Wireless Channel Model

Wireless Channel Model

Wha Sook Jeon

Mobile Computing and Communication Lab.

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





= d

K d P

P_{r t} 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

t

r P

P / ψ =











 −

−

= ₂

2 10

2

) log

10 exp (

2

10 ln / ) 10

(

dB dB

dB

p

ψ ψ

ψ σ

µ ψ ψ

σ ψ π

ψdB

ψ

ψdB

µ

ψdB

σ











 −

−

= ₂

2

2

) exp (

2 ) 1

(

dB dB

dB

dB

p dB

ψ ψ

ψ σ

µ ψ

σ ψ π

Shadowing (2)

constant.

n attenuatio an

is where

, )

(d e ^αd α

s = ⁻

∑

= _i

t d

d

i dt

d

t e e

d

s( ) = ^−α∑ = ^−α

 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

■

−

^d_t : 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 )

( log₁₀s d_t

ψdB

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

−

− : ψ

_dB

a Guassian r.v. with mean zero and variance

dB t

r

d K d

P dB

P = − γ +ψ

0 10

10 10 log

log 10 ) (

ψdB

σ

P_r/P_t (dB)

log d Very slow Slow

10logK

γ

−10

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)

P

min

Pmin

) , (P_min d p_out

) )

( ( )

,

(P_min d p P d P_min p_out = _r <











 − + −

−

=

<

dB

d d K

P Q P

P d

P

p _r ^t

σψ

γ log ( / ) 10

log 10 1 (

) )

(

( _min ^{min 10 10 0}

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 )

D D

D

f

v f t

π φ

λ θ φ

π

2

2 cos 1

=

∆ =

= ∆

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:

) (

) (

0

n j

N

n n

e n

c τ = α ^{− φ} δ τ −τ

∑

=

τ

−

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

0 t e t

t

c ^{N t} _n ^{j t} _n

n

n

δ τ τ

α

τ

=

∑

₌ ^{− φ} −

)

n(t τ

Dn

φ ⁿ

D n

c

n t πf τ t φ

φ ( )=2 ( )−

)

n(t α

)

n(t φ

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 s

_I

(t) and quadrature component s

_Q

(t)

{

^{( )}

}

^{( )cos(2 ) ( )sin(2 )}

Re )

(t u t e ² s t f t s t f t

s = ^{j πfct} = _I π _c − _Q π _c

) ( )

( )

(t s t js t

u = _I + _Q

Transmit & Receive Signal Models (2)

 Transmitted signal

−

 Received signal

−

−







 



 



 −

=







 



 



 





 − −

=







 



 



 − −

=







 





 −

=

∑

∑ ∫

∫ ∑

∫

=

−

=

∞

∞

−

−

∞

∞ =

−

∞

∞

−

t f j N(t)

n

n t

j n

t f j N(t)

n

n t

j n

t f j -

N(t)

n

n t

j n

t f j

c n

c n

c n

c

e t t

u e

t

e d

t u t e

t

e d t

u t e

t

e d t

u t c t

r

π φ

π φ

π φ

π

τ α

τ τ τ

τ δ α

τ τ τ

τ δ α

τ τ τ

2 0

) (

2 0

) (

2 0

) (

2

)) ( (

) ( Re

) ( )) ( (

) ( Re

) ( )) ( (

) ( Re

) (

) , ( Re

) (

{

^{u t ej fct}

}

t

s( ) = Re ( ) ^2π









−

=

∑

^{N t t e−j n t u t t ej fct}

t

r( ) Re ^{( )}α ( ) ^{φ ( )} ( τ ( )) ^2π

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:

T_m << 1B

i

i u t

t

u( −

τ

) ≈ ( ) for all

τ







 

 



=  ⁻

∑

=^{( ) ( )} 0

2 ( )

) ( Re )

( ^{j t}

t N

n n t

f

j _{c n}

e t e

t u t

r ^π α ^φ

channel

characteristic

{ }

^{e f t}

t

s( ) = Re ^j2πfct = cos2

π

_c

t

N( ) 

  

Narrowband Fading Model (2)

 Received Signal

−

− in-phase component:

− quadrature component:

 r

_I

(t) and r

_Q

(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

− r

_I

(t) and r

_Q

(t) are independent Gaussian process with the same autocorrelation , a mean of zero, and a crosscorrelation of zero

) ( cos ) ( )

(

) (

0

t t

t

r _n

t N

n n

I

∑

α φ

=

=

t f t

r t f t

r t

r( ) = _I( )cos2π _c + _Q( )sin2π _c

) ( sin ) ( )

(

) (

0

t t

t

r _n

t N

n n

Q

∑

α φ

=

=

)

n(t

α φn^(t)

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 360^o rotation for a small change

 The received signal r(t) is zero-mean Gaussian process

−

−

n

n D

n D n

n

n(t) ≈α ,τ (t) ≈τ , f (t) ≈ f α

)

n(t φ

t f f

t _{c n} ^D_n

n π τ π

φ ( ) = 2 −2

0 )]

( cos E[

] [ E )]

( [

E ^{r t} =

∑

_n ^{n t} =

n

I α φ

fc

τn

) 0 )]

( ) ( [ E

( r_I t r_Q t = 0

)]

( E[sin ]

[ E )]

( [

E ^{r t} =

∑

^{α φ t} =

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:

−

N n

n n θ π

θ = ∆ =2

N P_r

n] 2

[

E α² =

) sin(2

) ( )

2 cos(

) ( )]

( ) ( [ E )

( t r t r t t A t f t A _, t f t

A_{r r c r r c}

Q I

I ∆ ∆ − ∆ ∆

=

∆ +

=

∆ π π



 



 ∆

=

∆ +

=

∆

∑

^{n n}

n I

I r

t t v

t r t r t

AI θ

λ

α 2π cos cos

] [ E 5 . 0 )]

( ) ( [ E )

( ²



 



 ∆

−

=

∆ +

=

∆

∑

^{n n}

n Q

I r

r

t t v

t r t r t

AI Q θ

λ

α 2π cos sin

] [ E 5 . 0 )]

( ) ( [ E )

( ²

,

θ λ θ

π

π

^_^∆

 



 ∆ ∆

=

∆

∑

=

t n v t P

A

N

n r

r_I 2 cos

2 cos )

(

1

v

Autocorrelation and Cross Correlation (3)



) 2

cos(

) ( )

( t A t f t

A ∆ = ∆ π ∆

) 2

(

2 cos 2 cos

) (

0 2 0

t f J

P

t d v t P

A

D r

r r_I

∆

=



 



 ∆

=

∆

∫

π

θ λ θ

π π

π

0

2 cos 2 sin

)

( ²

, 0

=



 



 ∆

=

∆^{t P}_π

∫

^{π π}_λ^{v t θ dθ}

A_{r r} ^r

Q I

0

, ∆ →

∞

→

θ

N

λ 4 . 0

4 .

0 ⇒ ∆ =

=

∆t v t

f_D

Antenna spacing of 0.4λ:

the signal decorrelates over a distance of 0.4λ

π θ

θ π θ

d e j e

x

J_{n n} ^{2 jxcos jn} 2 0

) 1

( ⁼

∫

Power Spectral Density

)]

2 ( [

) (

)]

( )

( [

25 . ) (

0

π

_D

τ

r

c r

c r

r

f PJ

f S

f f

S f

f S f

S

I

I I

= F

+ +

−

=

f

_c

+f

_D

f

_c

S

_r

(f)

f

_c

-f

_D

Envelope and Power Distributions without LOS

 Signal envelope

−

− r

_I

and r

_Q

are Gaussian random variables with mean zero and variance

− z(t) is Rayleigh distributed

■

 Power

−

− The received signal power is exponentially distributed with mean

■

σ2

) ( )

( )

( )

( t r t r

²

t r

²

t

z = =

_I

+

_Q

0

, )

( ²

2

2

2 ≥

= z e⁻ z

z p

z Z

σ σ

1

^{− x}

2 2

) ( )

( t r t

z =

2σ 2

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

, )

( ² _{0 2}

) ( 2

2 2 2

 ≥



 



= 

− +

zs z I

z e z

p

s z

Z σ ^σ σ

2 2

0

2

( ) = + 2 σ

= ∫

^∞

x p x dx s

P

_{r Z}

LOS component power

NLOS component power

* I₀: the modified Bessel function

2 2

2σ K = s

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

1

2 ≥

= Γ ⁻ e⁻ m

P m

z z m

p ^Pr

mz m

r m m Z

) 1 2

( ) 1

( + ² +

= K K

m

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:

τ

1 τ₂ Bu

2 1

1 −

τ

>>

τ

T T_m >>

T T_m <<

 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.

−

)) ( (

) ( )

,

( ^{( )}

) (

0

t e

t t

c ^{j t} _n

t N

n n

n δ τ τ

α

τ = ^{− φ} −

∑

=

−

τ

t

) , ( t c

τ

)]

, ( ) , ( E[

) , , ,

( _{1 2} t t c^* ₁ t c ₂ t t

A_c τ τ ∆ = τ τ +∆

)]

, ( ) , ( E[

) , ,

( _{1 2} t c^* ₁ t c ₂ t t

A_c τ τ ∆ = τ τ +∆

τ

1 1

2

τ

τ

≠

) , ( )

( ) , ( )]

, ( ) , (

E[c^* τ₁ t c τ₂ t +∆t = A_c τ₁ ∆t δ τ₁ −τ₂ = A_c τ ∆t

) (

τ

=

τ

₂ =

τ

₁

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:

−

) , ( t c

τ

τ τ t e ^πτd c

t f

C

∫

−^∞∞ ^{j f}

= ( , ) − ²

) , (

) , (f t C

) , (f t C

)]

, ( ) , ( E[

) , , ,

( f₁ f₂ t t C^* f₁ t C f₂ t t

A_C ∆ = +∆

)]

, ( ) , ( E[

) , ,

(f₁ f₂ t C^* f₁ t C f₂ t t

A_C ∆ = +∆

[ ]

) , (

) ,

( ) , ( E

) ,

( )

, ( E

) , , (

) ( 2

2 1 2

2 2

1

*

2 2

- 2 1 2

- 1

* 2

1

1 2

2 2 1

1

2 2 1

1

d e

t A

d d e

e t t c t c

d e

t t c d

e t c

t f f A

f f j c

f j f j

f j f

j C

∆

=

∆ +

=



 +∆

=

∆

∞

−

∞ −

∞

−

−

∞ −

∞

−

∞

∞

−

∞ −

∞

∞

∞

∫

∫

∫ ∫

∫

∫

τ τ

τ τ τ

τ

τ τ

τ τ

τ π

τ π τ

π

τ π τ

π

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[

)

( = c^* t c t A ∆t ∆t =

A_c τ τ τ _c τ

τ

∫

∫

∞

∞

=

0 0

) (

) (

τ τ

τ τ µ τ

d A

d A

c c T_m

∫

∫

∞

∞ −

=

0 0

2

) (

) ( )

(

τ τ

τ τ µ

σ τ

d A

d A

c

c T

T

m m

Relative power

T A_c(τ) ≈ 0 for τ ≥

) 10 (

m

m s T

T

s T

T <<σ <σ

) 10 (

m

m s T

T

s T

T >>σ > σ

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:

c C

c A f f B

B such that (∆ ) ≈0 for all∆ ≥ )]

, (

) , ( E[

)

( f C^* f t C f f t

A_C ∆ = +∆

) ( and )

( _c τ

C f A

A ∆ T f

f A T

A_c( ) 0 for , then _C( ) 0 for 1

if τ ≈ τ > ∆ ≈ ∆ >

Tm

c k

B ^{≈ 1}T ⁼ σ

Bc

B <<

ISI negligible

1

1 >> ≈ ⇒

≈ c Tm

s B B

T σ

⇒

≈

<<

≈ σ ^c

B B >>

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

−

^S_c⁽ρ): the PSD of the received signal as a function of _ρ



Doppler Spread: B

)]

, ( ) , ( E[

)

( t C^* f t C f t t

A_C ∆ = +∆

c C

c A t t T

T such that (∆ ) ≈0 for ∆ ≥

t d e

t A

S_C =

∫

−^∞∞ _C ∆ ^{j t} ∆

∆

− πρ

ρ) ( ) ²

(

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/f_D

−

In general,



In summary,

c C

c

C t t T S T

A ( ) 0 for , then ( ) 0 for 1

if ∆ ≈ ∆ > ρ ≈ ρ>

) ( and )

( _C ρ

C t S

A ∆

c

D T

B ≈ 1

D

D v f

B ≤ λ = Maximum Doppler shift

).

( of shape on the

depends where

, _C ρ

c

D k T k S

B ≈

D c

T

c T B

B

m

1

1 , ≈

≈ σ