The ground reflection coefficient is given by [2, 14]
R =
sin
- Z
sin + Z
,
(2.10)
where
Z =
r
- cos
2
/
r
for vertical polarization
r
- cos
2
for horizontal polarization
,
(2.11)
and
r
is the dielectric constant of the ground, which for earth or road surfaces is approximately that of
a pure dielectric (
r
= 15).
We see from Figure 2.3 and (2.10) that for asymptotically large d, r + r
l d, 0, G
l
G
r
, and
R
-1. Substituting these approximations into (2.7) yields that, in this asymptotic limit, the received
signal power is approximately
P
r
G
l
4d
2
4h
t
h
r
d
2
P
t
=
G
l
h
t
h
r
d
2
2
P
t
,
(2.12)
or, equivalently, the dB attenuation is given by
P
r
(dBm) = P
t
(dBm) + 10 log
10
(G
l
) + 20 log
10
(h
t
h
r
)
- 40 log
10
(d).
(2.13)
Thus, in the asymptotic limit of large d, the received power falls off inversely with the fourth power
of d and is independent of the wavelength . The received signal becomes independent of since the
cancellation of the two multipath rays changes the effective area of the receive antenna. A plot of (2.13)
as a function of distance is illustrated in Figure 2.4 for f = 900MHz, R=-1, h
t
= 50m, h
r
= 15m, G
l
= 1,
G
r
= 1 and transmit power normalized so that the plot starts at 0 dBm. We see in this figure that up to a
certain critical distance d
c
, the wave experiences constructive and destructive interference of the two rays,
resulting in a wave pattern with a sequence of maxima and minima. At distance d
c
, the final maximum
is reached, after which the signal power falls off proportionally to d
-4
. At this critical distance the signal
components only combine destructively, so they are out of phase by at least . An approximation for
d
c
can be obtained by setting = in (2.9), obtaining d
c
= 4h
t
h
r
/. This critical distance, also
illustrated in Figure 2.4, would be a natural size for the cell radius since the path loss associated with
interference outside the cell is much larger than path loss for the desired signal inside the cell. However,
setting the cell radius to d
c
often results in very large cells, as illustrated in Figure 2.4 and in the next
example. Since smaller cells are more desirable to increase capacity and reduce transmit power, cell radii
are typically much smaller than d
c
, so that with a two-path propagation model power falloff within these
relatively small cells goes as distance squared.
Example 2.2: Determine the critical distance for the two-path model in an urban microcell (h
t
= 10m,
h
r
= 3 m) and an indoor microcell (h
t
= 3m, h
r
= .5 m) for f
c
= 2 GHz.
Solution: d
c
= 4h
t
h
r
/ = 800 meters for the urban microcell and 40 meters for the indoor system. A cell
radius of 800 m in an urban microcell system is a bit large: urban microcells today are on the order of
100 m to maintain large capacity. However, if we used a cell size of 800 m under these system parameters,
signal power would fall off as d
2
inside the cell, and interference from neighboring cells would fall off as d
4
,
and thus would be greatly reduced. Similarly, 40m is a bit large for the cell radius of an indoor system,
as there would typically be many walls the signal would have to go through for an indoor cell radius of
that size. So an indoor system would typically have a smaller cell radius, on the order of 10-20 m.
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