TLT-5906Lec4

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Digital Communication Through Fading Multipath Channels

Jukka RinneInstitute of Communications Engineering

Tampere University of Technology

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Introduction

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Characterization of Fading Multipath

Channels

• Let the transmitted signal be

• In the case of time-variant multipath propagation, the received signal may be expressed as

alternatively

t t 0=

t t 0 α+=

t t 1= t t 1 τ11+=

t t 2= t t 2 τ21+= t t 2 τ22+=

s t ( ) Re sl

t ( )e j2π f

ct

[ ]=

x t ( ) αn t ( )s t τn

t ( )–( )n∑=

x t ( ) Re αn t ( )e

j– 2π f cτ

n t ( )

sl

t τn t ( )–( )

n

∑ e j2π f

ct

=

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Channels (cont.)• The equivalent lowpass received signal is

• The time-variant channel response is given by

• For some kind of channels, when a continuum of multipath components is received, thereceived signal is

• The channel response is then

r l

t ( ) αn t ( )e

j– 2π f cτ

n t ( )

sl

t τn t ( )–( )

n

∑=

c τ t ;( ) αn

t ( )e j– 2π f

cτ

n t ( )

δ t τn

t ( )–( )n∑=

x t ( ) α τ t ;( )s t τ–( ) τd

∞–

∞

∫ =

Re α t τ;( )∞–

∞

∫ e j– 2π f

cτ

sl

t τ–( )d τ e j2π f

ct

=

c τ t ;( ) α τ t ;( )e j– 2π f

c

τ=

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Channels (cont.)• In the case of , we’ll have ,

where• It should be noted, that varies typically slowly w.r.t. time. Relatively fast changes

are possible in , since is typically large and small variations in will produce

large variations in .

• When the number of propagation paths is large and ‘s change in a random manner,

can be modeled as complex-valued gaussian random process.

• When is zero-mean complex gaussian then at any instant is Rayleigh dis-

tributed -> Rayleigh fading channel• In the case that there are fixed propagation paths is no longer zero-mean and

follows Rice distribution -> Ricean fading channel• Depending on the known scattering conditions, also other types of distributions may beused (Nakagami-m etc)

sl

t ( ) 1= r l

t ( ) αn

t ( )e j– 2π f

cτ

n t ( )

n∑ α

n t ( )e

j– θn

t ( )

n∑= =

θn t ( ) 2π f cτn t ( )=α

n t ( )

θn t ( ) f

c τn t ( )

θn

t ( )

τn t ( )

c τ t ;( )c τ t ;( ) c τ t ;( ) t

c τ t ;( )c τ t ;( )

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Channel Correlation Functions and Power

Spectra• The autocorrelation function of is defined by

• In typical radio communication path, the attenuation and phase shift of delay at are

uncorrelated with path delay at -> uncorrelated scattering

• In the case of uncorrelated scattering we’ll have

• In the case that , the autocorrelation is , which is called the mul-

tipath intensity profile or the delay power spectrum of the channel

• In practice, the function is measured by transmitting very narrow pulses andcross-correlating the signal with delayed version of itself

c τ t ;( )

φc τ1 τ2 t ∆;,( ) 1

2--- E c* τ1 t ;( )c τ2 t t ∆+;( )[ ]=

τ1

τ2

12--- E c* τ1 t ;( )c τ2 t t ∆+;( )[ ] φc τ1 t ∆;( )δ τ1 τ2–( )=

∆t 0= φc τ 0;( ) φ

c τ( )=

φc τ t ∆;( )

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Spectra (cont.)

• The measured correlation function may be like the one shown above. Here is called

the multipath spread of the channel and it is the range of ’s for which is essentially

nonzero

• The time-variant channel response, , can be analyzed in frequency domain by

• If is complex gaussian, then also follows the same statistics.

τ

φc τ( )

T m

T m

τ φc τ( )

c τ t ;( )

C f t ;( ) c τ t ;( )e j2π f τ– τd

∞–

∞

∫ =

c τ t ;( ) C f t ;( )

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Spectra (cont.)• If the channel can be assumed to be wide-sense stationary, the autocorrelation functioncan be defined as

• Furthermore, this can be written as

where

• is called the spaced-frequency, spaced-time correlation function of the chan-

nel.

• Note that .

φC f 1 f 2 t ∆;,( )

1

2--- E C *

f 1 t ;( )C f 2 t t ∆+;( )[ ]=

φC f 1 f 2 t ∆;,( ) 1

2--- E c* τ1 t ;( )c τ2 t t ∆+;( )[ ]e

j2π f 1

τ1

f 2

τ2

–( )τ1d τ2d

∞–

∞

∫ ∞–

∞

∫ =

… φc τ1 t ∆;( )e j2π∆ f τ

1–

τ1d

∞–

∞

∫ φC f t ∆;∆( )= = =

f ∆ f 2 f 1–=

φC

f t ∆;∆( )

φC

f t ∆;∆( ) F φc τ t ∆;( ) =

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Spectra (cont.)• If we suppose that , then and and simply

• The coherence bandwidth is a measure of frequency selectivity of the

channel. Depending on the bandwidth of transmitted signal, , the channel is either fre-

quency selective ( ) or frequency non-selective ( ).

t ∆ 0= φC f 0;∆( ) φC

f ∆( )= φc τ 0;( ) φc τ( )=

φC f ∆( ) F φc τ( ) =

kuva 14-1-3

f ∆( )c 1 T

m ⁄ ≈

W

W f ∆( )c» W f ∆( )c«

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Spectra (cont.)• The time variations of the channel are due to the Doppler effect, which causes broadeningof the spectrum or spectrum shifting.

• The Fourier transform of w.r.t. analyzes the spectral properties of the time

variations, i.e.,

• By reducing the case for , we can have

which is the Doppler power spectrum of the channel

• The range of over which is essentially nonzero is called the Doppler spread of

the channel, , which is related to the coherence time of the channel as

φC f t ∆;∆( ) t ∆

S C

f λ;∆( ) φC f t ∆;∆( )e j2πλ∆t – t ∆d

∞–

∞

∫ =

f ∆ 0=

S C λ( ) φC

t ∆( )e j2πλ∆t – t ∆d

∞–

∞

∫ =

λ S C λ( )

Bd

t ∆( )c

1

B

d

------≈

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Spectra (cont.)

• A slowly varying channel has large or correspondingly small

• Furthermore, it is possible to define a scattering function of the channel by double Fouriertransform of autocorrelation function as follows

kuva 14-1-4

t ∆( )c B

d

φC f t ∆;∆( )

S τ λ;( ) φC f t ∆;∆( )e j2πλ∆t – e j2πτ∆ f ∆t d ∆ f d

∞–

∞

∫ ∞–

∞

∫ =

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Spectra (cont.)• Relations between channel correlation functions and power spectra

Fig 14-1-5

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Spectra (cont.)• Typical scattering function of a medium-range tropospheric scatter channel

Fig 14-1-6

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Multipath Intensity Profiles of Mobile

Radio Channels (COST 207)

0 2 4 6 8 10 12 14 16 18 20−30

−25

−20

−15

−10

−5

0

0 2 4 6 8 10 12 14 16 18 20−30

−25

−20

−15

−10

−5

0

τ in µs

P o w e r i n d B

typ. suburban and urban

typ. bad hilly terrain

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The Jakes’ Model

• A widely used model for the Doppler power spectrum of a mobile channel is the so calledJakes’ model.

• In this model is given by

where is the zero-order Bessel function of the first kind and is the

maximum Doppler frequency.

• The Fourier transform of gives the Doppler power spectrum, i.e.,

C f t ;( )

φC

t ∆( ) 1

2

--- E C * f t ;( )C f t t ∆+;( )[ ]=

J 0 2π f m

t ∆( )=

J 0 ·( ) f

m v f

0 c ⁄ =

φC t ∆( )

S C λ( ) φC

t ∆( )e j2πλ∆t – t ∆d

∞–

∞

∫ J 0 2π f

m t ∆( )e j2πλ∆t – t ∆d

∞–

∞

∫ = =

1

π f m

----------1

1 λ f m ⁄ ( )2–

----------------------------------- λ f m≤( ),

0 elsewhere,

=

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The Jakes’ Model (cont.)

10−3

10−2

10−1

100

101

102

−0.5

0

0.5

1

J 0

2 π f m

t ∆

(

)

f m

t ∆

−5 −4 −3 −2 −1 0 1 2 3 4 5−0.5

0

0.5

1

f m

t ∆

J 0

2 π f

m

t ∆

(

)

S C λ( )

frequency f m

f m

–0

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Statistical Models for Fading Channels

• Several probability distributions are used in modeling the statistical characteristics of thechannel

• When the number of scatterers is large in the channel, using the central limit theoremyields to gaussian process model for the channel impulse response

• In the case that the process can be assumed to have zero-mean, the envelope follows

Rayleigh PDF

where and the phase is uniformly distributed between

• Two parameter models, such as Nakagami- and Rice PDFs are usable in cases when wewant to fit the model to observed signal statistics

• Nakagami- PDF is given by , where is the

fading figure (Rayleigh when )

• Rice PDF is given by , where is the noncentrality

parameter

p R

r ( ) 2r

Ω-----e r 2– Ω ⁄ = , r 0≥

Ω E R2( )= 0…2π

m

m p N

r ( ) 2mm

Γ m( )Ωm

---------------------- r 2m 1– e

m

Ω----– r 2

= r 0≥, m

m 1=

p r ( ) r

σ2------e r 2 s2+( ) 2σ2 ⁄ – I

0rs

σ2------

= s

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Statistical Models for Fading Channels (cont.)

• In many line-of-sight (LOS) radio links the number of multipath components is small-> simple channel models are possible

• In the case that only one multipath component exists, the impulse response of the channel

is

where and are the attenuation factors of specular component and fading

component, respectively. is the delay of reflected component

• If can be characterized as zero-mean gaussian random process, as is often the case,we can model the channel attenuation by using Ricean distribution

• The transfer function of this channel is

c τ t ;( ) αδ τ( ) β t ( )δ τ τ0 t ( )–( )+=

α β t ( )τ0

t ( )

β t ( )

C f t ;

( ) α β t

( )e

j2π f τ0

t ( )–+=

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Stat. Models for Fading Channels / Rummler

• A very similar model is used for fixed microwave LOS radio channels used for voice andvideo transmissions

• Based on the measurements by Rummler (1979) a three path model has been developed• The model has been reduced to a two path model since the delay differences between mul-tipath components have been found out to be relatively small

• The transfer function has a form

• The parameters and are characterized as random variables, which are nearly statisti-cally independent

• The distribution of has the form (hence the exact PDF will be

)

• Parameter is modeled by the lognormal distribution, i.e., is gaussian

• For , the mean of was found to be 25 dB and std 5 dB, for smaller values

of , the mean decreases to 15 dB

C f ( ) α 1 βe j2π f f

0–( )τ

0–

–[ ]=

α β

β 1 β–( )2.3

3.3 1 β–( )2.3 β 0 1,[ ]∈,α αlog

β0.5

>20

αlog–

β

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Stat. Models for Fading Channels / Rummler

• The magnitude of the frequency response in the case that is shown below.

• Typical BW is 30 MHz

τ0 6.3 ns=

Fig 14-1-9

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Propagation Models for Mobile

Radio Channels• In the free space, the attenuation is inversely proportional to ( distance btw tx and rx).

• In mobile channels, attenuation is typically inversely proportional to ( ).• For example in widely used classical model by Hata, the mean path loss in dB’s for a largecity in an urban area is given by

where (... -> )

• Another phenomenon in mobile channels is the effect of shadowing of the signal due tolarge obstructions.

• Shadowing is usually modeled as multiplicative, slowly time varying random process. In

the case that has been transmitted, the received signal, , can be modeled by

where is the mean path loss and represents the shadowing effect.

• is often modeled by lognormally distributed random process, i.e,

d 2 d

d p p 2 4,[ ]∈

69.55 26.16log10 f MHz 13.82log10ht – a h

r ( )– 44.9 6.55log10ht –( )log10d + +

a hr

( ) 3.2 log1011.75hr

( )2 4.97–= f 400 MHz≥ p 2.98 3.52,[ ]∈

s t ( ) r t ( )r t ( ) A

0g t ( )s t ( )=

A0

g t ( )

g t ( )

p g( )

1 2πσg( ) ⁄ g µ–( )ln( )2 2σ2 ⁄ –( )exp g 0≥0 elsewhere

=

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The Effect of Signal Characteristics on the

Choice of a Channel Model• By using the previously introduced notation, the received signal can be expressed as

• In the case that the signaling interval is , the bandwidth can be approximated by

• If is greater than the signal components will have different attenuations and

phase shifts over the bandwidth -> frequency-selective fading.

• When the signaling interval is selected as , we can write and the

channel is found to be frequency-nonselective.

• However in general, time-variant characteristics of the channel exists, i.e.,

giving . In this case the multipath components are not resolvable.

• Furthermore, we can write and if it is zero-mean complex gaussian

process, is Rayleigh and is uniformly distributed.

r l

t ( ) c τ t ;( )sl

t τ–( ) τd

∞–

∞

∫ C f t ;( )S

l f ( )e j2π ft f d

∞–

∞

∫ = =

T W 1

T ---=

W ∆ f ( )c

T T m» W

1

T m

-------« ∆ f ( )c≈

C 0 t ;( ) const.≠r

l t ( ) C 0 t ;( )s

l t ( )=

C 0 t ;( ) α t ( )e jφ t ( )–=

α t

( ) φ t

( )

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Choice of a Channel Model (cont.)• The speed of time variations of the channel can be determined from or and

hence the parameters or characterize the rapidity of the fading

• In a special case when and , channel is frequencynon-selective and basically “frozen” during signaling interval, i.e,

-> the spread factor -> channel is said to be underspread.

• If channel is overspread

φC t ∆( ) S

C λ( )

∆t ( )c B

d

W ∆ f ( )c« 1 T m ⁄ ≈ T ∆t ( )c« 1 Bd ⁄ ≈

Bd

W 1 T m

⁄ « « Bd

T m 1«→

T m

Bd 1<

T m Bd 1>

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Choice of a Channel Model (cont.)

Taulukko 14-2-1

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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 ≤ ≤

φ

αP

2 γ b( ) Q 2γ b( )=

γ b α2E b N 0 ⁄ =

P2 γ b( ) Q γ b( )=

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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 bE b

N 0

------- E α2( )=

P2

1

2--- 1

γ b1 γ

b+---------------–

, for binary PSK

1

2--- 1

γ b

2 γ b

+---------------–

, for binary FSK

=

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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, andafter calculation

.

P2 γ b( ) 1

2---e

γ b

–=

P21

2 1 γ b+( )-----------------------=

P2 γ b( )1

2---e

γ b

– 2 ⁄

=

P2

1

2 γ b+---------------=

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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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nel (cont.)

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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 ⁄ =

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