Weibull fading

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 eta>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/eta>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/eta} 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| as: Z = R^{2 / eta } .

tatistical model

The probability density function (pdf) of Z can be easily obtained as: f_Z( r ) = frac{eta }{Omega } , r^{eta - 1} , exp left( { - fracr^{eta} {Omega ight) with Omega = E { Z^eta } and E {} denoting the expected value. The above pdf follows the Weibull distribution with the fading parameter eta expressing the fading severity and Omega being the average fading power. As eta increases, the fading severity decreases, while for the special case of eta = 2, f_Z( r ) reduces to the well-known Rayleigh distribution. Moreover, for the special case of eta = 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^eta }{ Omega } ight) and: E left{ Z^n ight} = Omega^{n / eta} , Gamma left(1 + frac neta ight) 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) ight} is given by (Sagias 2005): M_Z ( s ) = frac1{ Omega, s^eta} , frac{ lambda^{eta} , sqrt{kappa /lambda{ ( sqrt{2 pi} )^{kappa+lambda-2 , G^{kappa ; lambda}_{lambda ; kappa} left [ left.frac{ lambda^lambda }{left( kappa , Omega , s^{eta} ight)^kappa} ight| ^{ (1- eta )/ lambda, , (2 - eta) / lambda, ldots, (lambda- eta ) / lambda }_{ 0/kappa, , 1 / kappa, ldots, (kappa-1) / kappa} ight] where "G" [] is the [http://mathworld.wolfram.com/MeijerG-Function.html 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 eta belongs to rationals, kappa and lambda are positive integers so that: frac{lambda}{kappa} = eta holds. Depending upon the specific value of eta, a set of minimum values of kappa and lambda can be properly chosen (e.g. for eta=3.5, kappa=2 and lambda =7). Moreover, for the special case where eta is an integer, kappa=1 and lambda=eta .

econd 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 ight) { m d}dot{r}where f_{dot Z, Z} left( cdot , cdot ight) 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: au(r)= frac{F_Z (r)}{N(r)} .

Average level crossing rate

The average LCR for the Weibull channel is given by: N ( ho ) = sqrt {2pi} , f_d ,left(frac{ ho}{sqrt{a ight)^{ { eta /2 expleft [- left(frac{ ho}{sqrt{a ight)^eta ight] where f_d is the maximum Doppler shift, ho = frac Z { Z_{ m rms, with Z_{ m rms} = sqrt{ Eleft{ Z^2 ight = frac{ Omega^{1/eta{ sqrt{a and a = frac1 { Gamma left(1+2/eta ight)} .

Average fade duration

The expression for the average fade duration is : au left( ho ight) = frac1 - exp left [-left({ ho/sqrt{a ight)^eta ight] {sqrt {2pi} ,f_d ,left({ ho/sqrt{a ight)^{ { eta /2 expleft [- left({ ho/sqrt{a ight)^eta ight] } .

The maximum value of the average LCR can be derived solving the equation which is obtained by differentiating N ( ho ) with respect to ho , setting the result equal to zero, i.e.,: left. frac{dN( ho)}{ d ho} ight|_{ ho= ho_{max = 0 and then replacing ho_{max} into N ( ho ) . It can be easily shown that the average LCR is maximized at : ho_{max} = 2^{-1/eta} , sqrt{a} as :N left( ho_{max} ight) = f_d , sqrt{frac pi e}. Note that N left( ho_{max} ight) is independent of eta 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+ frac2eta ight) , Omega^{2 /eta}. The average channel capacity, in Shannon's sense, is defined as :overline{C} = B_w , E left{log_2(1+gamma) ight} =B_w , int_0^infty log_2(1+gamma) , f_gamma(gamma) , { m d}gamma .

By following the above definition, the average channel capacity is given by: overline{C} = B_w , frac{eta left( a , overline{gamma} ight)^{-{eta/2}{2 , ln(2)} frac{sqrt{k} ,l^{-1{left( sqrt{2 pi} ight)^k+2l-3} , G^{k+2l,l}_{ 2l,k+2l} left [ left. frac{left(a , overline{gamma} ight)^{- {eta k/2}{k^k} ight
^{ mathrm{I} left(l ,-{eta/2} ight); , ; mathrm{I} left(l , 1-{eta/2} ight)}_{ mathrm{I}(k,0) ; , ; mathrm{I} left(l , -{eta/2} ight); , ; mathrm{I} left(l , -{eta/2} ight)} ight] where : mathrm{I}(n,xi) = frac xi n, , frac {(xi+1)}n, ldots, , frac {(xi+n-1)}nwith xi an arbitrary real value and "n" positive integer. Moreover, : frac l k = frac eta2 where "k" and "l" are positive integers. Depending upon the value of eta, 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 m var} (gamma )} {overline{gamma}^2} = frac{E{gamma^2{overline{gamma}^2} - 1is 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 /eta)}{Gamma^2( 1+2/eta)} -1 .

References

*Adawi, N.S. "et al". (1988). doi-inline|10.1109/25.42678|Coverage prediction for mobile radio systems operating in the 800/900 MHz frequency range, "IEEE Transactions on Vehicular Technology" 37, (1), 3–72
*Hashemi, H. (1993). doi-inline|10.1109/5.231342|The indoor radio propagation channel, "Proceedings IEEE" 81 (7) 943–968
*Yacoub, M.D.; (2002). [http://ieeexplore.ieee.org/search/wrapper.jsp?arnumber=1047298 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). doi-inline|10.1109/TIT.2005.855598|Gaussian class multivariate Weibull distributions: Theory and applications in fading channels, "IEEE Transactions on Information Theory" 51 (10), 3608-3619
*Sagias, N.C.; Zogas, D.A.; Karagiannidis, G.K.; & Tombras, G.S; (2004). doi-inline|10.1109/LCOMM.2004.831319|Channel capacity and second order statistics in Weibull fading "IEEE Communications Letters", 8 (6) 377-379

External links

* [http://mathworld.wolfram.com/MeijerG-Function.html The Meijer G-function (definitions, examples, and properties).]


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