GENERAL FADING DISTRIBUTIONS

This paper presents two general fading distributionsthe K-!L Distribution and the T)-!L Distribution. The K-!L Distribution includes the Rice and the Nakagarnim distnbutions as special cases. The T)-!L Distribution includes the Hoyt and the Nakagarni-m distributions as special cases. Therefore, in both fading distributions, the One-Sided Gaussian and the Rayleigh distributions also constitute special cases and the Lognormal distribution may be well-approximated. Preliminary results show that these new distributions provide a very good fitting to experimental data.


INTRODUCTION
The propagation of energy in a mobile radio environment is characterized by incident waves interacting with surface irregularities via diffraction, scattering, reflection, and absorption.The interaction of the wave with the physical structures generates a continuous distribution of partial waves [1], with these waves showing amplitudes and phases varying accoFding to the physical properties of the surface.The propagated signal then reaches the receiver through multiple paths.If the waves are not resolvable within the available bandwidth or if an appropriate signal treatment is not carried out, the result is a combined signal that fades rapidly, characterizing the short term fading.For surfaces assumed to be of the Gaussian random rough type, universal statistical laws can be derived in a parameterized form [1].
A great number of distributions exist that well describe the statistics of the mobile radio signal.Extensive field trials have been used to validate these distributions and the results show a very good agreement between measurements and theoretical formulas.The long term signal variation is well characterized by the Lognormal distribution whereas the short term signal variation is described by several other distributions such as Rayleigh, Rice, Nakagarni-m, and Weibull, though to the latter, originally derived for reliability study purposes, little attention has been paid.It is generally accepted that the path strength at any delay is characterized by the short term distributions over a spatial dimension of a few hundred wavelengths, and by the Lognormal distribution over areas whose dimension is much larger [2].Three other distributions attempt to describe the transition from the local distribution to the global distribution of the path strength, thus combining both fast and slow fading.These composite (or mixed) distributions assume the local mean, which is the mean of the fast fading distribution, to be lognormally distributed.The best-known composite distributions are Rayleighlognormal, also known as Suzuki, Rice-lognormal, and Nakagarni-m-lognormal.
In fact, the Rayleigh distribution constitutes a special case of the Rice, Nakagarni-m, Weibull, and of the composite distributions and can be obtained in an exact manner by appropriately setting the parameters of these distributions.Nakagarni-m and Rice are found to approximate each other by some simple equations relating the physical parameters associated to each distribution.
Among these, the Nakagarni-m distribution has been given a special attention for its ease of manipulation and wide range of applicability [3].Although, in general, it has been found that the fading statistics of the mobile radio channel may well be characterized by the Nakagarni-m, situations are easily found for which other distributions such as Rice and even Wei bull yield better results [ 4,5].More importantly, situations are encountered for which no distributions seem to adequately fit experimental data, though one or another may yield a moderate fitting.Some researches [5] even question the use of the Nakagarni-m distribution because its tail does not seem to yield a good fitting to experimental data, better fitting being found around the mean or median.
The well-known fading distributions have been derived assuming a homogeneous diffuse scattering field, resulting from randomly distributed point scatterers.With such an assumption, the central limit theorem leads to complex Gaussian processes with in-phase and quadrature Gaussian distributed variables x and y having zero means and equal standard deviations.In case one cluster of multipath wave is considered then the Rayleigh distribution can be obtained.If a specular component predominates over the scattered waves, then the Rice distribution is accomplished.The Nakagami signal can be understood as composed of clusters of multipath waves so that within any one cluster the phases of scattered waves are random and have similar delay times with delay-time spreads of different clusters being relatively large.The assumption of a homogeneous diffuse scattering field is certainly an approximation because the surfaces are spatially con-elated characterizing a non-homogeneous environment [1].This paper presents two general fading distributions -the 1<-!1-Distribution and the 1)-!1-Distribution.The 7<:-!1-Distribution includes the Rice and the Nakagami-m • distributions as special cases.The 1)-!1-Distribution includes the Hoyt and the Nakagami-m distributions as special cases.Therefore, in both fading distributions, the One-Sided Gaussian and the Rayleigh distributions also constitute special cases and the Lognormal distribution may be wellapproximated.Prelintinary results show that these new distributions provide a very good fitting to experimental data.

PHYSICAL MODEL: K-J.I. DISTRIBUTION
The fading model for the I<-f.i.Distribution considers a signal composed of clusters of multipath waves propagating in an non-homogeneous environment.Within any one cluster, the phases of the scattered waves are random and have similar delay times with delay-time spreads of different clusters being relatively large.The clusters of multipath waves are assumed to have the scattered waves with identical powers but within each cluster a dominant component is found that presents an arbitrary power.

DERIVATION OF THE IC-J.I. DISTRIBUTION
Given the physical model for the K:-!1-Distribution the envelope, the envelope r can be written in terms of the inphase and quadrature components of the fading signal as ?" ( ,.- where xi and Yi are mutually independent Gaussian processes with E(xi)=E(yi)=O Therefore Var~• 2 )= 4no.4 + 4a 2 L;'= 1 c{ .
We define """ 2 .:: "' ::c;-::!1-,:c,' --• (4) that In the Note that x: is the ratio between the total power of the dominant components and the total power of the scattered waves.Then (1 +x: )2 n X -7(1_+_2-"-x:' ) (5) Note from ( 5) that n may be totally expressed in terms of physical parameters such as mean squared value, variance of the power, and the ratio of the total power of the dominant components and the total power of the scattered waves of the fading signal.Note also that whereas these physical parameters are of a continuous nature, n is of a discrete nattue.It is plausible to presume that if these parameters are to be obtained by field measurements, their ratios, as defined in (5), will certainly lead to figures that may depart from the exact n.Several reasons exist for this.One of them, probably the most meaningful one, is that, although the model proposed here is general, it is in fact an approxinlate solution to the so-called random phase problem, as are approxinlate solution to the random phase problem all the other well-lmown fading models.The limitation of the model can be made less sttingent by defining f.l as 1+2x: X ' (1 +x: )- x: 2 exp(j!x:) In the same way, the probability density function of the power is given as (9) Equations ( 8) and (9) in their normalized forms are respectively given by (1) and (2).

OTHER FADING DISTRIBUTIONS
The X:-JL Distribution is a general fading distribution that includes the best !mown fading distributions, namely Rice and Nakagami-m disuibutions.Note that both Rice and Nakagami-m include the Rayleigh distribution and the Nakagami-m includes the One-Sided Gaussian.Therefore, these distributions can also be obtained from the K-J1 Distribution.The Lognormal distribution may also be weii approximated by the K-J.L Distribution.

RICE AND RAYLEIGH
The Rice distribution describes a fading signal with one cluster of multipath waves in which one specular component predominates over the scattered waves.

p(p)
which is the Rice probability density function for the normalized envelope.In titis case, the parameter K coincides with the weii-known Rice parameter 1c.Now setting K; 0 in (10) (therefore, J.L; 1 and K -7 0 in the K-J.L Distribution) the Rayleigh distribution can be obtained in an exact manner.Moreover, for K ; m -1 + Jm! m -1) in (10) (therefore J.L;l and K;m-l+Jm(m-1) in the K•J.L Distribution), where m is the Nakaganti parameter, the Nakaganti-m distribution can be obtained in an approximate manner.

NAKAGAMI-M, RAYLEIGH, AND ONE-SIDED GAUSSIAN
The Nakagami-m signal can be understood as composed of clusters of multipath waves with no dominant components within any cluster.Therefore, by setting K; 0 in the K-J.L Distribution it should be possible to obtain the Nakaganti-m distribution.We note, however, that, apart from the case f1 ; 1, wltich has been explored in the previous subsection, the introduction of K; 0 in the K•J.L Distribution leads to indeterminacy (zero divided by zero).For small arguments of the Bessel function the relation I J.f-l (z)= (z/2)J.!-I /r(J.L) holds [6, page 375, Eq. 9.6.7].
Using tltis in (1), and after some algebraic manipulation,

p(p) (11)
As K -7 0 (11) reduces to which is the exact Nakaganti-m density function for the normalized envelope.In tltis case, the parameter J.L coincides with the well-known Nakagami parameter m.Now setting J.L; 1 in (12) (therefore, J.L; 1 and K -7 0 in the K•J.L Distribution) the Rayleigh distribution can be 4 obtained in an exact manner.In the same way, by setting J.L; 0.5 in (12) (therefore, f1; 0.5 and K -7 0 in the K-J.L Distribution) the One-Sided Gaussian distribution can be obtained in an exact manner.Moreover, for J.L;(l+k)Z/(1+2k) in (12) (therefore K->0 and where k is the Rice parameter, the Rice distribution can be obtained in an approximate manner.The Lognormal distribution, given as a function of m in (12) of [7], can also be approximated by the K•J.L Distribution for e -I ~ p ~ e, and for K-> 0 and J.L ;m.

APPLICATION OF THE ~J.I. DISTRIBUTION
The K•J.L Distribution, as implied in its name, is based on two parameters, K and f1 .Its use involves .aprocedure similar to that of ti1e other distributions, as explained next.From ( 6), it can be seen that the two parameters K and f1 can be expressed in terms of the ratio between ti1e mean squared value and the variance of the power, which is usually defined as m.In other words For a given m, the parameters K and f.1 are chosen that yield the best fitting.Note, on the other hand, that, for a given m, the parameter f.1 shall lie within the range m and 0, obtained for K ; 0 and K -7 =,respectively.Therefore, for a given m (14) The parameter J.L is then chosen within the range of (14).Given that J.L has been chosen, then K must be calculated as so that the relation as in (13) be kept.

DISTRIBUTION
This section shows some plots of the K•J.L Distribution.Fig. 1 and Fig. 2, respectively, depict a sample of the various shapes of the K-J.L probability density function p(p) and probability distribution function P(p) as a function of the normalized envelope p for the same Nakaganti parameter m; 0.5 .Fig. 3 and Fig. 4 do the same but for m; 0.75; and Fig. 5 and Fig. 6, for m; 1.0; and Fig. 7 and Fig. 8, for m; 1.25; and Fig. 9 and Fig. 10, for m; 1.5.
The curves for which K ---> 0 coincide with the Nakagami-m curve, in which case 11 = m .11le curves for which J1. = 1 coincide with the Rice curve for which K=k.
It can be seen that, although the normalized variance (parameter m) is kept constant for each Fignre, the curves are substantially different from each other.And this is particularly relevant for the distribution function, in which case the lower tail of the distribution may yield differences in the probability of some orders.This feature renders the K-Jl.Distribution very flexible and this flexibility can be used in order to adjust the curves to practical data.

COMMENTS ON THE K-p. DISTRIBUTION
A new general fading distribution -the K-Jl.Distribution -has been presented.It models a signal composed of clusters of multipath waves propagating in a nonhomogeneous environment.Within any one cluster, the phases of the scattered waves are random and have similar delay times• with delay-time spreads of different clusters being relatively large.The clusters of multipath waves are assumed to have the scattered waves with identical powers but within each cluster a dominant component is found that presents an arbitrary power.The distribution includes the One-Sided Gaussian, the Rayleigh, and, more generally, the Nakagami-m and the Rice distributions as special cases and offers a higher degree of freedom.

Michel Daoud Yacoub
General Fading Distributions

THE 17"J.l DISTRIBUTION
The 11-11 distribution is a general fading distribution that can be used to better represent the small scale variation of the fading signal.For a fading signal whose envelope is r and whose envelope p normalized with respect to the nns value is given by p = r/r, r = J E~2 ), the 11-11 probability density function p(p) is wtitten as equivalently, J1 = ~x ( 1 +~: ) , rO is the Gamma VariP-J l+7]J function, I v (.) is the modified Bessel function of the first kind and arbitrary order v ( v real), J1 2: 0 and 0 s 71 $ 1.(In fact, the distribution is symmetrical for 1 S 71 < = , or Therefore, more generally, we may write 0 $ 71 < = and H =[11-n-1 [j4.But, due to the symmetry around 1, it suffices to consider 71 within one of the ranges only, the range 0 $ 71 $ 1 being preferable for its compactness.)For a fading signal with power w = r 2 /2 and normalized power (J) = wfw, where w=E(w), the 71-Jl probability density function p((J)) is given by p((J)) • 1 a! .

DERIVATION OF THE Tf•J.l DISTRIBUTION
Given the physical model for the 11-11 Distribution the envelope r can be written in terms of the in-phase and quadrature components of the fading signal as where w = r 2 /2 and w; = r?/2.We proceed to find the density of 'l .This can be carried by following the standard, but long and tedious, procedure so that where 71 = 0' ?;j 0' ~ and I 0 (.) is the modified Bessel function of the first kind order zero.Note that 0 $ 71 $ 1 defines the region within which a.~::; a~, whereas.We note, however, that w=E(w)=nw 0 .Therefore wp(w) The corresponding density of the envelope is found to be (20) From Equation 18we find that £~2 )= n(l +1) )a~ aud Var~2 )= 2n(l +1) 2 )a~ .Thus (21) Note from Equation 21 that n/2 may be totally expressed in terms of physical parameters such as meau squared value, variance of the power, aud power of the inphase aud quadrature components of the fading signal.Note also that whereas these physical parameters are of a continuous nature, n/2 is of a discrete nature (integer multiple of 1/2 ).It is plausible to presume that if these parameters are to be obtained by field measurements, their ratios, as defined in Equation 21, will certainly lead to figures that may depart from the exact n/2 .Several reasons exist for this.One of them, probably the most significant one, is that, although the model proposed here is general, it is in fact au approximate solution to the so-called random phase problem, as are approximate solution to the random phase problem all the other well-known fading models.The limitation of the model cau be made less stringent by defining J.1.as (22) J.l. being the real extension of n/2.Values of J.1.that differ from multiples of I/2 account for a) non-zero correlation among the clusters of multipath components aud b) non-Guassiauity of the in-phase aud quadrature components of the fading signal.C:We note that in derivation of the N akagami model [7], the parameter n , which describes the number of "component signals", therefore discrete, is also written in terms of the Nakagami continuous parameter m as m = n/2 .)It has been observed experimentally by Nakagami [7] that Therefore, for the 1)-j.l.Distribution (23) with 0 :> 1) :>I (24) (or equivalently 0:;; 1)-I :;; 1 ).Being of au experimental nature [7], the constraint of Equation 23 does not necessarily need to be observed.In fact, the distribution can be used for ,u assuming any real value J.1.;o, 0 and 0 :;; 1) :;; I , as already observed.
The probability density function of the envelope can be written as 'z;(r) In the same way, the probability density function of the power is obtained as which, in the normalized form, are given as in Equations 16 and 17, respectively.

THE TJ•J.L DISTRIBUTION AND THE OTHER FADING DISTRIBUTIONS
The TJ• f.l Distribution is a general fading distribution that includes the Hoyt, the One-Sided Gaussian, the Rayleigh, and, more generally, the Nakagami distributions as special cases.Rice and Lognormal distributions may also be wellapproximated by the TJ-J.L Distribution.We note that the One-Side Gaussian and the Rayleigh distributions can be obtained from the Nakagami distribution by setting the Nakagami parameter m = 0.5 and m = 1, respectively.Therefore, in order to relate the TJ•J.L Distribution with these two distributions it suffices to relate it with the Nakagami one.

HOYT, ONE-SIDED, AND RAYLEYGH
The Hoyt distribution can be obtained from the TJ•f.lDistribution in an exact manner by setting f.l = Yz .From the Hoyt distribution the One-Sided Gaussian is obtained when 1J --> 0 .In the same way, from the Hoyt distribution the Rayleigh distribution is obtained when TJ = 1 .

NAKAGAMI•M, RAYLEIGH, AND ONE-SIDED GAUSSIAN
The Nakagami distribution can be obtained in an exact manner from the TJ-f.lDistribution for f.l = m and TJ --7 0 (or equivalently TJ --7 =) or, in the same way, for f.l = m/2 and TJ --> I .This result is not straightforwardly seen from the densities here derived.We observe, nonetheless, that for these conditions all the Gaussian variates present identical variances and the fading model proposed here deteriorates into that of [8], where the Nakagami distribution is obtained in an exact manner.For intermediate values of TJ the TJ•f.ldistribution and tlte Nakagami distribution relate to each other for l'(l+~'f = m.This is a very interesting result l+ry which shows that an infinite number of curves of the TJ•f.ldistribution can be found that presents the same Nakagami parameter m, conditioned on the fact that the constraints and TJ are satisfied.The m Lognormal distribution, given as a function of m in Equation 13 of [7], can also be approximated by the TJ-J.L Distribution for e -! ~ p ~ e , and for TJ , f.l , and m satisfying the relations given above for the Nakagami case. In the same way, an infinite number of curves of the TJ•f.lDistribution can be found that presents the same Nakagami parameter for the Lognormal distribution.The Rice distribution can be approximated by the TJ-JL distribution for   The curves for which TJ --7 I and TJ --7 0 coincide with each other and also with the Nakagami one, as indicated in the Figures.In such cases, we have f.l = m/2 and f.l = m , respectively.It can be seen that, although the normalized variance (parameter m) is kept constant for each Figure, the curves are substantially different from each other.And this is particularly noticeable for the distribution function, in which case the lower tail of the distribution may yield differences in the probability of some orders.Moreover, the curves present a very interesting feature, as described next.
For the same value of m and departing from the condition for which TJ --> I , as TJ diminishes the curves depart from that for which TJ --7 1 , initially keeping a similar shape.As TJ dinainishes the shapes of the curves change substantially.
As '7 dinainishes even further and as TJ _, 0 , the curves merge with that of the initial shape but such curves present shapes very different from those obtained as TJ --7 0 .This feature renders the TJ•f.lDistribution very flexible and this flexibility can be used in order to adjust the curves to practical data.

COMMENTS ON THE ry•JL DISTRIBUTION
A new general fading distribution -the T]-Jl.Distribution -has been presented.It models a signal composed of clusters of multipath waves propagating in a nonhomogeneous environment.Within any one cluster, the phases of the scattered waves are random and have similar delay times with delay-time spreads of different clusters being relatively large.The clusters of multipath waves are assumed to have the scattered waves with different powers and no dominant component is found.The distribution includes the One-Sided Gaussian, the Rayleigh, the Hoyt and, more generally, the Nakagami-m distributions as special cases and offers a higher degree of freedom.

2 )
being the real extension of n.Non-integer values of the parameter f.l account for a) non-zero correlation among the clusters of multi path components and b) non-Guassianity of the in-phase and quadrature components of the fading signal.(We note that in derivation of the Nakagami-m model[7], the parameter n, which describes the number of ''component signals"[7], therefore discrete, is also vvritten in terms of the Nakagami continuous parameter m as m = n/2 .)It has been observed experimentally byNakagami [x: ~ 0 and Jl ~ 0 .Being of an experinlental nature[7], the constraint of Equation7does not necessarily need to be observed.In fact, the distribution can be used for Jl assuming any real value Jl ~ 0 and x: ~ 0, as already observed.Using the definitions and the considerations as above and by means of a transformation of variables and a series of algebraic manipulations, the x:-JL probability density function of the envelope can be written from (4) as , () 2JL(1+x:)";' (r)p.( ( 1 {r)

Figure 1 .
Figure 1.A sample of the vru.ious shapes of the K-Jl.probability density function for the same Nakagami parameter m = 0.5 .
MODEL: 11"J.lDISTRIBUTION The fading model for the 11-11 Distribution considers a signal composed of clusters of multipath waves propagating in an non-homogeneous environment.Within any one cluster, the phases of the scattered waves are random and have similar delay times with delay-time spreads of different clusters being relatively large.The in-phase and 8 quadrature components of the fading signal within each cluster are assumed to have different powers.

Figure 16 .Figure 18 .
Figure 16.A sample of the various shapes of the T]-Jl.probability distribution function for the same Nakagami parameter m = I. 0 .

Now we form the process r? = xl + yf, so 2 n ?
n that r = L If .In the same way, we may write w = L w; , i=l i=l