fading multipath radio channels
TRANSCRIPT
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Fading multipath radio channels
• Narrowband channel modelling• Wideband channel modelling• Wideband WSSUS channel (functions, variables & distributions)
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Low-pass equivalent (LPE) signal
Real-valued RF signal Complex-valued LPE signal
RF carrier frequency
In-phase signal component Quadrature component
2Re cj f ts t z t e
j tz t x t j y t c t e
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Spectrum characteristics of LPE signal
f
f
magnitude
phase
Real-valued time domain signal
(e.g. RF signal)
Signal spectrum is Hermitian 0
0
Complex-valued LPE time domain signal
Signal spectrum is not Hermitian
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Radio channel modelling
Narrowband modelling Wideband modelling
Deterministic models(e.g. ray tracing,
playback modelling)
Stochastical models(e.g. WSSUS)
Calculation of path loss e.g. taking into account - free space loss - reflections - diffraction - scattering
Basic problem: signal fading
Basic problem: signal dispersion
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Signal fading in a narrowband channel
magnitude of complex-valued LPE radio signal
distance
propagation paths fade <=> signal
replicas received via different propagation paths cause destructive interference
TxRx
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Fading: illustration in complex plane
in-phase component
quadrature phase
component
TxRx
Received signal in vector form: resultant (= summation result) of “propagation path vectors”
Wideband channel modelling: in addition to magnitudes and phases, also path delays are important.
path delays are not important
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Propagation mechanisms
CA
D
BReceiverTransmitter
A: free spaceB: reflectionC: diffractionD: scattering
A: free spaceB: reflectionC: diffractionD: scattering
reflection: object is large compared to wavelengthscattering: object is small or its surface irregular
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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)
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Bit interleaving
Transmitter Channel Receiver
Bits are interleaved ...
Fading affects many adjacent
bits
After de-interleaving of bits, bit errors are spread!
Bit errors in the receiver
... and will be de-interleaved in the receiver (better for FEC)
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Channel Impulse Response (CIR)
time t h(,t)
zero excess delay
delay spread Tm
delay
Channel is assumed linear!
Channel presented in delay / time domain (3 other ways possible!)
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CIR of a wideband fading channel
path delaypath attenuation path phase
LOS path
1
0
, iL
j ti i
i
h t a t e
The CIR consists of L resolvable propagation paths
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Received multipath signal
kk
s t b p t kT
,r t h t s t h t s t d
1
0
iL
j ti i
i
a t e s t
0 0f t t t dt f t
pulse waveformcomplex symbol
Transmitted signal:
Received signal:
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Received multipath signalThe received multipath signal is the sum of L attenuated, phase shifted and delayed replicas of the transmitted signal s(t)
T
Tm
00 0
ja e s t
11 1
ja e s t
22 2
ja e s t
Normalized delay spread D = Tm / T
:
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Received multipath signal
The normalized delay spread is an important quantity.When D << 1, the channel is
- narrowband - frequency-nonselective - flat
and there is no intersymbol interference (ISI).
When D approaches or exceeds unity, the channel is- wideband- frequency selective - time dispersive
Important feature has many names!
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BER vs. S/N performance
Typical BER vs. S/N curves
S/N
BER
Frequency-selective channel (no equalization)
Flat fading channelGaussian channel(no fading)
In a Gaussian channel (no fading) BER <=> Q(S/N)erfc(S/N)
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BER vs. S/N performance
Typical BER vs. S/N curves
S/N
BER
Frequency-selective channel (no equalization)
Flat fading channelGaussian channel(no fading)
Flat fading (Proakis 7.3): BER BER S N z p z dzz = signal power level
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BER vs. S/N performance
Typical BER vs. S/N curves
S/N
BER
Frequency-selective channel (no equalization)
Flat fading channelGaussian channel(no fading)
Frequency selective fading <=> irreducible BER floor
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BER vs. S/N performance
Typical BER vs. S/N curves
S/N
BER
Flat fading channelGaussian channel(no fading)
Diversity (e.g. multipath diversity) <=>
Frequency-selective channel(with equalization)
improved performance
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Time-variant transfer function
1
22
0
, , i i
Lj t j fj f
ii
H f t h t e d a t e e
1
0
, iL
j ti i
i
h t a t e
Time-variant CIR:
Time-variant transfer function (frequency response):
In a narrowband channel this reduces to:
1
0
, iL
j ti
i
H f t a t e
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Example: two-ray channel (L = 2)
1 21 1 2 2
j jh a e a e
1 1 2 22 21 2
j j f j j fH f a e e a e e
1 2constructiveH f a a
1 2destructiveH f a a
At certain frequencies the two terms add constructively (destructively) and we obtain:
f
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Deterministic channel functions
Time-variant impulse response
Time- variant transfer function
Doppler-variant transfer function
Doppler- variant impulse response
,h t
,H f t ,d
,D f
(Inverse) Fourier
transform
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Stochastical (WSSUS) channel functions
Channel intensity profile
Frequency time
correlation function
Channel Doppler spectrum
Scattering function
;h t
;H f t ;hS
;HS f
h
HS H f
H t Td
Bm
Tm
Bd
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Stochastical (WSSUS) channel variables
Maximum delay spread: Tm
Maximum delay spread may be defined in several ways. For this reason, the RMS delay spread is often used instead:
22
h h
h h
d d
d d
h
Tm
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Stochastical (WSSUS) channel variables
Coherence bandwidth of channel:
1m mB T
H f
Bm
f0Implication of coherence bandwidth:If two sinusoids (frequencies) are spaced much less apart than Bm , their fading performance is similar.
If the frequency separation is much larger than Bm , their fading performance is different.
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Stochastical (WSSUS) channel variables
Maximum Doppler spread:
The Doppler spectrum is often U-shaped (like in the figure on the right). The reason for this behaviour is the relationship (see next slide):
Bd
0
Bd
HS
cos cosd
Vf
Task: calculate p() for the case where p() = 1/2(angle of arrival is uniformly distributed between 0 and 2)
HS p
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Physical interpretation of Doppler shift
Varriving pathdirection of receiver
movementRx
cos cosd
Vf
Maximum Doppler shift
Angle of arrival of arriving path with respect to
direction of movement
V = speed of receiver = RF wavelength
Doppler frequency shift
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Delay - Doppler spread of channel
Doppler shift
delay
0
L = 12 components in delay-Doppler domain
Bd
1
2
0
, i iL
j ti i
i
h t a t e
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Fading distributions (Rayleigh)
2 222
aap a e
12
p
ij t j ti
i
c t a t e x t j y t a t e
In a flat fading channel, the (time-variant) CIR reduces to a (time-variant) complex channel coefficient:
When the quadrature components of the channel coefficient are independently and Gaussian distributed, we get:
Rayleigh distribution Uniform distribution
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Fading distributions (Rice)
In case there is a strong (e.g., LOS) multipath component in addition to the complex Gaussian component, we obtain:
From the joint (magnitude and phase) pdf we can derive:
Rice distributionModified Bessel function of
first kind and order zero
0 0
ij t j ti
i
c t a a t e a a t e
22 20 2 0
02 2
a aa aap a e I
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Representation in complex plane
iy
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
Complex Gaussian distribution is centered around the “strong path” => magnitude is Rice distributed, probability of deep fade is extremely small
,p x yiy
0a
Bell-shaped function
x x
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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.