Weibull fading

The Weibull fading can be used as a simple statistical model of fading (named after Waloddi Weibull). In wireless communications, the Weibull fading distribution seems to exhibit good fit to experimental fading channel measurements for both indoor (Adawi 1988) and outdoor (Hashemi 1993) environments.

Introduction

The fading model for the Weibull distribution considers a signal composed of clusters of one multipath wave, each propagating in a nonhomogeneous environment (Yacoub 2002).

Within any one cluster, the phases of the scattered waves are random and have similar delay times with delaytime spreads of different clusters being relatively large.

The clusters of the multipath wave are assumed to have the scattered waves with identical powers. The resulting envelope is obtained as a nonlinear function of the modulus of the multipath component $h$ .

The nonlinearity is manifested in terms of a power parameter $beta>0$ , such that the resulting signal intensity is obtained not simply as the modulus of the multipath component, but as this modulus to a certain given power $2/beta>0$ (Sagias 2004).

Hence, for the Weibull fading channel model, the complex envelope $h$ can be written as a function of the Gaussian in-phase $X$ and quadrature $Y$ elements of the multipath components

h = (X + j , Y )^{2/beta}

where j = √ -1 is the imaginary operator.

Let $Z$ be the fading envelope of $h$ , i.e., $Z = |h|$ . Then, $Z$ can be expressed as a power transformation of a Rayleigh distributed random variable (rv)

R = |X + j , Y|

Z = R^{2 / beta } .

Statistical model

The probability density function (pdf) of

Z

can be easily obtained as

f_Z(r ) = frac{beta }{Omega } , r^{beta - 1} , exp left({ - frac{{r^{beta} }}{Omega }} right)

with

Omega = E { Z^beta }

and

E {}

denoting the expected value. The above pdf follows the Weibull distribution with the fading parameter

beta

expressing the fading severity and

Omega

being the average fading power. As

beta

increases, the fading severity decreases, while for the special case of

beta = 2

f_Z(r )

reduces to the well-known Rayleigh distribution. Moreover, for the special case of

beta = 1

f_Z(r )

reduces to the well-known negative exponential distribution.

The corresponding cumulative distribution function and the n-th order moment of rv $Z$ can be expressed as

F_Z(r ) = 1 - exp left(- frac{ r^beta }{ Omega } right)

and

E left{ Z^n right} = Omega^{n / beta} , Gamma left(1 + frac nbeta right)

respectively, where

Gamma

is the Gamma function and n is a positive integer.

The moment generating function of fading envelope Z

M_Z (s ) = E left{ exp (-s , Z) right}

is given by (Sagias 2005)

M_Z (s ) = frac1{ Omega, s^beta} , frac{ lambda^{beta} , sqrt{kappa /lambda}}{ (sqrt{2 pi} )^{kappa+lambda-2}} , G^{kappa ; lambda}_{lambda ; kappa} left[left.

frac{ lambda^lambda }{left(kappa , Omega , s^{beta} right)^kappa} right| ^{ (1- beta ) / lambda, , (2 - beta) / lambda, ldots, (lambda- beta ) / lambda }_{ 0/kappa, , 1 / kappa, ldots, (kappa-1) / kappa} right] where G[] is the Meijer's G-function Note that the Meijer's G-function is included as a built-in function in most popular mathematical software packages. Additionally, G[] can be expressed in terms of more familiar generalized hypergeometric functions. By assuming that

beta

belongs to rationals,

kappa

and

lambda

are positive integers so that

frac{lambda}{kappa} = beta

holds. Depending upon the specific value of

beta

, a set of minimum values of

kappa

and

lambda

can be properly chosen (e.g. for

beta=3.5

kappa=2

and

lambda =7

). Moreover, for the special case where

beta

is an integer,

kappa=1

and

lambda=beta

Second order statistics

The average level crossing rate (LCR) is defined as the average number of times per unit duration that the envelope of a fading channel crosses a given value in the negative direction and it can be evaluated as

N(r) = int_0^infty dot{r} , f_{dot Z, Z}left(dot{r} , r right) { rm d}dot{r} where

f_{dot Z, Z} left(cdot , cdot right)

is the joint pdf of Z and its time derivative

dot Z

The AFD corresponds to the average length of time the envelope remains under a certain value once it crosses it in the negative direction and can be obtained as

tau(r)= frac{F_Z (r)}{N(r)} .

Average level crossing rate

The average LCR for the Weibull channel is given by

N (rho ) = sqrt {2pi} , f_d ,left(frac{rho}{sqrt{a}}right)^{ { beta /2}} exp

left[- left(frac{rho}{sqrt{a}} right)^betaright] where

f_d

is the maximum Doppler shift,

rho = frac Z { Z_{rm rms}}

, with

Z_{rm rms} = sqrt{ Eleft{ Z^2 right}} = frac{ Omega^{1/beta}}{ sqrt{a}}

and

a = frac1 { Gamma left(1+2/beta right)}

Average fade duration

The expression for the average fade duration is

tau left(rho right) = fracright)^beta right]}}{sqrt {2pi} ,
f_d ,left({rho/sqrt{a}}right)^{ { beta /2}} exp
left[- left({rho/sqrt{a}} right)^betaright]} .

The maximum value of the average LCR can be derived solving the equation which is obtained by differentiating $N (rho )$ with respect to $rho$ , setting the result equal to zero, i.e.,

left. frac{dN(rho)}{ drho} right|_{rho=rho_{max}} = 0

and then replacing

rho_{max}

into

N (rho )

. It can be easily shown that the average LCR is maximized at

rho_{max} = 2^{-1/beta} , sqrt{a}

N left(rho_{max} right) = f_d , sqrt{frac pi e}

Note that

N left(rho_{max} right)

is independent of

beta

and

Omega

Average channel capacity

We consider a signal transmission of bandwidth

B_w

and symbols energy

E_s

. The instantaneous signal-to-noise ratio (SNR) per symbol is given by

gamma = Z^2 , frac{E_s} { N_0}

with

N_0

the double-sided noise power spectral density of the additive white Gaussian noise (AWGN), while the corresponding average value can be written as

overline{gamma} = frac{ E_s}{ N_0} , Gamma left(1+ frac2beta right) , Omega^{2 / beta} . The average channel capacity, in Shannon's sense, is defined as

overline{C} = B_w , E left{log_2(1+gamma) right} =B_w , int_0^infty log_2(1+gamma) , f_gamma(gamma) , {rm d}gamma .

By following the above definition, the average channel capacity is given by

overline{C} = B_w , frac{beta left(a , overline{gamma} right)^{-{beta/2}}}{2 , ln(2)} frac{sqrt{k} , l^{-1}}{left(sqrt{2 pi} right)^} , G^{k+2l,l}_{ 2l,k+2l} left[left. frac{left( a , overline{gamma} right)^{- {beta k/2}}}{k^k} right| ^{ mathrm{I} left(l ,-{beta/2} right); , ; mathrm{I} left(l , 1-{beta/2} right)}_{ mathrm{I}(k,0) ; , ; mathrm{I} left(l , -{beta/2} right); , ; mathrm{I} left(l , -{beta/2} right)} right] where

mathrm{I}(n,xi) = frac xi n, , frac {(xi+1)}n, ldots, , frac {(xi+n-1)}n with

xi

an arbitrary real value and n positive integer. Moreover,

frac l k = frac beta2 where k and l are positive integers. Depending upon the value of

beta

, a set with minimum values of k and l can be properly chosen.

Amount of fading

The amount of fading (AoF), defined as

A_F = frac{{rm var} (gamma )} {overline{gamma}^2} = frac{E{gamma^2}}{overline{gamma}^2} - 1 is a unified measure of the severity of fading (var() denoted variance). Typically, this performance criterion is independent of the average fading power. As the AoF increases, the severity of fading also increases.

The AoF for the Weibull fading channel can be expressed as

A_F = frac{Gamma(1 + 4 /beta)}{Gamma^2(1+2/beta)} -1 .

References

Adawi, N.S. et al. (1988). , IEEE Transactions on Vehicular Technology 37, (1), 3–72
Hashemi, H. (1993). , Proceedings IEEE 81 (7) 943–968
Yacoub, M.D.; (2002). The $alpha$ - $mu$ distribution: A general fading distribution, Proc. IEEE International Symposium on Personal, Indoor, Mobile Radio Communications Lisbon, Portugal
Sagias, N.C.; & Karagiannidis G.K; (2005). , IEEE Transactions on Information Theory 51 (10), 3608-3619
Sagias, N.C.; Zogas, D.A.; Karagiannidis, G.K.; & Tombras, G.S; (2004). IEEE Communications Letters, 8 (6) 377-379