Batool Talha presentation

Channel Models for Mobile-to-Mobile CommunicationSystems in Cooperative Networks

Batool Talha

Faculty of Engineering and ScienceUniversity of Agder

P. O. Box 509, NO-4898 Grimstad, Norway

E-mails: [email protected]

Homepage: http://ikt.hia.no/mobilecommunications/

Joint OptiMO and M2M Project Meeting, 14 August 2009, UiA1/26

Contents

1. Project Goals

2. Summary of Promised Output and Accomplishments

3. Overview of Studied Mobile-to-Mobile (M2M) Fading Channels

4. Equal Gain Combining (EGC) over Double Rayleigh Fading Channels

5. Statistical Properties of Double Rayleigh Fading Channels with EGC

6. Simulation Results

7. Summary

8. Future Plans


1. Project Goals

WP1.1: Modeling, Analysis, and Simulation of M2M Fading Channels

• Mobile-to-mobile (M2M) communication in cooperative wireless networks is an emergingtechnology promising enhanced quality of service (QoS) with increased mobility support.

• To cope with the problems faced within the development and performance investigation offuture cooperative M2M communication systems, a solid knowledge of the underlying multi-path fading channel characteristics is essential.

• So far M2M fading channels have been modeled only for certain specific communicationscenarios either assuming non-line-of-sight (NLOS) or partial line-of-sight (LOS) propagationconditions.

• Studies pertaining to the statistical properties of the M2M fading channels are limited to theprobability density function ignoring the analysis of second-order statistics of the M2M fadingchannel.

Aim of WP1.1:

To develop and to analyze new reference and simulation models for cooperative M2Mchannels under realistic propagation conditions.



• Promised output: 1 book section, 3 journal papers, 7 conference papers

• Conference Papers till August 2009: 7 conference papers

• B. Talha and M. Patzold, On the Statistical Properties of Mobile-to-Mobile Fading Chan-nels in Cooperative Networks Under Line-of-Sight Conditions, in Proc. 10th InternationalSymposium on Wireless Personal Multimedia Communications, WPMC 2007, Jaipur, In-dia, Dec. 2007, pp. 388-393.

• B. Talha and M. Patzold, On the Statistical Properties of Double Rice Channels, in Proc.10th International Symposium on Wireless Personal Multimedia Communications, WPMC2007, Jaipur, India, Dec. 2007, pp. 517-522.

• B. Talha and M. Patzold, A Novel Amplify-and-Forward Relay Channel Model for Mobile-to-Mobile Fading Channels Under Line-of-Sight Conditions, in Proc. IEEE InternationalSymposium on Personal, Indoor and Mobile Radio Communications, PIMRC 2008, Cannes,France, Sept. 2008. DOI 10.1109/PIMRC.2008.4699733

• B. Talha and M. Patzold, Level-Crossing Rate and Average Duration of Fades of the Enve-lope of a Mobile-to-Mobile Fading Channel in Cooperative Networks Under Line-of-SightConditions, in Proc. IEEE Global Communications Conference, IEEE GLOBECOM 2008,New Orleans, LA, USA, Nov./Dec. 2008. DOI 10.1109/GLOCOM.2008.ECP.860



• B. Talha and M. Patzold, A Geometrical Channel Model for MIMO Mobile-to-Mobile FadingChannels in Cooperative Networks, in Proc. IEEE 69th Vehicular Technology Conference,VTC2009-Spring, Barcelona, Spain, Apr. 2009.

• B. Talha and M. Patzold, On the Statistical Characterization of Mobile-to-Mobile FadingChannels in Dual-Hop Distributed Cooperative Multi-Relay Systems, in Proc. 12th IEEEInternational Symposium on Wireless Personal Multimedia Communications, WPMC 2009,Sendai, Japan, Sept. 2009. accepted for publication.

• B. Talha and M. Patzold, Level-Crossing Rate and Average Duration of Fades of the En-velope of Mobile-to-Mobile Fading Channels in K-Parallel Dual-Hop Relay Networks, inProc. IEEE International Conference on Wireless Communications & Signal Processing,Nanjing, China, Nov. 2009. submitted for publication.



• Journal Papers till August 2009: 2 journal papers

• B. Talha and M. Patzold, Statistical Modeling and Analysis of Mobile-to-Mobile FadingChannels in Cooperative Networks Under Line-of-Sight Conditions, Wireless PersonalCommunications, Special Issue on “Wireless Future”, published online, April, 2009.

• B. Talha and M. Patzold, Mobile-to-Mobile Fading Channels in Amplify-and-Forward Re-lay Systems Under Line-of-Sight Conditions: Statistical Modeling and Analysis, Annals ofTelecommunications, submitted for publication.

• Book chapters till August 2009: 1 book chapter

• B. Talha and M. Patzold, A Novel Amplify-and-Forward Relay Channel Model for Mobile-to-Mobile Fading Channels Under Line-of-Sight Conditions, Radio Communications byINTECHWEB, ISBN 978-953-7619-X-X, accepted for publication, 2009.


3. Overview of Studied M2M Fading Channels

Overview of Studied M2M Fading Channels Under NLOS Propagation Conditions

• The double Rayleigh process:Complex notation: ςdRayleigh (t) = µ(2) (t) µ(3) (t)

Envelope: ΞdRayleigh (t) = |ςdRayleigh (t)| =∣

∣µ(2) (t) µ(3) (t)∣

∣

Phase: ΘdRayleigh (t) = arg ςdRayleigh (t)Discussion: • ΞdRayleigh (t) describes the fading envelope of the

overall SMS-DMS link via the MR.

• The NLOS second-order scattering (NLSS) process:Complex notation: ςNLSS (t) = µ(1) (t) + AMR µ(2) (t) µ(3) (t)

Envelope: ΞNLSS (t) = |ςNLSS (t)| =∣

∣µ(1) (t) + AMR µ(2) (t) µ(3) (t)∣

∣

Phase: ΘNLSS (t) = arg ςNLSS (t)Discussion: • ΞNLSS (t) describes the fading envelope of the


2t

Source

mobile station

Destination

mobile station

3t

Mobile relay

1t

2t

Source

mobile station

Destination

mobile station

3t

Mobile relay



Overview of Studied M2M Fading Channels Under LOS Propagation Conditions

• The double Rice process:Complex notation: ςdRice (t) = AMR µ

(2)ρ (t) µ

(3)ρ (t)

Envelope: ΞdRice (t) = |ςdRice (t)| =∣

∣

∣AMR µ

(2)ρ (t) µ

(3)ρ (t)

∣

∣

∣

Phase: ΘdRice (t) = arg ςdRice (t)Discussion: • ΞdRice (t) describes the fading envelope of the


• The single-LOS double-scattering (SLDS) process:Complex notation: ςSLDS (t) = m(1) (t) + AMR µ(2) (t) µ(3) (t)

Envelope: ΞSLDS (t) = |ςSLDS (t)| =∣

∣m(1) (t) + AMR µ(2) (t) µ(3) (t)∣

∣

Phase: ΘSLDS (t) = arg ςSLDS (t)Discussion: • ΞSLDS (t) describes the fading envelope of the


2t

3t

Sourcemobile station

Destinationmobile station

Mobile relay

m t

2t

Source

mobile station

Destination

mobile station

3t

Mobile relay



• The single-LOS second-order scattering (SLSS) process:Complex notation: χSLSS (t) = µ

(1)ρ (t) + AMR µ(2) (t) µ(3) (t)

Envelope: ΞSLSS (t) = |χ (t)| =∣

∣

∣µ

(1)ρ (t) + AMR µ(2) (t) µ(3) (t)

∣

∣

∣

Phase: ΘSLSS (t) = arg χ (t)Discussion: • ΞSLSS (t) describes the fading envelope of the


• The multiple-LOS second-order scattering (MLSS) process:Complex notation: χMLSS (t) = µ

(1)ρ (t) + AMR µ

(2)ρ (t) µ

(3)ρ (t)

Envelope: ΞMLSS (t) = |χ (t)| =∣

∣

∣µ

(1)ρ (t) + AMR µ

(2)ρ (t) µ

(3)ρ (t)

∣

∣

∣

Phase: ΘMLSS (t) = arg χ (t)Discussion: • ΞMLSS (t) describes the fading envelope of the


1t

2t

Source

mobile station

Destination

mobile station

3t

Mobile relay

Mobile relay

1t

2t

3t

Source

mobile station

Destination

mobile station



• Scenario:

Source

mobile station

Destination

mobile station

(1)( )t (2)( )t

(4)( )t(3)( )t

(2 1)( )k t (2 )( )k t

Mobile relay #1

Mobile relay #2

Mobile relay #K

• Scattered component: µ(i) (t) = µ(i)1 (t) + jµ

(i)2 (t), i = 1, 2, 3, . . . , 2K

Discussion: • µ(i) (t) ≈ CN(

0, 2σ2µ(i)

)

, i = 1, 2, 3, . . . , 2K

• Each µ(i) (t) models the scattered components of the M2M fading channels inthe S-Ri and Ri-D links.



• Double scattered component: ς (k) (t) = AR(k) µ(2k−1) (t) µ(2k) (t) , k = 1, 2, . . . , K

Discussion: • ς (k) (t) = ς(k)1 (t) + jς

(k)2 (t) → zero-mean complex double Gaussian process.

• Each ς (k) (t) models the overall M2M fading channel in the kth S-Rk-D link.• AR(k) is the relay gain associated with the kth relay.

• Received signal envelope at the output of EG combiner:

Ξ (t) =K∑

k=1

χ(k) (t) =K∑

k=1

∣

∣ς (k) (t)∣

∣ =∣

∣ς (1) (t)∣

∣ +∣

∣ς (2) (t)∣

∣ + · · · +∣

∣ς (K) (t)∣

∣

Discussion: • χ(k) (t) =∣

∣ς (k) (t)∣

∣ → double Rayleigh process.



Statistical properties of interest: • Probability Density Function (PDF)• Cumulative Distribution Function (CDF)• Level-Crossing Rate (LCR)• Average Duration of Fades (ADF)

Derivation of the Probability Density Function (PDF) and Cumulative Distribution Function (CDF)

Starting point: Computation of the characteristic function (CF) ΦΞ (ω) associated with Ξ (t).

Since Ξ (t) =K∑

k=1

χ(k) (t) =K∑

k=1

∣

∣ς (k) (t)∣

∣ thus, the CF ΦΞ (ω) can be given as

the product of the CFs φχ(k) (ω) of the constituent processes χ(k), i.e.,

ΦΞ (ω) =K∏

k=1

φχ(k) (ω)

where φχ(k) (ω) = 43

1(1−jωA

R(k)σµ(2k−1)σµ(2k))2 2F1

[

2, 12;

52;

−1−jωAR(k)σµ(2k−1)σµ(2k)

1−jωAR(k)σµ(2k−1)σµ(2k)

]

.

∀ k = 1, 2, · · · , K. Here 2F1 [·, ·; ·; ·] is the Gauss hypergeometric function.

Considering χ(k) (t) are independent and identically distributed (i.i.d.) randomvariables, then φχ (ω) = φχ(1) (ω) = φχ(2) (ω) = · · · = φχ(K) (ω).

⇒ ΦΞ (ω) = φχ (ω)K =

43

1(1−jωARσ

µ(1)σµ(2))2 2F1

[

2, 12; 5

2;−1−jωARσ

µ(1)σµ(2)

1−jωARσµ(1)σµ(2)

]K

.



Problem: The CF ΦΞ (ω) associated with Ξ (t) process shown is quite complicated even if theconstituent processes χ(k) (t) are assumed to be i.i.d. Thus, the inverse Fouriertransform of the CF ΦΞ (ω) is not trackable.

Proposed solution: Approximate the target PDF (i.e., the PDF pΞ (z) of the process Ξ (t) inour case) using Laguerre series expansion assuming the processes χ(k) (t)

are i.i.d.

Laguerre series: pΞ (z) =∞∑

n=0bn exp (−z) zαlL

(αl)n (z)

where L(αl)n (z) = exp (z) z(−αl)dn

z!dz [exp (−z)zn+αl] , αl > −1 are the generalizedLaguerre polynomials,

bn = n!Γ(n+αl+1)

∞∫

0

L(αl)n (z) pχ (z) dz, z = y

βl, and Γ (·) is the Gamma function.

Required parameters: The parameters αl and βl are defined through the mean and thevariance of Ξ (t).



Approximation procedure: • By definition, the n moments of χ(k) (t) can be calculated withthe help of the CF Φχ (ω) as

mn = (−j)n dn

dωnΦχ (ω) |ω=0

• Mean of χ(k) (t) = the first moment of χ(k) (t) = m1

= the first cumulant of χ(k) (t) = κ1

=A

R(k)σµ(2k−1)σµ(2k)π

2

• The second moment of χ(k) (t) = m2 = 4A2R(k)σ

2µ(2k−1)σ

2µ(2k)

• Variance of χ(k) (t) = the second cumulant of χ(k) (t) = κ2

= m2 − m21 = 1

4A2R(k)σ

2µ(2k−1)σ

2µ(2k)

(

16 − π2)

• The parameter αl of the Laguerre series can be given as

αl =κ2

1κ2

− 1 and the parameter βl can be expressed as βl = κ2κ1

• The first term in the Laguerre series can now be expressedas follows

p0 (z) = z(αl+1)

β(αl+1)l

Γ(αl+1)exp

(

− zβl

)

which is a Gamma distribution.

PDF pΞ (z) of Ξ (t) =K∑

k=1

χ(k) (t) pΞ (z) ≈ z(αl+1)

β(αl+1)l

Γ(αl+1)exp

(

− zβl

)

where αl and βl are calculated using κ1 and κ2 of Ξ (t).• Note that the nth cumulant of Ξ (t) is related to the nth

cumulant of χ(k) (t) as κΞn = κ

χn

K(n−1)



CDF FΞ− (r) of Ξ (t) =K∑

k=1

χ(k) (t) FΞ− (r) =r∫

0

pΞ (z) dz = 1 −∞∫

r

pΞ (z) dz

≈ 1 − 1Γ(αl+1)

Γ(

αl,rβl

)

where Γ (·, ·) is the upper incomplete Gamma function.

Derivation of the Level-Crossing Rate (LCR) and Average Duration of Fades (ADF)

LCR NΞ (r) of Ξ (t) =K∑

k=1

χ(k) (t) NΞ (r) =∞∫

0

zpΞΞ(r, z) dz

Starting point: Computation of the joint PDF pΞΞ (r, z) of Ξ (t) and its corresponding timederivative Ξ (t) at the same time t. Since we have shown that the PDF pΞ (z)

of Ξ (t) can be approximated with the help of Gamma distribution (i.e., pΓ (z)).Thus, it is logical to approximate the joint PDF pΞΞ (r, z) of Ξ (t) and itscorresponding time derivative Ξ (t) at the same time t with the joint PDFpΓΓ (x, x) of a Gamma distributed process and its corresponding time derivativeat the same time t, i.e.,

pΞΞ (r, z) ≈ pΓΓ (r, z)



Next step: Computation of the joint PDF pΓΓ (x, x) of a Gamma distributed process and itscorresponding time derivative at the same time t with the help of the joint PDFpNN (y, y) of a Nakagami-m distributed process and its corresponding time derivative.This can be done by applying the concept of transformation of random variables as

pΓΓ (x, x) = 14xpNN

(√x, x

2√

x

)

= 12√

2πxσx(m−1)

(Ω/m)mΓ(m)exp

(

− x(Ω/m)

− x8σ2x

)

where m, Ω, and σ are the parameters associated with the Nakagami-m distribution.

We can now easily write pΓΓ (x, x) in terms of αl and βl as

pΓΓ (x, x) = 12√

2πβxxαl

β(αl+1)l

Γ(αl+1)exp

(

− xβl− x

8βx

)

where β = 2(

ARσµ(1)σµ(2)

)2(

f 2Smax

+ 2f 2Rmax

+ f 2Dmax

)

Numerical investigations show that pΞΞ (r, z) ≈ 1√3pΓΓ (r, z)

Final result: NΞ (r) =∞∫

0

zpΞΞ (r, z) dz ≈∞∫

0

z√3pΓΓ (r, z) dz ≈

√

23πAR

√rβ rαl

β(αl+1)l

Γ(αl+1)exp

(

− rβl

)

ADF TΞ− (r) of Ξ (t) =K∑

k=1

χ(k) (t) TΞ− (r) =FΞ−(r)

NΞ(r)

where FΞ− (r) and NΞ (r) are the CDF and the LCR of Ξ (t),respectively.




k=1

χ(k) (t)

Expression: pΞ (z) ≈ x(αl+1)

β(αl+1)l

Γ(αl+1)exp

(

− xβl

)

0 1 2 3 4 5 6 7 80

0.2

0.4

0.6

0.8

1

1.2

x

Pro

bab

ility

den

sity

funct

ion,pΞ

(x)

TheorySimulation

N = 1

AR = 1

N = 3

N = 2

N = 10




k=1

χ(k) (t)

Expression: pΞ (z) ≈ x(αl+1)

β(αl+1)l

Γ(αl+1)exp

(

− xβl

)

0 2 4 6 8 10 12 140

0.1

0.2

0.3

0.4

0.5

0.6

x

Pro

bab

ility

den

sity

funct

ion,pΞ

(x)

TheorySimulation

N = 2

AR = 1

AR = 2

AR = 3




k=1

χ(k) (t)

Expression: FΞ− (r) ≈ 1 − 1Γ(αl+1)

Γ(

αl,rβl

)

0 1 2 3 4 5 6 7 80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Level, r

Cum

ula

tive

dis

trib

ution

funct

ion,F

Ξ−

(r)

TheorySimulation

N = 1

N = 2

N = 3

N = 10

AR = 1




k=1

χ(k) (t)

Expression: FΞ− (r) ≈ 1 − 1Γ(αl+1)

Γ(

αl,rβl

)

0 5 10 150

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Level, r

Cum

ula

tive

dis

trib

ution

funct

ion,F

Ξ−

(r)

Theory

Simulation

AR = 1

N = 2

AR = 2 AR = 3




k=1

χ(k) (t)

Expression: NΞ (r) ≈√

23πAR

√rβ rαl

β(αl+1)l

Γ(αl+1)exp

(

− rβl

)

10−2

10−1

100

101

102

10−3

10−2

10−1

100

101

102

103

Level, r

Lev

el-c

ross

ing

rate

,N

Ξ(r

)TheorySimulation

N = 1

N = 2

N = 3 N = 10

AR = 1




k=1

χ(k) (t)

Expression: NΞ (r) ≈√

23πAR

√rβ rαl

β(αl+1)l

Γ(αl+1)exp

(

− rβl

)

10−2

10−1

100

101

102

10−3

10−2

10−1

100

101

102

103

Level, r

Lev

el-c

ross

ing

rate

,N

Ξ(r

)TheorySimulation

AR = 1

AR = 2

AR = 3

N = 2




k=1

χ(k) (t)

Expression: TΞ− (r) =FΞ−(r)

NΞ(r)

10−1

100

101

102

10−4

10−3

10−2

10−1

100

101

102

103

104

Level, r

Aver

age

dura

tion

offa

des

,T

Ξ−

(r)

TheorySimulation

N = 1

N = 2

N = 3N = 10

AR = 1




k=1

χ(k) (t)

Expression: TΞ− (r) =FΞ−(r)

NΞ(r)

10−1

100

101

102

10−4

10−3

10−2

10−1

100

101

102

103

104

Level, r

Aver

age

dura

tion

offa

des

,T

Ξ−

(r)

TheorySimulation

AR = 3AR = 1

AR = 2N = 2


7. Summary

• A brief overview of several M2M fading channels in cooperative networks under both NLOSand LOS propagation conditions has been presented.

• In double Rayleigh fading environment, the statistical properties such as the PDF, CDF, LCR,and ADF of the received signal envelope at the output of the EG combiner were investigated.

• We have shown that the PDF of the received signal envelope at the output of the EG com-biner where each diversity branch has double Rayleigh fading, can be approximated as theGamma distribution.

• Based on this approximation, we have obtained and presented closed-form expressions forthe previously mentioned statistical quantities.

• The approximated analytical expressions fit nicely to the simulations, confirming their correct-ness.


8. Future Plans

• Performance analysis of EGC over double Rayleigh/ double Nakagami-m fading channels incooperative networks.

• Performance analysis of LDPC codes for M2M fading channels in cooperative networks.

• Thesis writing.


Documents

Batool Talha presentation