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Examples of Large-Scale Path Loss Models

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Free Space Propagation Model

• The free space power received by a receiver antenna which isseparated from a radiating transmitter antenna by a distanced, isgiven by the Friis free space equation

Pr(d) =PtGtGrλ

2

(4π)2d2L

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Free Space Propagation Model (continued)

• In the Friis free space equation of the previous slide

⋆ Pt = transmitted power

⋆ Pr(d) = received power which is a function of the T-R separation

⋆ Gt = transmitter antenna gain

⋆ Gr = receiver antenna gain

⋆ L = system loss factor not related to propagation(L ≥ 1)

⋆ λ = wavelength in meters

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Path Loss for the Free Space Model

• The path loss for the free space model when the antenna gains are

included is given by

PL(dB) = 10 log

Pt

Pr

= −10 log

[

GtGrλ2

(4π)2 d2

]

• When the antenna are assumed to have unity gain, the path lossfor

the free space model is given by

PL(dB) = 10 log

Pt

Pr

= −10 log

[

λ2

(4π)2d2

]

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Example: Free Space Model

A mobile receiver is located at the cell boundary which isR km from the

base station. The base station transmits at the carrier frequency is

2000MHz with 1W power. Transmitting antenna gainGt = 1.64 and

receiving antenna gainGr = 1. Assume the speed of propagation for

electromagnetic waves =3 × 108m/s, miscellaneous loss is equal to 1 and

minimum power required by the mobile to operate is -90dBm. The

heights of the transmitting and receiving antennas are 50m and 1.5m,

respectively.Please find the cell radius,R (km), when path loss obeys the

Free Space model.

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Ground Reflection (2-Ray) Model

• At large values ofd, the received power is

Pr(d) =PtGtGrh

2

th2

r

d4

where

⋆ ht = transmitter antenna height

⋆ hr = receiver antenna height

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Path Loss for the 2-Ray Model

• The path loss for the 2-ray model (with antenna gains) can be

expressed in dB as

PL(dB) = 10 log

Pt

Pr

= 10 log

Pt

PtGtGr

h2

th2

r

d4

= 10 log

d4

GtGrh2t h

2r

= 40 log d − 10 log (GtGr) − 20 log (hthr)

• Comments:Both Free Space and 2-ray models are based on theory

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Example: 2-Ray Model

A mobile receiver is located at the cell boundary which isR km from the

base station. The base station transmits at the carrier frequency is

2000MHz with 1W power. Transmitting antenna gainGt = 1.64 and

receiving antenna gainGr = 1. Assume the speed of propagation for

electromagnetic waves =3 × 108m/s and minimum power required by the

mobile to operate is -90dBm. If the heights of the transmitting and

receiving antennas are 50m and 1.5m, respectively,please find the cell

radiusR km, when the path loss obeys 2-ray model.

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Log-distance Path Loss Model

• It indicates thataverage received signal powerdecreases

logarithmically with distance

PL(d) = PL(d0) + 10n log

d

d0

⋆ n = path loss exponent which indicates the rate at which the path

loss increases with distance

⋄ In free spacen = 2, in urban areasn = 2.7 − 4

⋆ d0 = close-in reference distance which is determined from

measurements close to the transmitter

⋆ d = T-R separation distance

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Limitations of Previous Path Loss Models

• The previous models ignore the fact that the surrounding

environment clutter may be vastly different at two different locations

having the same T-R separation distance

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Log-normal Shadowing Model

• This model takes into account the fact that at any value ofd, the path

lossPL(d) at a particular location israndom and distributedlog-normally (normal in dB) about the mean distance dependent

value

• According to this model path loss at a T-R separation distanced is

expressed as

PL(d)[dB] = PL(d) + Xσ = PL(d0) + 10n log

d

d0

+ Xσ

• Xσ ∼ N (0, σ) , i.e. Xσ is a zero-mean Gaussian distributed random

variable (in dB) with standard deviationσ (also in dB)

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Received Signal under Log-normal Shadowing

• The received signal for this model is expressed as

Pr(d) = Pt − PL(d)

=

[

Pt − PL(d0) − 10n log

d

d0

]

− Xσ

= Pr(d) + Xσ

• Comments:Pr(d) ∼ N(

Pr(d), σ)

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QoS at Distanced

• The probability that the received signal level at distanced will exceed

a certain value (threshold)γ can be calculated as

Prob[Pr(d) > γ] = Q

γ − Pr(d)

σ

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Example: Log-normal Shadowing

• The base station transmits at the carrier frequency is 2000MHz with 1W

• The speed of propagation for electromagnetic waves =3 × 108m/s

• The heights of the transmitting and receiving antennas are 50m and 1.5m,

respectively

• Transmitting antenna gainGt = 1.64 and receiving antenna gainGr = 1

• The measured power at reference distanced0 = 0.1 km from the base station

follows the Free Space model

• At distanced > d0, the mean of the received power is proportional to1/d4

and lognormally distributed with standard deviation 10dB

• Minimum power required by the mobile to operate is -90dBm, which must

be guaranteed at the cell boundary with probability 0.75

• Please find the radius,R, of the cell

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Conclusions of Today’s Class

• As the environment becomes challenging,for a given transmitted

power, the cell coverage area reduces

• To handle the above situation,the transmitter power should be

adjusted based on the environment and the target coverage area

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