• The major technical challenges of wireless communications

Development of Complex Curricula for Molecular Bionics and Infobionics Programs within a consortial* framework**

Consortium leader

PETER PAZMANY CATHOLIC UNIVERSITY

Consortium members

SEMMELWEIS UNIVERSITY, DIALOG CAMPUS PUBLISHER

The Project has been realised with the support of the European Union and has been co-financed by the European Social Fund ***

**Molekuláris bionika és Infobionika Szakok tananyagának komplex fejlesztése konzorciumi keretben

***A projekt az Európai Unió támogatásával, az Európai Szociális Alap társfinanszírozásával valósul meg.

Ad hoc Sensor Networks

Wireless channel characterization and models

Érzékelő mobilhálózatok

Vezeték nélküli csatorna jellemzése és modelljei

Dr. Oláh András

Lecture 2 review

• The major technical challenges of wireless communications

• Noise- and interference-limited systems

• Components of the noise

• Link budget

• The cellular principle

Outline

• Signal propagation overview

• Path loss models

• Log Normal Shadowing

• Narrowband Fading Model

• Wideband Multipath Channels

• Components:

• Path loss (including avarage shadowing)

• Shadowing (due to obstructions)

• Multipath fading

• Basic propagation mechanisms

• Free-space loss

• Reflection and transmission

• Multipath propagation

• Diffraction

• Scattering

• Waveguiding effect

Signal propagation overview

Modelling methods

• Stored channel impulse responses

– Realistic

– Reproducible

– Hard to cover all scenarios

• Deterministic channel models

– Based on Maxwell’s equations – Site specific

– Computationally demanding

• Stochastic channel models

– Describe the distribution of the field strength – Mainly used for design and system comparisons

Why channel modelling?

• The performance of a radio system is ultimately determined by the radio channel. Without reliable channel models, it is hard to design radio systems that work well in real environments.

• The channel models are the basis for

– system design – algorithm design – antenna design,…

• The channel modelling is more than just a loss calculation, we need to answer to the following questions:

– behavior in time and place;

– behavior in frequency;

– behavior in delay;

– directional properties;

– bandwidth properties.

• Path loss defined as

L_P = P_TX/P_RX or L_P^[dB]= P_TX^[dB]- P_RX^[dB]

• Maxwell’s equations

– Complex and impractical

• Free space path loss model

– Too simple

• Ray tracing models

– Requires site-specific information

• Empirical Models

– Cannot be easily generalized to other environments

• Simplified power falloff models

• We assume RX antenna to be isotropic:

L_P = (4πd/λ)² and the receiver power is

P_RX= (λ /4πd)² P_TX

• Received power, with antenna gains:

P_RX= G_TX G_RX (λ /4πd)² P_TX

P_RX^[dB] = P_TX^[dB] + G_TX^[dB] + G_RX^[dB] –10lg (4πd/λ)²

• It’s valid in the far field only (d >> d_R, and d >> λ), where the Rayleigh distance is:

d_R=2 L_a²/ λ,

and L_a is the largest dimension of the antenna (e.g.

for λ/2-dipole L_a= λ/2, thus d_R= λ/2, and for parabolic antenna L_a= 2r, thus d_R= 8r² /λ, where r is the radius).

• Power falls off proportional to d² and to λ² _(inversely

Free-space (line-of-sight) modelling

Ray Tracing Approximation

• Represent wavefronts as simple particles

• Geometry determines received signal from each signal component

• It typically includes reflected rays and it can also include scattered and diffracted rays.

• It requires site parameters

– Geometry

– Dielectric properties

• Received power, with antenna gains:

P_RX= G_TX G_RX (h_TXh_RX /d²)² P_TX

P_RX^[dB] = P_TX^[dB] + G_TX^[dB] + G_RX^[dB] +20lg (h_TXh_RX ) – 40lg (d ) .

• For distances greater than (d >d_BREAK):

d_BREAK=4 h_TX h_RX/ λ .

• Power falls off proportional to d² (small d), proportional to d⁴ (d >d_BREAK) and independent of λ.

Two-path (or two-ray) model

• It models all signal components:

– Reflections – Scattering – Diffraction .

• It requires detailed geometry and dielectric properties of site (similar to

General ray tracing

Simplified Path Loss Model

P_RX= P_TX · K(d₀)·(d₀ /d)^γ , where

– d₀ is a reference distance for antenna far-field (it is tipically assumed to be 1-10m indoors and 10-100m outdoors);

– K is a constans depending on the G_TX, G_RX and average channel attenuation at d₀.

• It is used when path loss dominated by reflections.

• Most important parameter is the path loss exponent γ, determined empirically.

Empirical Models

• Okumura model

– Empirically based (site/freq specific) – uses graphs

• Hata model

– Analytical approximation to Okumura model

• COST 136 Model:

– Extends Hata model to higher frequency (2 GHz)

• Walfish/Bertoni (COST 231):

– Cost 136 extension to include diffraction from rooftops.

• Motley-Keenan model:

– For indoor environments

• Extensivensive measurement campaign in Japan in the 1960’s.

• Parameters varied during measurements:

– Frequency: 100-3000MHz – Distance: 1-100km

– Mobile station height 1-10m – Base station height: 20-1000m

– Environments: medium-size city, large city, etc.

Empirical Models: Okumura model

h_TX=200m h_RX=3m

Example: 900 MHz, 30 km

Empirical Models: Hata model

• The path loss can be calculated as:

L_P= A + B log(d^[km]) + C, where

A = 69.55 + 26.16 log(f^[MHz]) − 13log(h_TX) − a(h_RX) B = 44.9 − 6.55log(h_TX)

Empirical Models: Motley-Keenan indoor model

• For indoor environments, the attenuation is heavily affected by the building structure, walls and floors play an important rule:

L

= L

+ 10γlog(d/d

) + F

+ F

,

where F

is sum of attenuations from walls (1-20 dB/wall) and F

is sum of attenuation from the floors (it is often larger than the wall attenuation)

• This model is site specific, since it is valid for only a particular

case

• It models attenuation from obstructions.

• It is random due to random number and type of obstructions.

• It typically follows a log-normal distribution

• Its dB value of power is normally distributed

• It is decorrelated over decorrelation distance, that tipically ranges from 50 to 100m

• Standard deviation σ

Lp≈ 4…10 dB (empirical)

Shadowing (large-scale fading): log-normal distribution

[ ]

(

)

P P

2

P P0

P 2

pdf( ) 1 exp

2 2

dB dB

dB

L L

L L

L πσ σ

 − 

 

= − 

 

Combined path loss and shadowing

[ ]^dB [ ]^dB

( )

RX RX 0 10 lg d

P P d

γ ^ d ^ ψ

= −  +

 

[ ]^{dB N}

(

)

ψ _∼ σ

[ ]^dB

( )

( )

RX 0 TX RX TX 20lg 4 0 /

P d = P +G +G − πd λ

[ ]^dB

( )

K d

Alternatively, K can be

determined by measurements.

An example for shadowing and outage probabilty

Assume that at a certain distance, we have a deterministic propagation loss L

=127 dB and large-scale fading with σ

Lp

= 7 dB. How large is the outage probabilty Pr

at that particular distance, if the system is designed to handle a maximal propagation loss of 135 dB?

Solution: on the blackboard.(for check: Pr

=0.127 )

Q(0.1) = 0.460172163 Q(1.0) = 0.158655254 Q(2.0) = 0.022750132 Q(3.0) = 0.001349898 Q(0.1) = 0.460172163 Q(1.1) = 0.135666061 Q(2.1) = 0.017864421 Q(3.1) = 0.000967603 Q(0.2) = 0.420740291 Q(1.2) = 0.115069670 Q(2.2) = 0.013903448 Q(3.2) = 0.000687138 Q(0.3) = 0.382088578 Q(1.3) = 0.096800485 Q(2.3) = 0.010724110 Q(3.3) = 0.000483424 Q(0.4) = 0.344578258 Q(1.4) = 0.080756659 Q(2.4) = 0.008197536 Q(3.4) = 0.000336929 Q(0.5) = 0.308537539 Q(1.5) = 0.066807201 Q(2.5) = 0.006209665 Q(3.5) = 0.000232629 Q(0.6) = 0.274253118 Q(1.6) = 0.054799292 Q(2.6) = 0.004661188 Q(3.6) = 0.000159109 Q(0.7) = 0.241963652 Q(1.7) = 0.044565463 Q(2.7) = 0.003466974 Q(3.7) = 0.000107800

( )

2

1 2

2

u

x

Q x e du

π

∞ −

=

∫

An example for outage probabilty and fading margin

A mobile communication system is to be designed to the following specifications: the instantaneous received amplitude must not drop below at the cell boundary for 90% of the time.

The signal experiences large-scale fading with σ

Lp

= 6 dB. Find the required fading margin for the system to work.

Solution: on the blackboard.(for check: M=7.69dB )

• Goal: fit model to data.

• Path loss (K^[dB](d₀), γ) at d₀ known:

– “Best fit” line through dB data – K^[dB](d₀) obtained from

measurements at d₀.

– γ

– Captures mean due to

shadowing

• Shadowing variance σ_Lp²:

– Variance of data relative to path loss model (straight line) with

Model parameters from empirical measurements

An example for model parameter fitting

Consider the set of empirical measurement of P_TX^[dB]–P_RX^[dB] for an indoor system at 900MHz given following :

– Find the path loss exponent γ that minimizes the MSE, assuming that d₀=1m and K^[dB](d₀) is determined from the free space path gain.

– Find the received power at d=150m for simplified path loss model with this γ and transmit power of P_TX=1mW.

– Find the variance of log-normal shadoing σ_Lp² based on these empirical measurements.

Solution: on the blackboard.(for check: K^[dB](d₀)= –31.54dB, γ=3.71, σ_Lp=3.65dB

)

Distance, d [m] 10 20 50 100 300

P_TX^[dB]–P_RX^[dB] -70dB -75dB -90dB -110dB -125dB

Path loss exponent comparison

Location Path loss exponent

Open 4.35

Suburban 3.84

Newark 4.31

Philadelphia 3.68

New York 4.8

Tokyo 3.05

Free space 2 Dhaka, India 2.98

Main points of path loss and shadowing

• Models vary in complexity and accuracy.

• Power falloff with distance is proportional to d

in free space, d

in two path model.

• Empirical models used in 2G simulations.

• Main characteristics of path loss captured in simple model P

= P

· K(d

) · (d

/d)

.

• Random attenuation due to shadowing modeled as log-normal (empirical parameters).

• Path loss and shadowing parameters are obtained from

empirical measurements.

• The canonical form of a band pass transmitted radio signal is

where e^j2πft is the carrier factor.

• The s(t) can be written as

• We will define the following

• The complex envelope of s(t) is now written as

( ) ( )

( ( ) )

{ ^{( )}

}

s t = A t ωt + Θ t = A t ^Θ

( ) ( )

( ( ) )

^{( ) ( )}

( ^{( )} )

^{( )}

s t = A t Θ t ωt − A t Θ t ωt

( )

s tɶ = +s s

( ) ( ) ( ( ) )

( ) ( ) ( ( ) )

I

Q

cos sin

s t A t t

s t A t t

= Θ

= Θ

y(t)=A(t)e^{j Θ(t)}

x(t)=A(t)e^jΘ(t) r(t)e^jθ(t) e^-2πft=A(t)r(t)e^{j(Θ(t)+θ(t))}

It is the behaviour of the channel r(t) attenuation and the θ(t) phase we are going to model.

A narrowband system described by complex variables

• Random number of multipath components, each with

– Random amplitude – Random phase

– Random Doppler shift – Random delay

• Response of channel at time t to impulse at time t-τ^:

– t is the time when the impulse response is observed – t-τ is the time when impulse is fed into the channel

– τ indicates how long ago impulse was fed into the channel prior to the current observation

Statistical multipath model (cont’)

j ( ) 0

( , ) ( )e ⁿ ( ( ))

N

t

n n

n

h τ t r t ^θ δ τ τ t

=

=

∑

• Received signal consists of many multipath components

• Amplitudes change slowly

• Phases change rapidly

• Assume delay spread max_m,n|τ_n(t)-τ_m(t)|<<1/B then u(t) ≈ ^u(t-τ).

• Received signal given by

• No signal distortion (spreading in time)

• Multipath affects complex scale factor in brackets.

• Characterizes scale factor by setting y(t)=δ(t) (channel sounding)

Narrowband model

( )

2 ( )

0

( ) Re ( ) ^c ( ) ⁿ

N t

j f t j t

n n

x t y t e ^π r t e ^φ

=

  

=   

 



∑

• In phase and quadrature signal components:

• For N(t) large, s_I(t) and s_Q(t) jointly Gaussian by Central Limit Theorem.

• Received signal characterized by its mean, autocorrelation, and cross correlation.

• If θ(t) uniform, the s_I(t) in-phase and quadrature components are mean zero,

In-Phase and Quadrature under CLT Approximation

( )

j ( ) I

0

( ) ( )e ⁿ cos(2 ),

N t

t n

n

s t r t ^θ π ft

=

=

∑

( )

( ) Q

0

( ) ( ) ⁿ sin(2 )

N t

j t n

n

s t r t e ^θ π ft

=

=

∑

• No dominant component (no LoS) which means

• The signal envelope can be calculated as

• CLT approximation leads to Rayleigh distribution (power is exponential):

– for x<0 the pdf is zero, as amplitudes are

Small scale fading: Rayleigh fading

(

)

I, Q 0,

s s _∼ N σ

( )

2

2 2

1 e

2

x

pdfs x ^σ

πσ

 

−

 

 

=

( )

2

2 2

2 e

x r

pdf x x ^σ

σ

 

−

 

 

= 0 ≤ < ∞x

2 2

I Q

r = s + s r =σ π2 ^{2 2}

2 RX

r = σ = P

( ) ( )

2

2 2

1

x r

r r

cdf x pdf u du e ^σ

 

−

 

 

−∞

=

∫

( )

RX

2 2

1 e 2

x

pdfP x ^σ

σ

 

− 

 

=

The Rayleigh distribution is widely used in wireless communications due to following reasons:

• It is an excellent approximation in practical scenarios, but it is not valid for some particular scenarios (eg.: in LoS, some indoor csenarious, wideband scenarious)

• It describes a worst case scenarios (no dominant signal component) which is useful for the design of robust systems.

• It depends on a single parameter, if P_RX mean received power is known, the complete signal statistics are known.

• Mathematical convenience: the computation of error probability can often be

Small scale fading: Rayleigh fading (cont’)

In case of LoS one component dominates.

• Assume it is aligned with the real axis:

• The received amplitude has now a Rician distribution instead of a Rayleigh:

where I₀(.) is the modified Bessel function:

• The ratio between the power of the LoS component and the others is called Rician

Small scale fading: Rician fading

(

)

Q 0,

s _∼ N σ

(

)

I ,

s _∼ N A σ

( )

2 2

2 2

2 e 0 2

x A r

x xA

pdf x ^σ I

σ σ

− + 

 

   

=  

 

( )

=

( )

0

0

1 e ^x

I x d

π θ θ

π ⁻

=

∫

To simplify discussion, we assume only movement of the RX with speed u.

where the Doppler shift is

In the case of α degree between u and k:

Small scale fading: Doppler shifts

( ) ( ) ( [ ] )

( ) ( [ ] )

cos 2

cos 2 /

s t A t ft k d ut

A t t f u kd

π

π λ

= − + =

= − −

u u

f c ν ^{= − = −}λ

( ) ( )

( )

cos cos cos

u u

f c

ν α α ν α

= −λ = − =

• Two components have different Doppler shift

• The Doppler shifts will cause a random frequency modulation

Small scale fading: Doppler shifts (cont’)

• Incoming waves from several directions (relative to movement of receiver)

• Spectrum of received signal when a fcarrier frequency signal is transmitted.

Small scale fading: Doppler shifts (cont’)

• We are intrested in the statistical distribution of received signal power (S_r(ν) power spectrum density) assuming:

– s_I and s_Q and consequently r is wide-sense stationarity (WSS) random process [→ see later]

– uniform scattering environment (pdf_θ(x)=1/2π)

• Jakes (os classical) power spectrum density:

Small scale fading: Doppler shifts (cont’)

( )

max r

S ν P

π ν ν

= − ν ν≤ ^max

Used to generate simulation values

• Autocorrelation functions are definied as

– For s_I(t) autocorrelation: Rs_I(t, τ)=E{s_I(t) · s_I(t+τ)}

– For s_Q(t) autocorrelation: Rs_Q(t, τ)=E{s_Q(t) · s_Q(t+τ)}

– For r(t) autocorrelation: R_r(t, τ)=E{r(t) · r(t+τ)}

• Crosscorrelation function is definied as

– Rs_Qs_I(t)=E{s_Q(t) · s_I(t)}

Small scale fading: correlation functions

• Autocorrelation functions

– Rs_Q(t, τ)= Rs_I(t, τ)= …~ I₀(2πν_maxτ) – R_r(t, τ)= Rs_I(t, τ)= …=P_RXI₀²(2πν_maxτ)

• Crosscorrelation function

Under uniform scattering

Decorrelates over roughly half a wavelength

• What about the length and the frequency of fading dips?

–Level Crossing Rate (CLR): the rate at which the fieldstrength goes below the considered r^* threshold.

–Average Fade Duration (AFD): the total percentage of time the fieldstrength is lower than the r^* threshold.

Small scale fading: fading dips

CLR and AFD for a Rayleigh fading amplitude and Jakes spectrum

Small scale fading: CLR and AFD

* 2

r r r^* r²

Narrowband channel model and simulation metholodgy

Channel model for static enviroment

Simplified

communication system

Wideband channel

Wideband vs. narrowband system

s

s

Wideband channel: system functions

• Time-variant impulse response h(t, τ):

–Due to movement, impulse response changes with time –Input-output relationship:

• Time-variant transfer function H(t,f)

–Perform Fourier transform with respect to τ –Input-output relationship:

( ) ( ) ( )

x t ^∞ y t τ h t τ τd

−∞

=

∫

( )

( )

H t f ^∞ t τ ^{− π τ}dτ

−∞

=

∫

( ) ( ) (

)

X f = ^{∞ ∞}

∫ ∫

Wideband channel: system functions (cont’)

• Further equivalent system functions:

– Since the impulse response depends on two variables (t, τ), Fourier transformation can be done with respect to each of them

• Four equivalent system descriptions are possible (Bello functions):

– Impulse response h(t, τ)

– Time-variant transfer function H(t,f) – Spreading function

– Doppler-variant spreading function

( )

( )

S ν τ ^∞ h t τ ^{− π} dt

−∞

=

∫

(

) ( )

B ν f = ^∞

∫

Wideband channel: system functions (cont’)

Return to stochastic description of wireless channel

• Complete description requires the multidimensional pdf of h(t, τ )

–Too complicated in practice

• Second-order description: Auto Correlation Function

–Input-output relationship:

–If the channel is non-Gaussian, then the first- and second- order statistics are

(

) { ( ) (

) }

R t th ^′ τ τ^′ = E h t^∗ τ h t^{′ ′}τ

( )

(

) (

)

xx yy h

R t t ^{∞ ∞} R t τ t τ R t t τ τ τ τd d

−∞ −∞

′ =

∫ ∫

Wideband channel: WSSUS model

• Wide-sense stationarity (WSS): the statistical properties of the channel do not change with time

• Uncorretated scatterers (US): the different delays are uncorrelated

• If WSSUS is valid, R

depends only two variables:

–Autocorrelation of impuse response function:

–Autocorrelation of transfer function:

(

) ( ) (

)

h h

R t t^′ τ τ^′ = δ τ τ− ^′ P ∆t τ

(

) (

)

H H

R t + ∆t f + ∆ =f R ∆ ∆t f

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

h h h

R t t ^′ τ τ ^′ = R t t + ∆ t τ τ ^′ = R ∆ t τ τ ^′

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

h h

R t t ^′ τ τ ^′ = δ τ τ − ^′ P t t ^′ τ

Wideband channel: WSSUS model (cont’)

• WSSUS channel can be presented as a tapped delay line

• The impulse response of the tapped time delay line model:

Wideband channel: Condensed parameter

• Correlation functions depend on two variables

• For concise characterization of channel, we want

–a function depending on one variable –a scalar parameter

• Most common condensed parameters

–Power delay profile, P(τ)

–Root mean square (rms) delay spread, S –Coherence bandwidth, B_coh

–(Doppler spread) –(Coherence time)

Wideband channel: power delay profile

• The power delay profile is the expected value of the received power at a certain delay:

• For tapped-delay line:

( )

{ ( )

} ^{( )}

P τ E h t τ ^∞ h t τ dt

−∞

= =

∫

( )

{ }

1 1

j ( ) 2

1

( )e ( ) ( ) e ( )

2 e ( )

n n

n

N N

t t

t n n t n n

n n

N

t

n n

n

P E r t ^θ E r t ^θ

θ

τ δ τ τ δ τ τ

σ δ τ τ

= =

=

 

 

=  −  = − =

 

 

= −

∑ ∑

∑

of tap n

Wideband channel: power delay profile (cont’)

• We can reduce the power delay profile into more compact descriptions of the channel:

–Total power:

–Average mean delay:

–Average rms delay spread:

• For tapped delay line channel:

–Total power:

–Average mean delay:

–Average rms delay spread:

( )

PRX ^∞ P τ τd

−∞

=

∫ _{( )}

- m

RX

P d

T P

τ τ τ

∞

=

∫

2 RX

1

2

N

n n

P σ

=

=

∑

1 m

RX

2

N

n n

T n

P τ σ

=

∑

( )

2 - 2

RX

m

P d

S T

P

τ τ τ

∞

=

∫

2 2

1 2

2

N

n n

n

S Tm

P τ σ

=

∑

Wideband channel: frequency correlation

• A property closely related to the power delay profile is the R

(0, ∆ f)frequency correlation of the channel:

• For tapped delay line model:

( ) ( )

-

e

R

f

P τ

dt

∞

∆ = ∫

( )

- 1

2 j2

2 ( ) e

2 e

N

f

H n n

n N

f n

R f

dt

π τ

σ δ τ τ σ

∞ − ∆

∞ =

− ∆

 

∆ =  −  =

 

=

∫ ∑

∑

( )

R

∆ f

∆ f

Wideband channel: coherence bandwidth

coh

1 B 2

π S

≥

( )

H

0

R

( )

H

0

2 R

( )

R

∆ f

3 dB bandwidth

( ) ( ) ( )

H H

( )

coh

0 H 0 H

1 arg max 0.5 arg max 0.5

2

0

0

R f R f

B

R

R

  ∆   ∆  

=   =  −  =  

   

 

• Tapped delay line model often used

• Often Rayleigh distributed tap, but might LoS and different distributions of the tap values

• Mean tap power determined by the power delay profile (eg.: GSM)

Wideband models

( ) ( )

j ( )

1 1

( , ) ( )e ⁿ ( )

N N

t

n n n n

n n

h τ t r t ^θ δ τ τ c τ δ τ τ

= =

=

∑

∑

COST 207 model for GSM

• Four specified power delay profile for different environments

Main points

• Received signal has random amplitude fluctuations

• Narrowband fading distribution depends on environment (Rayleigh, Ricean fading)

• Statistical multipath model leads to a time-varying channel impulse response

• The most frequently used assumption is WSSUS model.

• The WSSUS model can be represented as a tapped delay line.

• The coherence bandwidth determines the channel behaviour

(flat- or frequency selective channel) and the type of the system

(narrowband or wideband system).