Page 45 - Integrated Wireless Propagation Models

I n t r o d u c t i o n t o M o d e l i n g M o b i l e S i g n a l s i n W i r e l e s s C o m m u n i c a t i o n s 23

The pdf of r(t) can be expressed as4

r ( r2 )
p(r) = 2 exp - z (1.8.1.2)
cr 2cr

and the cdf of r(t), the probability that r is less than level A, is

(1.8.1.3)

where the mean square of r is

(1.8.1.4)

It is also called average power, and .fi cr is called RMS. Letting A = .fi cr in Eq. (1.8.1.3),
we find that

p(r :::; .fi cr) = 63% (1.8.1.5)

This means that 63 percent of the signal is equal or below average power (also equal or
below its RMS). The function of Eq. (1.8.1.3) is plotted on Rayleigh paper, as shown in
Fig. 1.8.1.1.
The mean value of r is
= r2 r2 ) /n
(
d
r =
E[r] = [rp(r) r = [ (Y2 exp - 2cr 2 d V2: cr (1.8.1.6)
=
1.8.2 Log-Normal Fading Model-Long-Term Fading Model
A lognormal distribution is used for studying a long-term fading. It is a continuous
probability distribution of a random variable x whose logarithm is normally distributed
in probability theory. This model is used to quantify the distribution of multiple reflec­
tions and diffractions rays between a transmitter and a receiver. The lognormal PDF
can be given as

[ln(x) mF}
1 {_
p(x) = exp � (1.8.2.1)
x-J2rccr 2 2cr

where m is the median value and cr is the standard deviation of the corresponding
normal distribution. From the measurement data, we take the local mean average (see
1
Sec . 6.3.1) to average out the short-term fading; the remaining data are called long­
term fading, which follows the lognormal fading model shown in Eq. (1.8.2.1). All the
prediction models are predicting the long-term fading signal. The long-term fading
1
signal is obtained from averaging the sampled data. 9 Therefore, the true long-term
fading signal has to be treated very carefully, as described in the next section.