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

Theoretical Modeling Considerations 11-13
11.4.1.3 Lognormal Density Function
Shadowing is a manifestation of the absorption and scattering of the incident direct wave
by roadside trees or other obstacles as it is transmitted via the line-of-sight between the
satellite and the mobile terminal. The cumulative distribution function of the received
power expressed in dB can often be fit to a straight line when plotted on a normal
probability scale. The voltage variation due to shadowing is then lognormal. The
lognormal density function for a random variable z can be derived from the normal
density function for x by using
( z)
x = ln . (11-34)
In this case the lognormal density of z has the form
f z
( z)
( ln(
z)
m)
1 ⎡ −
= exp⎢−
sz 2π
2
⎢⎣
2s
2 ⎤
⎥ , (11-35)
⎥⎦
where m and s are the mean and standard deviation of ln(z), respectively. Since the
power x is usually expressed in dB, the relation between x (in dB) and z is
or
x = 10log( z)
z = power (watts) (11-36)
x = 20log( z)
z = voltage (volts). (11-37)
The lognormal density function of power when z is the power in watts is
f z
( z)
( 10log(
z)
m)
4.
343 ⎡ −
= exp⎢−
sz 2π
2
⎢⎣
2s
where m and s are the mean and standard deviation of log(
z)
2 ⎤
⎥
⎥⎦
lognormal density function of power when z is voltage is
f z
( z)
( 20log(
z)
m)
8.
686 ⎡ −
= exp⎢−
sz 2π
2
⎢⎣
2s
2 ⎤
⎥
⎥⎦
where m and s are the mean and standard deviation of log(
z)
11.4.2 Loo's Distribution Model
z = power (watts), (11-38)
10 , respectively. The
z = voltage (volts), (11-39)
20 , respectively.
A statistical model for land mobile satellite propagation based on probability density
functions of multipath and shadowing propagation has been developed by Loo [1985;
1987]. The following assumptions are made: (a) the receiver voltage due to the diffusely