Free-Space Propagation Loss - Understanding GPS: Principles and Applications - page 672

Computation of the received power is commonly performed in decibels. Denot-
ing the quantities of units as subscripts allows (C.7) to be rewritten in decibels as

λ
π

() () ( ) ( )

P = P + G + G − L + 20 log

R dBW T T R dB dB 10

dBW dB

4 d

(C.8)

d

() () ()

= P + G + G − L − 2198 . − 20 log

T T R dB dB 10

dBW dB

λ

The latter expression is particularly simple, using a constant and the separation
between transmitter and receiver expressed as the number of wavelengths.
Finally, the generalized free-space propagation loss (which includes excess loss)
is found from (C.1) and (C.7) to be

2

4 π

λ

d

Λ= L (C.9)

with

PGG

P =

TTR

R

Λ

and

() () ( ) ( )

P = P + G + G − Λ

R dBW T T R dB dB

dBW dB

where

()

Λ = 10 log Λ

dB 10

While (C.9) is a very compact expression for free-space propagation loss, sim-
plistic interpretation of this expression leads to the faulty conclusion that, since
free-space propagation loss increases with frequency, there is a frequency-dependent
attenuation mechanism in free space. The correct interpretation is that the loss in
PFD (in W/m 2 ) with distance from the transmitter does not depend on frequency, as
seen in (C.4). However, free-space propagation loss is defined to include the effects
of a receive antenna having a gain (often unity) that remains constant over fre-
quency. Since an antenna of given gain has smaller effective area at higher frequen-
cies, the fixed-gain antenna collects a smaller fraction of the PDF at higher
frequencies, resulting in lower received power at higher frequencies.
Since the antenna area contributes to the free-space propagation loss as com-
monly defined, free-space propagation loss increases with frequency. If free-space
propagation loss were instead defined for fixed effective area of the receive antenna
rather than fixed gain of the receive antenna, (C.5) shows that the free-space propa-
gation loss would then be independent of frequency (but the antenna would become
increasingly directive at higher frequencies, since it would remain the same physical
size).