linear time variant channel

7/30/2019 linear time variant channel

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Linear time-variant channel model



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.





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,





Physical basis for

multipath propagation

Channel spectrum as a

function of time

Scatterers

Scatterers




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





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 −

=

= −∑





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 ∞

−∞

= −∫




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 ∞

∞

≠=

∞ =

=∫




Time-invariant impulse response (LTI) vs. Time-variant impulse response

LTV)




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

=

= ,,




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




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




Doppler shift




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

,,,

,,,




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.




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 ∞ ∞

−∞ −∞

= −∫ ∫




Relation between system functions



Channel correlation functions and

parameters




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.




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 ∆ = + ∆ ∆ ∆∫





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





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




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

∞− −

−∞

∞− ∆

−∞

∆ = ∆

= ∆

≡ ∆ ∆

∫

∫





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 ∆





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

~∆





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 ∆ ∆




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





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





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 ≈




Relations between channel correlations and parameters




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





Time-dispersive channels

T T

T

B

W

T T T cmm >><<>>>> and 1

but1

and d

Source: Lee





Frequency-dispersive channels

T T W

T T

BW B mmd <<<<>>>> and 1

but1

and d

Source: Lee,



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

⎩⎪⎪⎪⎨⎪⎪⎪⎧

=




• 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+---------------=


• 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⎩⎪

⎪⎨⎪⎪⎧

≈





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 ⁄ =

linear time variant channel

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