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PROPAGATION OFSIGNALS 1272 4 6 8 10 12 14 16 18 20051015202530Transmitter-Receiver separation kmExcess loss over n = 2 (dB)n=3n=42 4 6 8 10 12 14 16 18 20051015202530Transmitter-Receiver separation kmExcess loss over n = 2 (dB)n=3n=4Figure 4: The losses atn= 3 andn= 4 are compared with
the free space loss (n= 2 ).c being the velocity of light and f 0 the frequency. Note
thatGt, andGr, are dimensionless (i.e., numbers) andd
andλmust have the same units (centimeters, meters, or
kilometers). Assuming equal gain antennas (Gt=Gr), the
received power can be expressed as inversely proportional
to the square of the distancePd∝1
d^2.Conversely, we can say that the loss experienced by the
signal is directly proportional to the square of the dis-
tance. The path loss exponent or the path loss coefficient
determines the decay of the power as distance increases
and is denoted byn. In free space under LOS conditions,
the path loss exponentnis 2. Because there are no ob-
stacles in the path of the signal in LOS propagation in
free space, no reflection, diffraction, or scattering takes
place,n=2 will be the best case scenario expected in sig-
nal transmission. In a general case, where propagation
takes place in a region containing obstacles, the path loss
exponent will be larger than 2, pointing to higher path
loss asnincreases. The excess loss overn=2 is plotted in
Figure 4 as a function of the distance. The loss at a given
distance increases asngoes up. Note that the path loss
exponentnis also sometimes referred to as power decay
index, distance power gradient, or slope factor.
It must be noted that the low values ofnalso increase
cochannel interference, namely, the interference coming
from other cells using the same channel. As the value of
nincreases, the interference goes down, leading to an im-
provement in the capacity of the cellular communication
systems.Calculation of the Received Power at Any Distance
Power received at any distancedis inversely proportional
to thenth power of the distance,Pd∝1
dn. (2)
Equation 2 cannot be applied directly because of the
need to evaluate the proportionality factor. This is done
by applying Frii’s equation (Feher, 1995) at a very short
distanced 0 from the transmitter, where it can be assumed
that free space LOS conditions exist. IfPd 0 is the power at
a distance ofd 0 from the transmitter (Gt=Gr=1),Pd 0 =Pt(
λ
4 πd 0) 2. (3)
Using Equation 2 one can now writePd
Pd 0=dn 0
dn. (4)
Taking the logarithm, one arrives the expression for the
received power at a distance d (d>d 0 ) from the transmit-
ter to bePd(dBm)=Pd 0 (dBm)− 10 nlog 10(
d
d 0)
, (5)wherePd 0 (dBm)=Pt(dBm)+20 log 10(
λ
4 πd 0). (6)
Note that in Equations 5 and 6, the power is expressed
in decibel units (dBm). The question is now what value of
d 0 is appropriate. Typically, this value, known as theref-
erence distance(λ<d 0 ), is chosen to be 100 m in outdoor
environments and1minindoor environments (Pahlavan
& Levesque, 1995; Rappaport, 2002; Shankar, 2001).
The models described thus far can be applied for cal-
culating the received power at various operating frequen-
cies indoors and outdoors. A few points are in order.
The received power according to Frii’s equation (Equa-
tion 3) decreases as the wavelength decreases. This means
that as one moves from 900 MHz band to the PCS (per-
sonal communication systems) operating at the 1,800 to
2,000-MHz band, the received power decreases. As wire-
less communication systems move into the 4–6 GHz band,
the received power decreases further as the wavelength
decreases (Table 1). Another factor that becomes criti-
cal as the frequencies increase is the inability of the sig-
nal to penetrate buildings. For example, compared with
900-MHz signals, 2-GHz signals will not travel far from
the transmitter. Signals may even be blocked by a sin-
gle building between the transmitter and receiver. Thus,
the LOS propagation becomes the predominant means
by which the signal reaches the receiver. In the case of a
microwave signal, a truck obstructing the path can bring
down the received signal to extremely small levels (IEEE,
1988).
The approaches based on the path loss exponent are
not the only ones to estimate the loss suffered by the signal
as it reaches the receiver. The disadvantage of the path-
loss-based approach is that it does not directly take into
account a number of system dependent factors such the
heights of the transmitting and receiving antennas and
their locations. Models such as Lee’s (1986) model, Hata’s