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unsatisfactory.
To overcome these difficulties, particle filtering (PF) is utilized [Jian et al.,
2007].
Although particle filtering performs better than Kalman filtering in noisy environment, but
it also diverges in low SNRs and cannot be trustable in this range of SNR. In addition, this
method requires much more processing. We note that Kalman filtering, extended Kalman,
unscented Kalman and particle filtering are recursive in nature. The new procedure proposed
in this chapter is not recursive and can be used in the non-Gaussian, non-stationary noise,
and nonlinear target motion. In here, the target tracking is performed based on the estimated
time delay and Doppler. Since the accuracy of the time delay and Doppler estimation are high
enough even in the severe noise, the results in tracking are acceptable compared to other rival
approaches.
In section II the moment concept is reviewed and moment method is described as the base
item in our estimations. Section III provides a model for the received signal. This signal
has been influenced by unknown noise, delay and Doppler. It is shown in Section IV that
it is possible to estimate Doppler by using the moments of the received random signal. The
method is also useful for delay estimation. The noise power and its behavior play a prominent
role in our work. So some analysis in this field is presented in this section too. After the
parameter estimation, section V is devoted to explain about how the tracking a target is
done based on the estimated delay and Doppler. And finally, section VI contains results that
illustrate the effectiveness of the proposed method.
2. Moment concept
In probability theory, the moment method is a way in which the moments of a discrete
sequence are used to determine its distribution.
Suppose that
X
is a random variable, and
f
X
(
)
is the probability density function (PDF) of
this random variable. The moments of the random variable
X
is calculated from the following
equation:
x
∞
∞
X
n
x
n
f
X
(
x
n
dF
X
(
m
n
=
E
(
) =
x
)
dx
=
x
)
dx
,
(1)
−
∞
−
∞
which
F
X
(
)
(
)
x
is the cumulative distribution function (CDF) of the random variable
X
, and
E
.
is the expectation value.
On the other hand, the moment generating function (MGF) of this random variable is
calculated as follows:
E
e
uX
,
u
M
X
(
) =
∈
C
u
.
(2)
Note that the equation will be hold if the expectation value exists.
In here, to obtain the moments of a random variable, the relation between the moment and
the moment generating function is use instead of using equation (1).
This relation can be
demonstrated as follows:
E
e
uX
∞
e
ux
f
X
(
M
X
(
u
) =
=
x
)
dx
=
−
∞
∞
−
∞
(
u
2
x
2
2!
u
2
m
2
2!
1
+
ux
+
+
···
)
f
X
(
x
)
dx
=
1
+
um
1
+
+
···
.
(3)
