WIRELESS FADING CHANNELS

s [λ/8]

µ[ s]

Relative Power [dB]

0

-10

-20

-300.0 0.2

0.40.6 0.8

1.0 040

80120

160

τ

Free Space Loss

• Isotropic antenna: power is distributed homogeneously over surface area of a sphere.

Received power is power through effective antenna surface over total surface area of a sphere of radius d

Transmit antenna

Free Space Loss

The power density w at distance d is

where PT is the transmit power.

w PdT=

4 2π

R TP Ad

P=4 2π

The received power is

with A the `antenna aperture' or the effective receiving surface area.

FREE SPACE LOSS, continued

The antenna gain GR is related to the aperture A according to

Thus the received signal power is

GRA= 4

2πλ

d41

4GP = P 2

2

RTR ππλ ⋅⋅

Received power decreases with distance,PR :: d-2

Antenna Gain

• Antenna Gain GT (φ,θ) is the amount of power radiated in direction (φ, θ), relative to an isotropic antenna.

H: Magnetic Field

φθ

E: Electric FieldP: Poynting VectorP = E x H

Point Source

Space-wave approximation for UHF land-mobile communication

• Received field strength = LOS + Ground-reflected wave.

• The received signal power is

Rj

T T RP = 4 d P G Gλπ

12

+

Re ∆

Space-wave approximation

• The phase difference ∆ is found from Pythagoras.• Distance TX to RX antenna = √ ( ht - hr)2 +d2

• Distance mirrored TX to RX antenna = √ (ht + hr)2 + d2

{(ht - hr)2 +d2}

{(ht + hr)2 +d2}

hthr

Space-wave approximation

The difference ∆ is

At large a distance (d >> 5 ht hr), the received signal power is

where Rc is the ground reflection coefficient

( )

−

=∆

22

11d

hh+-d

h+h+d2)h-h(+d-)h+h(+d2= rtrt2

rt22

rt2

λπ

λπ

dhhxx tr

λπ4

211 ≈∆⇒±≈±

GGPdhhjR

d4 = P RTT

trcR

24exp1

+

λπ

πλ

Space-wave approximation

GGPdhhjR

d4 = P RTT

trcR

24exp1

+

λπ

πλ

The reflection coefficient approaches Rc → -1 for large propagation distances (d→∞) and low antenna heights, So ∆ → 0, and LOS and ground-reflected wave cancel!!

GGPdhh

d4GGPdhh

d4

GGPdhh

dhh

d4

GGPdhhj

dhh

d4 =

GGPdhhj

d4 = P

RTTtr

RTTtr

RTTtrtr

RTTtrtr

RTTtr

R

=

−

=

+

−

=

−−

−

λπ

πλ

λπ

πλ

λπ

λπ

πλ

λπ

λπ

πλ

λπ

πλ

2sin44cos22

4sin4cos1

4sin4cos1

4exp1

222

222

2

2

Two-ray model

• For Rc = -1, the received power is

Rr2

t2

4 T R TP h hd

P G G=

22

πλ

πr th hd

=

Macro-cellular groundwave propagation: For small D (dλ >> 4 hr ht), we approximate sin(x) » x:

An important turnover point occurs for

PGGdhh

d = P TRT

trR λ

ππλ 2sin4

42

2

Two-Ray Model

• 40 log d beyond a turnover point

• Attenuation depends on antenna height

• Turnover point depends on antenna height

• Wave interference pattern at short rangeFree space

ht = 100 meterht = 30 meterht = 2 meter

10 100 1000

Multipath Propagation

CA

D

BReceiverTransmitter

A: free spaceB: reflectionC: diffractionD: scattering

A: free spaceB: reflectionC: diffractionD: scattering

reflection: object is large compared to wavelength

scattering: object is small or its surface irregular

Low-pass equivalent (LPE) signal

RF carrier frequencyRF carrier frequency

Real-valued RF signalReal-valued RF signal Complex-valued LPE signalComplex-valued

( ) ( ){ }2Re cj f ts t z t e π=

LPE signal

( ) ( ) ( ) ( ) ( )j tz t x t j y t c t e φ= + =

In-phase signal componentIn-phase signal component Quadrature componentQuadrature component

Fading: illustration in complex plane

Received signal in vector form:resultant (= summation result) of “propagation path vectors”

quadraturephase

component

Rx

path delays are not important

Tx in-phase component

Wideband channel modelling: in addition to magnitudesand phases, also path delays are important.

Wireless and mobile channel impairments

– Multipath transmission due to reflections from (possibly moving) surrounding objects ⇒ multipath fading and inter-symbol interference (frequency-selectivity)

– Relative transmitter-receiver motion ⇒ Doppler effect (⇒ time-varying, correlated random fading)

– Attenuation of signal power from large objects ⇒ (slow) shadowing

– Interference between different wireless carriers, transmitters, and systems ⇒ inter-channel/inter-cell/inter-system interference

– Spreading of radiated electromagnetic power in space as function of distance ⇒ path loss

– Thermal noise and background noise ⇒ additive noise

Assumptions

• Bandwidth B [Hz] available for communications, on a carrier frequency fc [Hz].

• Digital communications with linear modulation (e.g. QAM, QPSK) is used. – Transmission of a sequence of complex-valued symbols,

modulating a sinusoidal carrier.

• Communications take place at Nyquist rate, i.e. 2B channel symbols are transmitted per second (highest possible rate where intersymbol-interference can be compensated for).

• Perfect synchronization in time and frequency is available [no timing errors or oscillator drift].

Assumptions, cont’d

• Thus, the complex baseband representation of transmitted signals can be used:

Transmitted waveform x(t) may be modelled as a sequence of complex-valued discrete samples x(k), where sampling

is done at Nyquist rate.

• Real part corresponds to in-phase (I) component of modulation symbol/waveform, and imaginary part corresponds to quadrature (Q) component.

Radio channel modelling

tt

t

h(τ)

t Narrowband

h(τ)

Wideband

Radio channel modelling

Narrowband modelling Narrowband modelling Wideband modelling Wideband modelling

Deterministic models(e.g. ray tracing,

playback modelling)

Deterministic models(e.g. ray tracing,

playback modelling)

Calculation of path loss e.g. taking into account

- free space loss- reflections- diffraction- scattering

Calculation of path loss e.g. taking into account

- free space loss- reflections- diffraction- scattering

Stochastical modelsStochastical models

Basic problem: signal fading

Basic problem: signal dispersion

Random flat-fading models

• The complex baseband model of a flat-fading channel becomes

y(k) = α(k)x(k) + w(k), k ∈ Z,

where y(k) is received symbol at discrete time instant k, α(k) is the complex-valued fading coefficient, x(k) is the transmitted information channel symbol, and w(k) is (complex-valued) AWGN.

• α(k) is modelled as a temporally correlated random variable.– Its distribution (pdf) is given by multipath model.– Its correlation properties are given by multipath model and

transmitter-receiver relative motion.

Random flat-fading models, cont’d

• Rayleigh fading: Assumes isotropic scattering conditions, no line-of-sight [most common model]– I- and Q-components of complex fading gain are complex, zero-

mean gaussian processes– thus the fading envelope follows a Rayleigh distribution

• Ricean (Rice) fading: Assumes line-of-sight component is also present.– I- and Q-components of complex fading gain are still complex

gaussian, but not zero-mean– thus the fading envelope follows a Rice distribution

• Nakagami-m fading: More general statistical model which encompasses Rayleigh fading as a special case, and can also approximate Ricean fading very well.

Relative transmitter-receiver motion

• Assume: Transmitter and receiver move relative to each other with a constant effective velocity v [m/s].

• This results in a Doppler shift in the carrier frequency fc by a maximal Doppler frequency of

fD = vfc/c [Hz]

where c = 3•108 m/s is the speed of light.• It results even in randomly time-varying fading envelope as

reflection and scattering conditions change with time as transmitter and receiver move.

• Random fading models used to describe this phenomenon.• How fast the fading varies depends on fD. The faster the

motion, the more rapid the fading variations.

Interference

• Electromagnetic disturbances from different sources within a frequency band may interfer with the desired information signal.

• These disturbances may come from other users (intra- or inter-cell), or from other systems sharing the same frequency band (may be problem in unlicensed bands).

• In our discussion we shall either disregard such interference, or model it simply as an increased noise floor (appropriate if there are many independent interference sources).– our “additive noise” term may encompass certain types of

interference (e.g., inter-user interference in a fully loaded cellular network).

Local Propagation Effects with Mobile Radio

• Slow fading– Arises from large reflectors and diffracting objects with distant

path from the small terminal. – With slow propagation changes, these factors contribute to the

median path losses between a fixed transmitter and receiver.– The statistical variation of these mean losses due to

• variation was modeled as lognormal distribution for terrestrial application.

Shadowing or lognormal Fading

Lognormal Shadowing

• Log-normal shadowing– Received signal is the product of many transmission factors --

In dB, it is the sum– Central Limit Theorem implies Gaussian

(ln )r

r −

12 2

2

2πσµ

σPr( ) exp ,r r= ⋅ − ≥ 0

Pr(r)

r

Local Propagation Effects with Mobile Radio

Fast fading is the – Rapid variation of signal levels when user terminal moves

short distances. – It is due to reflections of local objects and motion of

terminal. That is, the received signal is the sum of a number of signals reflected from local surfaces and signals sum in the constructive or destructive manner.

– The resulting phase relationships are dependent on relative path lengths to the local object, speed of motion and frequency of transmission

Rayleigh Fading

• The complex phasor of the N signal rays is given by

where En is the electric field strength of the nth path and θn is relative phase. Ẽ represent the multiplication effects.

• By applying the central limit theorem, we have

where Zr and Zi are real Gaussian random variables.

∞→+→∑=

NjZZeE ir

N

n

jn

n as 1

θ

∑=

=N

n

jn

neEE1

~ θ

Rayleigh Fading

• Considering one of the components of the sum, the expectation of each component is

• Mean of the complex envelope is given by

[ ] [ ] [ ][ ] 0

21 2

0

==

=

∫ θπ

πθ

θφ

deEE

eEEEeEE

jn

jn

jn

nn

[ ]

[ ] 0

~

1

1

==

=

∑

∑

=

=

N

n

jn

N

n

jn

n

n

eEE

eEEEE

θ

θ

Rayleigh Fading

• The variance (power) in the complex envelope is given by mean-square value

• Difference of two random phases is a random phase.• By symmetry, the power is equally distributed

between the real and imaginary parts of complex envelope.

[ ]

01

2

1 1

)(

11

2~

PE

eEEE

eEeEEEE

N

nn

N

n

N

m

jmn

N

m

jm

N

n

jn

mn

mn

==

=

=

∑

∑∑

∑∑

=

= =

−

=

−

=

θθ

θθ

Rayleigh Fading

• Since the complex envelope has zero mean, for σ2=P0/2, the probability density function of Zr is given by the Gaussian density function

• Define the amplitude of complex envelope as

• Rayleigh probability density function

22ri ZZR +=

( ) 22 22

σ

σr

R errf −=

( ) 22 2/

21 σ

σπr

r

zrZ ezf −=

Rayleigh Fading

( ) 22 22

σ

σr

R errf −=

0 2 4 6 8 1000.10.20.30.40.50.60.70.80.9

r

Rayleigh Fading

• Integrating the Rayleigh probability density function yields the corresponding cumulative probability distribution function:

• The mean value of Rayleigh distribution is given by

[ ]

2

)(0

πσ=

= ∫∞

drrrfRE R

( )22 2

0

1

)(

σR

R

R

e

drrfRrprob

−−=

=< ∫

Rayleigh Fading

Rayleigh Fading

• Mean-square value is given by

• The root-mean-square (rms) amplitude is

[ ]2

02

0

2

2

)(

rms

R

RP

drrfrRE

===

= ∫∞

σ

σ2=rmsR

Representation in the complex plane

Complex Gaussian distribution is centered around the “strong path” => magnitude is Rice distributed, probability of deep fade is extremely small

Complex Gaussian distribution is centered at the origin of the complex plane => magnitude is Rayleigh distributed, the probability of a deep fade is larger than in the Rician case

( ),p x y iyiy

x x

Bell-shaped functionBell-shaped function

Rician Fading

• For the direct ling-of-sight path in mobile radio channels and indoor wireless, the reflected paths tend to be weaker than the direct path and the complex envelope is

• For s2=|E0|2, Rician factor defined as

• Amplitude density function

where I0(.) is the modified Bessel Function of zeroth order.

∑=

= N

nEn

sK

1

2

2

( ) 0 202

2222

≥

=

+−

rrsIerrfsr

R σσ

σ

∑=

+=N

n

jn

neEEE1

0~ θ

N2

01

2 2σ==∑=

PEn

n

Rician Fading

( )

<

≥

=

+−

0 0

0 202

2222

r

rrsIerrf

sr

R σσ

σ

r r

Pdf

r 86420

0.6

0.5

0.4

0.3

0.2

0.1

0

K = 2K = 1K= 0 (Rayleigh)

σ = 1

K = 3

Rician Fading

Received Signal Power

shadow fading

Rayleigh fading

path loss

log (distance)

Received Signal Power (dB)

Implications – increases the required transmit power – causes bursts of errors

Countermeasures: narrowband fading

Diversity (transmitting the same signal at different frequencies, at different times, or to/from different antennas)

- will be investigated in later lectures- wideband channels => multipath diversity

Interleaving (efficient when a fade affects many bits or symbols at a time), frequency hopping

Forward Error Correction (FEC, uses large overhead)

Automatic Repeat reQuest schemes (ARQ, cannot be used for transmission of real-time information)

Bit interleaving

TransmitterTransmitter ChannelChannel ReceiverReceiver

After de-interleaving of bits, bit errors

are spread!

Bits are interleaved ...

Fading affects many adjacent

bits

... and will be de-interleaved in the receiver

Bit errors in the receiver (better for FEC)

wideband fading channel

channel Impulse Response (CIR)

delay spread TmChannel is assumed linear!

time t|h(τ,t) |

zero excess delay delay τ

Channel Impulse Response (CIR) of a wideband fading channel

τ

path delay

( ) ( ) ( ) ( )1

0

, iL

j ti i

i

h t a t e φτ δ τ τ−

=

= −∑

The CIR consists of L resolvable propagation paths

path delaypath attenuationpath attenuation path phasepath phase

LOS pathLOS path

delay spread Tm

Received multipath signal

( ) ( )kk

s t b p t kT∞

=−∞

= −∑Transmitted signal:

( ) ( ) ( ) ( ) ( ),r t h t s t h t s t dτ τ τ∞

−∞

= ∗ = −∫Received signal:

( ) ( ) ( )1

0

iL

j ti i

i

a t e s tφ τ−

=

= −∑ ( ) ( ) ( )0 0f t t t dt f tδ − =∫

pulse waveformpulse waveformcomplex symbolcomplex symbol

Received multipath signal

The received multipath signal is the sum of L attenuated, phase shifted and delayed replicas of the transmitted signal s(t)

( )00 0

ja e s tφ τ−

Tm

T

( )11 1

ja e s tφ τ−

( )22 2

ja e s tφ τ−:

Normalized delay spread D = Tm / T

BER vs. S/N performance

In a Gaussian channel (no fading) BER <=> Q(S/N)erfc(S/N)

Typical BER vs. S/N curves

S/N

BER

Frequency-selective channel(no equalization)

Flat fading channel

Gaussianchannel

(no fading)

BER vs. S/N performance

( ) ( )BER BER S N z p z dz= ∫z = signal power level

Flat fading

Typical BER vs. S/N curves

S/N

BER

Frequency-selective channel(no equalization)

Flat fading channel

Gaussianchannel

(no fading)

BER vs. S/N performance

Frequency selective fading <=> irreducible BER floor

Typical BER vs. S/N curves

S/N

BER

Frequency-selective channel(no equalization)

Flat fading channel

Gaussianchannel

(no fading)

BER vs. S/N performance

Diversity (e.g. multipath diversity) <=> improved performance

Typical BER vs. S/N curves

S/N

BER

Flat fading channel

Gaussianchannel

(no fading)

Frequency-selective channel(with equalization)

Time-variable transfer function

( ) ( ) ( ) ( )1

0, i

Lj t

i ii

h t a t e φτ δ τ τ−

=

= −∑Time-variant CIR:

Time-variant transfer function (frequency response):

( ) ( ) ( ) ( )1

22

0, , i i

Lj t j fj f

ii

H f t h t e d a t e eφ π τπ ττ τ∞ −

−−

=−∞

= =∑∫

Deterministic channel functions

( ),d τ ν

( ),D f ν

Time-variantimpulse response

Time-variantimpulse response

(Inverse) Fourier

transform ( ),h tτ

Time-varianttransferfunction

Time-varianttransferfunction

Doppler-variantimpulseresponse

Doppler-variantimpulseresponse

( ),H f t

Doppler-varianttransfer function

Doppler-varianttransfer function

Stochastic channel functions

( ) ( ) ( )[ ]

( ) ( )[ ] ( ) ( )

( ) ( )τφτφ

ττδτφττ

ττττφ

hh

h

h

ttththE

tththEt

≡

−∆=∆+

∆+=∆

0;

;;;

;;;,

21121

21*

21

Under the assumption of uncorrelated scatterers

Average power output of the channel as a function of τMultipath Intensity Profile

Stochastic channel functions

( ) ( )

( ) ( ) ( )[ ] ( ) ( )[ ]

( ) ( ) ( ) ( ) ( )

( ) ( )

( ) ( ) ( ) ( )∫ ∫

∫

∫∫ ∫

∫ ∫

∫

∞

∞−

∞

∞−

∆−∆−

∞

∞−

∆−

∞

∞−

−∞

∞−

∞

∞−

−

∞

∞−

∞

∞−

−−

∞

∞−

−

==∆=∆

∆∆=∆

=∆=−∆=

==∆+=∆

=

−−

12

12

1

12

1

12

1212

211

2122

2*

1*

21*

21

2

1

1

22112211

2211

0;0;

;;

;;

;;;;;,

;;

ττφττφφφ

φττφ

ττφττττδτφ

ττττφ

τ

τπτπ

τπ

ττπττπ

τπτπ

τπ

dedeff

tfdet

detddet

ddeeththEttfHtfHEtff

ethtfH

fjh

fjhHH

Hfj

h

ffjh

ffjh

fjfjH

fj

( )τφc( )fH ∆φ

Βm≈1/Tm

τ∆fTm :delay spread of the channel

Received multipath signal

The normalized delay spread is an important quantity.

When Tm << 1, the channel is

- narrowband- frequency-nonselective- flat

and there is no intersymbol interference (ISI).

When Tm approaches or exceeds unity, the channel is

- wideband- frequency selective- time dispersive

Important feature has many names!

Stochastic channel functions

( ) ( )

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )∫

∫∫

∫

∫

∞

∞−

∆−

∞

∞−

∆∞

∞−

∆−

∞

∞−

∆−

∞

∞−

∆−

∆∆==

∆∆=∆∆=

∆∆==

∆∆∆=∆

tdetSS

fdefStdetS

tdetSS

tdetffS

tjHhh

fjH

tjHh

tjHHH

tjHH

πν

πνπν

πν

πν

φνν

ντφντ

φνν

φν

2

22

2

2

;0

;;;

;0;0

;;

( )tH ∆φ ( )νHS

∆tΒd: Doppler Spread of Power SpectrumΤd:coherence time

ν

Stochastical channel functions

( );hS τ ν

( );HS f ν∆

Channel intensityprofile

Channel intensityprofile ( )hφ τ( )H tφ ∆ Tm

τσTd

( );h tφ τ ∆Frequency

timecorrelationfunction

Frequencytime

correlationfunction

Channel Dopplerspectrum

Channel Dopplerspectrum

Scatteringfunction

Scatteringfunction( );H f tφ ∆ ∆

( )H fφ ∆ ( )HS νBm Bd

Attenuation values for different materials

Material Loss (dB) FrequencyConcrete block 13-20 1.3 GHzPlywood (3/4”) 2 9.6 GHzPlywood (2 sheets) 4 9.6 GHzPlywood (2 sheets) 6 28.8 GHzAluminum siding 20.4 815 MHzSheetrock (3/4”) 2 9.6 GHzSheetrock (3/4”) 5 57.6 GHzTurn corner in corridor 10-15 1.3 GHz

Indoor channel measurment

TX

RX

1

garage

LOS

officeofficeofficeoffice

hall labslabs

labs

office

2

3NLOS

RX TXLOS

Measures @ 2GHz - 20 MHz bandwidth

Power Delay Profile: corridor LOS

h s( , )τ2

Power Delay Profile: corridor NLOS

Power Delay Profile: garage LOS

h s( , )τ2

s [λ/8]

µ[ s]

Relative Power [dB]

0

-10

-20

-300.0 0.2

0.40.6 0.8

1.0 040

80120

160

τ

Scattering Function

{ }Sc h ss( , ) ( , )τ υ τ= F 2

Modified Scattering Function: LOS corridor

FourierTransform

Doppler Spread

Scattering Function

{ }Sc h ss( , ) ( , )τ υ τ= F 2Modified Scattering Function: corridor NLOS

Scattering Function

{ }Sc h ss( , ) ( , )τ υ τ= F 2Modified Scattering Function: garage LOS

Physical interpretation of Doppler shift

Vαarriving patharriving path

direction of receivermovement

direction of receivermovementRx

Doppler frequency shiftDoppler frequency shiftV = speed of receiverλ = RF wavelength

V = speed of receiverλ = RF wavelength

ααλ

coscos Ddop fVf ==∆

Maximum Doppler shiftMaximum Doppler shift

Angle of arrival of arrivingpath with respect to

direction of movement

Angle of arrival of arrivingpath with respect to

direction of movement

Power spectrum & Doppler effect

effect.Doppler the todue fromfrequency different a has angle arrival of waveElementary

cfα

αcosDc fff +=

( ) [ ]is , range in the power received that so ddistributeuniformly is angle Arriving

dfffdffS +

( )( )[ ]

dfffff

df

ddf

dffS

ffff

ddf

DcD

D

cD

21

1212

cos ;sin

−−=

×=

−=−=

πα

π

ααα

Power SpectrumHz)135 km/h,50 GHz,5.1 (cf. === Dc fvf

Countermeasures: wideband systems

Equalization (in TDMA systems)- linear equalization- Decision Feedback Equalization (DFE)- Maximum Likelihood Sequence Estimation (MLSE) using Viterbi algorithm

Rake receiver schemes (in DS-CDMA systems)

Sufficient number of subcarriers and sufficiently long guard interval (in OFDM or multicarrier systems)

Interleaving, FEC, ARQ etc. may also be helpful in wideband systems.