path loss models

1. EE161 Handout #3
March 31, 2009
EE161 Spring 2009
Wireless Communications
Path Loss Models
The transmitted signal
s(t) = u(t)ej(2πft+φ0)
,
where u(·) is a complex baseband signal, f is the carrier frequency and φ0 is a
random initial phase (uniformly distributed between 0 and π).
Basic propagation mechanisms
1. Reflection
2. Diffraction
3. Scattering
Free space loss
The received signal
r(t) = u(t)
λ
√
GtGrej 2πd
λ
4πd
,
where d is the distance, Gt and Gr are the transmit and receive antenna power
gains and λ is the wavelength.
The received power
Pr = Pu
λ
4πd
2
GtGr.
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2. Ground reflection – Two-path model
See figure 2.4 of Goldsmith.
r(t) =
λ
4π
GtGr

u(t)ej 2πd
′
λ
d′ +
u(t + τ)Rej 2πd
′′
λ
d′′

 , (1)
where d
′
is the LOS distance, d
′′
is the ground reflected distance, τ = d
′′
−d
′
c is the
path delay and
R =
sin θ −
√
ǫr − cos2θ
sin θ +
√
ǫr − cos2θ
,
for horizontal polarization and
R =
sin θ −
√
ǫr − cos2θ/ǫr
sin θ +
√
ǫr − cos2θ/ǫr
,
where θ is the angle of reflection.
For d > dc = 4hthr
λ , we have
Pr ≈
GtGr(hthr)2
d4
Pu,
i.e., the signal decays as d−4. For all values of d a reasonable approximation is
Pr ≈
GtGrd2
0
d2(1 + (d/dc)2q)1/q
Pu,
for some values of q and d0.
General ray tracing
r(t) =
λ
4π
GtGr

u(t)ej 2πl
λ
l
+
i∈all paths
u(t + τi)Riej
2πli
λ
li

 .
Simplified path loss model
Pr = PuK
d0
d
γ
,
with γ often between two and six.
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3. Log-normal shadowing
10 log10
Pr
Pu
= 10 log10 K − 10γ log10
d
d0
+ ψdb,
where ψdb is a zero-mean Gaussian random variable.
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