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7/30/2019 linear time variant channel
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Linear time-variant channel model
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TLT-6206 Radio Propagation in Wireless Networks
Multipath propagation
In a wireless communication system, the propagation around a
single mobile occurs most of the time by scattering from local
objects/obstructions.
This is the case especially in built-up areas, where mobiles arelocated considerably lower than surrounding buildings, which
moreover prevent LOS conditions.
The received signal consists of several replicas of the originally
transmitted signal – these replicas are characterized by different
amplitudes, phases, polarisations, and angle of arrivals at the
mobile unit.
Since the received replicas are vectorially combined at the mobile
the received signal envelope has huge variations due to either
constructive or destructive combination.
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TLT-6206 Radio Propagation in Wireless Networks
Multipath propagation
The locations of local scatterers are continuously changing due to
mobile’s movement or movement of surrounding obstacles.
Fast fluctuation of the received signal level is called fast fading (o
short-term/small-scale fading).
Prediction of exact field strength in multipath environment is
demending as it would require exact knowledge of all scatterers.
Hence, statistical models are applied for modeling a mobile radio
channel.
Source: Parson,
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TLT-6206 Radio Propagation in Wireless Networks
Multipath propagation
Physical basis for
multipath propagation
Channel spectrum as a
function of time
Scatterers
Scatterers
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TLT-6206 Radio Propagation in Wireless Networks
Channel as a linear system
Radio channel can be visualized by a system element that
transforms the input signal into an output signal.
It is similar to a linear filter with the extension that the propagatio
channel is time-variant.
The radio channel can be modeled as a linear time-variant (LTV)
channel that can be characterized by four functions:
Time-variant impulse response (channel delay spread function)
Time-variant transfer function
Doppler-spread function
Delay-Doppler-spread function
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TLT-6206 Radio Propagation in Wireless Networks
Channel as a linear system
In a multipath environment of N distinct scatterers, every nth
scatterer is characterized by its amplitude, αn(t ), and associated
propagation delay τ n(t ).
If a narrowband signal x(t )
is transmitted at carrier frequency f c, in the absence of background
noise, the channel output is
with
( ) ( ){ }2Re c j f t
x t x t eπ =
( ) ( ){ } ( ) ( )( ) ( )( )22
1
Re Re c nc
N
j f t t j f t n n
n
r t r t e t x t t e π τ π α τ −
= = = − ∑
( ) ( ) ( )( ) ( )2
1
c n
N j f t
n n
n
r t t x t t eπ τ
α τ −
=
= −∑
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TLT-6206 Radio Propagation in Wireless Networks
Channel as a linear system
Since the complex envelopes, x(t ) and r (t ), are equivalent
representations of the transmitted and received narrowband signal
the channel can be hence characterized equivalently by its impuls
response at the baseband . Considering an impulse response of linear time-invariant channe
(LTI), the output signal can be given as a convolution (*) between
transmitted signal and channel impulse response h(t ):
For a LTV channel, h(t ) has to include the time instant when the
impulse is applied (initial time), and the time instant of observing
the channel output (final time):
( ) ( ) ( ) ( ) ( )r t h t x t h x t d
∞
−∞
= ∗ = −∫ τ τ τ
( ) ( ) ( ),r t h t x t d τ τ τ ∞
−∞
= −∫
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TLT-6206 Radio Propagation in Wireless Networks
Time-variant impulse response
The channel impulse response becomes then
with θ n(t )=2π f cτ n(t ) represents the carrier phase distortion by nth
scatterer.
Thus, θ n(t ) will change by 2π whenever τ n(t ) changes by 1/ f c.
This means that θ n(t ) have a far grater effect on the transmitted
signal than changes in αn(t ), as a small change (such as motion) inthe scatterer can cause a significant change in the phase but not on
the amplitude.
( ) ( ) ( ) ( )( )1
, n
N j t
n n
n
h t t e t θ
τ α δ τ τ −
=
= −∑
( )
( )-
Dirac delta func
0,
,
and
1
t t
t
t dt
δ
δ ∞
∞
≠=
∞ =
=∫
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TLT-6206 Radio Propagation in Wireless Networks
Time-invariant impulse response (LTI) vs. Time-variant impulse response
LTV)
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TLT-6206 Radio Propagation in Wireless Networks
Time-variant transfer function
Due to relationship of the time and frequency domains, the chann
impulse response h(t,τ ), can be represented in frequency domain
with Fourier transform F (·):
where H ( f ,t ) is the channel transfer function. Hence,
( ) ( )[ ] ( )
( ) ( )[ ] ( )
==
==
∫
∫∞
∞−
+−
∞
∞−
−
df et f H t f H F t h
d et ht hF t f H
f j
f
f j
τ π
τ π
τ
τ
τ τ τ
21
2
,,,
,,,
( ) ( ) 2,
j f t r t R f t e df
π
∞+
−∞
= ∫( ) ( ) ( ) ( )[ ]t xF f X
X t f H t f R
=
= ,,
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TLT-6206 Radio Propagation in Wireless Networks
Doppler shift
The time variations, or dynamic changes in the propagation path
lengths, can be related directly to the motion of the receiver and
indirectly to the Doppler effect.
The rate of change of phase due to motion is apparent as a Dopplefrequency shift of each propagation path.
Scatterer
α
A A’ d
Δl
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TLT-6206 Radio Propagation in Wireless Networks
Doppler shift
The incremental distance d =v Δt , which on the other hand, is
Δl=d cos(α) in path lenght difference.
The corresponding change in the received signal phase is
and in the change in the frequency
where f d is the Doppler shift.
Negative Doppler shift indicates that the mobile is travelling away
from the source and positive indicated that the mobile is heading
towards the source.
( )α λ
π
λ
π φ cos
22 t vl
∆−=∆−=∆
( ) d f v
t f ==
∆
∆−=∆ α
λ
φ
π cos
2
1
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TLT-6206 Radio Propagation in Wireless Networks
Doppler shift
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TLT-6206 Radio Propagation in Wireless Networks
Doppler spread function
The Doppler spread function H ( f ,v) affects the received signal
spectrum as
where v is a variable describing the Doppler shift.
The relation between the time-variant transfer function H ( f ,t ) and
the Doppler spread function H ( f ,v) is
( ) ( ) ( )
∫
∞
∞−
−−= dvvv f H v f X f R ,
( ) ( )[ ] ( )
( ) ( )[ ] ( )
==
==
∫
∫∞
∞−
+−
∞
∞−
−
dvev f H v f H F t f H
dt et f H t f H F v f H
vt j
v
vt j
π
π
τ
21
2
,,,
,,,
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TLT-6206 Radio Propagation in Wireless Networks
Doppler spread function
Hence, being time-variant in the time domain can be equivalently
described by having Doppler shifts in the frequency domain.
Doppler spread function describes the broadening of the spectrum
by the time rate of change of the radio channel.
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TLT-6206 Radio Propagation in Wireless Networks
Delay-Doppler spread function
The delay-Doppler spread function (also scattering function) is
the Fourier transform of the channel impulse response h(τ ,t ) with
respect to t :
It provides information how much signal power is found in signal
components with a certain propagation delay τ and Doppler shift v
Given the channel input signal x(t ), the output corresponds to
( ) ( )[ ] ( )∫∞
∞−
−== dt et ht hF v H vt j
t
π τ τ τ 2,,,
( ) ( ) ( ) 2, j vt r t x t H v e dvd
π τ τ τ ∞ ∞
−∞ −∞
= −∫ ∫
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TLT-6206 Radio Propagation in Wireless Networks
Relation between system functions
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Channel correlation functions and
parameters
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TLT-6206 Radio Propagation in Wireless Networks
Channel correlation functions
When the channel changes randomly with time, h(τ ,t ), H ( f ,t ),
H ( f ,v), H (τ ,v) are random processes and demanding to characteriz
Assuming the random processes have zero mean, the interest lies
correlation functions.
Assumptions
1) the channel impulse response is a wide-sense stationary (WSS) process
2) the channel impulse response at τ 1 and τ 2 are uncorrelated if τ 1 ≠ τ 2 for any
If both assumptions hold, the channel is said to be wide-sense
stationary uncorrelated scattering (WSSUS) channel.
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TLT-6206 Radio Propagation in Wireless Networks
Delay power spectral density
The correlation function of the channel impulse response is given
as a function of propagation delays and time difference:
Under assumption 2), the autocorrelation function becomes
where
At Δt =0, the multipath intensity profile of the channel is
( ) ( ) ( )*
1 2 1 2
1
, ; , ;2h t E h t h t t φ τ τ τ τ
∆ ≡ + ∆
( ) ( ) ( )
( ) ( ) ( )
1 2 1 1 2, ; ,
or
, ; ,
h h
h h
t t
t t
φ τ τ φ τ δ τ τ
φ τ τ τ φ τ δ τ
∆ = ∆ −
+ ∆ ∆ = ∆ ∆
( ) ( )0,τ φ τ φ hh ≡
( ) ( ), , ;h h
t t d φ τ φ τ τ τ τ ∆ = + ∆ ∆ ∆∫
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TLT-6206 Radio Propagation in Wireless Networks
Delay power spectral density
Combining the previous equations yields:
Hence, is the Fourier transform of the correlation function
and represents delay power spectral density (PSD). It measures th
average PSD at the channel output as a function of propagation
delay. From the delay PSD, statistical nth moment of the channel can be
found by
( ) ( )
( ) ( )
0
*
, ;
1, ;
2
h h Δt F t
F E h t h t
τ
τ
φ τ φ τ τ τ
τ τ τ
∆ =
∆
= + ∆ ∆
= + ∆
( )τ φ h
( )
( )∫∫=
τ τ φ
τ τ φ τ τ
d
d
h
h
n
n
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TLT-6206 Radio Propagation in Wireless Networks
Delay power spectral density
The mean propagation delay, or the first moment, is thus
and the rms (root-mean-square) delay spread is
( )
( )∫
∫=
τ τ φ
τ τ φ τ τ
d
d
h
h
( ) ( )
( )
2
h
h
d
d τ
τ τ φ τ τ
σ φ τ τ
−=
∫
∫ T m=σ τ
T m=rms multipath delay spre
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TLT-6206 Radio Propagation in Wireless Networks
Frequency and time correlation functions
If the channel impulse response, h(τ ,t ), is WSS, then also the
channel transfer function, H ( f ,t ) is WSS with respect to t .
The autocorrelation function of H ( f ,t ) is defined as:
Due to duality between time and frequency
( ) ( ) ( )*
1 2 1 2
1, ; , ;
2 H
f f t E H f t H f t t φ ∆ ≡ + ∆
( ) ( ) ( )
( )
( )
2 12
1 2
2
, , ,
,
,
j f f
H h
j f
h
H
f f t t e d
t e d
f t
π τ
π τ
φ φ τ τ
φ τ τ
φ
∞− −
−∞
∞− ∆
−∞
∆ = ∆
= ∆
≡ ∆ ∆
∫
∫
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TLT-6206 Radio Propagation in Wireless Networks
Frequency and time correlation functions
This is the time-frequency correlation function. Letting Δt =0, thefrequency domain representation can be written as a Fouriertransform of the delay PSD, i.e.,
which is called the frequency correlation function.
As the delay PSD portrays the time domain behaviour of the fadinchannel, the frequency correlation function portrays the frequencydomain behaviour.
The nominal width of is denoted by , which is calledthe channel coherence bandwidth.
( ) ( ) ( )
( )
*
2
1 , ,2
H
j f
h
f E H f t H f f t
e d π τ
φ
φ τ τ
∞− ∆
−∞
∆ ≡ + ∆
= ∫
( ) f H ∆φ ( )c
f ∆
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TLT-6206 Radio Propagation in Wireless Networks
Frequency and time correlation functions
Moreover, we have
meaning that a large multipath delay
results in a small coherence bandwidth.
The frequency correlation function states that frequencycomponents separated by a frequency width greater than thecoherence bandwidth are distorted in an uncorrelated manner.
The degree of fading depends on the relations between the channecoherence bandwidth and signal bandwidth: If (Δ f c) < W Frequency selective fading
If (Δ f c) >> W Frequency nonselective fading or flat fading
where W is the transmit signal bandwidth.
( )m
c
T
f 1
~∆
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TLT-6206 Radio Propagation in Wireless Networks
Frequency and time correlation functions
In , letting Δ f=0,
we will get the time correlation function.
It characterises how fast the channel transfer function changes wit
time at each frequency. The nominal width of is called the
channel coherence time, (Δt )c, and it informs the time after which
the fading is very much different.
If the channel coherence timeis more than symbol duration,
channel exhibits ”slow fading”.
( ) ( ) ( ) ( )[ ]t t f H t f H E t t H H ∆+=∆≡∆ ,,2
1,0
*φ φ
( )t H ∆φ
( ), H
f t φ ∆ ∆
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TLT-6206 Radio Propagation in Wireless Networks
Doppler power spectral density
The autocorrelation function of the Doppler spread function H( f ,v
is
which in WSSUS channel becomes
Moreover, from the Fourier transform of frequency-time
correlation function with respect to Δt , it can be obtained:
( ) ( )[ ]2211
*,,
2
1v f H v f H E
( ) ( )211
, vvv f H −∆Φ δ
( ) ( )
( ) ( ) ( )∫
∫∞
∞−
∆−
∞
∞−
∆−
∆∆=Φ≡Φ
∆∆∆=∆Φ
t d et vv
t d et f v f
t v j
H H H
t v j
H H
π
π
φ
φ
2
2
,0
,,
0=∆ f
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TLT-6206 Radio Propagation in Wireless Networks
Doppler power spectral density
is a PSD in terms of the Doppler shift, and it is called the
Doppler PSD.
The nominal width of the Doppler PSD, Bd , is called the Doppler
spread .
Relation between time correlation function and Doppler PSD
yields:
From the Doppler PSD, statistical n
th
moment of the channel can be found by
( )v H Φ
( )d
c B
t 1
~∆
( )
( )∫∫
Φ
Φ=
dvv
dvvvv
H
H
n
n
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TLT-6206 Radio Propagation in Wireless Networks
Doppler power spectral density
The mean Doppler shift, or the first moment, is
and the rms Doppler spread is
As an approximation, it is assumed that
( )
( )
H
H
v v dvv
v dv
Φ=
Φ
∫
∫
( ) ( )
( )
2
H
v
H
v v v dv
v dvσ
− Φ=
Φ
∫∫
vd B σ ≈
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TLT-6206 Radio Propagation in Wireless Networks
Relations between channel correlations and parameters
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TLT-6206 Radio Propagation in Wireless Networks
Classification of channels
In a dispersive medium, there are two kinds of spreads: Doppler
spread Bd and multipath delay spread T m (delay spread). The
Doppler spread is spreading in frequency where as multipath
spread is spreading in time. In a strict sense, all media are dispersive. We can classify
characteristics of a medium based on the symbol duration T and
the signal bandwidth W .
Non-dispersive but fading channel is observed if
In practical systems, the symbol duration and channel bandwidth
can be selected so that channel becomes nondispersive.
W BT T W
T T
B mmd <<>><<<< and or1 and 1c
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TLT-6206 Radio Propagation in Wireless Networks
Classification of channels
Time-dispersive channels
T T
T
B
W
T T T cmm >><<>>>> and 1
but1
and d
Source: Lee
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TLT-6206 Radio Propagation in Wireless Networks
Classification of channels
Frequency-dispersive channels
T T W
T T
BW B mmd <<<<>>>> and 1
but1
and d
Source: Lee,
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Frequency-Nonselective, Slowly Fading Chan-nel
• As was described earlier, the frequency-nonselective channel results multiplicative distor-
tion to the transmitted signal as
• If the fading is slow, the received signal can be written in the presence of noise as
• In the case that can be followed perfectly at the receiver, ideal coherent detection can be
applied -> now only the effect of has to be taken into account.
• In the case of binary PSK, the error rate in AWGN channel can be written as
where .
• For binary FSK the error rate is
r l t ( ) C 0 t ;( )sl t ( )=
r l t ( ) αe jφ– sl t ( ) z t ( )+= , 0 t T ≤ ≤
φα
P2 γ b( ) Q 2γ b( )=
γ b α2 E b N 0 ⁄ =
P2 γ b( ) Q γ b( )=
Frequency-Nonselective, Slowly Fading Chan-nel (cont.)
• In the case that changes randomly and hence also changes randomly, the average
error rate is
• In Rayleigh fading case, is Rayleigh distributed and , where
.
• After integration we’ll have
α γ b
P2 P2 γ b( ) p γ b( ) γ bd
∞–
∞
∫ =
α p γ b( ) 1
γ b-----e
γ b γ b ⁄ –=
γ b
E b
N 0------- E α2( )=
P2
1
2--- 1
γ b1 γ
b+
---------------–⎝ ⎠ ⎜ ⎟ ⎛ ⎞
, for binary PSK
1
2--- 1
γ b2 γ b+---------------–
⎝ ⎠ ⎜ ⎟ ⎛ ⎞
, for binary FSK
⎩⎪⎪⎪⎨⎪⎪⎪⎧
=
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Frequency-Nonselective, Slowly Fading Chan-nel (cont.)
• When the channel response varies more rapidly with respect to symbol duration, the phaseestimation must be carried out only for limited number of signaling intervals.
• In DPSK, only the phase difference of two consecutive symbols is detected and themethod is quite robust to phase changes in the channel.
• For DPSK in AWGN channel we have and hence the average error in
Rayleigh fading channel can be solved to be .
• FSK with noncoherent detection (envelope or sq. law) is even more robust technique than
DPSK in fast fading channels. For noncoherent orthogonal FSK, and
after calculation
.
P2 γ b( ) 1
2---e
γ b
–=
P21
2 1 γ b+( )-----------------------=
P2 γ b( ) 1
2---e
γ b
– 2 ⁄ =
P21
2 γ b+---------------=
Frequency-Nonselective, Slowly Fading Chan-nel (cont.)
• If we let grow very large for the discussed binary signaling methods, the resulting bit
error probabilities can be approximated as
• It can be seen that for large SNR, coherent PSK is 3 dB better than DPSK and 6 dB betterthan noncoherent FSK.
• Error rates are inversely proportional to SNR, whereas in AWGN-case the decrease isexponential.
γ b
P2
1 4γ b ⁄ for coherent PSK
1 2γ b ⁄ for coherent, orthogonal FSK
1 2γ b ⁄ for DPSK
1 γ b ⁄ for noncoherent, orthogonal FSK⎩⎪
⎪⎨⎪⎪⎧
≈
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Frequency-Nonselective, Slowly Fading Chan-nel (cont.)
Frequency-Nonselective, Slowly Fading Chan-nel (cont.)
Binary PSK with Nakagami- fading
is Nakagami- distributed , where
m
α m p γ ( ) mm
Γ m( )γ m--------------------γ m 1– e mγ γ ⁄ –= γ E α2( ) E b N 0 ⁄ =