86 ◾ Advances in Communications-Based Train Control Systems
απερ=
()
+
()
−
()
20
10
1
2
2
1
3/2
0
2
2
2
1/2
ln b
ff baff
ff
wg
c
c
c
(5.3)
where:
ε
0
is the dielectric constant of vacuum (the value is
8.85
10 /
12
×
−
Fm
)
ρ
wg
is the resistivity of the material of leaky waveguide (the value is
2.9
10
8
×
−
⋅Ω
m
)
a
and
b
are the dimensions of leaky waveguide
f
c
is the cut-o frequency of the leaky waveguide
As each slot is taken as a magnetic dipole, the electric eld radiated from a slot is
as follows [7]:
EE
r
jr
=
−
0
sin
e
θ
β
(5.4)
where:
E
0
is the electric eld amplitude of the slot
E
is the electric eld strength radiated from the slot
θ is the intersection angle between the line from the receiving point to the slot
and the plane of leaky waveguide
β is the propagation constant of free space
r
is the distance between the slot and the receiving point
erefore, the leaky waveguide can be taken as a magnetic dipole antenna array,
and then the radiated electric eld strength can be expressed as follows:
EE
r
k
N
jk d
k
k
jr
k
=
=
−
−
∑
1
0
(1)
e
sin
e
α
β
θ
(5.5)
where:
E
0
is the electric eld amplitude of the rst slot of leaky waveguide, normalized
as
1/Vm
N
is the total number of slots
α is the transmission loss of leaky waveguide (its unit is linear here, but not dB)
θ
k
is the intersection angle between the line from the receiving point to the
k
th
slot and the plane of leaky waveguide
r
k
is the distance between the receiving point to the
k
th slot
In order to theoretically prove the linear model of the large-scale fading, the equiva-
lent magnetic dipole method is applied as the supplement to the measurements. e
simulation results are shown in Figure 5.4, which are approximately straight line
and almost parallel to the tting lines. We use MATLAB® to implement the simu-
lation of the equivalent magnetic dipole method. e length of leakywaveguide
Modeling of the Wireless Channels with Leaky Waveguide ◾ 87
is
300m
, and the distance between adjacent slots is set as
60mm
. e receiv-
ing antenna is set as
320mm
right above the leaky waveguide, and it can receive
electromagnetic waves leaked from slots based on Equation 5.5. e operation fre-
quency is
2.412
GHz. Considering the transmission loss of leaky waveguide, the
sum of electric strength from dierent slots is obtained. For ease of calculation, the
electric eld amplitude of the rst slot is normalized. Moreover, in simulations, it
is dicult to consider the insertion loss, the feeder line loss, and the splicing loss,
which generally cannot be ignored in measurements. As a result, there is dierence
between the simulation results and the tting lines. However, the simulation shows
that the distribution of electric eld strength is approximately linear and the slope
is close to the transmission loss.
From the derivative of Equation 5.5 with respect to
d
, we can nd that the
path loss exponent is mostly dependent on the value of transmission loss α.
e transmission loss of the leaky waveguide can be calculated according to
Equation5.3, which is
0.0139
/dB
m
denoted as
n
t
. Comparing the parameters
n
with
n
t
, we nd that
n
t
is almost equal to
n
, which means that we can use
the transmission loss as the path loss exponent of the radio channel with leaky
waveguide. And it will be convenient and eective for practical engineering,
especially the link budget.
0 50 100 150 200 250 30
0
−50
−45
−40
−35
−30
−25
−20
−15
−10
−5
0
Signal strength (dBm)
e location of the receiving antenna (m)
Simulation results
Fitting lines of several measurements
Figure5.4 Simulation results of the equivalent method and the tting lines of
several measurements.
88 ◾ Advances in Communications-Based Train Control Systems
In Equation 5.1, the path loss exponent
n
has been obtained, and
PL(0)
is an
inherent parameter of one kind of leaky waveguide. Next, we will discuss the dis-
tribution model of small-scale fading.
5.4.3 Determination of the Small-Scale Fading
e statistical model of the small-scale fading needs to be determined to show how
much the signal level can vary on the basis of the large-scale fading. A number
of dierent distributions have been proposed for small-scale amplitude fading in
indoor and outdoor environments, including Rice, Rayleigh, Nakagami, Weibull,
and log-normal distributions. In order to determine the distribution of the small-
scale fading of the channel with leaky waveguide, we rst need to remove the eects
of the large-scale fading. And the length of small-scale area (SSA) should be deter-
mined as
20λ
to get enough data to estimate the model precisely for one SSA[8].
As the empirical decorrelation distance is
λ/2
, we select one sample from the mea-
surement results in each
λ/2
section. And the samples in dierent decorrelation
distance areas are spatial independent (uncorrelated).
We select ve kinds of classic distributions mentioned in the preceding text as
the candidate models of the small-scale fading for the channel with leaky wave-
guide. ese models are also used in [8], where the AICc is used. AICc is also called
the second-order AIC, which is dened as follows [9]:
AIC
AICc AIC
e
ij
n
N
i
ij
in j
ij ij
j
lx k
kk
,
=1
,
,
,,
2((|)) 2
2(
=− +
=+
∑
log
θ
jj
ij
Nk
+
−−
1)
1
(5.6)
where:
i
means the
i
th SSA
j
means the
j
th candidate model
θ
ij
,
means the estimated parameters of the
j
th candidate model for the
i
th SSA
using the maximum likelihood estimator (MLE)
x
in
,
is the
n
th sample of the
i
th SSA
k
j
is the number of parameters of the
j
th candidate model
N
i
is the total number of samples of the
i
th SSA
For each SSA, we calculate the value of AICc of each candidate model and select the
best-t model with the lowest value of AICc. Figure 5.5 shows that the log-normal
distribution provides the best t in a majority of the cases, about
54.7
%. is is valid
for 20 measurements in Beijing Subway Yizhuang Line. erefore, we can see that the
log-normal model is the best parametric t to the distribution of the small-scale fading.
In addition, we show the plots of the empirical cumulative distribution func-
tions(CDFs) of measurement data and the candidate models in Figure5.6, where
Modeling of the Wireless Channels with Leaky Waveguide ◾ 89
0.0
Nakagami
Weibull
Log-normal
Rician
Rayleigh
0.1
0.2
0.3
Relative frequency
0.4
0.5
Figure5.5 Relative frequencies of AICc selecting a candidate distribution as the
best t to the distribution of small-scale fading amplitudes.
0.0 0.5 1.0 1.5 2.0 2.5
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Cumulative probability
Measurement data
Log-normal
Nakagami
Rician
Rayleigh
Weibull
Figure5.6 Sample empirical CDFs of the small-scale fading amplitudes and their
theoretical model ts.
90 ◾ Advances in Communications-Based Train Control Systems
the
x
-axis is the fading amplitude whose unit is linear. e same observation can be
obtained through CDFs. e log-normal distribution provides the closest t to the
measurement results. e expression of log-normal distribution is
px
x
x
x
(;,)
1
2
,>
0
2
2
µσ
σπ
µσ
=
−−
(
)
e
ln /
(5.7)
where:
μ and
σ
are the mean and standard deviations of the variable’s natural loga-
rithm, respectively
As we can obtain μ
dB
and
σ
dB
through the MLE method at every SSA with the exact
location information, the relationship between the parameters of log-normal distribu-
tion and the location of the receiving antenna can be built. We get the mean values of
μ
dB
and
σ
dB
of dierent measurements at the same SSA, as shown in Figure5.7, where
the
x
-axis is the distance from the beginning of the leaky waveguide to the midpoint
of each SSA. We build the relationship between the parameters of log-normal distri-
bution and the location of the receiver. e average of μ
dB
is 0 and the average of
σ
dB
is almost
−10
. At the beginning and the end of the leaky waveguide, the value of
σ
dB
increases quickly, which means that the amplitudes uctuate wildly. e reason is
that the beginning of the leaky waveguide is out of the radiation coverage of the rst
0 50 100 150 200 250 30
0
−12
−10
−8
−6
−4
−2
0
2
e middle position of SSA
dB
μ
dB
σ
dB
Figure5.7 Variance of µ
dB
and σ
dB
with different receiving points.
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