Path-loss and Shadowing (Large-scale Fading)

Path-loss and Shadowing (Large-scale Fading)

PROF. MICHAEL TSAI 2015/03/27

Multipath

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3

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Friis Formula

TX Antenna

EIRP=
__



Power spatial density ^

^

41  

×

×

RX Antenna

⇒
_=
__ 4

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Antenna Aperture

• Antenna Aperture=Effective Area

• Isotropic Antenna’s effective area _, ≐ _^

• Isotropic Antenna’s Gain=1

•  = ^_ _

• Friis Formula becomes:  =  _!^_^ = _^_!^_

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 =    4

Friis Formula

• ^



! ^ is often referred as “Free-Space Path Loss” (FSPL)

• Only valid when d is in the “far-field” of the transmitting antenna

• Far-field: when ! > !_#, Fraunhofer distance

• !_# = ^$_^, and it must satisfies !_# ≫ $ and !_# ≫ 

• D: Largest physical linear dimension of the antenna

• &: Wavelength

• We often choose a !_' in the far-field region, and smaller than any practical distance used in the system

• Then we have  ! =  !_' ^!_!^{' }

 =   ^

! ^

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Received Signal after Free-Space Path Loss

 = (   )* − ,!

! -. )* ,#_/

phase difference due to propagation distance

Free-Space Path Loss

Complex envelope

Carrier (sinusoid)

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Example: Far-field Distance

• Find the far-field distance of an antenna with

maximum dimension of 1m and operating frequency of 900 MHz (GSM 900)

• Ans:

• Largest dimension of antenna: D=1m

• Operating Frequency: f=900 MHz

• Wavelength:  = _#^/ = _3''×1'^0×1'2₄ = '. 00

• !_# = ^6_^ = _'.00^ = 4. '4 (m)

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Example: FSPL

• If a transmitter produces 50 watts of power, express the transmit power in units of (a) dBm and (b) dBW.

• If 50 watts is applied to a unity gain antenna with a 900 MHz

carrier frequency, find the received power in dBm at a free space distance of 100 m from the antenna. What is the received power at 10 km? Assume unity gain for the receiver antenna.

• Ans:

• 1' 8-_1' 9' = 1: !;< = : =>?

• Received Power at 100m

(1''A) = 9' × 1 × 1 × 0 × 1'3'' × 1'^{2 4}



 × 1'' ^ = 0. 9 × 1'^C4 <

= −9. 9 (!;<)

• Received Power at 10km

 1'DA =  1''A 1'' 1''''



= 0. 9 × 1'^C1' <

= −3. 9 (!;<) 11

Two-ray Model

TX Antenna

 E

F

G G′

RX Antenna ℎ_

ℎ_

_J

_K

_L

_M

N_CJO P = QR &

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_J_LST P exp − X2F&

F + Q _K_MST P − [ exp − X2 G + G& ^\

G + G^\ exp X2]_KP

Delayed since x+x’ is longer. [ = (G + G^\ − F)/_

R: ground reflection coefficient (phase and amplitude change) 12

Two-ray Model:

Received Power

•  =  _^{  `}₈^a + ^/!_)d)\^{)* C,bc }

• The above is verified by empirical results.

• bc = () + )^\ − 8)/

• ) + )^\ − 8 = e + e_ + !^ − e − e_ + !^

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l x

x’

x’

x

Two-ray Model: Received Power

• When ! ≫ e + e, bc = ^{ )d)}_ ^fC8 ≈ ^e_!^e

• For asymptotically large d, ) + )^\ ≈ h ≈ !, E ≈ ', i_ji_k ≈

_/!, l ≈ −1 (phase is inverted after reflection)

• ₈^{`a</sup> + <sup>/!</sup><sub>)d)\</sub><sup>)* C,bc </sup> ≈ <sup></sup><sub>!</sub><sup>`a } 1 + )* −,bc ^

• 1 + )* −,bc ^ = 1 − / bc ^ + m^bc =  −

nop bc = m^(^bc_ ) ≈ bc^

•  ≈ ^_!^{`a  e</sup><sub>!</sub><sup>e </sup> q<sub>r</sub> = <sup>`}_!^a_^{e e } q_r

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Independent of & now

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_K = 4ℎ_ℎ_

&

ℎ_

Could be a natural choice of cell size

Indoor Attenuation

• Factors which affect the indoor path-loss:

• Wall/floor materials

• Room/hallway/window/open area layouts

• Obstructing objects’ location and materials

• Room size/floor numbers

• Partition Loss:

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Partition type Partition Loss (dB) for 900-1300 MHz

Floor 10-20 for the first one,

6-10 per floor for the next 3, A few dB per floor afterwards.

Cloth partition 1.4

Double plasterboard wall 3.4

Foil insulation 3.9

Concrete wall 13

Aluminum siding 20.4

All metal 26

Simplified Path-Loss Model

• Back to the simplest:

• _s: reference distance for the antenna far field (usually 1-10m indoors and 10-100m outdoors)

• t: constant path-loss factor (antenna, average channel attenuation), and sometimes we use

• u: path-loss exponent

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_ =
_t _s



v

t = &

4_s

Some empirical results

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Measurements in Germany Cities

Environment Path-loss

Exponent

Free-space 2

Urban area cellular radio 2.7-3.5 Shadowed urban cellular

radio

3-5

In building LOS 1.6 to 1.8 Obstructed in building 4 to 6 Obstructed in factories 2 to 3

Empirical Path-Loss Model

• Based on empirical measurements

• over a given distance

• in a given frequency range

• for a particular geographical area or building

• Could be applicable to other environments as well

• Less accurate in a more general environment

• Analytical model: / is characterized as a function of distance.

• Empirical Model: / is a function of distance including the effects of path loss, shadowing, and multipath.

• Need to average the received power measurements to remove multipath effects
Local Mean Attenuation (LMA) at distance d.

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Example: Okumura Model

• Okumura Model:

_w ! !; = w #_/, ! + _x #_/, ! −  e −  e − _(y

• w #_/, ! : FSPL, _x #_/, ! : median attenuation in addition to FSPL

•  ℎ_ = 20 log_~s(_ss^€ ) ,  ℎ_ =  10 log_~s ^_ƒ^‚ , ℎ_ ≤ 3†, 20 log_~s ^_ƒ^‚ , 3† < ℎ_ < 10†.

:antenna height gain factor.

• _(y: gain due to the type of environment

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Example: Piecewise Linear Model

• N segments with N-1

“breakpoints”

• Applicable to both

outdoor and indoor channels

• Example – dual-slope model:

• t: constant path-loss factor

• u_~: path-loss exponent for _s~_K

• u: path-loss exponent after _K 21

_  =
_t _s



v_‰

_t _K



v_‰ 

_K

v_

_s ≤  ≤ _K,

 > _K.

Shadow Fading

• Same T-R distance usually have different path loss

• Surrounding environment is different

• Reality: simplified Path-Loss Model represents an

“average”

• How to represent the difference between the average and the actual path loss?

• Empirical measurements have shown that

• it is random (and so is a random variable)

• Log-normal distributed

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Log-normal distribution

• A log-normal distribution is a probability distribution of a random variable whose logarithm is normally

distributed:

• G: the random variable (linear scale)

• —, ˜: mean and variance of the distribution (in dB)

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]_™ G; —, ˜ = 1

G˜ 2exp − log G − —  2˜

logarithm of the random variable

normalized so that the integration of the pdf=1

Log-normal Shadowing

• Expressing the path loss in dB, we have

• ›_œ: Describes the random shadowing effects

• ›_œ~(', œ^)

(normal distribution with zero mean and œ^ variance)

• Same T-R distance, but different levels of clutter.

• Empirical Studies show that œ ranges from 4 dB to 13dB in an outdoor channel

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_ž  Ÿ =
_ž  +  _¡ =
_ž _s + 10u log 

_s +  _¡

Why is it log-normal distributed?

• Attenuation of a signal when passing through an object of depth d is approximately:

• ¢: Attenuation factor which depends on the material

• If £ is approximately the same for all blocking objects:

• _ = ∑ _{¥ ¥}: sum of all object depths

• By central limit theorem, ! ~(x, œ^) when the number of object is large (which is true).

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¦  = exp −¢

¦ _ = exp −¢ § _¥

¥

= exp −¢_

Path Loss, Shadowing, and Multi-Path

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Cell Coverage Area

• Cell coverage area: expected percentage of locations within a cell where the received power at these

locations is above a given minimum.

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Some area within the cell has received

power lower than

_¥¨ Some area outside of

the cell has received power higher than

_¥¨

Cell Coverage Area

• We can boost the transmission power at the BS

• Extra interference to the neighbor cells

• In fact, any mobile in the cell has a nonzero

probability of having its received power below _Am.

• Since Normal distribution has infinite tails

• Make sense in the real-world:

in a tunnel, blocked by large buildings, doesn’t matter if it is very close to the BS

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Cell Coverage Area

• Cell coverage area is given by

• _ ≐ *   > _Am

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© = y 1

(^ ª 1   > _Am m ! !

/88 ``

= 1

(^ ª y 1   > _Am m ! !

/88 ``

1 if the statement is true, 0 otherwise.

(indicator function)

© = 1

(^ ª _!

/88 `` = 1

(^ ª ª ⁽ _

' !



' !«

Cell Coverage Area

• Q-function:

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_¬ = ­
® N ≥
_{ ¥¨} = °
_¥¨ −
_ −
_ž N

˜

° ± ≐ ­   > ± = ª 1 2

²

³ exp − ´

2 ´

Log-normal distribution’s standard deviation

z

Cell Coverage Area

• Solving the equations yield:

• ` = ^AmC_œ ⁽ , a = ^1'µ8-_œ^{1' }

• If _Am =  (

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¶ = ° · + exp 2 − 2·¸

¸ ° 2 − ·¸

¸

average received power at cell boundary (distance=R)

¶ = 1

2 + exp 2

¸ ° 2

¸

Example

• Find the coverage area for a cell with

• a cell radius of 600m

• a base station transmission power of 20 dBm

• a minimum received power requirement of -110 dBm.

• path loss model:
_() =
_t ^M_M^{¹ v}, u = 3.71, t = −31.54 Ÿ, _s = 1, shadowing standard deviation ˜ = 3.65 dB

• Ans:

•  ( = ' − 01. 9 − 1' × 0. :1 × 8-_1' 4'' = −11. 4 !;A

• ` = ^C11'd11.4_0.49 = 1. 4, a = 1'×0.:1×'.0

0.49 = . 1

• © = ¾ 1. 4 + )* −'. 4 ¾ −'. 2': = '. 4 (not good)

• If we calculate C for a minimum received power requirement of -120 dBm

• C=0.988!

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¶ = ° · + exp 2 − 2·¸

¸ ° 2 − ·¸

¸

` = ^AmC_œ ⁽ , a = ^1'µ8-_œ^{1' }

Example: road corners path loss

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5 m 40 m

10 m Radio: Chipcon CC2420

IEEE 802.15.4, 2.4 GHz TX pwr: 0 dBm

8 dBi peak gain omni-directional antenna

Intel-NTU Connected Context Computing Center

• Compare the path-loss exponent of three different locations:

1. Corner of NTU_CSIE building 2. XinHai-Keelong intersection 3. FuXing-HePing intersection

Link Measurements –

Path loss around the corner building

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1 2 3 Passing-by

vehicles

Occasionally Frequently Frequently Buildings

Around

No Few

buildings

Some high buildings

Intersection Narrow Wide Wide

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Quiz 3

• Out of all 4G (LTE) cellular service providers in Taiwan, if we consider only FSPL (1) which provider would give you the best SNR? (2) what’s the difference (in dB) of SNR comparing signals from the “best” and the

“worst” providers?

• You can assume that all the other factors are the same (base station antennas and cellular phone antennas)

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