Page 71 - ITU Journal: Volume 2, No. 1 - Special issue - Propagation modelling for advanced future radio systems - Challenges for a congested radio spectrum

ITU Journal: ICT Discoveries, Vol. 2(1), December 2019

In the above, ( ) is the Fresnel reflection Furthermore, the in Equation (12) are the SPM

coefficient for polarization as defined by: polarization factors with
2 ( −1)
( ) = − √ − , ℎℎ = 2 2 cos( − ),

ℎℎ
2
+ √ − (cos +√ −sin )(cos +√ −sin )
2
and ℎ = −( −1)√ −sin sin( − ),

2
2
( cos +√ −sin )(cos +√ −sin )
2
( ) = − √ − . (10) ( −1)√ −
2

2
+ √ − ℎ = 2 2 sin( − ),

(cos +√ −sin )( cos +√ −sin )
Moreover, ( − ) and are the Dirac and and

Kronecker delta functions respectively:
2
2
( −1)( sin sin −√ −sin √ −sin cos( − ) )
1, = . (14)
( − ) = { (10a) = 2

2
0, ℎ ( cos +√ −sin )( cos + √ −sin )
and For Gaussian correlation functions, Equation (12)
reduces to:
1, =
2
= { (10b) 2 2 2 ℓ 2 }. (15)
)
0, ℎ ( ̂ , ̂ ) = (2 ℓ cos cos | | exp {− 4

Analytical expressions for the diffuse bi-static Fig. 2 depicts the value for the backscattering
scattering coefficient components are provided coefficient based on SPM, Equation (15).
below, based on the three different models with
values of cross-polarized bi-static scattering
coefficients

ℎ ( , ), and ℎ ( , )
̂ ̂
̂ ̂

within the plane of incidence based on each model.
3. DIFFUSE BI-STATIC COEFFICIENT
BASED ON THE SMALL
PERTURBATION METHOD (SPM)
The small perturbation method (SPM) applies for
slightly rough surfaces having variance and
correlation length governed by equations (35) and
(36) defined in [4]. Those equations are recalled as
Equation (11).
Fig. 2 – Vertically backscattering coefficient (ε_r=20)
< 0.3 ,
Case (1): kσ=0.1, kl=1.0, Case (2): kσ=0.2, kl=2.0, Case (3):
ℓ > 4.71 . (11) kσ=0.3, kl=3.0
In order to examine values of cross-polarized
The diffuse bi-static scattering coefficient backscattering coefficients in the plane of incidence
̂ ̂
( , ) based on SPM can be written as set

(Equation (22) of [5], (21-67) of [6], (31) of [4], and − = 0 , ± . (16)
(37) of [7])
2
̂ ̂
2
2
( , ) = 8 ( cos cos ) | | ( ). (12) Then introduce Equation (16) into Equation (14)

and the resultant into Equation (12) yielding null
where ( ) is the surface height spectral density values for the cross-polarized bi-static scattering

function of Equation (4) calculated at roughness coefficients within the plane of incidence. Those
wavenumber equal to . null values indicate that SPM is not capable of

predicting the cross-polarized bi-static scattering
2 2 ) (13)
= √sin + sin − 2 sin sin cos( − coefficients within the plane of incidence.

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