Digital Signal Processing Reference
In-Depth Information
velocity relative to the radar,
σ
n
is the noise term. Here we consider two typical Radar
signal models.
Complex circular multivariate Gaussian model
. Gaussian distributions are widely
used to model radar signals due to its simplicity. For the radar signal
Z
n
=
z
n
]
T
modeled by a complex circular multivariate Gaussian distribution with
zero mean, the probability density function of
Z
n
is given by the formula
[
z
1
z
2
···
exp
Tr
1
R
n
R
−
1
p
(
Z
n
|
R
n
)
=
−
(11.2)
n
n
π
|
R
n
|
T
where
(
·
)
denotes the transpose and conjugate transpose of a matrix,
|·|
represents
E
R
n
=
the determinant value of a matrix,
Tr
represents the trace of a matrix ,
R
n
=
E
Z
n
Z
n
is an
n
×
n
Hermitian Symmetric Positive Definite Matrix
⎡
⎣
⎤
⎦
,
r
11
r
12
···
r
1
n
E
z
i
z
j
r
21
r
22
···
r
2
n
.
···
.
.
.
.
r
21
r
22
···
R
n
=
r
ij
=
(11.3)
r
2
n
H
where
(
·
)
denotes the conjugate transpose of a matrix.
Complex
autoregressive
model
.
For
a
sequence
of
complex
Radar
signals
{
from
N
coherently transmitted pulses, the Doppler information
can be obtained from the complex autoregressive modeling parameters by using
Burg's maximum entropy algorithm. In the AR model of order
n
,thevalueof
z
m
can
be predicted from a linear combination of the preceding
n
samples of the sequence
Z
m
=[
z
0
,
z
1
,
···
,
z
N
−
1
}
T
z
m
−
1
z
m
−
2
···
z
m
−
n
]
as described in the following equation
A
n
Z
m
(
z
m
=
n
,
m
−
n
≤
m
≤
N
)
(11.4)
n
,
m
is the prediction error which is supposed to be a complex white noise,
where
=[
···
a
n
,
n
]
is a sequence of
n
coefficients defining the model which
is determined by minimizing the mean square errors of
A
n
a
n
,
1
a
n
,
2
n
,
m
E
z
m
+
A
n
Z
n
,
m
∗
z
m
+
A
n
Z
n
,
m
E
n
,
m
n
,
m
=
2
ε
m
=
:
A
n
C
n
A
n
=
c
0
+
C
n
+
A
n
+
R
n
A
n
(11.5)
E
z
m
z
m
−
k
is the complex autocorrelation coefficient,
C
n
where
c
k
=
=[
c
1
c
2
···
c
n
]
,
R
n
is an Toeplitz Hermitian covariance matrix
