4 Radio Propagation - Fading

Personal & Mobile Communication – Spring 2008 Dr Daniel So

1

Lecture 4 Radio Wave Propagation:

Fading and Multipath


2

Lecture Aims

• Understand the theory of multipath fading channel• Know how to calculate the parameters for fading

channel• Learn different types of multipath fading channel• Some common fading channel models


3

Backgrounds (I)

• Large-scale propagation (Lecture 2)– Predicts mean received signal strength at large Tx-Rx

distance • Hundreds or thousands of meters

– Path loss, shadowing etc– Importance

• Proper site planning

• Small-scale propagation– Characterize the rapid fluctuations over short distance

or time– Fading– Importance

• Proper receiver design to handle the rapid fluctuations


4

Backgrounds (II)


5

Factors Affecting Fading (I)

• Multipath propagation – Signal arrives at Rx through different paths

• Reflection, Diffraction, Scattering

– Paths could arrive with different gains, phase, & delays– Small dist variation can have large amplitude variation

• e.g. 2 paths with perfect reflector (Plan earth model)– At 900MHz, 0-to-0 within 30cm

signal at sender

LOS pulsesmultipathpulses

⎟⎠⎞

⎜⎝⎛ Δ

=λ

π dEE LOSTOT sin~2~


6

Factors Affecting Fading (II)

• Speed of mobile/surrounding objects– The mobile can be in motion and the environment can

also be varying (cars, pedestrians, etc) • Induces Doppler shift

• Doppler Shift– The change of frequency due to movement

• Phase change

• Frequency change

• v = speed of mobile, λ = carrier wavelength• +/-ve when moving towards/away the wave

θλπ

λπφ cos22 tvl Δ

=Δ

=Δ

θλ

φπ

cos21 v

tfd =

ΔΔ

=


7

Factors Affecting Fading (III)

• Signal transmission bandwidth– If the signal bandwidth is wider than the channel

“bandwidth”, the received signal will be distorted– Consider a 2-ray model

• Second ray arrive at one symbol period later than the first ray

( ) ( ) ( )sTh ++= τδατδτ 2

( ) sfTjefH πα 221 −+=


8

Multipath Channel Impulse Response (I)

• The multipath channel can be modelled as a filter– A summation of all multipath– Each multipath contain its gain, phase and delay– Varies with time and distance

• If the mobile is moving, distance is also related to time d = vt– Two time-related variables

• t is the time variation due to motion• τ is the time variation due to multipath delay

– Excess delay - relative delay compared to the first arriving path

– τi(t) = i-th path excess delay at time t– N = total number of arriving paths

( ) ( ) ( )[ ] ( )( )∑−

=

−=1

0

,exp,,N

iiiip ttjtth ττδτθτατ


9

Multipath Channel Impulse Response (II)


10

Multipath Channel Impulse Response (III)

• Received signal

• x(t) = passband transmitted signal• hp(t,τ) = channel impulse response due to motion and excess delay

• Baseband equivalent representation– Signal processing is done in baseband– Need to obtain a baseband representation of the

channel hb(t,τ)– Consider the transmitted passband signal x(t)

• Let s(t) be the baseband signal with s(t) = sI(t) + jsQ(t)

( ) ( ) ( )ττ ,, thtxty p⊗=

( ) ( ) ( ) ( ) ( ) ( ){ }tfjcQcI

cetstftstftstx πππ 2Re2sin2cos =−=


11

Multipath Channel Impulse Response (IV)

– Similarly, baseband received signal r(t)

– Baseband equivalent channel impulse response h(t,τ)

– Hence

• ½ come from down-conversion from passband to baseband• To retrieve the in-phase component, multiply with carrier

• After LPF:

( ) ( ){ }tfj cetrty π2Re=

( ) ( ){ }tfjp

cethth πττ 2,Re, =

( ) ( ) ( )τ,21 thtstr ⊗=

( ) ( ) ( ) ( ) ( ) ( ) ( )tftftstftstftx ccQcIc ππππ 2sin2cos2cos2cos 2 −=

( ) ( )( ) ( ) ( )tftstfts cQcI ππ 4sin214cos1

21

−+=

( )tsI21

=


12

Multipath Channel Impulse Response (V)

• Discrete-time baseband impulse response– Divide multipath delay into discrete segments called

excess delay bins– i-th excess delay =

• Δτ = delay bin width; L = maximum resolvable delay path

• φi(t,τ) = i-th path random phase shift at time t• a(t,kΔτ) = baseband real amplitude of k-th bin at time t• θ(t,kΔτ) = phase shift due of k-th bin at time t

{ }1,,0 −=∀Δ= Liii Kττ

( ) ( ) ( ) ( )( )[ ] ( )( )∑−

=

−+=1

0

,2exp,,N

iiiici tttfjtath ττδτφτπττ

( ) ( )[ ] ( )∑−

=

Δ−ΔΔ=1

0

,exp,L

k

kktjkta ττδτθτ


13

Multipath Channel Impulse Response (VI)


14

• If the excess delay bin equals to the symbol period, the channel can be modelled as a tap delay-line filter– Each tap delay is exactly 1 symbol period

• st = baseband signal, nt = noise, rt = received baseband signal

Tap Delay Line Model

ISIz-1 z-1 z-1 z-1st

nt

rt

00

θjea 11

θjea 22

−−

LjL ea θ 1

1−

−Lj

L ea θ


15

Problems with Multipath Channel

• Several paths arrive within one delay bin (Δτ)– These paths cannot be resolved (N≠L)– Different gains and phase will be combined – Constructive and destructive interference can occur– Fading

• The received signal power changes rapidly from bin to bin• When a symbol is in deep fade (all bins in that symbol period have

small values), it cannot be detected correctly• Diversity technique required

– If the excess delay is longer than one symbol period• Inter-symbol interference (ISI) occurs• Equalization techniques required


16

WSSUS Channel

• Assume channel is wide-sense stationary (WSS)– The autocorrelation function is dependent on the time

difference• i.e. Autocorrelation function is the same at any time

• Assume all paths are uncorrelated– Uncorrelated Scattering (US)

• Rh=0 unless τ1=τ2

( ) ( )[ ] ( )212122*

11 ,;,, ττττ ttRththE h −=

( ) ( ) ( )211212121 ;,; ττδτττ −−=− ttRttR hh


17

Channel Autocorrelation Function

• Let

– where is the autocorrelation function of the discretised channel bin

• Note that τ1=τ2, i.e. uncorrelated among different bins

( ) ( ) ( )[ ]22*

112121 ,,,; ττττ ththEttRh =−

( ) ( ) ( )[ ]τθττ ΔΔ= ktjktatak ,exp,,~

( ) ( ) ( ) ( )⎥⎦

⎤⎢⎣

⎡−−= ∑∑

−

=

−

=

1

0222

*1

0111 ,~,~ L

iii

L

kkk tataE ττδτττδτ

( ) ( )[ ] ( ) ( )∑∑−

=

−

=

−−=1

0

1

02122

*11 ,~,~L

k

L

iikik tataE ττδττδττ

( ) ( ) ( )∑−

=

−−=1

0211121 ;,

L

kka ttR ττδττδτ

( ) ( ) ( )[ ]12*

11121 ,~,~;, τττ tataEttR ika =


18

Power Delay Profile

• Characterise the power distribution against the excess delays– Average of |h(t,τ)|2 over time t

– For discrete-time model

– Pk = power at k-th delay bin• Common delay profiles

– Uniform: Pk being constant over k– Exponential: Pk = cexp(-kΔτ/c)

( ) ( ) ( )τττ ;0,1lim0

2h

T

TRdtth

TP == ∫∞→

( ) ( ) ( ) ( )∑∑−

=

−

=

Δ−=Δ−Δ=1

0

1

0

2,L

kk

L

k

kPkktaP ττδττδττ


19

Time Dispersion Parameters (I)

• Determined from the power delay profile– Treat the power delay profile as a probability mass function– Calculate the mean, second moment, and standard deviation for it

• Mean excess delay Second moment

• RMS delay spread

• Maximum excess delay (X dB) = τX - τ0– τ0 = first arriving signal delay– τX = max excess delay within X dB of the strongest path

[ ] ( ) ∑ ∑∑⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛===

kk

kk

k

kk a

akprobE ττττ 2

2

[ ] ( ) ∑ ∑∑⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛===

kk

kk

k

kk a

akprobE 22

222

___2 ττττ

2___

2 ττστ −=


20

Time Dispersion Parameters (II)


21

Time Dispersion Parameters (III)


22

Coherence Bandwidth

• Coherence bandwidth Bc– Frequencies separated by less than this bandwidth will

have their fades highly correlated• Flat frequency spectrum within Bc

– Signals will be affected differently with the frequency separation goes beyond Bc

– Frequency correlation higher than 0.9 and 0.5

• RMS delay spread ↑, Coherence bandwidth ↓• Frequency correlation ↑, Coherence bandwidth ↓

τσ501

≈cBτσ5

1≈cB

( ) 9.0;21 >Δ− τffRh ( ) 5.0;21 >Δ− τffRh


23

Doppler Spread and Coherence Time

• Parameters to describe the time varying nature of a channel• Doppler spread BD

– Measure of spectral broadening due to time variation– BD = 2fd-max

• fd-max = max Doppler shift = v/λ

• Coherence Time Tc– Time duration that the fading parameters remain fairly constant– Coherence time for correlation above 0.5:

– Mobile speed ↑, Doppler spread ↑, Coherence time ↓

max169

−

≈d

c fT

π

( ) ( ) 5.00; >Δ=Δ tRtR ah


24

Types of Fading (Delay Spread)

• Flat fading– Signal BW BS < Coherence BW BC

– Delay spread στ < Symbol period TS

– A deep fade can degrade the system performance significantly


25

Types of Fading (Delay Spread)

• Frequency selective fading– BS > BC , στ > TS

– ISI occurs => equalization technique required– With proper equalization, frequency diversity can be

achieved


26

Types of Fading (Doppler Spread)

• Fast fading (Time selective fading)– TS > Coherence time TC , BS < Doppler spread BD

– Channel impulse response changes within the symbol duration

– Occurs for very low data rates– With packetised transmission, fast fading is now

commonly referred to rapid channel changes within one packet or frame

– With proper system design, time diversity can be obtained


27

Types of Fading (Doppler Spread)

• Slow fading– TS << TC , BS >> BD

– Channel changes at a rate much slower than the symbol duration

– Very common, especially in high data rate applications• GSM900 at 200km/h, TC ≈ 1ms, User frame = 576.92 μs

• Quasi-static slow fading– Channel is static within a frame but varies

independently from frame to frame– Used in simulation to provide an averaged

performance over many channel realisations


28

Types of Fading

– Fast/slow and frequency flat/selective fading is not mutually exclusive


29

Common Channel Models - Rayleigh

• Consider the channel gain at k-th bin with Nk arriving paths

– I and Q is the in-phase and quadrature phase component of channel gain

• Assumptions– Infinite arrival paths at the same time– All paths have zero mean and similar variance (i.e. no dominant

path)– All path gains are statistically independent

• By central limit theorem, the I and Q are Gaussian distributed– Rayleigh distribution = envelope of the sum of 2 quadrature

Gaussian source (x, y)

Qk

Ik

N

i

Qik

Iik

N

i

jik

jkk jaajaaeaeaa

kkikk +=+=== ∑∑

−

=

−

=

1

0,,

1

0,

,~ θθ


30

Common Channel Models - Rayleigh

• Rayleigh fading– A commonly used model for no line-of-sight (N-LOS)

channels

• I & Q component akI and ak

Q is Gaussian distributed N(0, σ2)• Magnitude ak is Rayleigh distributed • Phase θk is uniformly distributed over 2π

– Rayleigh distribution pdf

• E[ak2] = 2σ2 = average channel power

( )⎪⎩

⎪⎨

⎧

<

∞≤≤⎟⎟⎠

⎞⎜⎜⎝

⎛−=

00

02

exp 2

2

2

k

kkk

k

a

aaaap σσ

22 Qk

Ikk aaa +=kj

kQk

Ikk eajaaa θ=+=~ ⎟⎟

⎠

⎞⎜⎜⎝

⎛= −

Ik

Qk

k aa1tanθ


31

Rayleigh Fading Channel Simulator


32

Common Channel Models - Rician

• Rician fading– A channel with a dominant path and numerous “weak” multipath

• i.e. with line-of-sight path– Channel fading statistics is Ricean distributed

• When the dominant component fades away, the statistics degenerates to Rayleigh

– PDF

• A = peak amplitude of the dominant signal• I0(.) = zero-order Bessel function of the first kind• Often described by K = A2/2σ2

( )⎪⎩

⎪⎨⎧

<

≥≥⎟⎠⎞

⎜⎝⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛ +−=

00

0,02

exp 202

22

2

r

rAArIArrrp σσσ


33

Common Channel Models

• Example of Rayleigh and Ricean distribution


34

Common Channel Models – Clarke Model (I)

• Assuming all rays arrive in horizontal direction and at the same time– Channel gain with N arriving paths

• When the mobile is moving, each ray experience different Doppler shifts

• where fi and ψi is the Doppler shift and direction of travel for path i• fD-max is the maximum Doppler shift

∑−

=

=1

0

~ N

i

ji

ieaa θ

( ) ( )∑−

=

+=1

0

2~ N

i

tfji

iieata θπiDi ff ψcosmax−=


35

Clarke Model (II)

• Consider the channel autocorrelation function

– As all paths arrive at the same time (τi= τk ∀i, k)• τ can be removed and L=1

– Let t=t1, t2=t1+Δt ( ) ( )tRtR ah Δ=Δ τ;

( ) ( ) ( ) ( )∑−

=

−−=−1

02111212121 ;,,;

L

kkah ttRttR ττδττδτττ

( ) ( ) ( )[ ] ( ) ( )( )⎥⎦

⎤⎢⎣

⎡=Δ+=Δ ∑∑

−

=

+Δ+−

=

+1

0

21

0

2*~~ N

k

ttfji

N

i

tfjia

kkii eaeaEttataEtR θπθπ

[ ] [ ] [ ]∑∑−

=

Δ−−

=

Δ− ==1

0

221

0

22N

i

tfji

N

i

tfji

ii eEaEeaE ππ

[ ] [ ]∑−

=

Δ− −=1

0

cos22 max

N

i

tfji

iDeEaE ψπ


36

Clarke Model (III)

– Computing the expectation for the second term• Assuming the angle of arrival is uniformly distributed [-π, π]

• where I0(x) is the zeroth order Bessel function of the first kind

• Pav is the average channel power

( ) [ ]∑ ∫−

=−

Δ− ⎟⎠⎞

⎜⎝⎛=Δ −

1

0

cos22 max

21N

ii

tfjia deaEtR iD ψ

ππ

π

ψπ

[ ] ( ) ( )tfIPtfIaE Dav

N

iDi Δ=Δ= −

−

=−∑ max0

1

0max0

2 22 ππ

( ) ∫ −=π ψ ψ

π 0

cos0

1 dexI jx

[ ]∑ −

==

1

02N

i iav aEP


37

Clarke Model (IV)

• Doppler Spectrum – The power spectral density (PSD) is the Fourier transform of the

autocorrelation function

• Significance– Convolved with signal spectrum

• Spectrum will be smeared– This is why Doppler spread BD = 2fd-max

• Receiver must be able to handle this widened bandwidth– Infinity at fD-max because of uniform arrival assumption

( ) ( ){ } ( ){ }tfIPtRfS Davhh Δℑ=Δℑ= −max0 2; πτ

( )⎪⎩

⎪⎨

⎧

>

<−=

−

−

−

max

max2max

01

D

D

D

av

ff

ffff

P


38

Simulating Doppler Spread


39

Summary

• Multipath channel impulse response• Parameters of multipath channel

– Time dispersion parameters: mean, rms delay spread, max excess delay

– Coherence bandwidth– Coherence time and Doppler spread

• Different types of fading– Frequency flat/selective fading– Fast/slow fading

• Common fading channel models– Rayleigh/Ricean fading– Clarke Model


40

Tutorial Question

• Consider the provided power delay profile– Calculate the mean excess delay,

rms delay spread, and the max excess delay (10dB) for the power delay profile provided

– Estimate the 50% coherence bandwidth of the channel

– Would this channel be suitable for AMPS (30kHz) or GSM (200kHz) service without the use of an equalizer?

• If you are travelling at 100km/h and the system carrier freq is 900MHz– What is the maximum Doppler shift?– What is the 50% coherence time?– What is Doppler Spread?


41

Tutorial Question – Solution

• Max excess delay (10dB) = 5μs• Mean excess delay

• Second moment

• RMS delay spread

• Coherence bandwidth

• AMPS do not need an equalizer (30kHz BW) but GSM does (200kHz BW)

( )( ) ( )( ) ( )( ) ( )( )( ) sμτ 38.4

11.01.001.0001.021.011.051=

++++++

=

( )( ) ( )( ) ( )( ) ( )( )( )

22222

2 07.2111.01.001.0

001.021.011.051 sμτ =++++++

=

sμστ 37.138.407.21 2 =−=

( ) kHzBc 1461037.15

15

16 =

×=≈ −

τσ


42

Tutorial Question – Solution

– f = 900MHz – v = 100km/hr = 100*1000/3600m/s = 27.778m/s– Maximum Doppler shift

– 50% Coherence time

– Doppler Spread BD = 2*fD-max = 166.67Hz

HzcvfvfD 333.83

10310900778.27

8

6

max =×

××===− λ

msf

TD

c 149.216

9

max

==−π

4 Radio Propagation - Fading

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