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Personal & Mobile Communication – Spring 2008 Dr Daniel So
1
Lecture 4 Radio Wave Propagation:
Fading and Multipath
Personal & Mobile Communication – Spring 2008 Dr Daniel So
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
Personal & Mobile Communication – Spring 2008 Dr Daniel So
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
Personal & Mobile Communication – Spring 2008 Dr Daniel So
4
Backgrounds (II)
Personal & Mobile Communication – Spring 2008 Dr Daniel So
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~
Personal & Mobile Communication – Spring 2008 Dr Daniel So
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
=
Personal & Mobile Communication – Spring 2008 Dr Daniel So
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 −+=
Personal & Mobile Communication – Spring 2008 Dr Daniel So
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
Personal & Mobile Communication – Spring 2008 Dr Daniel So
9
Multipath Channel Impulse Response (II)
Personal & Mobile Communication – Spring 2008 Dr Daniel So
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 =−=
Personal & Mobile Communication – Spring 2008 Dr Daniel So
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
=
Personal & Mobile Communication – Spring 2008 Dr Daniel So
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
Personal & Mobile Communication – Spring 2008 Dr Daniel So
13
Multipath Channel Impulse Response (VI)
Personal & Mobile Communication – Spring 2008 Dr Daniel So
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
Personal & Mobile Communication – Spring 2008 Dr Daniel So
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
Personal & Mobile Communication – Spring 2008 Dr Daniel So
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
Personal & Mobile Communication – Spring 2008 Dr Daniel So
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 =
Personal & Mobile Communication – Spring 2008 Dr Daniel So
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
Personal & Mobile Communication – Spring 2008 Dr Daniel So
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 −=
Personal & Mobile Communication – Spring 2008 Dr Daniel So
20
Time Dispersion Parameters (II)
Personal & Mobile Communication – Spring 2008 Dr Daniel So
21
Time Dispersion Parameters (III)
Personal & Mobile Communication – Spring 2008 Dr Daniel So
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
Personal & Mobile Communication – Spring 2008 Dr Daniel So
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
Personal & Mobile Communication – Spring 2008 Dr Daniel So
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
Personal & Mobile Communication – Spring 2008 Dr Daniel So
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
Personal & Mobile Communication – Spring 2008 Dr Daniel So
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
Personal & Mobile Communication – Spring 2008 Dr Daniel So
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
Personal & Mobile Communication – Spring 2008 Dr Daniel So
28
Types of Fading
– Fast/slow and frequency flat/selective fading is not mutually exclusive
Personal & Mobile Communication – Spring 2008 Dr Daniel So
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
Personal & Mobile Communication – Spring 2008 Dr Daniel So
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
Personal & Mobile Communication – Spring 2008 Dr Daniel So
31
Rayleigh Fading Channel Simulator
Personal & Mobile Communication – Spring 2008 Dr Daniel So
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
• 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
Personal & Mobile Communication – Spring 2008 Dr Daniel So
33
Common Channel Models
• Example of Rayleigh and Ricean distribution
Personal & Mobile Communication – Spring 2008 Dr Daniel So
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−=
Personal & Mobile Communication – Spring 2008 Dr Daniel So
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
Personal & Mobile Communication – Spring 2008 Dr Daniel So
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
Personal & Mobile Communication – Spring 2008 Dr Daniel So
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
Personal & Mobile Communication – Spring 2008 Dr Daniel So
38
Simulating Doppler Spread
Personal & Mobile Communication – Spring 2008 Dr Daniel So
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
Personal & Mobile Communication – Spring 2008 Dr Daniel So
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?
Personal & Mobile Communication – Spring 2008 Dr Daniel So
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
Personal & Mobile Communication – Spring 2008 Dr Daniel So
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