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
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.