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

11-12
Usually, the K factor is quoted in terms of dB. That is,
K(
dB)
⎛
10log⎜
⎜
⎝
Propagation Effects for Vehicular and Personal Mobile Satellite Systems
⎞
2
P′
los ⎛ ⎞
⎟
a
= ⎜ ⎟
′ ⎟
10log
. (11-28)
Pmp
⎠ ⎝ 2σ
⎠
= 2
It is apparent from (11-27) that the lower the relative level of the multipath power, the
larger the K value, and conversely. Further normalizing P′ los such that P ′ los = 1,
reduces
the Ricean density (11-24) to a single parameter density function of the voltage, which
can be written as a function of K by
where
2 ( − K(
z + 1)
) I ( 2Kz)
fz ( z)
= 2Kz
exp
0 , (11-29)
1
K = . 2
(11-30)
2σ
11.4.1.2 Rayleigh Density Function
The Rayleigh density is a special case of the Ricean distribution and arises when no lineof-sight
power is received. Setting a = 0 in (11-24) results in
⎡ 2
z z ⎤
fz ( z)
= exp⎢−
2
2
⎥ . (11-31)
σ ⎢⎣
2σ
⎥⎦
Even though no direct signal is received, the Rayleigh density can also be defined in
terms of a K -factor as
1
K ≡ . 2
(11-32)
2σ
Substituting (11-32) into (11-31) gives
2 ( − K )
fz ( z)
= 2Kz
exp z . (11-33)
Note that the Rayleigh distribution has but a single parameter (namely, σ or K ). For
Rayleigh scattering, the average scattered power is variable, but the standard deviation on
a dB scale is constant and equal to 5.57 dB. As a rule of thumb, based on the Central
Limit Theorem, at least six random scattering sources are required to produce a Rayleigh
(or Ricean) distribution [Papoulis, 1965].