K distribution: an appropriate substitute for Rayleigh-lognormal. distribution in fading-shadowing wireless channels

Transcription

1 K distribution: an appropriate substitute for Rayleigh-lognormal distribution in fading-shadowing wireless channels A. Abdi and M. Kaveh Indexing terms: Rayleigh-lognormal distribution, K distribution, fading, shadowing, wireless channel Rayleigh-lognormal distribution, proven useful for modeling fading-shadowing wireless channels, has a complicated integral form. In this paper we have accurately approximated it by the K distribution. This distribution is simpler and thus more appropriate for analysis and design of wireless communication systems. Introduction: Rayleigh-lognormal distribution is a mixture of Rayleigh and lognormal distributions [1]: fu ( u) = f ( uv = v) f v dv u UV V ( ) 0 (1) 0 where f ( uv = v ) is the Rayleigh distribution with mode v and f v UV V ( ) is the lognormal distribution with parameters μ and λ : ( ln v μ) u u 1 f ( uv = v) = exp f v UV V ( ) = exp v v πλ v λ v 0 () The Rayleigh-lognormal distribution in eqn. 1 appropriately describes local and global 1

2 spatial variations of signal envelope, and/or short-term and long-term temporal fluctuations of signal envelope in fading-shadowing wireless channels []. The lack of general acceptance of this distribution may have been caused by its complicated integral form. Note that f U ( u) is very similar to the Suzuki distribution [3]. But in Suzuki distribution instead of v, v has the lognormal distribution. Although both definitions have already been used as mixtures of Rayleigh and lognormal distributions [], we focus on the former. However, it should be mentioned that Suzuki distribution can be approximated by a K distribution as well. K distribution, extensively used for modeling diverse scattering phenomena such as tropospheric propagation of radio waves, various types of radar clutter, optical scintillation from the atmosphere, etc., is a mixture of Rayleigh and gamma distribution [4]: X XY Y 0 f ( x) = f ( xy = y) f ( y) dy = β+ 1 x K x β x 0 α > 0 β > 1 (3) αγ( β + 1) α α where f ( xy = y ) is the Rayleigh distribution with mode y, f y XY Y ( ) is the gamma distribution with parameters α and β : β y y fy ( y) = ( ) ( + ) exp β+ 1 α Γ β 1 α y 0 (4) Γ(.) is the gamma function, and K β (). is the modified Bessel function of the second kind and order β. Comparison of the moment generating functions of lnu and ln X : It is proven that lognormal and gamma distributions can closely approximate each other [5] [6]. This fact, together with the mixture representations in eqns. 1 and 3, motivated us to see how similar

3 Rayleigh-lognormal and K distributions are. The moment generating function of lnu, φ ln ( t U ) = E [exp( t ln U )], can be obtained using EU k k [ V= v] = ( v) Γ( 1+ k ) and n EV [ ] = exp( nμ+ n λ ) [6]: t t μ λ φ ln U () t = Γ1+ exp t + 8 t (5) k k Since EX [ ] = ( α) Γ( 1+ k) Γ( 1+ β+ k) Γ ( 1+ β) [4], φ ln () t can be written as: t t t φln X () t = ( α) + β ( β) + + Γ 1 Γ 1 Γ 1+ (6) Clearly ln φlnu ( t) = ln Γ( 1+ t ) + (ln + μ) t + λ t 8, while substitution of power expansion of ln Γ( 1+ β + t ) into the logarithm of eqn. 6 gives: ( ) ( ) ln φ ( t t β β ln X ) = ln + Ψ ln( α) + t t + Ψ 1 + Γ 1 8 Ψ ( 1 + β) 3 Ψ ( 1 + β) 4 + t + t +... (7) where Ψ(.), Ψ (.), Ψ (.),and Ψ (.) are psi function and its derivatives, respectively [7]. By neglecting t 3, t 4,... in eqn. 7 and then comparing it with ln ( t ),wegettherequired X φ ln U relationship between ( μ, λ) and ( α, β) that yields approximate equivalence of U and X: μ = ln( α ) + Ψ( 1+ β), λ = Ψ ( 1+ β) (8) Discussion: According to eqn. 7, α has no effect on the accuracy of approximation. In fact, k k upon application of the relations in eqn. 8, EU [ ] EX [ ] will be independent of α.sothe ( ) accuracy of approximation depends on β. In eqn. 7 Ψ m ( s), m= 1,,..., is a decreasing ( ) function of s such that Ψ m ( s) as s 0 + ( ) and Ψ m ( s) ( m 1)! s m for large 3

4 positive s. Therefore f ( U u ) and f X ( x ) coincide more and more as β increases. This fact is depicted in Fig. 1. Moreover, we observe that roughly speaking, tails of both distributions are the same, independent of the value of β. Conclusion: Based on our results, Rayleigh-lognormal and K distributions are similar, but the latter has a simpler form. In addition, existence of numerous analytic results about Bessel functions make it possible to obtain closed-form solutions in the calculation of bit error rates, diversity effects, etc. using K distribution. One can also take advantage of numerous available methods, specially in radar literature, for its simulation and parameter estimation. Another important supporting fact for the K distribution is its theoretical explanation: it is a limit solution to a general scattering problem modeled as two-dimensional random walks [8]. A. Abdi and M. Kaveh (Department of Electrical and Computer Engineering, University of Minnesota, Minneapolis, MN 55455, USA) abdi@ece.umn.edu References 1 HANSEN, F., and MENO, F. I.: Mobile fading-rayleigh and lognormal superimposed, IEEE Trans. Vehic. Technol., 1977, 6, (4), pp STUBER, G. L.: Principles of mobile communication (Kluwer, Boston, Massachusetts, 1996) 4

5 3 SUZUKI, H.: A statistical model for urban radio propagation, IEEE Trans. Commun., 1977, 5, (7), pp RAGHAVAN, R. S.: A model for spatially correlated radar clutter, IEEE Trans. Aerospace Electronic Syst., 1991, 7, (), pp CLARK, J. R., and KARP, S.: Approximations for lognormally fading optical signals, Proc. IEEE, 1970, 58, (1), pp JOHNSON, N. L., and KOTZ, S.: Distributions in statistics: continuous univariate distributions (Wiley, New York, 1970) 7 MAGNUS, W., OBERHETTINGER, F., and SONI, R. P.: Formulas and theorems for the special functions of mathematical physics (Springer, New York, 3rd ed., 1966) 8 JAKEMAN, E., and PUSEY, P. N.: Significance of K distribution in scattering experiments, Phys. Rev. Lett., 1978, 40, (9), pp

6 0.3 Densities of U and X (i) (ii) (iii) Fig. 1 f U ( u ) ( ) and f X ( x ) ( ) for and different values of (i) 0. (ii) 03. (iii) 0.9