Channel Modeling for Wideband MIMO Vehicle-to-Vehicle Channels

MIMO V2V Channel Model
Channel Simulator
MIMO Channel Sounder
MIMO V2V Channel Measurement Experiments
Conclusion
Channel Modeling for Wideband MIMO Vehicle-to-Vehicle Channels
A 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

Channel Simulator
Conclusion
Introduction
• Vehicular networks are a key component of future intelligent
transportation systems.
• Vehicular networks consist of vehicles communicating with each
other (V2V) as well as with roadside stations (V2R).
• Building a reliable physical layer requires an awareness with the
channel 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.

Channel Simulator
Conclusion
Introduction
The Big Picture

Channel Simulator
Conclusion
Outline
1 MIMO V2V Channel Model
Scattering Model
Channel Impulse Response
Model Characterization
Statistical Properties of the Channel Coeﬃcients
Numerical Example
2 Channel Simulator
Stationary Channel Coeﬃcients Matrix Generator
Birth/Death Process Generator
3 MIMO Channel Sounder
CIR Measurement Technique
Measurement System and Its Implementation
Experimental Results
4 MIMO V2V Channel Measurement Experiments
Measurements
Analysis of the Measurements and Parameters Extraction
5 Conclusion

Channel Simulator
Conclusion
Scattering Model
Numerical Example
Outline
Scattering Model
Numerical Example
2 Channel Simulator
Measurements
5 Conclusion

Channel Simulator
Conclusion
Scattering Model
Numerical Example
Geometric Elliptical Scattering Model

Channel Simulator
Conclusion
Scattering Model
Numerical Example
Scattering Modeling- Cont’d
• We use a birth/death process to account for the appearance and
disappearance of scatterers in each ellipse.
• We do not account for the drift of scatterers into a diﬀerent delay
bin.
• We express the state of the slot, whether it is occupied or not, using
a Markov chain.
• The Markov chain has 2 states namely {0,1} which stand for the
absence and the existence of a scatterer in the nth slot of the mth
ellipse at time t, respectively.

Channel Simulator
Conclusion
Scattering Model
Numerical Example
Scattering Model- Cont’d
• The Markov chain can change its state every Ts seconds.

Channel Simulator
Conclusion
Scattering Model
Numerical Example
Scattering Model- Cont’d
• Ts is the sampling rate of the channel impulse response which is
determined by the maximum frequency of the transmitted
baseband-equivalent signal of the system under consideration.
• The state transition probabilities of the Markov chain reﬂect the
degree of nonstationarity of the environment.
Example
as the relative velocity of the vehicles increases, the probability that the
Markov chain will make a transition from 0 to 1 or from 1 to 0 will
increase.

Channel Simulator
Conclusion
Scattering Model
Numerical Example
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

Channel Simulator
Conclusion
Scattering Model
Numerical Example
Outline
Scattering Model
Numerical Example
2 Channel Simulator
Measurements
5 Conclusion

Channel Simulator
Conclusion
Scattering Model
Numerical Example
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 nth
scatterer in the mth ellipsoid which is deﬁned as
zn,m [q] =
1 if the scatterer is in the nth slot in the mth ellipse
0 if the scatterer is not in the nth slot in the mth ellipse

Channel Simulator
Conclusion
Scattering Model
Numerical Example
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 kth
transmit antenna to lth receive element and scattered via the nth
scatterer slot in the mth ellipse and received at an angle φ
(m)
R at the
receive array.

Channel Simulator
Conclusion
Scattering Model
Numerical Example
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 and
0 otherwise.

Channel Simulator
Conclusion
Scattering Model
Numerical Example
We can write the coeﬃcient g
(n,m)
kl (φ
(m)
R ,q) that represents the tuple (lth
transmit 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
e
j2π 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 ).

Channel Simulator
Conclusion
Scattering Model
Numerical Example
Outline
Scattering Model
Numerical Example
2 Channel Simulator
Measurements
5 Conclusion

Channel Simulator
Conclusion
Scattering Model
Numerical Example
• Signal parameters: the sampling frequency Ts, the wavelength of the
RF signal λ.
• Transmit (receive) array geometry: the number of elements MT
(MR), the tilt angle of the array αT (αR), and the inter-element
spacing δT (δR).
• Propagation environment parameters: the delay-spread MTs, Doppler
frequency fD, distance between transmitter and receiver 2 f.

Channel Simulator
Conclusion
Scattering Model
Numerical Example
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 .

Channel Simulator
Conclusion
Scattering Model
Numerical Example
Outline
Scattering Model
Numerical Example
2 Channel Simulator
Measurements
5 Conclusion

Channel Simulator
Conclusion
Scattering Model
Numerical Example
Model Assumptions
• Channel coefficients that account for different delays are
uncorrelated.
• 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].

Channel Simulator
Conclusion
Scattering Model
Numerical Example
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

Channel Simulator
Conclusion
Scattering Model
Numerical Example
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]

Channel Simulator
Conclusion
Scattering Model
Numerical Example
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
.

Channel Simulator
Conclusion
Scattering Model
Numerical Example
S
(n,m)
kl [q] = En,me− j2K0a
φ
(n,m)
maxˆ
φ
(n,m)
min
e
j 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 |2
φ
(n,m)
maxˆ
φ
(n,m)
min
e
j 2π fD cos φ
(m)
R −αv pTs
dφ
(m)
R

Channel Simulator
Conclusion
Scattering Model
Numerical Example
We can write the temporal correlation sequence of the mth channel
coeﬃcient 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
e
j 2π fD cos φ
(m)
R −αv pTs
dφ
(m)
R
the channel coeﬃcients are nonstationary as it depends on the time index
q. The nonstationarity is introduced via the Markov process zn,m[q] which
accounts for the persistence of the scatterer in the nth slot of the mth
ellipse.

Channel Simulator
Conclusion
Scattering Model
Numerical Example
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 become
stationary and the temporal correlation function of the mth channel
coefficient 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
e
j 2π fD cos φ
(m)
R −αv pTs
dφ
(m)
R

Channel Simulator
Conclusion
Scattering Model
Numerical Example
Spatial Correlation
The spatial correlation function of the mth channel coeﬃcient is deﬁned
as a function of the geometry of the transmit array Tx and receive array
Rx 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



.

Channel Simulator
Conclusion
Scattering Model
Numerical Example
Spatial Correlation - Cont’d
Using the modeling assumptions and going through the same steps as in
the 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 time
instant, 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 )
e
jθn,m φ
(m)
R
dφ
(m)
R

Channel Simulator
Conclusion
Scattering Model
Numerical Example
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 function
of 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

Channel Simulator
Conclusion
Scattering Model
Numerical Example
Outline
Scattering Model
Numerical Example
2 Channel Simulator
Measurements
5 Conclusion

Channel Simulator
Conclusion
Scattering Model
Numerical Example
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 100 km/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◦].

Channel Simulator
Conclusion
Scattering Model
Numerical Example
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 σ
indicates that the scatterers exist in the slot for 83.33% of the time. The
parameters of the simulation correspond to a highway environment with
high mobility.

Channel Simulator
Conclusion
Scattering Model
Numerical Example
Temporal correlation function of the channel versus time
and delay

Channel Simulator
Conclusion
Scattering Model
Numerical Example
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
Z
n,m
(τ) is not present

Channel Simulator
Conclusion
Scattering Model
Numerical Example
Spatial correlation between channel coeﬃcients h11 and h22

Channel Simulator
Conclusion
Outline
Scattering Model
Numerical Example
2 Channel Simulator
Measurements
5 Conclusion

Channel Simulator
Conclusion
MIMO Channel Coeﬃcients Matrix
We can write the mth coeﬃcient of the time varying channel between the
lth 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]








Channel Simulator
Conclusion
The Structure of the Simulator

Channel Simulator
Conclusion
Spatio-temporal Correlation Function
• The temporal correlation function
˜r(n,m)
[p] =| En,m |2
ˆ
Φ
(n,m)
R
ej2πF
(m)
D pTs
dφ
(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 are
independent
Spatio-temporal Correlation Function
˜Rn,m[p] = ˜r(n,m)
[p]Cn,m.

Channel Simulator
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

Channel Simulator
Conclusion
Spatial Correlation Filter
y(n,m)
[q] = Ln,mx(n,m)
[q]
Cn,m = Qn,mΓn,mQH
n,m,
Ln,m = Qn,mΓ
1
2
n,m.

Channel Simulator
Conclusion
Interpolator
• The sampling frequency associated with the CIR, 1/Ts, is typically
much larger than the one associated with the complex path
generator 1/T.
• In our case, the sampling frequency 1/Ts is much larger than that of
1/T. In order to reduce the computational complexity of the
interpolation.
• We use this interpolator in our simulator

Channel Simulator
Conclusion
Outline
Scattering Model
Numerical Example
2 Channel Simulator
Measurements
5 Conclusion

Channel Simulator
Conclusion
• Generates the sequence {zn,m[q]}n,m.
• These processes take only two values
{1,0} and are generated according to
the probability transition matrices
{Λ(n,m)
}n,m such that {zn,m[q]}n,m = 1
with probability
p1[q] = p1[q−1]×λ11 + p0[q−1]×λ01.
• The process zn,m[q] is multiplied by the
output of the stationary channel
coeﬃcient matrix generator.

Channel Simulator
Conclusion
The Structure of the Simulator

Channel Simulator
Conclusion
PSD of the channel coeﬃcients for the 1stslot versus the
frequency 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)
MagnitudeSpectralDensity(dB)
|H(e
j2πf
)|
2
S(2πf)

Channel Simulator
Conclusion
PSD of the channel coeﬃcients for the 2ndslot versus the
frequency 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)
MagnitudeSpectralDensity(dB)
|H(e
j2πf
)|
2
S(2πf)

Channel Simulator
Conclusion
Time varying channel coefficients between the first transmit
antenna 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)
MagnitudeofchannelcoefficientsoftheTDPs
path 0

Channel Simulator
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
ãH(φT ,φR)ˆR
−1
H ã(φT ,φR)
,
where
ã(φT ,φR) = aT (φT )⊗ aR (φR),
⊗ denotes the Kronecker product, aT (φT ) and aR (φR) are respectively
the normalized steering vectors of the transmit and receive arrays in the
directions φT and φR. The MT MR × MT MR matrix ˆRH is the sample
covariance matrix which is calculated as
ˆRH =
1
NT
NT −1
∑
q=0
vec{H [q]}vecH
{H [q]} (1)

Channel Simulator
Conclusion
Angular power spectra of the MIMO channel generated from
the simulator
ReceiveAzimuthangle(degrees)
Transmit Azimuth angle (degrees)
0 50 100 150 200 250 300 350
0
50
100
150
200
250
300
350
−340
−330
−320
−310
−300
−290
−280
−270
−260
−250
−240
−230

Channel Simulator
Conclusion
Outline
Scattering Model
Numerical Example
2 Channel Simulator
Measurements
5 Conclusion

Channel Simulator
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 high
amplitude, short duration signal.
• We will avoid that by using a Pseudo-Noise (PN) sequence.

Channel Simulator
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)

Channel Simulator
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)

Channel Simulator
Conclusion
Outline
Scattering Model
Numerical Example
2 Channel Simulator
Measurements
5 Conclusion

Channel Simulator
Conclusion
System Description
correlation with
code 1
correlation with
code 2
code 1
code 2
correlation with
code 1
correlation with
code 2
h11
h22
h21
h12
h11
h21
h12
h22

Channel Simulator
Conclusion
Channel Sampling
• We don’t need all the
samples of the channel.
• A sample within the order of
the channel coherence time
is suﬃcient.
Examples
for v = 100Km/hr at 5GHz we
havefc = 0.5KHz then one CIR
each 1ms is suﬃcient.

Channel Simulator
Conclusion
System Description
ADC I
ADC Q
concate
nate
I/Q
Rx
Buffer SRAM
RF Amplifier
Down Conversion
BBAmpllifier
RF Board FPGABoard Off Chip Memory

Channel Simulator
Conclusion
System Description - Cont’d

Channel Simulator
Conclusion
Outline
Scattering Model
Numerical Example
2 Channel Simulator
Measurements
5 Conclusion

Channel Simulator
Conclusion
An Indoor Experiment

Channel Simulator
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
AbsolutevalueofCIR

Channel Simulator
Conclusion
Power Delay Proﬁle of the Channels between diﬀerent
transmit and receive antennas
0 1 2 3 4 5
0
0.005
0.01
0.015
0.02
0.025
0.03
Tap index
Powerdelayprofile
Channel from Tx1 to Rx1
0 1 2 3
0
0.005
0.01
0.015
0.02
0.025
0.03
Tap index
Powerdelayprofile
0 1 2 3
0
0.005
0.01
0.015
0.02
0.025
0.03
Tap index
Powerdelayprofile
0 1 2 3 4
0
0.005
0.01
0.015
0.02
0.025
0.03
Tap index
Powerdelayprofile

Channel Simulator
Conclusion
Measurements
Outline
Scattering Model
Numerical Example
2 Channel Simulator
Measurements
5 Conclusion

Channel Simulator
Conclusion
Measurements
Parameters of the Measurement Experiment Setup
These are the parameters of the measurement setup
Operating Frequency 5.8 GHz
Measurement Bandwidth 10 MHz
Test signal length 25.6 µs
Snapshot time 51.2 µs
Number of snapshots 256
Snapshots inter-duration 1ms
Maximum transmit power 19 dBm
Number of Tx antenna elements 2
Number of Rx antenna elements 2
Tx antenna separation 2.5 Cm
Rx antenna separation 2.5 Cm
Tx antenna high 26.5 Cm
Rx antenna high 26.5 Cm

Channel Simulator
Conclusion
Measurements
The Cars We Used in the Experiment

Channel Simulator
Conclusion
Measurements
Measurements Experiments Scenarios

Channel Simulator
Conclusion
Measurements
Outline
Scattering Model
Numerical Example
2 Channel Simulator
Measurements
5 Conclusion

Channel Simulator
Conclusion
Measurements
Spatial Parameters Extraction Phase
The spatial correlation function of the mth channel coeﬃcient is deﬁned
as a function of the geometry of the transmit array Tx and receive array
Rx 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 in
the rest of this presentation. To compute this correlation matrix, one
needs
• Evaluate the inner integration.
• Find and estimate for the inner terms |En|2
π
(n)
1 ∀n = 0,···Nc

Channel Simulator
Conclusion
Measurements
Spatial Parameters Extraction Phase - Cont’d
Evaluating the inner integration
The inner integration can be evaluated numerically given the knowledge
of the set of receive azimuth angles ΦRs. This set can be estimated from
the Capon spectrum.
ReceiveAzimuthangle(degrees)
Transmit Azimuth angle (degrees)
0 50 100 150 200 250 300 350
0
50
100
150
200
250
300
350
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08

Channel Simulator
Conclusion
Measurements
Resolving the Ambiguity of Linear Arrays
• We do our measurements using 2 × 2 linear array; this cause an
ambiguity in the clusters locations.
• This ambiguity can be resolved by choosing the best match to an
ellipse structure.

Channel Simulator
Conclusion
Measurements
Estimating 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

Channel Simulator
Conclusion
Measurements
Temporal Parameters Extraction Phase
We want to ﬁnd an estimate for the probability transition matrix Λ(n) for
every cluster.
The Temporal Correlation function of the mth channel coeﬃcient
˜r[p] =
Nc
∑
n=0
Λ
(n)p
(2,2)
π
(n)
1 | En |2
ˆ
Φ
(n)
R
ej(2πFD pTs)
dφR
=
Nc
∑
n=0
Λ
(n)p
(2,2)
π
(n)
1 | En |2
Eθn

Channel Simulator
Conclusion
Measurements
Temporal Parameters Extraction Phase - Cont’d
A 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
+
T2
n
4
− Dn
ζ
(n)
1 =
Tn
2
−
T2
n
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

Channel Simulator
Conclusion
Measurements
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
(un
2)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
p
0
0 ζ
(n)
2
p



 u
(n)
1
H
un
2
H



Channel Simulator
Conclusion
Measurements
Temporal Parameters Extraction Phase - Cont’d
Least Squares Fitting Problem
Thus we can write the ﬁtting problem as an optimization of a least
sequares problem
Least Squares Fitting Problem
min
x,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 the
elements of the transition matrix Λ(n)

Channel Simulator
Conclusion
Measurements
Theoretical temporal correlation versus temporal correlation
computed 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))
Measurements
Theoretical

Channel Simulator
Conclusion
Measurements
Channel Coeﬃcients Distribution
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
0
5
10
15
20
25
0 0.5 1 1.5 2 2.5 3
0
0.5
1
1.5
2
2.5
3
x 10
4

Channel Simulator
Conclusion
Conclusion

Channel Simulator
Conclusion
Any Questions ?

Channel Simulator
Conclusion
Thank You :-)

Channel Modeling for Wideband MIMO Vehicle-to-Vehicle Channels

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