Introduction to Propagation Modeling - Aaltogeta.aalto.fi/en/courses/propagation_modeling_slides-002.pdf · Introduction to Propagation Modeling ... • Free space attenuation model

Introduction to Propagation

Modeling Dr. Jussi Salmi

GETA Winter School 2012

13.2.2012

Outline

• Motivation

• Physical Propagation Mechanisms – Free space, reflection, diffraction, scattering

• Statistical Descriptions – Path loss, shadowing and small-scale fading

• Deterministic Abstractions – Bandwidth and Signal duration, Tapped Delay Line model, and

Double Directional Propagation Path model

• Application example: UWB MIMO Radar

13.2.2012 Jussi Salmi

Motivation

• Radiowave propagation is a key factor in any wireless

localization system

– Line-of-sight (LOS) vs. Non-line-of-sight (NLOS)

– Path loss modeling and fading due to multipath

• The scope of this talk is to

– Introduce typical physical propagation mechanisms

– Familiarize with statistical modeling concepts

– Introduce the concept of double-directional propagation paths

– Application example: UWB MIMO Radar


Physical Description vs. Statistical (or

Deterministic) Abstractions • Physical description of propagation mechanisms include

– Free space ”attenuation”

– Reflection

– Diffraction

– Scattering

– Waveguiding

• Statistical descriptions include – Path loss

– Shadow (slow) fading

– Small-scale (fast) fading

• Deterministic Abstraction – Tapped delay line model

– Double directional propagation path modeling


Physical Propagation Mechanisms

• Introductory description of

– Free space ”attenuation”

– Reflection

– Diffraction

– Scattering

– Waveguiding

• This part is mostly based on:

A. F. Molisch, ”Wireless Communications”,

2nd edition, Wiley 2011


Physical propagation mechanisms:

Free Space Attenuation • Tx antenna in free space

– Transmitted power 𝑃𝑇𝑥 is distributed into free space

– For an isotropic radiator, the power density per m2 at distance 𝑑

is given by 𝑃𝑇𝑥 4𝜋𝑑2

– An integral of the power density over any closed surface

surrounding the Tx antenna must be equal to 𝑃𝑇𝑥



Free Space Attenuation

• Tx and Rx in free space

– For an isotropic radiator, the power density per m2 at distance 𝑑

is given by 𝑃𝑇𝑥 4𝜋𝑑2

– And the received power, for an antenna with an ”effective area”

of 𝐴𝑅𝑥 as a function of distance is

𝑃𝑅𝑥(𝑑) = 𝑃𝑇𝑥

1

4𝜋𝑑2𝐴𝑅𝑥


• Non-isotropic Tx

– If the Tx antenna is not isotropic, but has gain 𝐺𝑇𝑥 towards the

direction of the Rx, then we have

𝑃𝑅𝑥(𝑑) = 𝑃𝑇𝑥𝐺𝑇𝑥

1

4𝜋𝑑2𝐴𝑅𝑥




• Rx Gain vs. effective area

– Rx gain can be expressed as

– i.e. for fixed antenna area, antenna gain increases with

frequency

𝐺𝑅𝑥 =4𝜋

𝜆2𝐴𝑅𝑥




• Friis’ law

– Solving for effective area of the Rx antenna yields

– Inserting to equation of received power

– which is known as the Friis’ law

𝐴𝑅𝑥 = 𝐺𝑅𝑥

𝜆2

4𝜋

𝑃𝑅𝑥 𝑑 = 𝑃𝑇𝑥𝐺𝑇𝑥

1

4𝜋𝑑2𝐴𝑅𝑥 = 𝑃𝑇𝑥𝐺𝑇𝑥𝐺𝑅𝑥

𝜆

4𝜋𝑑

2




• Note that the ”attenuation” in free space does not increase with frequency – However Friis’ equation seems to indicate that

– The spreading of the signal energy over a sphere can not

depend on the frequency

• The seeming contradiction is caused by the fact that the antenna gain 𝐺𝑅𝑥 is assumed to be independent of the frequency – The size of a similar type of antenna (having the same gain)

decreases; thus the effective area decreases

𝑃𝑅𝑥 𝑑 = 𝑃𝑇𝑥𝐺𝑇𝑥𝐺𝑅𝑥

𝑐

4𝜋𝑑𝑓

2



𝑓 = 𝑐/𝜆


• If on the other hand the effective area of the Rx antenna

is independent of the frequency, then the received

power would be independent of the frequency too

• Note also that the validity of the Friis’ law is restricted to

the far field of the antenna, requiring

– Moreover, (at least) the 1st Fresnel zone needs to be free of

obstacles

𝑑 >2𝐿𝑎

2

𝜆 𝑑 ≫ 𝜆




• Free space attenuation model as such is applicable only in very few terrestrial radio services – Fixed radio links or relays ar an example of such, but these

would not be of interest from localization point of view

• Free space attenuation provides a baseline for other path loss models

• Note that also the antenna gains 𝐺𝑇𝑥 and 𝐺𝑅𝑥 may have significant influence on received signal strength (RSS) based positioning techniques





Reflection

• Examples of reflection from conductive material

– Round conductive object -> another source of a (semi)spherical

wavefront (within a limited angle)

– Flat conductive object

-> reflection at a specific direction

Incoming planewave



Reflection

• Dielectric surface

– Characterized by complex dielectric constant

Where 𝜖 is the dielectric constant, and 𝜎𝑒 is conductivity (relative

magnetic permeability is assumed to be 𝜇𝑟 = 1)

• Radiowave is partially reflected, partially transmitted

– Reflection angle is equal to the incidence angle Θ𝑟 = Θ𝑒

– Transmission angle is given by Snell’s law:

𝛿 = 𝜖 − 𝑗𝜎𝑒

2𝜋𝑓𝑐

sin(Θ𝑟)

sin (Θ𝑒)=

𝛿1

𝛿2



Reflection

• The amplitude and phase of the reflected

and transmitted wave depend on

– Dielectric constants of the media

– Angle of incidence

– Polarization

• Example, concrete at 1 GHz

– 𝛿𝑟 = 7 − 𝑗0.85

Reflection and transmission from dielectric interface

(from A. F. Molisch, Wireless Communications, 2nd ed.)

0 10 20 30 40 50 60 70 80 90

-180

-90

0

e

arg

( )

(deg

)

TM

TE

0 10 20 30 40 50 60 70 80 900

0.2

0.4

0.6

0.8

1

e

| |

TM

TE



Diffraction

• Radiowaves do propagate ”around the corner”

• Models exist for several typical cases

– Knife-edge – single or multiple (conductive) edges

– Lossy wedge

– Polarization also plays a role



Diffraction

• The magnitude behind a single edge is given by

where 𝐹 𝑣𝐹 is the Fresnel integral, and the Fresnel

parameter is obtained by

𝜃𝑑

𝑑𝑅𝑥 𝑑𝑇𝑥 Tx Rx

Knife-edge 𝑣𝐹 = 𝜃𝑑

2𝑑𝑇𝑥𝑑𝑅𝑥

𝜆(𝑑𝑇𝑥+𝑑𝑅𝑥)

1

2−

exp𝑗𝜋4

2𝐹 𝑣𝐹



Diffraction

• Example: Single Knife-edge

– Field strength at 1 GHz in a given point behind knife-edge

-10 0 10 20 30 40 500

5

10

15

20

25

30

x (m)

y (

m)

-25

-20

-15

-10

-5

0



Scattering

• Wave interaction with ”rough” surfaces

– Sometimes ”roughness” in wireless channel modeling is also

used to describe irregularities in building structures e.g. bookshelves, windowsills etc.

𝜌𝑠𝑚𝑜𝑜𝑡ℎ 𝜌𝑟𝑜𝑢𝑔ℎ



Scattering

• Example: Kirchoff theory of scattering from rough

surfaces

– Reflection coefficient is modified

as

– Where 𝜎ℎ is the standard deviation

of the height distribution

• Generalization for including spatial correlation and

”effective dielectric constant” exist (perturbation theory)

𝜌𝑟𝑜𝑢𝑔ℎ = 𝜌𝑠𝑚𝑜𝑜𝑡ℎ ∙ exp (−2 ∙2𝜋

𝜆𝜎ℎ sin Θ𝑒

2

)



Waveguiding

• Typical description of propagation in e.g. street canyons,

corridors and tunnels

• Refers to both

– Theoretical waveguiding modes, specific to the shape and

dimension of the environment (while being in the order of

wavelength or so)

– Geometric optics approximation of waves being ”guided” along a

pathway

• Waveguiding may yield a path loss exponent smaller

than 2, i.e. that of free space propagation

– Typically irregularities and lossy materials increase attenuation



Conclusion

• Free-space attenuation forms a baseline for path loss

modeling

– Frequency dependency is only due to assumption about the

antenna type!

• Material interactions may have frequency dependency

– Excluding reflection from a conductive surface…

– Material parameters as well as polarization have significant

influence on reflection, diffraction, scattering, waveguiding


Statistical Propagation Descriptions

• Path loss

• Shadow (slow) fading

• Small-scale (fast)fading

• Doppler

– This part is mostly based on:

L. J. Greenstein, A. F. Molisch, and M. Shafi, ”Propagation

issues for Cognitive Radio”, to appear in a book.

101

102

103

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-80

-70

-60

-50

-40

Distance (m)

Po

wer

(d

B)

Received power

Shadow fading

Median Path Gain


Statistical Propagation Descriptions:

Path Loss vs. Shadow Fading vs. Fast Fading

101

102

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Distance (m)

Pow

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dB

)

Received signal strength (RSS)

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102

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Distance (m)

Pow

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dB

)

Received power

"Slow" fading

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-50

-40

Distance (m)

Pow

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dB

)

Received power

Shadow fading

Median Path Gain



Path Loss

• Power gain, 𝐺, of a radio link is defined as the average

power over

– The frequency interval of interest

𝐹 ∈ 𝑓𝑐 −𝑊

2, 𝑓𝑐 +

𝑊

2

of bandwidth 𝑊 around center frequency 𝑓𝑐

– The local spatial domain around the potential coordinates of the

Tx and Rx positions, denoted by 𝒙

• If either link end is fixed, no averaging over that local

neighbourhood is necessary

𝐺 = 𝐴𝑣𝑒𝒙1

𝑊 𝐻(𝑓, 𝒙) 2 𝑑𝑓𝐹

=𝐴𝑣𝑒𝒙1

𝑊 ℎ(𝜏, 𝒙, 𝐹) 2 𝑑𝜏𝜏



Path Loss

• The impulse response can then be written as

– Where 𝐴𝑣𝑒𝒙1

𝑊 ℎ (𝜏, 𝒙, 𝐹)

2𝑑𝜏

𝜏= 1

• 𝐺 varies in large-scale manner, both in terms of – location (tens of wavelengths)

– frequency (hundreds of MHz)

• ℎ (𝜏, 𝒙, 𝐹) varies in small-scale, i.e., potentially within – fraction of the wavelength or

– within communication signal bandwidth (tens of MHz or smaller)

ℎ 𝜏, 𝒙, 𝐹 = 𝐺 ∙ ℎ (𝜏, 𝒙, 𝐹)


• Path loss (in dB) is defined as

𝐿 = −10 ∙ log 10(𝐺)


Path Loss

101

102

103

-130

-120

-110

-100

-90

-80

-70

-60

-50

-40

Distance (m)

Po

wer

(d

B)

Received power

Shadow fading

Median Path Gain

101

102

103

55

60

65

70

75

80

85

90

95

100

Pat

h L

oss

(d

B)

Distance (m)


• A generic formula for the power gain of a wireless link:

– 𝑃𝑅 denotes average over signal BW and local area

– 𝑑0 is a reference distance, e.g. 1 m

– 𝑎 and 𝛾 are dimensionless model parameters

– For example, for free space path attenuation with 𝐺𝑅 = 𝐺𝑇 = 1:

𝐺(𝑑) =𝑃𝑅

𝑃𝑇~𝑎

𝑑

𝑑0

−𝛾


Path Loss

𝑎 =𝜆

4𝜋𝑑0

2 and 𝛾 = 2


• Note that typically 𝛾 is greater than 2

– However, also 𝛾 < 2 is possible as mentioned, due to

waveguiding effects in some environments

• The above relation is not accurate for all distances

– Hence it should be rather treated as the median power gain:

𝐺(𝑑) ~𝑎𝑑

𝑑0

−𝛾


Path Loss

𝐺𝑚𝑒𝑑 𝑑 = 𝑎𝑑

𝑑0

−𝛾


• A more accurate model can be achieved by

• Turning this into path loss in dB

– With 𝐴 = −10 ∙ log 10 𝑎 and 𝑆 = −10 ∙ log10 𝜉

– Hence we have: 𝑃𝐿𝑚𝑒𝑑 = 𝐴 + 𝛾 ∙ 10 ∙ log 10𝑑

𝑑0


Path Loss

𝐺 𝑑 = 𝑎𝑑

𝑑0

−𝛾

∙ 𝜉 = 𝐺𝑚𝑒𝑑 𝑑 ∙ 𝜉

𝐿 𝑑 = 𝐴 + 𝛾 ∙ 10 ∙ log 10𝑑

𝑑0+ 𝑆

(𝐿 = −10 ∙ log 10(𝐺))


• An example of a model for the median path loss is the

Hata-Okumura

– Urban environment, user terminal at 1.5 m, and 𝑑0=1.5 km, and

base station height ℎ𝑏 (in m), carrier frequency 𝑓𝑐 (in MHz)

• The frequency dependency stems partly from aperture effect and

partly due to diffraction loss increasing with frequency


Path Loss

𝑃𝐿_𝑚𝑒𝑑 = 𝐴 + 𝛾 ∙ 10 ∙ log 10𝑑

𝑑0

𝐴 = 69.55 + 26.16 ∙ log10 𝑓𝑐 − 13.82 ∙ log10 ℎ𝑏

𝛾 = 4.49 − 0.655 ∙ log10 ℎ𝑏


• The variable 𝑆 represents random variation about the

median path loss

• Empirical studies have indicated that 𝑆 (in dB) is best

modeled as a zero mean normal distributed RV

– with the standard deviation 𝜎𝑠 depending on the environment


Shadow Fading

𝐿 𝑑 = 𝐴 + 𝛾 ∙ 10 ∙ log 10𝑑

𝑑0+ 𝑆

𝑆~𝑁(0, 𝜎𝑆2)


• The term shadow fading stems from history of cellular

network modeling, as buldings were ”shadowing” the

signal between BS and MS


Shadow Fading

101

102

103

-130

-120

-110

-100

-90

-80

-70

-60

-50

-40

Distance (m)

Pow

er (

dB

)

Received power

Shadow fading

Median Path Gain

101

102

103

-5

-4

-3

-2

-1

0

1

2

3

4

5

Sh

ado

w f

adin

g (

dB

)

Distance (m)


• Typical values for 𝜎𝑆 range between 3-12 dB

• Spatial correlation and dynamics of shadowing are less

studied

– These would be useful for localization modeling purposes too!

• One potential model is given by

– Where 𝑋𝑐 is so-called correlation distance of shadow fading

• Values can range from 10 m in urban microcells to 500 m in rural

macrocells

– Often too simple, but easy to implement via an AR(1) process


Shadow Fading

𝜌 Δ𝑥 = 𝑆 𝑥 𝑆(𝑥 + Δ𝑥) = 𝜎𝑆2 ∙ exp (−

Δ𝑥

𝑋𝑐)


• Recall

– With 𝐺 𝑑 = 10−𝑃𝐿𝑚𝑒𝑑+𝑆

10 and 𝐴𝑣𝑒𝒙1

𝑊 ℎ (𝜏, 𝒙, 𝐹)

2𝑑𝜏

𝜏= 1


Small-scale Fading

101

102

103

-130

-120

-110

-100

-90

-80

-70

-60

-50

-40

Distance (m)

Pow

er (

dB

)

Received power

Shadow fading

Median Path Gain

0 100 200 300 400 500 600 700 800 900 1000-25

-20

-15

-10

-5

0

5

10

15

Distance (m)

Pow

er (

dB

)

Fast fading

ℎ 𝜏, 𝒙, 𝐹 = 𝐺 ∙ ℎ (𝜏, 𝒙, 𝐹)

|ℎ (𝜏, 𝒙, 𝐹)|


• Demo for WLAN 802.11g

– 20 MHz, 52 OFDM subcarriers

– 10 random reflectors + LOS, Rx moves at 1 m/s


Small-scale Fading

-100 -50 0 50 100-100

-80

-60

-40

-20

0

20

40

60

80

100

x (m)

y (

m)

Time: 10 s

2395 2400 2405 2410 2415 2420-70

-60

-50

-40

-30

-20

-10

0

Frequency (MHz)

20lo

g 10(|

H(f

)|)

(dB

)


fading1.exe

• ”Ideal” model was used, where the channel is given by

where 𝑑𝑝denotes the path length of the pth multipath component


Small-scale Fading

-100 -50 0 50 100-100

-80

-60

-40

-20

0

20

40

60

80

100

x (m)

y (

m)

Time: 10 s

2395 2400 2405 2410 2415 2420-70

-60

-50

-40

-30

-20

-10

0

Frequency (MHz)

20lo

g 10(|

H(f

)|)

(dB

)

ℎ 𝑓, 𝑡 = 𝑑𝑝−1exp (−𝑗2𝜋

𝑓

𝑐𝑑𝑝)

𝑝


fading1.exe

• Demo for WLAN 802.11g

– 20 MHz, 52 OFDM subcarriers

– 10 random reflectors + LOS, Rx moves at 1 m/s


Small-scale Fading

0 2 4 6 8 10

5

10

15

20

25

30

35

40

45

50

Time (s)

Subca

rrie

r in

dex

-50

-45

-40

-35

-30

-25

-20

0 2 4 6 8 10-55

-50

-45

-40

-35

-30

-25

-20

-15

Time (s)

20lo

g 10(|

H(f

)|)

(dB

)


• Small-scale fading results from the constructive and

destructive interplay of signal arrivals via different paths,

i.e., multipath propagation


Small-scale Fading

The phase of the signal at

reception point depends on

the path length and

interactions, e.g. phase

change during reflection etc.


• Special case: Rayleigh fading

– Signal is a superposition from many multipath components (MPCs)

with similar magnitude -> Central limit theory applies, and the

signal can be modeled as a complex Gaussian

– Phase has uniform pdf over (−𝜋, 𝜋], and magnitude 𝑟 has a

Rayleigh pdf:

where σ is the standard deviation of the underlying Gaussian process


Small-scale Fading

𝑝𝑑𝑓 𝑟 =𝑟

𝜎2 ⋅ exp −𝑟2

2𝜎2 0 ≤ 𝑟 < ∞


• Rayleigh fading cont.

– The squared magnitude 𝑝 of a Rayleigh fading has a decaying

exponential pdf:

where 𝑝 𝑟 = 2𝜎2 is the mean power (local spatial average)

– Dynamic range between the 1st and 99th percentile is 26.6 dB!

• Fading at a given frequency can be very significant!


Small-scale Fading

𝑝𝑑𝑓 𝑝 =1

𝑝 𝑟⋅ exp −

𝑝

𝑝 𝑟 0 ≤ 𝑝 < ∞


• The spread of the multipaths influences the ”fastness” of

the fading

– If MPCs arrive at similar directions, fading is not that ”fast”

– ”Fastest” fading occurs when MPC arrive from all directions


Small-scale Fading


• Examples: 100 reflectors, evenly spread


Small-scale Fading

-100 -50 0 50 100-100

-80

-60

-40

-20

0

20

40

60

80

100

x (m)

y (

m)

Time: 10 s

0 1 2 3 4 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

r

pdf

Empirical

Rayleigh


• Examples: 10 reflectors, randomly spread


Small-scale Fading

-100 -50 0 50 100-100

-80

-60

-40

-20

0

20

40

60

80

100

x (m)

y (

m)

Time: 10 s

0 1 2 3 4 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

r

pdf

Empirical

Rayleigh


• Examples: 10 reflectors, in similar direction, no LOS


Small-scale Fading

-100 -50 0 50 100-100

-80

-60

-40

-20

0

20

40

60

80

100

x (m)

y (

m)

Time: 10 s

0 1 2 3 4 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

r

pdf

Empirical

Rayleigh


fading1_tight.exe

• Notes about Rayleigh distribution

– Excellent approximation in a large number of scenarios

• However, many scenarios exists (LOS etc.) where Rayleigh fails

– Depends on a single parameter – the mean received power – only

– Mathematically convenient: Often enables analytical analysis in

closed form

– Spatial (temporal) correlation needs to be modelled separately


Small-scale Fading


• More general case: Ricean fading

– The magnitude distribution is modelled Ricean, if there is a

dominant component with power 𝑝 𝑑

– Given total average received power 𝑝 𝑟, the average power of the

weaker components is 𝑝 𝑠 = 𝑝 𝑟 − 𝑝 𝑑

– The Ricean K factor is defined as the ratio

and the pdf of the received amplitude is defined as:


Small-scale Fading

𝐾 =𝑝 𝑑𝑝 𝑠

𝑝𝑑𝑓 𝑟 = 2𝑟exp (−𝐾)

𝑝 𝑠⋅ exp −

𝑟2

𝑝 𝑠⋅ 𝐼0

4𝑟2𝐾

𝑝 𝑠


• Ricean fading cont.

– 𝐾 = 0 equals Rayleigh

– For large K, approximates

to Gaussian distribution

• Mean depends on the

dominant component

• Spread depends on the

variance of the weaker

components

– 𝐾 = ∞, single component,

i.e. no fading (amplitude is

deterministic)


Small-scale Fading

0 1 2 3 4 5 6 7 80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

pd

f

r

K=0

K=1

K=10


• Notes about Ricean distribution

– Also the phase of the Ricean signal has a non-uniform distribution

– Ricean fading model is a good example of a mathematical tool

whose invention had no relation to wireless systems

• Solution to the problem of having deterministic phasor added to a

zero-mean complex Gaussian distribution


Small-scale Fading


• Another popular fading model is the Nakagami distribution

– Does not have as clear physical interpretation

– Has been succesfully fitted to numerous empirical data

– Parameterized by the Nakagami m-factor, defined as


Small-scale Fading

𝑚 =𝑝 𝑟2

< 𝑟2 − 𝑝 𝑟2 >


• The aforementioned description of small scale fading is

related to spatial domain

– Channel variations only occur if there is movement

• Temporal variation is a result of physical motion in the

environment

– If nothing in the propagation environment is varying – the channel

is constant


Doppler


• Considering the simulation model introduced earlier

– Complex bandpass signal (phase only) of a single path

– After downconversion (e.g. multiplied by exp −𝑗2𝜋𝑓𝑐𝑡 )


Doppler

𝑓𝑐

ℎ𝑝,𝑏𝑝 𝑓, 𝑡 = exp 𝑗2𝜋𝑓(𝑡 − 𝜏𝑝) = exp(𝑗2𝜋(𝑓𝑐 + Δ𝑓)(𝑡 − 𝜏𝑝))

With 𝜏𝑝 =𝑑𝑝

𝑐 and 𝑓 = 𝑓𝑐 + Δ𝑓

ℎ𝑝,𝑙𝑝 𝑓, 𝑡 = exp 𝑗2𝜋 𝑓𝑐 + Δ𝑓 𝑡 − 𝜏𝑝 exp −𝑗2𝜋𝑓𝑐𝑡

= exp (−𝑗2𝜋 𝑓𝑐𝜏𝑝 − Δ𝑓𝑡 + Δ𝑓𝜏𝑝 )


– After ”equalization”, e.g. multiplying by exp −𝑗2𝜋Δ𝑓𝑡 , the single

path phase term reduces to

– On the other hand, if something is moving, then

𝜏𝑝 = 𝜏𝑝 𝑡 =𝑑𝑝(𝑡)

𝑐=

𝑑𝑝(0)

+ 𝑣𝑝,𝑟 ⋅ 𝑡

𝑐

• Where 𝑣𝑝,𝑟 denotes the relative velocity, i.e., the rate of change of the

path length (depends on the direction of signal vs. that of motion etc.)

– A couple ways to proceed from here…


Doppler

ℎ𝑝 𝑓, 𝑡 = exp (−𝑗2𝜋 𝑓𝑐𝜏𝑝 + Δ𝑓𝜏𝑝 )


• Approach #1:

• Approach #2:


Doppler

ℎ𝑝 𝑓, 𝑡 = exp −𝑗2𝜋 𝑓𝜏𝑝 = exp −𝑗2𝜋 𝑓𝑑𝑝

(0)+ 𝑣𝑝,𝑟 ⋅ 𝑡

𝑐

= exp(−𝑗2𝜋 𝑓 ⋅ 𝜏𝑝0 +

𝑣𝑝,𝑟

𝑐𝑓 ⋅ 𝑡

A initial phase Doppler frequency

ℎ𝑝 𝑓, 𝑡 = exp −𝑗2𝜋 𝑓𝑐𝜏𝑝 + Δ𝑓𝜏𝑝

= exp −𝑗2𝜋 𝑓𝑐 ⋅ 𝜏𝑝0 +

𝑣𝑝,𝑟

𝑐𝑓𝑐 ⋅ 𝑡 exp −𝑗2𝜋 Δ𝑓𝜏𝑝

Only model Doppler here

Doppler w.r.t. carrier only (or Δ𝑓 = 0) A initial phase (at the carrier)


• The former loses the notion of absolute delay, i.e., path

length over time

– However, Doppler may be infered from frequency samples

• The latter maintains parameterization of the absolute

delay as well

– Doppler estimation requires sampling over time


Doppler


• Real wireless channel comprises of many paths

– Each path has unique Doppler (depends on the relative velocity,

i.e. the direction of signal vs. movement)

– The paths have different amplitudes

– Other aspects influence the phase (evolution) as well, such as

material interactions)

• All this supports Doppler to be described as a spectrum


Doppler Spectrum


• Example from the simulator (𝑓𝑐 = 2.4 GHz)

– 𝑣𝑚𝑎𝑥 = 1 m/s ⇒ 𝑓𝐷,𝑚𝑎𝑥 =𝑣𝑚𝑎𝑥

𝑐𝑓𝑐 = 8 Hz

– 100 reflectors randomly placed


Doppler Spectrum

-80 -60 -40 -20 0 20 40 60 80

-80

-60

-40

-20

0

20

40

60

80

x (m)

y (

m)

Time: 10 s

-10 -5 0 5 10-20

-15

-10

-5

0

5

10

15

20

fD

(Hz)

PS

D




𝑐𝑓𝑐 = 8 Hz

– 10 reflectors randomly placed


Doppler Spectrum

-80 -60 -40 -20 0 20 40 60

-60

-40

-20

0

20

40

60

x (m)

y (

m)

Time: 10 s

-10 -5 0 5 10-35

-30

-25

-20

-15

-10

-5

0

5

fD

(Hz)

PS

D




𝑐𝑓𝑐 = 8 Hz

– 100 reflectors on a circle with equal amplitudes


Doppler Spectrum

-80 -60 -40 -20 0

-50

-40

-30

-20

-10

0

10

20

30

40

x (m)

y (

m)

Time: 10 s

-10 -5 0 5 1020

25

30

35

40

45

50

55

fD

(Hz)

PS

D


• The Doppler power spectrum depends on the movement,

and direction distribution of the signal components

• A classical example is the Clarke-Jakes spectrum, which

is the case when signals arrive uniformly from all

directions (on the plane) at equal amplitudes

– As seen from the

previous examples,

this rarely holds


Doppler Spectrum

-10 -5 0 5 10-30

-20

-10

0

10

20

fD

(Hz)

PS

D (

dB

)

-10 -5 0 5 100

2

4

6

8

10

12

fD

(Hz)

PS

D (

lin

ear)

𝑆𝑑 𝑓𝐷 =𝑝 𝑟

𝜋 𝑓𝐷,𝑚𝑎𝑥2 − 𝑓𝐷

2


• The Fourier transform of the Doppler Spectrum is the auto-correlation

function (ACF)

– Assuming stationary movement and uniform angular distribution on

horizontal plane, the ACF for the Jakes spectrum is

– Different ACF can be derived for

different angular distributions

such as von Mises distribution,

see e.g.


Doppler Spectrum

𝐴𝐶𝐹 Δ𝑡 = 𝐽0(2𝜋𝑓𝐷,𝑚𝑎𝑥Δ𝑡)

0 0.5 1 1.5 2 2.5 3 3.5 40

0.5

1

d/

AC

F

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.5

1

t (s) (vmax

= 1 m/s)

AC

F

Small scale fading

decorrelated at 0.4𝜆

J. Salmi, et al., "Incorporating diffuse scattering

in geometry-based stochastic MIMO channel

models," EUCAP 2010.


• Modeling the K-factor (for Ricean fading) or m-factor

(Nagakami fading) is sufficient for statistical

characterization of the small scale fading

– Doppler spectrum and related ACF provide means for modeling

spatial correlation

• Combined with path loss modeling (including shadowing)

one can have a complete statistical description for

independent channel realizations

– Also spatial correlation of shadowing is of interest


Conclusion


Deterministic Propagation Abstractions

• Signal Bandwidth and Duration

• Tapped Delay Line Model

• Double Directional Propagation Modeling

– Description of the channel through the (deterministic) geometry


Deterministic Propagation Abstractions:

Signal Bandwidth and Duration • Signal bandwidth and duration play important role in wireless

channel estimation

– Hence, they are also crucial for e.g. time-of-arrival based positioning

systems

From A. F. Molisch, Wireless Communications, 2nd ed., Wiley 2011

Wideband system Narrowband system


• WLAN example cont. – Nc = 52, W=(52-1)/64*20 MHz, fc= 2.4 GHz

– Δ𝑓 =𝑊

𝑁𝑐= 312.5 kHz

– 𝜏𝑚𝑎𝑥 =1

Δ𝑓= 3.2 μs 3.2 ⋅ 10−6 s

– Δ𝜏 =𝜏𝑚𝑎𝑥

𝑁𝑐=

1

𝑊≈ 63 ns (63 ⋅ 10−9 s)

• Translated to propagation distance

– Δ𝑥 = Δ𝜏 ⋅ 𝑐 ≈ 19 m

– 𝑥𝑚𝑎𝑥 = 𝜏𝑚𝑎𝑥 ⋅ 𝑐 ≈ 960 m


Signal Bandwidth and Duration

2400 2402 2404 2406 2408 2410 2412 2414 2416 2418 24200

0.5

1

|H(f

)|

2400 2402 2404 2406 2408 2410 2412 2414 2416 2418 2420-1

0

1

arg

(H(f

))

Frequency (MHz)

0 2 4 6 8 10 120

0.5

1

Delay (s)

Re/

Im

Re

Im


• WLAN example cont., number of carriers fixed, smaller bandwidth – Nc = 52, W=(52-1)/64* 5 MHz, fc= 2.4 GHz

– Δ𝑓 =𝑊

𝑁𝑐= 78.125 kHz

– 𝜏𝑚𝑎𝑥 =1

Δ𝑓= 12.8 μs

– Δ𝜏 =𝜏𝑚𝑎𝑥

𝑁𝑐=

1

𝑊≈ 246 ns

• Translated to propagation distance

– Δ𝑥 = Δ𝜏 ⋅ 𝑐 ≈ 74 m

– 𝑥𝑚𝑎𝑥 = 𝜏𝑚𝑎𝑥 ⋅ 𝑐 ≈ 3.84 km

2400 2402 2404 2406 2408 2410 2412 2414 2416 2418 24200

0.5

1

|H(f

)|

2400 2402 2404 2406 2408 2410 2412 2414 2416 2418 2420-1

0

1

arg

(H(f

))

Frequency (MHz)

0 2 4 6 8 10 120

0.5

1

Delay (s)

Re/

Im

Re

Im




• WLAN example cont. – Nc = 52, W=(52-1)/64* 20 MHz, fc= 2.4 GHz

– 3 paths: LOS (51 m), S1 (175 m), S2 (280 m)

-100 -50 0 50 100-100

-50

0

50

100

x (m)

y (

m)

Time: 20 s

2395 2400 2405 2410 2415 2420-70

-60

-50

-40

-30

-20

-10

0

Frequency (MHz)

20lo

g 10(|

H(f

)|)

(dB

)

0 200 400 600 800-100

-80

-60

-40

-20

0

Path length, x (m)2

0lo

g 10(|

h(x

)|)

(dB

)

Δ𝑥 = Δ𝜏 ⋅ 𝑐 ≈ 19 m





– The same 3 paths: LOS (51 m), S1 (175 m), S2 (280 m)

-100 -50 0 50 100-100

-50

0

50

100

x (m)

y (

m)

Time: 20 s

2395 2400 2405 2410 2415 2420-70

-60

-50

-40

-30

-20

-10

0

Frequency (MHz)

20lo

g 10(|

H(f

)|)

(dB

)

0 1000 2000 3000-100

-80

-60

-40

-20

0

Path length, x (m)

20lo

g 10(|

h(x

)|)

(dB

)

Δ𝑥 = Δ𝜏 ⋅ 𝑐 ≈ 74 m





– 2 paths with path lengths: d1=140 m, d2= 1230 m

• The longer path gets aliased and is falsely observed at 1230 m -960 m = 270 m with

the 20 MHz channel!

-100 -50 0 50 100

-600

-500

-400

-300

-200

-100

0

100

x (m)

y (

m)

Time: 20 s

2395 2400 2405 2410 2415 2420-70

-60

-50

-40

-30

-20

-10

0

Frequency (MHz)

20

log 1

0(|

H(f

)|)

(dB

)

0 200 400 600 800-100

-90

-80

-70

-60

-50

-40

-30

-20

-10

Path length, x (m)

20

log 1

0(|

h(x

)|)

(dB

)

No window

Hann window

𝑥𝑚𝑎𝑥 = 𝜏𝑚𝑎𝑥 ⋅ 𝑐 ≈ 960 m

-100 -50 0 50 100

-600

-500

-400

-300

-200

-100

0

100

x (m)

y (

m)

Time: 20 s

2395 2400 2405 2410 2415 2420-70

-60

-50

-40

-30

-20

-10

0

Frequency (MHz)

20

log 1

0(|

H(f

)|)

(dB

)

0 1000 2000 3000-100

-90

-80

-70

-60

-50

-40

-30

-20

-10

Path length, x (m)

20

log 1

0(|

h(x

)|)

(dB

)

No window

Hann window

20 MHz channel 5 MHz channel




• If a bandlimited 𝑓𝑐 −𝑊

2, 𝑓𝑐 +

𝑊

2 signal passes through a

number of paths with complex gains 𝑎𝑝 at delays 𝜏𝑝, the

channel impulse response is mathematically given by

– 𝜉 𝑥 = 𝑊 ⋅ sinc(𝑥) is

called interpolation

function with null-to-null

width 2/W

– Note that 𝜏𝑝 may be

irregularly spaced -100 -50 0 50 100-100

-50

0

50

100

x (m)

y (

m)

Time: 20 s

0 200 400 600 800 1000-15

-10

-5

0

5x 10

-3

Path length, x (m)

h(

)

Re

Im

ℎ 𝜏 = 𝑎𝑝 ⋅ 𝜉 𝜏 − 𝜏𝑝

𝑃

𝑝=1


Tapped Delay Line Model


• If 𝑊𝜏𝑃,𝑚𝑎𝑥 ≪ 1 ⇔ 𝜏𝑃,𝑚𝑎𝑥 ≪1

𝑊= Δ𝜏, the channel is flat

fading

• If 𝑊𝜏𝑃,𝑚𝑎𝑥 > 1, channel consists of several ”delay

resolution bins”, channel is considered wideband

• Sampling at 1

𝑊= Δ𝜏 interval yields tapped delay line as

– Where

ℎ 𝜏 = 𝑐𝑚 ⋅ 𝜉 𝜏 − 𝑚 ⋅ Δ𝜏

𝑀

𝑚=1

𝑐𝑚 = 𝑎𝑝 ⋅ 𝜉 𝑚 ⋅ Δ𝜏 − 𝜏𝑝𝑝




• Time varying tapped delay line

– The coefficients 𝑐𝑚(𝑡) are typically fading

• They are comprised of a superposition of multipaths

– However, the path coefficients 𝑎𝑝 𝑡 = 𝑎𝑝 𝑡 ⋅ exp 𝑗𝜙𝑝 𝑡 do

not suffer from multipath fading

ℎ 𝜏, 𝑡 = 𝑐𝑚 𝑡 ⋅ 𝜉 𝜏 − 𝑚 ⋅ Δ𝜏

𝑀

𝑚=1

𝑐𝑚 𝑡 = 𝑎𝑝 𝑡 ⋅ 𝜉 𝑚 ⋅ Δ𝜏 − 𝜏𝑝𝑝




• Demo: Comparison of a few well distinct paths vs. dense

multipath environment

-100 -50 0 50 100-100

-80

-60

-40

-20

0

20

40

60

80

100

x (m)

y (

m)

Time: 20 s

2395 2400 2405 2410 2415 2420-70

-60

-50

-40

-30

-20

-10

0

Frequency (MHz)

20lo

g 10(|

H(f

)|)

(dB

)

0 200 400 600 800-100

-90

-80

-70

-60

-50

-40

-30

Path length, x (m)

20lo

g 10(|

h(x

)|)

(dB

)

No window

Hann window

𝜏𝑖 − 𝜏𝑗 > Δ𝜏

𝜏𝑖 − 𝜏𝑗 < Δ𝜏

-100 -50 0 50 100-100

-80

-60

-40

-20

0

20

40

60

80

100

x (m)

y (

m)

Time: 20 s

2395 2400 2405 2410 2415 2420-70

-60

-50

-40

-30

-20

-10

0

Frequency (MHz)

20

log 1

0(|

H(f

)|)

(dB

)

0 200 400 600 800-100

-90

-80

-70

-60

-50

-40

-30

-20

-10

Path length, x (m)

20

log 1

0(|

h(x

)|)

(dB

)

No window

Hann window




fading_3path.exe

fading_manypaths_pdp.exe

Power Delay Profile, RMS Delay Spread

and Frequency Correlation • Power Delay profile (PDP) and the RMS delay spread (DS)

– Evaluation using sinc interpolator for ℎ 𝑡, 𝜏 (rectangular window in

frequency domain) may not be wise

• sinc2 has discontinuous derivative, and the second central moment does

not converge

– Frequency correlation function (FCF) is given by the Fourier transform

of the PDP

• Coherence bandwidth is inversely proportional to 𝜏𝑟𝑚𝑠 (exact value depends

on the PDP shape)

𝑃 𝜏 =1

𝑊𝐸𝑡 ℎ 𝑡, 𝜏 2 𝜏𝑟𝑚𝑠 = 𝑃 𝜏 𝜏2𝑑𝜏

∞

0

− 𝑃 𝜏 𝜏𝑑𝜏∞

0

2


Summary of System vs. Correlation

Functions and Their Special Cases

From R. Katterbach, Characterisierung zeitvarianter Indoor Mobilfunkkanäle mittels ihrer System- und Korrelationsfunktionen, Dissertation an der Universität GhK Kassel, Aachen 1997. Here copied from A. F. Molisch, Wireless Communications, 2nd ed., Wiley 2011.

These relations hold for WSSUS (ercodig) channel, i.e., defining correlation functions w.r.t. time and frequency differences only.


Double Directional Propagation Path Model

• Description of the radiowave

propagation between two

points in space as a

superposition of discrete

”paths”

– Going ”behind” the tapped

delay line

– Independent of the application

(may depend on frequency)

• Mapping those paths into a

MIMO Channel transfer

function between the input and

output ports

– System specific


Influence of the Tx and Rx

- Obtained by calibration

Influence of the antennas

- Calibration and parametric model required

Propagation channel parameters

- Depend only on the channel itself

)(),(),()()()(1

,,

T

,,

,,

,, fGafGf T

P

p

pTpTT

pVVpHV

pVHpHH

pRpRRpfR

AAH



Jussi Salmi

P

p

pTpTTpRpRRpf fffaf1

,,

T

,, ),,()(),,()()( AΓAH

• Narrowband model

– Antenna responses and RF interactions (electric properties of interacting

materials) are constant within the signal band

• (Semi-)wideband model

– Antenna responses are frequency dependent

• Obtained by calibration (or analytically)

– Frequency dependency of RF interactions?

• More difficult and complex to model

P

p

pTpTT

pVVpHV

pVHpHH

pRpRRpfaf1

,,

T

,,

,,

,, ),(),()()(

AAH



Jussi Salmi

• Double directional model (DDM) vs. tapped delay line (TDL)

– Typically TDL is unable to resolve individual path components, i.e.,

each tap, or delay bin is a superposition of many and thus fading

– DDM can resolve individual paths as long as they differ in at least one

of the parameter dimensions, i.e., Rx direction, Tx direction, or delay

P

p

pTpTTpRpRRpf fffaf1

,,

T

,, ),,()(),,()()( AΓAH

ℎ 𝜏 = 𝑐𝑚 ⋅ 𝜉 𝜏 − 𝑚 ⋅ Δ𝜏

𝑀

𝑚=1



Jussi Salmi

• Tensor formulation (narrowband) RTf MMM

CH

)2jexp()()( fa f

f

– are the antenna responses, modeled

by e.g., the EADF (or analytically)

– is the delay-dependent phase shift of

each frequency

– is the complex path weight, i.e.,

attenuation and phase

),(),,( )()(

TT

T

tRR

R

r aa

jja ee

)(),(),(}{ )()()(

,, f

fTT

T

tRR

R

rrtfS aaa H



Jussi Salmi

• Measurement model may also include stochastic

component for diffuse scattering and noise

Superposition of specular-like

propagation paths

Diffuse scattering

),(~ DD RNH C 0

Measurement noise

)( 2,~ INH C 0N

RTf MMM

NDS

CHHHH

'

1'

',',

)(

',',

)(

'

)(

' )( ),(),()(P

p

pRpR

R

pTpT

T

p

f

pS aaa H



Jussi Salmi

Property SC DMC

Fading type N/A Rayleigh

% of power indoors 10-50 50-90

% power outdoors 20-80 20-80

Suitable representation Discrete delta-functions Continuous power distribution

DMC – Dense Multipath Component*

Jussi Salmi

• Bandwidth and signal duration (as well as other

measurement system ”apertures”) influence how the

wireless medium is perceived

– Delay resolution is inversely proportional to bandwidth and

frequency interval is inversely proportional to ”delaywidth”

• Same principles could be applied to directional perceivance

• Double directional propagation path modeling can

resolve paths beyond tapped delay line

– Resolvability in any dimension suffices

Jussi Salmi


Conclusion

Application Example:

Real-Time UWB MIMO Radar Testing

• Was conducted in September 2011 at USC, Los Angeles

Jussi Salmi



Jussi Salmi

• Multitone (OFDM-like) sounding signal was employed



Jussi Salmi

• The signal can be thought of as a training sequence in

communications

– In fact, a measured template of the signal is used as a reference

• The channel frequency response (CFR) is obtained

through channel estimation

• The CFR can be used for estimating the double

directional propagation path parameters, providing delays

and angles

– Also PDP, Power-angular-delay profiles (PADP), or Doppler

spectrum analysis can be done for the CFRs

– These can be used for passive localization, human detection etc.



Jussi Salmi

• Interleaved rectangular UWB array, University of

Southern California

– Planar monopole design

– 2-7 GHz




2.5 GHz

5.5 GHz

7.5 GHz






• Two balls swinging in

mid-air couple meters

from the radar

• Video


balls_swing.avi

• Example of two

persons breathing

side-by-side




Notes About the Measurements

• System can observe a human moving within a room

(potentially behind obstacles, or a wall)

– Relies on the double directional wideband model

– Goal would be to detect the precense and estimate location

passively, potentially even observe vital signs such as breathing

• Spherical modeling for the wideband array is being

developed

– Potential for increased resolution in the near field

– Will UWB radar be the next ”Kinetic”?


• M. Z. Win, A. Conti, S. Mazuelas, Y. Shen, W. M. Gifford, and D. Dardari, “Network localization and navigation via cooperation,” IEEE Commun. Mag., vol. 49, no. 5, pp. 56–62, May 2011.

• D. Dardari, A. Conti, U. J. Ferner, A. Giorgetti, and M. Z. Win, “Ranging with ultrawide bandwidth signals in multipath environments,” Proc. IEEE, vol. 97, no. 2, pp. 404–426, Feb. 2009, special issue on Ultra-Wide Bandwidth (UWB) Technology & Emerging Applications, Invited Paper.

• H. Wymeersch, J. Lien, and M. Z. Win, “Cooperative localization in wireless networks,” Proc. IEEE, vol. 97, no. 2, pp. 427–450, Feb. 2009, special issue on Ultra-Wide Bandwidth (UWB) Technology & Emerging Applications, Invited Paper.


Some Other Research on UWB-based

Localization

Conclusions for Propagation Modeling

from Localization Perspective • Several propagation mechanisms have influence on the

wireless channel – Understanding propagation is crucial for interpreting the channel

properties

• Statistical models are important especially for Received Signal Strength (RSS) –based localization

• Deterministic models can be used to enhance delay (ToA) or direction (AoA) based localization – Bandwidth has significant influence


Main References

• A. F. Molisch, Wireless Communications, 2nd ed., Wiley 2011.

• L. Greenstein, A. F. Molisch, M. Shafi, ”Propagation issues for Cognitive Radio”,

to appear in a book

– See meanwhile: A.F. Molisch, L.J. Greenstein, M. Shafi, , "Propagation Issues for Cognitive Radio,"

Proceedings of the IEEE , vol.97, no.5, pp.787-804, May 2009.

• J. Salmi, A. F. Molisch,, "Propagation Parameter Estimation, Modeling and

Measurements for Ultrawideband MIMO Radar," IEEE Transactions on Antennas

and Propagation, vol.59, no.11, pp.4257-4267, Nov. 2011.

– Upcoming: S. Sangodoyin, J. Salmi, S. Niranjayan, A. F. Molisch, ”Real-Time Ultrawideband

MIMO Channel Sounding”, EuCAP 2012, Prague, Mar. 2012.


Documents

Introduction to Propagation Modeling - Aaltogeta.aalto.fi/en/courses/propagation_modeling_slides-002.pdf · Introduction to Propagation Modeling ... • Free space attenuation model