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

6-8
6.5.1 Joint Probabilities
Propagation Effects for Vehicular and Personal Mobile Satellite Systems
In Figure 6-6 are shown a family of cumulative fade distribution functions derived from
the above mentioned simulation. The curve labeled d = 0 represents the single terminal
cumulative fade distribution corresponding to data acquired from over 400 km of driving
in Australia. The curves labeled d = 1 to 10 m represent the individual joint probability
cumulative fade distribution for the indicated antenna separations (in the direction of
vehicle motion). Such a distribution represents the joint probability that two antennas
spaced a distance d mutually exceed the abscissa value of fade. Finally, the curve labeled
“independent fading” corresponds to the joint distribution of two links with single
terminal fading, assuming that the two are independent. We note that the joint
probabilities tend to coalesce with increasing antenna separation at about 2/3 the dBdistance
to the independent fading case. That is, the fade distributions for 8 m and 10 m
separations have insignificant differences.
6.5.2 Diversity Improvement Factor, DIF
The DIF is defined as
Po ( A)
DIF( A, d)
= , (6-5)
P ( A)
d
where P0(A) represents the single terminal probability distribution at the fade depth A,
and Pd(A) represents the joint probability distribution for an antenna spacing d assuming
the same attenuation A is exceeded. These probabilities may be obtained from Figure 6-
6.
Employing the above results, a least square estimate of DIF was derived given by,
[ 0.
2 ln( ) 0.
23]
DIF ( A,
d)
= 1+
A d + , (6-6)
where d is the antenna separation expressed in m and A is the fade depth in dB. In Figure
6-7 are plotted a family of curves of DIF as a function of fade depth for antenna
separations between 1 and 10 m. We note, for example, that DIF(8,1) ≈ 3. This implies
that when the antennas are separated 1 m, the distance over which the signal is received
above noise is three times greater for diversity operation relative to the single terminal
case assuming an 8 dB fade margin. At the larger separations for any given fade depth,
the rate at which DIF increases is shown to diminish rapidly.