Handbook of Propagation Effects for Vehicular and ... - Courses

Theoretical Modeling Considerations 11-11
In the following section is given an overview of the density functions used in
modeling procedures. A further characterization is given by the ITU-R [ITU-R, 1986a
(Report 1007)].
11.4.1 Density Functions Used In Propagation Modeling
11.4.1.1 Ricean or Nakagami-Rice Density Function
The voltage phasors from all the reflection sources can be combined into two
independent orthogonal vectors x and y, the in-phase and quadrature components, having
normal envelopes and uniform phase distributions. When received together with a direct
signal voltage a, the two-dimensional probability density of the received voltage can be
expressed as
2 2
1 ⎡ ( x − a)
+ y ⎤
f xy ( x,
y)
= exp⎢−
2
2
2πσ
⎥ , (11-22)
⎣ 2σ
⎦
where σ is the standard deviation of the voltage. The signal envelope represents the
length of the voltage vector z, given by
z +
2 2
= x y , (11-23)
from which we derive the Ricean density fz(z) [Papoulis, 1965]
⎡ 2 2
z ( z + a ) ⎤ ⎛ za ⎞
fz ( z)
= exp⎢−
⎥I0
⎜ ⎟ , (11-24)
2
2 2
σ ⎢⎣
2σ
⎥⎦
⎝σ
⎠
where I0 is the 0 th order modified Bessel function.
The normalized line-of-sight power is given by
2
Plos ′ = a
(11-25)
and the average (normalized) multipath power is given by
2
P ′mp = 2σ , (11-26)
where we denote the powers by a prime to distinguish it from probability. The ratio of
these two powers defines the K value that characterizes the influence of multipath
scattering on the signal distribution. Hence,
2
P′
los a
K = = . 2
(11-27)
P′
2σ
mp