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G
=
the effective power is the same as the radiated
power), in the direction of the receiving antenna, by the transmitter; and the second
term in the curly brackets is the inverse-square law, a manifestation of the principle
of conservation of energy, which states that the energy is evenly spread over the surface
of a sphere whose center is at the transmitter and the surface contains the receiver at
a radial distance
d
from the transmitter.
It is common practice for engineers to work using logarithmic units (decibels).
The conversion of a power ratio from linear units to decibels is
1
(since for an isotropic antenna
tx
P
(W)
P
r
=
2
Þ=
R
(dB)
10 log
r
=
10 log
2
10
10
P
(W)
10 log (
P
1
1
Þ=
R
P
(W)) 10 log (
-
P
(W))
10
2
10
1
The convention is to take the unit of power out of the logarithm by subtracting
and adding
(1 W)
and expressing terms such as
10 log
10
æö
P
10 log (
P
(W)) 10 log (1(W)) 10 log
-
=
2
(dBW) 10 log (
=
P
)(dBW).
ç
èø
10
2
10
10
10
2
1
where now we introduce the “unit” of dBW, or decibel relative to 1 W. Decibels
relative to 10
−3
W = 1 mW are expressed as dBm, or decibels relative to 1 mW.
Taking logarithms on both sides of the earlier equation for conservation of
energy, yields
2
æ
4
π
λ
d
ö
P
(dBW)
=
P
(dBW)
+
G
(dBi)
+
G
(dBi) 10 log
-
ç
÷
rx
tx
tx
rx
10
è
ø
where we have normalized the logarithmic antenna gains relative to the gain of an
isotropic antenna, which is by definition equal to unity, and introduced the unit of
dBi, or decibel relative to an isotropic antenna. The last term on the right-hand side
is identified to be the free-space path loss,
= πλ
.
Empirical-statistical propagation models are a generalization of the above
expression and typically take the following form [
7
]:
L
(dB)
20 log (4
d
/
)
0
10
P
(dBW)
=
P
(dBW)
+
G
(dBi)
+
G
(dBi)
-
L
(dB)
rx
tx
tx
rx
where
P
is
the corresponding received power,
{tx,rx}
(dBi
G
is the transmit/receive antenna gain
in the direction of the receiver/transmitter measured in decibels relative to an
idealized reference isotropic antenna, and
(dB
L
is the path loss between the two
antennas in decibels. As we shall see shortly, for nearly all propagation environ-
ments of any practical interest we have highly incomplete knowledge of
(dB
L
,
which we separate into an empirical, deterministic component and one or more
stochastic components.
In the case of free-space path loss, we have been considering above, the free-
space loss is deterministic and can be expressed simply as
is the transmitted power in decibels relative to 1 W,
P
tx
(dBW)
rx
(dBW)
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