Weibull fading

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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.

1 Introduction
2 Statistical model
3 References
4 External links

[edit] 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 $β > 0$ , 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 $2 / β > 0$ (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 / β .

[edit] 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 $Ω = E {Z β}$ and $E {}$ denoting the expected value. The above pdf follows the Weibull distribution with the fading parameter $β$ expressing the fading severity and $Ω$ being the average fading power. As $β$ increases, the fading severity decreases, while for the special case of $β = 2$ , $f Z (r)$ reduces to the well-known Rayleigh distribution. Moreover, for the special case of $β = 1$ , $f Z (r)$ 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

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

respectively, where $Γ()$ 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 $β$ belongs to rationals, $κ$ and $λ$ are positive integers so that

$\frac{\lambda}{\kappa} = \beta$

holds. Depending upon the specific value of $β$ , a set of minimum values of $κ$ and $λ$ can be properly chosen (e.g. for $β = 3.5$ , $κ = 2$ and $λ = 7$ ). Moreover, for the special case where $β$ is an integer, $κ = 1$ and $λ = β$ .

[edit] 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

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

[edit] Average level crossing rate

The average LCR for the Weibull channel is given by

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

where $f d$ is the maximum Doppler shift, $\rho = \frac Z { Z_{\rm rms}}$ , with $Z_{\rm rms} = \sqrt{ E\left\{ Z^2 \right\}} = \frac{ \Omega^{1/\beta}}{ \sqrt{a}}$ and $a = \frac1 { \Gamma \left(1+2/\beta \right)}$ .

[edit] Average fade duration

The expression for the average fade duration is

$\tau \left( \rho \right) = \frac{{1 - \exp \left[- \left({\rho/\sqrt{a}}\right)^\beta \right]}}{\sqrt {2\pi} \, f_d \,\left({\rho/\sqrt{a}}\right)^{ { \beta /2}} \exp \left[- \left({\rho/\sqrt{a}} \right)^\beta\right]} .$

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

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

and then replacing $ρ m a x$ into $N (ρ)$ . 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 $β$ and $Ω$ .

[edit] 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

$\overline{\gamma} = \frac{ E_s}{ N_0} \, \Gamma \left(1+ \frac2\beta \right) \, \Omega^{2 / \beta}$ .

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

$\overline{C} = B_w \, \frac{\beta \left( a \, \overline{\gamma} \right)^{-{\beta/2}}}{2 \, \ln(2)} \frac{\sqrt{k} \, l^{-1}}{\left( \sqrt{2 \pi} \right)^{{k+2l-3}}} \, G^{k+2l,l}_{ 2l,k+2l} \left[ \left. \frac{\left( a \, \overline{\gamma} \right)^{- {\beta k/2}}}{k^k} \right| ^{ \mathrm{I} \left(l ,-{\beta/2} \right)\; , \; \mathrm{I} \left(l , 1-{\beta/2} \right)}_{ \mathrm{I}(k,0) \; , \; \mathrm{I} \left(l , -{\beta/2} \right)\; , \; \mathrm{I} \left(l , -{\beta/2} \right)} \right]$

where

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

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

[edit] 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

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

[edit] References

Adawi, N.S. et al. (1988). "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). "The indoor radio propagation channel". Proceedings IEEE 81 (7) 943–968
Yacoub, M.D.; (2002). "The $α$ - $μ$ distribution: A general fading distribution". Proc. IEEE International Symposium on Personal, Indoor, Mobile Radio Communications Lisbon, Portugal
Sagias, N.C.; & Karagiannidis G.K; (2005). "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). "Channel capacity and second order statistics in Weibull fading". IEEE Communications Letters, 8 (6) 377-379