Wireless Channel Modelocw.snu.ac.kr/sites/default/files/NOTE/10340.pdf · 2018. 1. 30. · Path loss as a function of distance: − sometimes simple model can captures the essence

Wireless Channel Model

Wha Sook Jeon Mobile Computing and Communication Lab.

Mobile Computing and Communication Lab. 1

Radio Channel (1)

Signal Fading − large-scale component: Path Loss − medium-scale slow varying component:

Shadowing − small-scale fast varying component:

Multipath fading


Radio Channel (2)


LOS Path vs. NLOS Path


Path Loss & Shadowing

Path Loss − caused by dissipation of the power radiated by the

transmitter − depends on the distance between transmitter and receiver

Shadowing − caused by obstacles between the transmitter and receiver

that absorb power.


Multipath fading

short-term fluctuation of the received signal caused by multipath propagation

when mobile is moving Fading becomes fast as a mobile moves faster delay-power profile (delay spread)

Average power (dB)

Delay

Narrowband fading

Wideband fading

Rayleigh distribution


Channel Model


Simplified Pass Loss Model

Path loss as a function of distance: − sometimes simple model can captures the essence of signal propagation

without resorting to complicated path loss models. − Free-space, two-ray, Hata, COST extension to Hata are all of the same form − K : constant which depends on antenna characteristics and the average channel

attenuation ■ Free space path gain at distance d0 assuming omni-directional antennas ■ Empirical measurements at d0

− d0: reference distance ■ generally valid only at d> d0

■ d0 : 1-10m (indoor), 10-100m (outdoor) − : path loss exponent

■ at higher frequencies tend to be higher and at higher antenna heights tend to be lower

γ

= ddKPP tr

0

γ


Empirical Path Loss Models

Piecewise Linear Model

Indoor Attenuation Factors − partition loss

− floor loss − the building penetration loss − It is difficult to find generic

models


Shadowing (1) Statistical models

− The transmitted signal experiences random variation due to blockage from objects in the signal path and changes in reflecting surfaces and scattering objects.

Log-normal shadowing − Ratio of transmit-to-receive power is a random variable

with a log-normal distribution

− Distribution of (the dB value of ) is Gaussian with mean and standard deviation

tr PP /=ψ

−−= 2

210

2)log10(

exp2

10ln/10)(dB

dB

dB

pψ

ψ

ψ σµψ

ψσπψ

dBψ ψdBψµ

dBψσ

−−= 2

2

2)(

exp2

1)(dB

dB

dB

dBdBp

ψ

ψ

ψ σµψ

σπψ


Shadowing (2)

constant.n attenuatioan is where,)( ααdeds −=∑= it dd

ti ddt eeds αα −− =∑=)(

Justification for the Gaussian model as the distribution of − when shadowing is dominated by the attenuation from blocking object − attenuation of a signal as traveling through an obstacle with depth d

■ − attenuation of a signal as it propagates through the region

■ − dt : Gussian r.v. (by the Central Limit Theorem) − : Gussian r.v.

Decorrelation distance:

− the distance at which autocovariance equals 1/e of its maximum value − on the order of the size of the blocking objects or clusters of objects

)(log10 tds

dBψ


Combined Path Loss and Shadowing

Combined model − average dB path loss: the path loss model − shadow fading with mean of 0 dB: variations about the path loss

Simplified path loss with log-normal shadowing −

− : a Guassian r.v. with mean zero and variance dBψ

dBt

r

ddKdB

PP ψγ +−=

01010 log10log10)(

dBψσ

Pr/Pt

(dB)

log d

Very slow

Slow 10logK

γ10−


Outage Probability

under the path loss and shadowing − : the minimum received power level − outage probability:

■ the probability that the received power at a given distance d falls below

■ ■ for the combined path loss and shadowing (at slide 23)

minP

minP

),( min dPpout

))((),( minmin PdPpdPp rout <=

−+−−=<

dB

ddKPPQPdPp tr

ψσγ )/(log10log10(1))(( 01010min

min


Multipath Multipath fading

− Constructive and destructive addition of different multipath components introduced by a channel

Time-varying channel impulse response − If a single pulse is transmitted over a multipath channel, the received

signal appears as a pulse train

− A multipath channel is modeled by a channel impulse response. Characteristic of the multipath channel

− time delay spread ■ Time delay between the arrivals of the first received signal component

and the last received component associated with a single transmitted pulse − time-varying nature due to moving

Narrowband fading model Wideband fading model


Example of Multipath


Multipath Component (1)

Nonresolvable components: these are combined into a single component.

Resolvable components: if the delay difference between two components significantly exceeds the inverse signal bandwidth


Multipath Component (2)


Doppler Shift

When the transmitter is moving, the received signal has a Doppler shift

Doppler effect is on the order of 100 Hz for typical vehicle speed (75km/hr) and frequencies (about 1GHz)

DD

D

f

vt

f

πφ

θλ

φπ

2

cos21

=

=∆∆

=


Time-varying Channel Impulse Response Equivalent lowpass time-varying channel impulse response at time t to an

impulse at time :

− N(t): the number of multipath components − For the nth component

■ : delay ■ : phase shift - ■ : Doppler phase shift ■ : amplitude

Time-invariant channel:

)()(

0n

jN

nn

nec ττδατ φ −= −

=∑

τ−t ))(()(),( )()(

0tettc n

tjn

tN

nn ττδατ φ −= −

=∑

)(tnτ

nDφnDncn tft φτπφ −= )(2)(

)(tnα

)(tnφ


Transmit & Receive Signal Models (1)

Complex baseband representation − The transmitted or received signals are actually real sinusoids − The complex representations are used to facilitate analysis

Transmitted signal − −

■ complex baseband signal with in-phase component sI(t) and quadrature component sQ(t)

{ } )2sin()()2cos()( )(Re)( 2 tftstftsetuts cQcItfj c πππ −==

)()()( tsjtstu QI +=


Transmit & Receive Signal Models (2) Transmitted signal

− Received signal

−

−

−=

−−=

−−=

−=

∑

∑ ∫

∫ ∑

∫

=

−

=

∞

∞−

−

∞

∞=

−

∞

∞−

tfjN(t)

nn

tjn

tfjN(t)

nn

tjn

tfj

-

N(t)

nn

tjn

tfj

cn

cn

cn

c

ettuet

edtutet

edtutet

edtutctr

πφ

πφ

πφ

π

τα

ττττδα

ττττδα

τττ

2

0

)(

2

0

)(

2

0

)(

2

))(()(Re

)())(()(Re

)())(()(Re

)(),(Re)(

{ }tfj cetuts π2)(Re)( =

−= −

=∑ tfj

ntj

tN

nn

cn ettuettr πφ τα 2)()(

0))(()(Re)(

Narrowband Fading Model


Narrowband Fading Model (1)

Narrowband fading assumption − Delay spread: − The LOS and all multipath components are typically nonresolvable.

Received Signal − −

In order to characterize the random scale factor caused by the

multipath, u(t)=1 − Transmitted signal: − Received signal:

BTm1<<

ii tutu ττ allfor )()( ≈−

= −

=∑ )(

)(

0

2 )()(Re)( tjtN

nn

tfj nc etetutr φπ α

channel characteristic

{ } tfets ctfj c ππ 2cosRe)( 2 ==

tftrtftreettr cQcItfjtj

tN

nn

cn ππα πφ 2sin)(2cos)( )(Re)( 2)()(

0+=

= −

=∑


Narrowband Fading Model (2)

Received Signal −

− in-phase component:

− quadrature component:

rI(t) and rQ(t) can be approximated as Gaussian random

processes − if N(t) is large, and and are independent for different

components. − when for small N(t) each ray has a Rayleigh distributed envelope

and uniform phase − rI(t) and rQ(t) are independent Gaussian process with the same

autocorrelation , a mean of zero, and a crosscorrelation of zero

)(cos)()()(

0tttr n

tN

nnI φα∑

=

=

tftrtftrtr cQcI ππ 2sin)(2cos)()( +=

)(sin)()()(

0tttr n

tN

nnQ φα∑

=

=

)(tnα )(tnφ


Autocorrelation and Cross Correlation (1) Assumption

− There is no dominant LOS component − The amplitude, multipath delay and Doppler frequency shift are

changing slowly enough to be considered constant over the time interval of interest

■ ■

− is uniform distributed on [-π, π] ■ this is reasonable because is large and hence can go through a

360o rotation for a small change The received signal r(t) is zero-mean Gaussian process

− − −

nn DDnnnn ftftt ≈≈≈ )(,)(,)( τταα

)(tnφ

tfft nDncn πτπφ 22)( −=

0)](cosE[ ][E)]([E n ==∑ ttr nn

I φα

cfnτ

)0)]()([E ( =trtr QI

0)]([E =tr

0)](E[sin ][E)]([E n ==∑ ttr nn

Q φα


Autocorrelation and Cross Correlation (2)

Autocorrelation

−

■

■

− A uniform scattering environment assumption ■ The channel consists of many scatters densely packed with respect to angle ■

− Each component has the same received power: −

Nnnn πθθ 2=∆=

NPrn 2][E 2 =α

)sin(2 )()2cos()()]()([E)( , tftAtftAttrtrtA crrcrr QII∆∆−∆∆=∆+=∆ ππ

∆

=∆+=∆ ∑ nnn

IIrtvttrtrtA

Iθ

λπα cos2cos][E5.0)]()([E)( 2

∆

−=∆+=∆ ∑ nnn

QIrrtvttrtrtA

QIθ

λπα cos2sin][E5.0)]()([E)( 2

,

θθλπ

π∆

∆

∆=∆ ∑

=

ntvPtAN

n

rrI

cos2cos2

)(1

v


Autocorrelation and Cross Correlation (3)

)2cos()()( tftAtA crr I∆∆=∆ π

)2(

cos2cos2

)(

0

2

0

tfJP

dtvPtA

Dr

rrI

∆=

∆

=∆ ∫π

θθλπ

ππ

0

cos2sin2

)(2

0,

=

∆

=∆ ∫ θθλπ

ππ

dtvPtA rrr QI

0 , →∆∞→ θN

λ4.0 4.0 =∆⇒=∆ tvtfD

Antenna spacing of 0.4λ: the signal decorrelates over a distance of 0.4λ

θπ

θθπdee

jxJ jnjx

nncos2

021)( ∫=


Power Spectral Density

)]2([)(

)]()([25.)(

0 τπ Dr

crcrr

fPJfS

ffSffSfS

I

II

F=

++−=

fc+fD fc

Sr(f)

fc-fD


Envelope and Power Distributions without LOS

Signal envelope − − rI and rQ are Gaussian random variables with mean zero and

variance − z(t) is Rayleigh distributed

■

Power

− − The received signal power is exponentially distributed with mean

■

2σ

)()()()( 22 trtrtrtz QI +==

0 ,)( 2

2

22 ≥=

−zezzp

z

Zσ

σ

0 x,2

1)( 2

22

2 ≥=−

σ

σ

x

Z exp

22 )()( trtz =22σ


Signal Envelope over Channel having a LOS

Signal envelope − The received signal equals the superposition of a LOS component

and a complex Gaussian component − The signal envelope z(t) has a Rician distribution.

Average received power − − Fading parameter :

■ K = 0: Rayleigh fading ■ K = ∞: no fading (only a LOS component) ■ the smaller K, the severer fading

0 ,)( 202

)(

22

22

≥

=

+−

zzsIezzpsz

Z σσσ

22

0

2 2)( σ+== ∫∞

sdxxpxP Zr

LOS component power

NLOS component power

* I0: the modified Bessel function

2

2

2σsK =


Nakagami Fading Channel

Nakagami fading distribution − More general fading distribution whose parameters (m) can be

adjusted to fit a variety of empirical measurements

− m = 1: Rayleigh fading − m = ∞: no fading − : approximately Rician fading

5.0 ,)(

2)(2

12

≥Γ

=−−

mePm

zmzp rPmz

mr

mm

Z

)12()1( 2 ++= KKm

Wideband Fading


Intersymbol Interference

Narrowband fading: delay spread

- There is little interference with a subsequently transmitted pulse.

Wideband fading:

- The resolvable multipath components interfere with subsequently transmitted pulses: intersymbol interference (ISI)

1τ 2τuB121 >>−ττ

TTm >>

TTm <<

Two multipath components with delay and are resolvable if


Autocorrelation: Time Domain

Equivalent lowpass time-varying channel impulse response at time t to an impulse at time − : a complex zero-mean Gaussian random process

The statistical characterization of is determined by its autocorrelation function

For the WSS channel, the autocorrelation is independent of t. −

For the WSS channel with uncorrelated scattering − The channel response associated with a multipath component of delay

is uncorrelated with the response associated with a component of a different delay because they are caused by different scatters.

−

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

0tettc n

tjtN

nn

n ττδατ φ −= −

=∑

τ−t

),( tc τ)],(),(E[),,,( 21

*21 ttctcttAc ∆+=∆ ττττ

)],(),(E[),,( 21*

21 ttctctAc ∆+=∆ ττττ

1τ

12 ττ ≠

),()(),()],(),(E[ 21121* tAtAttctc cc ∆=−∆=∆+ τττδτττ )( 12 τττ ==


Autocorrelation: Frequency Domain

Characteristics of the time-varying multipath channel at time t − in time domain: − in frequency domain:

■ A complex zero-mean Gaussian process ■ is completely characterized by its autocorrelation.

Autocorrelation of −

− the WSS channel:

−

),( tc τττ τπ detctfC fj∫

∞

∞−

−= 2),(),(

),( tfC),( tfC

)],(),(E[),,,( 21*

21 ttfCtfCttffAC ∆+=∆)],(),(E[),,( 21

*21 ttfCtfCtffAC ∆+=∆

[ ]

),(

),(

),(

),(),(E

),(),(E),,(

2

)(2

2122

21*

22

2-12

1-

*21

12

2211

2211

tfA

detA

detA

ddeettctc

dettcdetctffA

C

fjc

ffjc

fjfj

fjfjC

∆∆=

∆=

∆=

∆+=

∆+=∆

∆−∞

∞−

−−∞

∞−

−−∞

∞−

∞

∞−

−∞

∞

∞

∞

∫∫∫ ∫

∫∫

ττ

ττ

ττττ

ττττ

τπ

τπ

τπτπ

τπτπ

ττ τπ deAfA fjcC

∆−∞

∞−∫=∆ 2)()(0=∆t


Power Delay Profile Power delay profile

− − Average power associated with a given multipath delay

Delay spread − Average delay spread:

− rms delay spread:

− the delay associated with a given multipath component is weighted by its relative power

The delay spread of the channel is roughly by the time delay T where − : the system experiences significant ISI − : ISI can be negligible

)0 when ),(( )],(),(E[)( * =∆∆= ttAtctcA cc τττττ

∫∫

∞

∞

=

0

0

)(

)(

ττ

τττµ

dA

dA

c

cTm

∫∫

∞

∞−

=

0

0

2

)(

)()(

ττ

ττµτσ

dA

dA

c

cTT

m

m

Relative power

TAc ≥≈ ττ for 0)()10(

mm TsTs TT σσ <<<)10(

mm TsTs TT σσ >>>

symbol period


Coherence Bandwidth (1)

− describes the autocorrelation of the time-varying multipath channel in

frequency domain (coherence) Coherence bandwidth of the channel

− − By the Fourier transform relationship between

Flat fading and frequency selective fading − signal bandwidth: B − Flat fading:

■ − Frequency selective fading:

■

cCc BffAB ≥∆≈∆ allfor 0)(such that

)],(),(E[)( * tffCtfCfAC ∆+=∆

)( and )( τcC AfA ∆

TffATA Cc 1for 0)( then ,for 0)( if >∆≈∆>≈ ττmT

ck

TB σ=≈ 1

cBB <<ISI negligible 11 ⇒≈>>≈

mTcs BBT σ

ISIt significan 11 ⇒≈<<≈mTcs BBT σ

cBB >>


Coherence Bandwidth (2)


Coherence Time and Doppler Power Spectrum (1)

Time variation of the channel which arise from transmitter or receiver motion cause a Doppler shift

− describe the autocorrelation of the channel at a frequency over the time

(channel coherence over the time)

Coherence time −

Doppler power spectrum − − ρ : Doppler shift − Sc(ρ): the PSD of the received signal as a function of ρ

Doppler Spread: BD

− the range of ρ such that

)],(),(E[)( * ttfCtfCtAC ∆+=∆

cCc TttAT ≥∆≈∆ for 0)(such that

tdetAS tjCC ∆∆= ∫

∞

∞−

∆− πρρ 2)()(

0)( >ρCS


Coherence Time and Doppler Power Spectrum (2)

Relationship between the coherence time and the Doppler spread − By the Fourier transform relationship between

If the transmitter and reflector are stationary and the receiver is moving with velocity v − − Remind that in the narrowband fading model the signal decorrelates over the time

of 0.4/fD

− In general,

In summary,

cCcC TSTttA 1for 0)( then ,for 0)( if >≈>∆≈∆ ρρ

)( and )( ρCC StA ∆

cD TB 1≈

DD fvB =≤ λMaximum Doppler shift

).( of shape on the depends where, ρCcD SkTkB ≈

Dc

Tc BTB

m

1 , 1 ≈≈ σ

Documents

Wireless Channel Modelocw.snu.ac.kr/sites/default/files/NOTE/10340.pdf · 2018. 1. 30. · Path loss as a function of distance: − sometimes simple model can captures the essence