b

2

)c+1

p

(c+1

2

)−1

Kc−1(b√

p)

U (p) (56)

where the average power is given by

〈P〉K = c

(2

b

)2

. (57)

Another model for shadowed fading channels makes use of the similarity between the log-normal pdf and the inverse Gaussian pdf [24,25]. The density function of the average poweris considered to be given by the inverse Gaussian pdf,

f I G (z) =√

λ

2π z3 exp

(−λ (y − θ)2

2θ2z

)U (z) (58)

where the two parameters θ and λ can be related to µ and σdB. Using the first and secondmoments of the pdf in Eq. 51, we have

θ = 〈Z〉I G = exp

(µ

A0+ 1

2

σ 2dB

A20

)(59)

θ2(θ

λ+ 1

)= exp

(2µ

A0+ 2σ 2

dB

A20

). (60)

The pdf of the received signal power when the shadowing is treated in terms of an inverseGaussian distribution is obtained by putting Eq. 58 into (50) resulting in [25]

fNIG (p) =4mm

√λ

2π

(λθ2

)m+ 12

� (m)[√

g (p)]m+ 1

2

exp

(λ

θ

)p2m−1 Km+ 1

2

(√g (p)

)U (p) (61)

where

g (p) = 2λ

θ2

(mp2 + λ

2

)(62)

Another model for shadowed fading channels can be created by taking a different look at thegeneralized gamma pdf for short term fading. It was suggested that the generalized gammapdf can also model the received power in shadowed fading channels [4]. Let us go back toEq. 29. The pdf in Eq. 29 was arrived on the basis of a simple scaling of the power so thatthe scaled power v is given by

v = p1s , s > 0 (63)

It is possible to treat the scaling in Eq. 63 as akin to an exponential multiplication, similarto what was described in the section on lognormal fading. Thus, scaling should produce ashadowed fading channel. In other words, Eq. 63 is a case where a short term faded signalis scaled to produce a shadowed fading case. While s could take any positive value in theabsence of shadowing, we will now see that treating Eq. 29 as the case of a shadowed fadingchannel will lead to limits on the value of s. Note that v in Eq. 63 is a dummy variable and

123

P. M. Shankar

we can go back and replace v with p so that the pdf of the signal power under the shadowedfading channel using the generalized gamma model becomes

fGG (p) = spms−1

� (m) Pmg

exp

(− ps

Pg

)U (p) , 0 < s <?. (64)

Equation 64 is identical to Eq. 29 except for the change in the condition on s indicating thatthe upper limit is yet to be determined as shown by the question (?) mark in Eq. 64. To obtainthe relationship between the parameters of the generalized gamma pdf for shadowed fadingchannels and those of the Nakagami-lognormal pdf, we can proceed along the lines of theG K distribution. Comparing the moments of the generalized gamma pdf and Nakagami-log-normal, we get the relationship among the parameters of the Nakagami-lognormal and thegeneralized gamma as

A20ψ

′ (m)+ σ 2dB = A2

0

s2 ψ′ (m) . (65)

Thus, the scaling factor s is related to both the Nakagami parameter m and the standarddeviation of shadowing σdB. Examining Eq. 65, it is clear that if σdB = 0 (no shadowing),the scaling parameter s is equal to unity and the GG pdf becomes the Nakagami pdf. If σdB

goes to ∞ (extreme shadowing), s approaches zero. Thus, the scaling parameter s must bein the range 0 and 1 in shadowed fading channels for the GG pdf. When used to describe theshadowed fading channels, the generalized gamma distribution in Eq. 64 takes the form

fGG (p) = spms−1

� (m) Pmg

exp

(− ps

Pg

)U (p) , 0 < s < 1 (66)

Now that we have a few pdfs for the received signal power in shadowed fading channels, wecan now use the quantitative measure of AF to compare the power fluctuations that wouldbe observed in those channels. Using the moments of the pdf in Eq. 51 for the Nakagami-lognormal shadowing, the amount of fading is

AFNLN =(

m + 1

m

)exp

(σ 2

dB

A20

)− 1 (67)

Using the moments of the G K distribution given in Eq. 54, the amount of fading is

AFGK = 1

m+ 1

c+ 1

mc(68)

Using the moments of the Nakagami-inverse Gaussian distribution in Eq. 61, the amount offading becomes

AFNIG =[

m + 1

m

] [θ + λ

λ

]− 1 =

[m + 1

m

] [2 sinh

(σ 2

dB

2A20

)exp

(σ 2

dB

2A20

)]− 1 (69)

The amount of fading in shadowed fading channels having the generalized gamma pdf forthe power is still given by Eq. 32 with s < 1. The upper limit of 1 of s is also intuitivelyobvious if we examine the plot of AF given in Fig. 3. Since the amount of fading in shadowedfading channels is worse than in channels with fading alone, amount of fading in shadowedfading channels with the generalized gamma pdf will be above the curve for the gammafading alone. This limits the value of s to lie in the range of 0–1.

123


Fig. 4 Cluster based model is sketched. The transmitted signal can reach the signal either by passing throughthe two clusters (solid line) or by passing through a single cluster (dotted line)

Note that while exponential (Rayleigh envelope), gamma (Nakagami envelope), Weibull(which can be obtained from gamma pdf), lognormal densities for the received power are sup-ported by theoretical and experimental observations, the other distributions such the inverseGaussian is based on pure empirical matching. This also suggests that the Nakagami-log-normal and the generalized K distribution can also be justified on the basis of experimentaland theoretical observations. A strong case of G K distribution is based on a very generalmodel which can describe short term fading, long term fading and concurrent instances ofboth (shadowed fading channels). This is discussed in the next section where we go back tothe clustering model used earlier to justify the Nakagami distribution for fading and take ita few steps in a different direction.

6 General Model for Fading, Shadowing, and, Shadowed Fading

Let us go back to the case of a wireless channel which is modeled as consisting of a numberof clusters of scattering (or reflecting, diffracting, etc.) centers. The concept is illustrated inFig. 4 which shows three clusters, with each cluster having a number of scattering/reflect-ing/diffracting centers consisting of buildings, trees, people, vehicles, etc. To arrive at thecase for the Nakagami or Rayleigh pdf, we had assumed that the clusters are separated sothat the signal from each of these clusters arrives at the receiver independently and makesup the total signal at the receiver. Now, let us make the channel a bit more closely packedso that there is a likelihood that the signals from the clusters could only reach the receiverafter multiple scattering among them instead them arriving independently. The transmittedsignal is shown to reach the receiver after passing through two clusters (solid line in Fig. 4).Let the signal power from each of the cluster be Ci , i = 1, 2, …. The received signal powerP can now be expressed as [3,26–28]

P = C0 + α1C1 + α12C1C2 + α123C1C2C3 + · · · (70)

Equation 70 needs some explanation. If there is a possibility that a direct path can existbetween the transmitter and receiver, the power contributed that component is C0. If thereis only a single cluster and hence there is no chance of multiple scattering, the receivedpower will come from the second term, α1C1 (dotted line in Fig. 4). If there are at least twoclusters and the chance of multiple scattering exists, the received signal will come from thethird term, α12C1C2, and so on. Note that α‘s are scaling factors and can be made equal tounity. Equation 70 further assumes that each of those processes (i.e. power from each term)is independent of the other.

123

P. M. Shankar

Let us first look at the second term, α1C1. If there is no multiple scattering, and, thereis only a single cluster, we will now treat the scatterers within that single cluster acting asminiclusters. This case similar to the cluster model used to explain the Nakagami or Rayleighfading in connection with the pdf in Eq. 23. Therefore, the second term in Eq. 70 will leadto pure short term fading with a Nakagami or Rayleigh pdf.

Now consider the third term α12C1C2. There are two clusters, and the received poweris expressed as the product of the powers from the two clusters. If each cluster power canbe described in terms of a gamma pdf (Nakagami pdf for the envelope), the received powerbecomes (treating α12 to be unity)

P = C1.C2 (71)

where C1 and C2 are gamma distributed. We will now look at a few special cases of Eq. 71.If we assume that both clusters result in Rayleigh pdf for the envelopes, pdf of the received

power in Eq. 71 can obtained using transformation techniques for obtaining density functionsof products [5]. This leads to

fDR (p) = 2(u

2

)2K0

(u√

p)

U (p) (72)

where

〈P〉DR =(

2

u

)2

. (73)

This model for shadowed fading channels is often referred to as the double Rayleighmodel and is identified by the subscript (DR) [27].

If we assume that both clusters result in identical Nakagami pdf for the envelopes, prob-ability density function of the received power in Eq. 71 can once again obtained usingtransformation techniques. This leads to [28]

fDN (p) = 2

[� (m)]2

(v2

)2mpm−1 K0

(v√

p)

U (p) (74)

where

〈P〉DN =(

2m

v

)2

. (75)

The pdf in Eq. 75 is called the double Nakagami (DN) or double gamma distribution. Wenow consider the most general case where C1 and C2 are Nakagami pdfs for envelopes withnon-identical values of the Nakagami m-parameters, say m1 and m2. We can use Eq. 53 ifwe replace m by m1 and c by m2. We get,

fNd N (p) = 2

� (m1) � (m2)

(w2

)(m1+m2)

p

(m1+m2

2

)−1

Km1−m2

(w

√p)

U (p) (76)

where

〈P〉Nd N = m1m2

(2

w

)2

. (77)

In Eq. 77, the pdf has a subscript (NdN) indicating that the two Nakagami pdfs have identicalpowers, but different levels of fading through m1 and m2. It can be seen that that Eq. 76 isidentical to the G K pdf in Eq. 53. The K distribution of Eq. 56 is a special case of Eq. 76when C1 comes from an exponential pdf and C2 comes from a gamma pdf.

123


Note that shadowing occurs only in conjunction with short term fading and therefore, thereis no need to look for modeling shadowing as a stand alone process. It is possible to createand analyze more complex shadowed fading channels by considering the received power asrising out of the product of three or more clusters. We can assume that the condition underwhich only a direct path exists are not realistic and C0 can be put equal to zero. This raisesthe issue of the case of Rician fading channels. To understand how Rician conditions canbe included in this model, we can go back to the case of a single cluster which gave riseto a Rayleigh or Nakagami fading channel where we assumed that the single cluster can beconsidered to be made up of several miniclusters. Once that premise is accepted, a singlecluster can given rise to Rayleigh, Nakagami or Rician when one of the ‘miniclusters’ istreated as contributing the direct path. Another way to visualize pure Rayleigh or Nakagamifading is to reclassify the third term C1C2 in Eq. 71 by arguing that C2 is a deterministicscalar quantity and C1 corresponds to the case of an exponential or gamma pdf. Thus, evenmultiple scattering can result in Rayleigh or Nakagami channels if all but one of the multiplescattering components is deterministic and thus, they will only provide a scaling factor. Thisnotion can now be extended to the other terms in Eq. 70 so that regardless of the number ofproduct terms, we can always get the case of Rayleigh, Nakagami, double Rayleigh, doubleNakagami or K or generalized K channels.

The representation of the channel in terms of Eq. 71 permits us to create any number of dif-ferent short term fading or shadowed fading channels such as those based on the generalizedgamma pdf, Weibull pdf and each of these separately give rise to other pdfs described asgamma-Weibull or Weibull-Weibull channels for modeling shadowed fading channels. Therepresentation of fading in this manner also makes it unnecessary even to consider the log-normal shadowing since the random variations in the channel can be described in termsof a single cluster (Rayleigh or Nakagami fading) or two clusters (double Rayleigh, doubleNakagami, or generalized K fading). Since the evidence of Rayleigh and Nakagami fading inwireless channels is overwhelming, use of double Rayleigh, double Nakagami, generalizedK fading, is well justified to characterize the statistical fluctuations observed in wirelesschannels.

7 Concluding Remarks

A broad overview of the probability density functions occurring in the study of wireless chan-nels was presented. The list of the probability density functions for fading is summarizedin Table 1. It also contains the expressions for the amount of fading (AF). Table 2 lists theexpressions for pdfs in shadowed fading channels and the amount of fading (AF).

As mentioned earlier, the amount of fading provides a quantitative measure of fadingpresent in wireless channels. Note that the definition of the amount of fading depends onlyon two moments, namely the first and the second moments of the power. Thus, it is possibleto get the same value of AF from several other probability density functions as well. In otherwords, the quantification of the level of fading in Eq. 7 does not fully validate the measuredstatistics of the fading channel, i.e., the density function of the received power, matches thepdf employed to model the channel. Another way to quantify the characteristics of the fadingchannels is through the use of outage probabilities. Whenever the signal power goes below athreshold which depends on the data rate, coding, modulation, demodulation, etc., the chan-nel goes into outage. The outage probability associated with fading, shadowing or shadowedfading can be expressed as

123

P. M. Shankar

Table 1 The probability density functions of the received power in fading channels and the amount of fading

Probability density function(p ≥ 0)

Amount of fading (AF) Additionalinformation

fR (p) = 1P0

exp(− P

P0

)1

fRi (p)= K0+1PRi

exp[−K0− (K0+1) p

PRi

]1+2K0(1+K0)

2

PRi = 2σ 2 + a20

K0 = a20

2σ2

×I0

(2√

K0(K0+1)PRi

p)

fN (p) = mm pm−1

Pm0 �(m)

exp(−m p

P0

)1m m ≥ 1

2

fG (p) = pm−1

Pmg �(m)

exp(− p

Pg

)1m m > 0; 〈P〉G = m Pg

fGG (p) = spms−1

�(m)Pmg

exp(− ps

Pg

) �(

m+ 2s

)�(m)

[�

(m+ 1

s

)]2 − 1

m > 0; 0 < s < ∞

〈P〉GG = �(

m+ 1s

)

�(m) P

(1s

)

g

fW (p) = mwPg

pmw−1 exp(− pmw

Pg

) �(

1+ 2mw

)

[�

(1+ 1

mw

)]2 − 1mw > 0

〈P〉W = �(

1 + 1mw

)P

(1

mw

)

g

Special conditions on the parameters are also provided. Note that all the pdfs exist only for p ≥ 0. Thesubscripts with the pdfs and the moments indicate the names associated with the pdfs: Rayleigh (R), Rician(Ri), Nakagami (N), gamma (G), generalized gamma (GG), Weibull (W)

Pout =PT∫

0

f (p)dp = F (PT ) (78)

where PT is the threshold power. The density function of the power is f(p) and F(.) isthe cumulative distribution function of the power evaluated at p = PT . Table 3 lists theexpressions for the outage probabilities for the different density functions which describethe fading and shadowed fading channels. It is seen that neither the Nakagami-lognormalnor the Nakagami inverse Gaussian distribution leads to a closed form expression for theoutage probability while all the other density functions possess an analytical expressions forthe outage probabilities. The outage probability in a Nakagami inverse Gaussian channelcan be expressed as an infinite sum [25]. For the case of the pdfs containing the modifiedBessel function K (), the outage probabilities are in terms of the modified Bessel, MeijerG,and, hypergeometric functions. All these functions are available in most of the computationalsoftware packages such as Matlab (Mathworks, Natick, MA, USA), Maple (Maplesoft, 615Kumpf Drive, Waterloo, ON, Canada) and Mathematica (Wolfram Research Inc, 100 TradeCenter Drive, Champaign, IL, USA). The outage probabilities of associated with all the fad-ing distributions described in the section on the general model are plotted in Figs. 5 and 6for two sets of values of the parameters. In Fig. 5, it is seen that the double Rayleigh has thehighest outage probabilities among the group. The Rayleigh, Nakagami (m = 0.95), doubleNakagami (m = 1.5), K distribution (c = 7.2) are all very close together while for m = 1.5and c = 7.2, the G K distribution results in the lowest outage probabilities. For another set ofparameters, Fig. 6 shows that the outage probabilities are the lowest for the case of Rayleighfading.

The pdfs arising out of the cluster model to permits us describe all types of statistical fluc-tuations present in the channel and plots of the outage probabilities demonstrate the range

123


Tabl

e2

The

prob

abili

tyde

nsity

func

tions

(pdf

)of

the

rece

ived

pow

erin

shad

owed

fadi

ngch

anne

lsan

dth

eam

ount

offa

ding

(AF

)ar

eta

bula

ted

Prob

abili

tyde

nsity

func

tion

(p≥

0)A

mou

ntof

fadi

ng(A

F)

Add

ition

alin

form

atio

n

f NL

N(p )

=∫ ∞ 0

mm

pm−1

zm�(m)

exp

( −m

p z

)A

0√

2πσ

2 dBz2

exp

[ −(1

0lo

g 10

z−µ)2

2σ2 dB

] dz

( m+1 m

)ex

p

(σ

2 dB A2 0

)−

1A

0=

10lo

g e(1

0 )

f GK(p )

=2

�(m)�(c)

( b 2

) c+m

p(c+

m 2

) −1K

c−m

( b√p) U

(p )

1 m+

1 c+

1 mc

σ2 dB

=A

2 0ψ

′ (c )

;〈P

〉 GK

=m

c(

2 b

) 2

f K(p )

=2�(c)

( b 2

) c+1

p(c+

12

) −1K

c−1( b√

p) U(p )

1+

2 cG

Kpd

fw

ithm

=1;σ

2 dB=

A2 0ψ

′ (c )

;〈P

〉 K=

c(

2 b

) 2

f DR(p )

=2

( u 2) 2

K0( u

√p) U

(p )

3G

Kpd

fw

ithm

=c

=1;

〈 P〉 D

R=

(2 u

) 2

f DN(p )

=2

[�(m)]

2

( v 2) 2

mpm

−1K

0( v

√p) U

(p )

2 m+

1 m2

GK

pdf

with

c=

m;〈

P〉 D

N=

( 2m v

) 2

f Nd

N(p )

=2

�(m

1)�(m

2)

( χ 2) (

m1+m

2)

p(m

1+m

22

) −1K

m1−m

2

( χ√

p) U(p )

1 m1

+1 m2

+1

m1

m2

GK

pdf

with

m=

m1;c

=m

2;〈

P〉 N

dN

=m

1m

2

(2 χ

) 2

f GG(p )

=sp

ms−

1

�(m)P

m gex

p( −

ps

Pg

)�

( m+

2 s

) �(m)

[ �( m

+1 s

)]2

−1

0<

s<

1;A

2 0ψ

′ (m)+σ

2 dB=

A2 0

s2ψ

′ (m)

f NIG(p )

=(

1H

P 0

) m+

1 2√

P 02π

H

4mm

exp(

1 H

)

�(m)[ √

g (p )

] m+

1 2p2m

−1K

m+

1 2

( √g(p )

)[ m

+1 m

]H

−1

g(p )

=2

HP 0

( mp2

+P 0 2H

)H

=2

sinh

(σ

2 dB2

A2 0

)ex

p

(σ

2 dB2

A2 0

)

Nak

agam

i-lo

gnor

mal

dist

ribu

tion

(NL

N),

gene

raliz

edK

dist

ribu

tion

(GK

),K

dist

ribu

tion

(K),

doub

leR

ayle

igh

dist

ribu

tion

(DR

),do

uble

Nak

agam

idis

trib

utio

n(D

N),

non-

iden

tical

doub

leN

akag

amid

istr

ibut

ion

(NdN

),ge

nera

lized

gam

ma

dist

ribu

tion

(GG

)an

dN

akag

amii

nver

seG

auss

ian

dist

ribu

tion

(NIG

)

123

P. M. Shankar

Tabl

e3

The

outa

gepr

obab

ilitie

sav

aila

ble

inan

alyt

ical

form

sar

eta

bula

ted

infa

ding

and

shad

owed

fadi

ngch

anne

ls

Prob

abili

tyde

nsity

func

tion

f(p),

p≥

0O

utag

epr

obab

ility

[Eq.

78]

Add

ition

alin

form

atio

n

Fadi

ngch

anne

lsR

[Eq.

6]1

−ex

p(−

PT/

P 0)

N[E

q.19

]γ

( m,

mP

TP 0

)[�(m)]

−1;m

≥1 2

γ(.

,.)is

the

inco

mpl

ete

gam

ma

func

tion

[19,

8-35

0]

G[E

q.25

]γ

( m,

PT Pg

)[�(m)]

−1;m

>0

GG

[Eq.

29]

γ

( m,

Ps T Pg

)[�(m)]

−1

W[E

q.36

]1

−ex

p( −P

mw

T/

Pg

),mw>

0

Shad

owed

fadi

ngch

anne

ls

GK

[Eq.

53]

�(m

−c)

�(m)�(c

+1)

1F

2

( c,[1

−m

+c,

1+

c ],

PT

b2

4

)(

PT

b2

4

) c

+�(c

−m)

�(m

+1)�(c)

1F

2

( m,[1

−c

+m,1

+m

],P

Tb2

4

)(

PT

b2

4

) m1

F2(.,[.,. ],. )

isth

ehy

perg

eom

etri

cfu

nctio

n[1

9,6-

592,

9-13

7]

GG

[Eq.

66]

γ

( m,

Ps T Pg

)[�(m)]

−1

DR

[Eq.

72]

[ 1−

2u√

PT 2

K1( u

√P

T)]

K1()

isth

em

odifi

edB

esse

lfun

ctio

nof

the

seco

ndki

nd

DN

[Eq.

74]

[�(m)]

−2M

eije

rG( [[1

],[]],

[[m,m

],[0]],

1 4P

Tv

2)

Mei

jerG

func

tion

[19,

9-30

1]

K[E

q.56

]1

−2�(c)

( b√P

T 2

) cK

c( b√

PT)

Ray

leig

h(R

);N

akag

ami

(N);

gene

raliz

edK

dist

ribu

tion

(GK

),K

dist

ribu

tion

(K),

doub

leR

ayle

igh

dist

ribu

tion

(DR

),do

uble

Nak

agam

idi

stri

butio

n(D

N),

gene

raliz

edga

mm

adi

stri

butio

n(G

G),

Wei

bull

(W)

123


Fig. 5 The outage probabilities for a set of parameters are plotted for the pdfs arising out of the cluster model.The axis is the normalized threshold PT /<P> in dB. The parameter values are indicated

Fig. 6 The outage probabilities for a set of parameters are plotted for the pdfs arising out of the cluster model.The axis is the normalized threshold PT /< P > in dB. The parameter values are indicated

and versatility of the cluster model. The results and descriptions provided here should assistthe instructors and students to easily comprehend the statistical characteristics of the wirelesschannel.

123

P. M. Shankar

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Author Biography

P. M. Shankar received his M.Sc. (1972) in Physics from KeralaUniversity, India, M.Tech. (1975) in Applied Optics and Ph.D. inElectrical Engineering (1980) from Indian Institute of Technology,Delhi, India. He was a visiting scholar at the School of ElectricalEngineering, University of Sydney, Australia, from 1981 to 1982. Hejoined Drexel University in 1982 and is currently the Allen Roth-warf Professor of Electrical and Computer Engineering. He is theauthor of the textbook Introduction to Wireless Systems, published byJohn Wiley & Sons, 2002. His research interests are in Statistical sig-nal processing for medical applications, Modeling of fading channels,Wireless communications, Medical ultrasound, and Optical fibersensors.

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