Channel Modeling for Wideband MIMO Vehicle-to-Vehicle Channels

MIMO V2V Channel ModelChannel Simulator

MIMO Channel SounderMIMO V2V Channel Measurement Experiments

Conclusion

Channel Modeling for Wideband MIMO Vehicle-to-Vehicle ChannelsA Thesis Presented to Nile University in Partial Fulfillment of the Requirements for the Degree of

Master of Science in Communication and Information Technology - Wireless Technologies

Ahmad Amr ElMoslimany, B.Sc.

Wireless Intelligent Networks Center (WINC)Nile University, Cairo, Egypt

Thesis Advisers:

Dr. Amr ElKeyi

Dr. Yahya Mohasseb

Ahmad Amr ElMoslimany, B.Sc. MIMO-V2V Channels Model 1/81



Conclusion

Introduction

• Vehicular networks are a key component of future intelligenttransportation systems.

• Vehicular networks consist of vehicles communicating with eachother (V2V) as well as with roadside stations (V2R).

• Building a reliable physical layer requires an awareness with thechannel model.

• Models are classified according to the way you develop your model• Analytical Models.

• Simulation Models.• Empirical Models.

• We followed the analytical approach in our channel model.




Conclusion

IntroductionThe Big Picture




Conclusion

Outline

1 MIMO V2V Channel ModelScattering ModelChannel Impulse ResponseModel CharacterizationStatistical Properties of the Channel CoefficientsNumerical Example

2 Channel SimulatorStationary Channel Coefficients Matrix GeneratorBirth/Death Process Generator

3 MIMO Channel SounderCIR Measurement TechniqueMeasurement System and Its ImplementationExperimental Results

4 MIMO V2V Channel Measurement ExperimentsMeasurementsAnalysis of the Measurements and Parameters Extraction

5 ConclusionAhmad Amr ElMoslimany, B.Sc. MIMO-V2V Channels Model 4/81



Conclusion

Scattering ModelChannel Impulse ResponseModel CharacterizationStatistical Properties of the Channel CoefficientsNumerical Example

Outline








Conclusion


Geometric Elliptical Scattering Model




Conclusion


Scattering Modeling- Cont’d

• We use a birth/death process to account for the appearance anddisappearance of scatterers in each ellipse.

• We do not account for the drift of scatterers into a different delaybin.

• We express the state of the slot, whether it is occupied or not, usinga Markov chain.

• The Markov chain has 2 states namely {0,1} which stand for theabsence and the existence of a scatterer in the nth slot of the mthellipse at time t, respectively.




Conclusion


Scattering Model- Cont’d

• The Markov chain can change its state every Ts seconds.




Conclusion


Scattering Model- Cont’d

• Ts is the sampling rate of the channel impulse response which isdetermined by the maximum frequency of the transmittedbaseband-equivalent signal of the system under consideration.

• The state transition probabilities of the Markov chain reflect thedegree of nonstationarity of the environment.

Example

as the relative velocity of the vehicles increases, the probability that theMarkov chain will make a transition from 0 to 1 or from 1 to 0 willincrease.




Conclusion


Scattering Model - Cont’d

The steady state probabilities of the Markov chain are determined by the

ratio λ (n,m)01 /λ (n,m)

10 and can be obtained as

π (n,m)0 =

λ (n,m)10

λ (n,m)10 +λ (n,m)

01

π (n,m)1 =

λ (n,m)01

λ (n,m)10 +λ (n,m)

01




Conclusion


Outline








Conclusion


Channel Impulse Response

hkl [p,q] =M

∑m=0

N(m)c

∑n=0

zn,m[q]

φ (n,m)maxˆ

φ (n,m)min

g(n,m)kl (φ (m)

R ,q)dφ (m)R

δ (p−m)

zn,m[q] is a multiplicative process that models the persistence of nthscatterer in the mth ellipsoid which is defined as

zn,m [q] =

{

1 if the scatterer is in thenth slot in themth ellipse

0 if the scatterer is not in thenth slot in themth ellipse




Conclusion


Channel Impulse Response - Cont’d

hkl [p,q] =M

∑m=0

N(m)c

∑n=0

zn,m[q]

φ (n,m)maxˆ

φ (n,m)min

g(n,m)kl (φ (m)

R ,q)dφ (m)R

δ (p−m)

g(n,m)kl (φ (m)

R ,q) is the contribution of the ray transmitted from the kthtransmit antenna to lth receive element and scattered via the nthscatterer slot in the mth ellipse and received at an angle φ (m)

R at thereceive array.




Conclusion



hkl [p,q] =M

∑m=0

N(m)c

∑n=0

zn,m[q]

φ (n,m)maxˆ

φ (n,m)min

g(n,m)kl (φ (m)

R ,q)dφ (m)R

δ (p−m)

δ (p−m) is the Dirac-delta function which is equal to 1 when p = m and0 otherwise.




Conclusion



We can write the coefficient g(n,m)kl (φ (m)

R ,q) that represents the tuple (lthtransmit antenna, nth scattering slot, kth receive antenna) for the mth

ellipse as:

g(n,m)kl (φ (m)

R ,q) = En,m(φ(m)R )e

jθn,m

(

φ (m)R

)

e− jK0Dn,m

(

φ (m)R

)

ej2π fD cos

(

φ (m)R −αv

)

qTs

D(n,m)kl (φ (m)

R ) = D(l,n,m)T (φ (m)

R )+D(n,k,m)R (φ (m)

R )

D(n,m)kl (φ (m)

R ) is the distance from the lth element in the transmitter to the

scatterer in the nth slot of the mth ellipse at the angle φ (m)R and

analogously D(n,k,m)R (φ (m)

R ).




Conclusion


Outline








Conclusion


Model Characterization

• Signal parameters: the sampling frequency Ts, the wavelength of theRF signal λ .

• Transmit (receive) array geometry: the number of elements MT

(MR), the tilt angle of the array αT (αR), and the inter-elementspacing δT (δR).

• Propagation environment parameters: the delay-spread MTs, Dopplerfrequency fD, distance between transmitter and receiver 2 f .




Conclusion










Conclusion










Conclusion


Model Characterization - Cont’d

• Vehicular environment parameters:

• Density of scatterers on the road: This is reflected in the parameter

N(m)c which determines how many scattering slots exist in the mth

ellipse. Also, the ratio λ (n,m)01 /λ (n,m)

10 determines the ratio between the

long-run proportion of time in which the scatterers will occupy the

nth scattering slot and that in which they will be absent.• Speed of the transmitting or receiving vehicle: which determines

how fast the scatterers appear and disappear. This is reflected in the

state transition probabilities λ (n,m)01 and λ (n,m)

10 .




Conclusion


Model Characterization - Cont’d

• Vehicular environment parameters:

• Density of scatterers on the road: This is reflected in the parameter

N(m)c which determines how many scattering slots exist in the mth

ellipse. Also, the ratio λ (n,m)01 /λ (n,m)

10 determines the ratio between the

long-run proportion of time in which the scatterers will occupy the

nth scattering slot and that in which they will be absent.• Speed of the transmitting or receiving vehicle: which determines

how fast the scatterers appear and disappear. This is reflected in the

state transition probabilities λ (n,m)01 and λ (n,m)

10 .




Conclusion


Outline








Conclusion


Model Assumptions

• Channel coefficients that account for different delays areuncorrelated.

• Scattering from different slots within the same delay is uncorrelated.

• En,m

(

φ (m)R

)

is independent of φ (m)R for each slot, i.e,

En,m

(

φ (m))

= En,m.

• The scattering phase angles θn,m

(

φ (m)R

)

are independent for

different n, m, and φ (m)R and independent of the process zn,m[q].




Conclusion


Temporal Correlation

r(m)kl [p,q] = Ezn,m,θn,m

N(m)c

∑n=0

zn,m[q]

φ (n,m)maxˆ

φ (n,m)min

g(n,m)kl (φ (m)

R ,q)dφ (m)R

N(m)c

∑n=0

zn,m[q+ p]

φ (n,m)maxˆ

φ (n,m)min

(

g(n,m)kl (φ (m)

R ,q+ p))∗

dφ (m)R

let

S(n,m)kl [q] =

φ (n,m)maxˆ

φ (n,m)min

g(n,m)kl (φ (m)

R ,q)dφ (m)R




Conclusion


Temporal Correlation - Cont’d

r(m)kl [p,q] =

N(m)c

∑n=0

Ezn,m {zn,m [q]zn,m [q+ p]}Eθn,m

{

S(n,m)kl [q]

(

S(n,m)kl [q+ p]

)∗}

where,

zn,m [q]zn,m [q+ p] =

{

1 p = βnm [p,q]

0 p = 1−βn,m [p,q]

and,Ez {zn,m [q]zn,m [q+ p]}= βn,m [p,q]




Conclusion



βn,m [p,q]

Using the Chapman-Kolmogorov equation,

βn,m [p,q] = P{zn,m [q+ p] = 1|zn,m [q] = 1}P{zn,m [q] = 1}

= Λ(n,m)p

(2,2) π (n,m)1 [q]

Λ(n,m) =

(

1−λ (n,m)01 λ (n,m)

01

λ (n,m)10 1−λ (n,m)

10

)

where Λ(n,m)p

(i, j) is the i, jth entry of the p-step state transition matrix

Λ(n,m)p.




Conclusion



S(n,m)kl [q] = En,me− j2K0a

φ (n,m)maxˆ

φ (n,m)min

ej(

2π fD cos(

φ (m)R −αv

)

qTs+θn,m

(

φ (m)R

))

dφ (m)R

Eθn,m

{

S(n,m)kl [q]

(

S(n,m)kl [q+ p]

)∗}

Eθn,m

{

S(n,m)kl [q]

(

S(n,m)kl [q+ p]

)∗}

=|En,m |2φ (n,m)

maxˆ

φ (n,m)min

ej(

2π fD cos(

φ (m)R −αv

)

pTs

)

dφ (m)R




Conclusion



We can write the temporal correlation sequence of the mth channelcoefficient as

r(m) [p,q] =N(m)

c

∑n=0

Λ(n,m)p

(2,2) π (n,m)1 [q] | En,m |2

φ (n,m)maxˆ

φ (n,m)min

ej(

2π fD cos(

φ (m)R −αv

)

pTs

)

dφ (m)R

the channel coefficients are nonstationary as it depends on the time indexq. The nonstationarity is introduced via the Markov process zn,m[q] whichaccounts for the persistence of the scatterer in the nth slot of the mthellipse.




Conclusion



As time progresses (q increases), the effect of zn,m[q] diminishes as

π (n,m)1 [q] approaches π (n,m)

1 . In this case, the channel coefficients becomestationary and the temporal correlation function of the mth channelcoefficient is given by

r(m) [p] =N(m)c

∑n=0

Λ(n,m)p

(2,2) π (n,m)1 | En,m |2

φ (n,m)maxˆ

φ (n,m)min

ej(

2π fD cos(

φ (m)R −αv

)

pTs

)

dφ (m)R




Conclusion


Spatial Correlation

The spatial correlation function of the mth channel coefficient is definedas a function of the geometry of the transmit array Tx and receive arrayRx as

r(m)kl,k′l′ (q,Tx,Rx) = E

N(m)c

∑n=0

zn,m[q]ˆ

Φ(n,m)R

g(n,m)kl (q)dφ (m)

R

N(m)c

∑n′=0

zn′ ,m[q]ˆ

Φ(n′ ,m)R

g(n′ ,m)∗

k′ l′ (q)dφ (m)R

.




Conclusion


Spatial Correlation - Cont’d

Using the modeling assumptions and going through the same steps as inthe previous subsection, we can write

r(m)kl,k′ l′ (q,Tx,Rx) =

N(m)c

∑n=0

Ezn,m

{

zn,m [q]zn,m [q]}

Eθn,m

{

S(n,m)kl [q]S(n,m)∗

k′l′ [q]}

The first term in the above equation is given by π (n,m)1 [q].

For simplicity, we will evaluate the second term at q = 0. At this timeinstant, we can write

S(n,m)kl [0] =

φ (n,m)maxˆ

φ (n,m)min

En,me− jK0D(n,m)kl (φ (m)

R )ejθn,m

(

φ (m)R

)

dφ (m)R




Conclusion


Spatial Correlation - Cont’d

One can write D(n,m)kl (φ (m)

R ) as

D(n,m)kl (φ (m)

R ) = 2a−dTl cos(

φ (m)T −αTl

)

−dRk cos(

φ (m)R −αRk

)

Using the fourth assumption, we can write the spatial correlation functionof the mth channel coefficient at q = 0 as

r(m)kl,k′l′(Tx,Rx) =

N(m)c

∑n=0

|En,m |2 π(n,m)1 [q]

ˆ

Φ(n,m)R

C(n,m)ll′ (δT )K

(n,m)kk′ (δR)dφ (m)

R




Conclusion


Outline








Conclusion


Simulation Parameters

We consider a vehicular channel with the following parameters

• Center frequency, fc equals 5.8 GHz.

• System bandwidth, BW equals 10 MHz.

• Inclination angle of the velocity vector is αv = 0.

• The relative velocity between the two vehicles is given by 100km/hr.

• The lengths of the major and minor axes as 2a = 20 and 2b = 12.

• We have two scattering slots, Nc = 2, that extend over the angular

intervals Φ(1)R = [37◦,51◦] and Φ(2)

R = [280◦,298◦].




Conclusion


Simulation Parameters - Cont’d

The state transition matrix for the nth scattering slot is generated such

that the ratio σ = λ (n)01 /λ (n)

10 = 5, i.e.,

Λ(n) =

(

1−σρn σρn

ρn 1−ρn

)

where ρ1 = 10−3 and ρ2 = 10−3. Note that the selected value for σindicates that the scatterers exist in the slot for 83.33%of the time. Theparameters of the simulation correspond to a highway environment withhigh mobility.




Conclusion


Temporal correlation function of the channel versus timeand delay




Conclusion


Temporal correlation of a channel coefficient sequence

0 0.5 1 1.5 2 2.5 3 3.5 4−0.5

0

0.5

1

τ(ms)

r(τ)

Z

n,m(τ) is present

Zn,m

(τ) is not present




Conclusion


Spatial correlation between channel coefficients h11 and h22




Conclusion

Stationary Channel Coefficients Matrix GeneratorBirth/Death Process Generator

Outline








Conclusion


MIMO Channel Coefficients Matrix

We can write the mth coefficient of the time varying channel between thelth transmitter and kth receiver as

h(m)kl [q] =

N(m)c

∑n=0

zn,m[q]h(n,m)kl [q]

where

h(n,m)kl [q] =

φ (n,m)maxˆ

φ (n,m)min

g(n,m)kl (φ (m)

R ,q)dφ (m)R

Thus, The MIMO channel matrix can be written as

H(m) [q]=

h(m)11 [q] h(m)

12 [q] . . . h(m)1MT

[q]

h(m)21 [q] h(m)

22 [q] . . . h(m)2MT

[q]...

.

.

.. . .

.

.

.

h(m)MR1 [q] h(m)

MR2 [q] . . . h(m)MRMT

[q]




Conclusion


The Structure of the Simulator




Conclusion


Spatio-temporal Correlation Function

• The temporal correlation function

r(n,m) [p] = | En,m |2ˆ

Φ(n,m)R

e j2πF(m)D pTsdφ (m)

R .

• The spatial correlation function can be written as

r(n,m)kl,k′ l′(Tx,Rx) =|En,m |2

ˆ

Φ(n,m)R

e− jK0

(

D(n,m)kl (φ (m)

R )−D(n,m)

k′ l′(φ (m)

R ))

dφ (m)R

• Since the temporal correlation and spatial correlation areindependent

Spatio-temporal Correlation Function

Rn,m[p] = r(n,m) [p]Cn,m.




Conclusion


Time Correlation Shaping Filter

y(n,m)kl [q] = h(n,m)

kl [qT/Ts]

r(n,m)y [pT ] = r(n,m) [pT/Ts] .

S(n,m)

y (ω)=Ns

∑q=−Ns

r(n,m)y [q]e− jωqT




Conclusion


Spatial Correlation Filter

y(n,m)[q] = Ln,mx(n,m) [q]

Cn,m = Qn,mΓn,mQHn,m,

Ln,m = Qn,mΓ12n,m.




Conclusion


Interpolator

• The sampling frequency associated with the CIR, 1/Ts, is typicallymuch larger than the one associated with the complex pathgenerator 1/T .

• In our case, the sampling frequency 1/Ts is much larger than that of1/T . In order to reduce the computational complexity of theinterpolation.

• We use this interpolator in our simulator




Conclusion


Outline








Conclusion


• Generates the sequence {zn,m[q]}n,m.

• These processes take only two values{1,0} and are generated according tothe probability transition matrices{Λ(n,m)}n,m such that {zn,m[q]}n,m = 1with probabilityp1[q] = p1[q−1]×λ11+ p0[q−1]×λ01.

• The process zn,m[q] is multiplied by theoutput of the stationary channelcoefficient matrix generator.




Conclusion


The Structure of the Simulator




Conclusion


PSD of the channel coefficients for the 1stslot versus thefrequency response of the designed filter

−1.5 −1 −0.5 0 0.5 1 1.5−40

−35

−30

−25

−20

−15

−10

−5

0

5

10

Frequency (KHz)

Mag

nitu

de S

pect

ral D

ensi

ty (

dB)

|H(ej2πf)|2

S(2πf)




Conclusion


PSD of the channel coefficients for the 2ndslot versus thefrequency response of the designed filter

−1.5 −1 −0.5 0 0.5 1 1.5−50

−40

−30

−20

−10

0

10

Frequency (KHz)

Mag

nitu

de S

pect

ral D

ensi

ty (

dB)

|H(ej2πf)|2

S(2πf)




Conclusion


Time varying channel coefficients between the first transmitantenna element and the first receive antenna element

0 0.5 1 1.5 2 2.5 3 3.5 4

−6

−4

−2

0

2

4

Time (msec)

Mag

nitu

de o

f cha

nnel

coe

ffici

ents

of t

he T

DP

s

path 0




Conclusion


Joint DoD/DoA APS of the channel coefficient matrix

The angular PSD is calculated using the Capon beamformer as follows

Capon Beamformer

PCapon (φR,φT ) =1

aH(φT ,φR)R−1H a(φT ,φR)

,

wherea(φT ,φR) = aT (φT )⊗ aR (φR) ,

⊗ denotes the Kronecker product, aT (φT ) and aR (φR) are respectivelythe normalized steering vectors of the transmit and receive arrays in thedirections φT and φR. The MT MR ×MT MR matrix RH is the samplecovariance matrix which is calculated as

RH =1

NT

NT −1

∑q=0

vec{H [q]}vecH{H [q]} (1)




Conclusion


Angular power spectra of the MIMO channel generated fromthe simulator

Rec

eive

Azi

mut

h an

gle

(deg

rees

)

Transmit Azimuth angle (degrees)

0 50 100 150 200 250 300 3500

50

100

150

200

250

300

350

−340

−330

−320

−310

−300

−290

−280

−270

−260

−250

−240

−230




Conclusion

CIR Measurement TechniqueMeasurement System and Its ImplementationExperimental Results

Outline








Conclusion


Channel Impulse Response (CIR) Measurement Technique

Channel h( k )

n ( k ) y( k )

• To measure the CIR we need to excite the system with a highamplitude, short duration signal.

• We will avoid that by using a Pseudo-Noise (PN) sequence.




Conclusion


CIR Measurement Technique - Cont’d

The Idea of the Measurement Technique

y(k) = h(k)⊛ x(k)⇒ y(k) = δ (k)⊛ h(k)⇒ y(k) = h(k)

instead ifx(k) was a signal with an impulsive autocorrelation

φny (k) = φnn (k)⊛ h(k)




Conclusion


CIR Measurement Technique - Cont’d

−600 −400 −200 0 200 400 600−0.2

0

0.2

0.4

0.6

0.8

1

1.2

k

φ nn(k

)




Conclusion


Outline








Conclusion


System Description

correlation with code 1

correlation with code 2 code 1

code 2



h11

h22

h21

h12

h11

h21

h12

h22




Conclusion


Channel Sampling

• We don’t need all thesamples of the channel.

• A sample within the order ofthe channel coherence timeis sufficient.

Examples

for v = 100Km/hr at 5GHz wehave fc = 0.5KHz then one CIReach 1ms is sufficient.




Conclusion


System Description

ADC I

ADC Q

concate nate I/Q

Rx Buffer SRAM

RF Amplifier

Down Conversion

BB Ampllifier

RF Board FPGA Board Off Chip Memory




Conclusion


System Description - Cont’d




Conclusion


Outline








Conclusion


An Indoor Experiment




Conclusion


Absolute Value of the CIR versus time

0

5

10

15

20

0

1

2

3

4

5

0

0.05

0.1

0.15

0.2

time (msec)

Tap index

Abs

olut

e va

lue

of C

IR




Conclusion


Power Delay Profile of the Channels between differenttransmit and receive antennas

0 1 2 3 4 50

0.005

0.01

0.015

0.02

0.025

0.03

Tap index

Pow

er d

elay

pro

file

Channel from Tx1 to Rx1

0 1 2 30

0.005

0.01

0.015

0.02

0.025

0.03

Tap index

Pow

er d

elay

pro

file


0 1 2 30

0.005

0.01

0.015

0.02

0.025

0.03

Tap index

Pow

er d

elay

pro

file


0 1 2 3 40

0.005

0.01

0.015

0.02

0.025

0.03

Tap index

Pow

er d

elay

pro

file





Conclusion

MeasurementsAnalysis of the Measurements and Parameters Extraction

Outline








Conclusion


Parameters of the Measurement Experiment Setup

These are the parameters of the measurement setup

Operating Frequency 5.8 GHzMeasurement Bandwidth 10 MHz

Test signal length 25.6 µsSnapshot time 51.2 µs

Number of snapshots 256Snapshots inter-duration 1msMaximum transmit power 19 dBm

Number of Tx antenna elements 2Number of Rx antenna elements 2

Tx antenna separation 2.5 CmRx antenna separation 2.5 Cm

Tx antenna high 26.5 CmRx antenna high 26.5 Cm




Conclusion


The Cars We Used in the Experiment




Conclusion


Measurements Experiments Scenarios




Conclusion


Outline








Conclusion


Spatial Parameters Extraction Phase

The spatial correlation function of the mth channel coefficient is definedas a function of the geometry of the transmit array Tx and receive arrayRx as

r(m)kl,k′ l′(Tx,Rx) =

N(m)c

∑n=0

|En,m |2 π (n,m)1 [q]

ˆ

Φ(n,m)R

C(n,m)ll′ (δT )K

(n,m)kk′ (δR)dφ (m)

R

we will focus on one channel tap, and hence, we will drop the index m inthe rest of this presentation. To compute this correlation matrix, oneneeds

• Evaluate the inner integration.

• Find and estimate for the inner terms{

|En|2 π (n)

1

}

∀n = 0, · · ·Nc




Conclusion


Spatial Parameters Extraction Phase - Cont’dEvaluating the inner integration

The inner integration can be evaluated numerically given the knowledgeof the set of receive azimuth angles ΦRs. This set can be estimated fromthe Capon spectrum.

Rec

eive

Azi

mut

h an

gle

(deg

rees

)

Transmit Azimuth angle (degrees)

0 50 100 150 200 250 300 3500

50

100

150

200

250

300

350

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08




Conclusion


Spatial Parameters Extraction Phase - Cont’dResolving the Ambiguity of Linear Arrays

• We do our measurements using 2×2 linear array; this cause anambiguity in the clusters locations.

• This ambiguity can be resolved by choosing the best match to anellipse structure.




Conclusion


Spatial Parameters Extraction Phase - Cont’dEstimating the Average Power

To find an estimate for{

|En|2 π (n)

1

}

∀n = 0, · · ·Nc, we will compute the

average power of the clusters in the Capon spectrum. Thus we can write{

|En|2 π (n)

1

}

as

Average Power of Capon Spectrum

|En|2 π (n)

1 =

‹

Φ(n)R ,Φ(n)

T

Pcapon

4∗ pi2(φR,φT )dφRdφT ∀n = 0, · · ·Nc




Conclusion


Temporal Parameters Extraction Phase

We want to find an estimate for the probability transition matrix Λ(n) forevery cluster.

The Temporal Correlation function of the mth channel coefficient

r [p] =Nc

∑n=0

Λ(n)p

(2,2)π(n)1 | En |

2ˆ

Φ(n)R

e j(2πFD pTs)dφR

=Nc

∑n=0

Λ(n)p

(2,2)π(n)1 | En |

2 Eθn




Conclusion


Temporal Parameters Extraction Phase - Cont’dA Closed Form Expression for the Probability Factor

The probability transition matrix Λ(n) can be written as

Λ(n) =

(

xn 1− xn

yn 1− yn

)

The eigenvalues of this matrix are given by

ζ (n)1 =

Tn

2+

√

T 2n

4−Dn

ζ (n)1 =

Tn

2−

√

T 2n

4−Dn

where Tn and Dn are the trace and the determinant of the matrix Λ(n)

respectively which can be written as

Tn = 1+ xn − yn

Dn = xn − yn




Conclusion


A Closed Form Expression for the Probability Factor - Cont’d

and the eigenvectors can be written as

u(n)1 =

[

ζ (n)1 − (1− yn)

yn

]

u(n)2 =

[

ζ (n)2 − (1− yn)

yn

]

Hence, one can write Λ(n) as

Λ(n) =U (n)Z

(n)(

U (n))H

=[

u(n)1 u(n)2

]

[

ζ (n)1 0

0 ζ (n)2

][ (

u(n)1

)H

(un2)

H

]

Thus one can write(

Λ(n))p

(

Λ(n))p

=U (n)Z

(n)(

U (n))H

=[

u(n)1 u(n)2

]

(

ζ (n)1

)p0

0(

ζ (n)2

)p

(

u(n)1

)H

(

un2

)H




Conclusion


Temporal Parameters Extraction Phase - Cont’dLeast Squares Fitting Problem

Thus we can write the fitting problem as an optimization of a leastsequares problem

Least Squares Fitting Problem

minx,y

∥

∥

∥

∥

∥

rmeasured−Nc

∑n=0

(

Λ(n)(2,2)

)p(xn,yn)Eθn

∥

∥

∥

∥

∥

2

s.t. 0≤ x ≤ 1

0≤ y ≤ 1

where x = [x0,x1, · · · ,xNc ] and y = [y0,y1, · · · ,yNc ] are vectors contain theelements of the transition matrix Λ(n)




Conclusion


Theoretical temporal correlation versus temporal correlationcomputed from measurements

0 20 40 60 80 100 120 140−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

τ(ms)

real

(r(τ

))

MeasurementsTheoretical




Conclusion


Channel Coefficients Distribution

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

5

10

15

20

25

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

2.5

3x 10

4




Conclusion

Conclusion




Conclusion

Any Questions ?




Conclusion

Thank You :-)


Engineering

Channel Modeling for Wideband MIMO Vehicle-to-Vehicle Channels