Digital Signal Processing Reference
In-Depth Information
(
,
)
sample time
n
(and of
f
k
, due to the normalization), and we write it as
z
j
n
k
.
(
,
,
)
m
n
,where
√
1
m
2
due to the normalization. There are
of concern for us here.
Taking the phase factors into account, we can model the received signal vector
2
,
m
are direction cosines, and
n
=
−
−
x
(
n
,
k
)
as
Q
q
=
1
a
q
(
n
,
k
)
s
q
(
n
,
k
)+
n
(
n
,
k
)
x
(
n
,
k
)=
(8)
(
,
)
where
a
q
is called the “array response vector” for the
q
th source, consisting
of the phase multiplication factors, and
n
n
k
(
,
)
n
k
is an additive noise vector, due to
(
,
)
(
,
)
thermal noise at the receiver. We will model
s
q
as baseband com-
plex envelope representations of zero mean wide sense stationary white Gaussian
random processes sampled at the Nyquist rate.
With the above discussion, the array response vector is modeled (for an ideal
receiver) as
n
k
and
n
n
k
T
p
q
e
−
j
Z
(
n
,
k
)
a
q
(
n
,
k
)=
,
Z
(
n
,
k
)=[
z
1
(
n
,
k
)
,···,
z
J
(
n
,
k
)]
.
(9)
For convenience of notation, we will in future usually drop the dependence on the
frequency
f
k
(index
k
) from the notation.
acquisition process, where the time index
m
corresponds to the
m
th short term
integration interval (STI), such that
(
m
−
1
)
N
≤
n
≤
mN
. Due to Earth rotation,
the vector
a
q
(
changes slowly with time, but we assume that within an STI it
can be considered constant and can be represented, with some abuse of notation,
by
a
q
(
n
)
m
)
. In that case,
x
(
n
)
is wide sense stationary over the STI, and a single STI
autocovariance is defined as
=
n
N
x
H
R
m
=
E
{
x
(
n
)
(
n
)
},
m
(10)
where
R
m
has size
J
J
. Each element of
R
m
represents the interferometric
correlation along the baseline vector between the two corresponding receiving
elements. It is estimated by STI sample covariance matrices
×
R
m
}
=
our stationarity assumptions imply
E
R
m
.
If we consider only a single signal from direction
p
q
and look at entry
{
(
i
,
j
)
of
R
m
, then its value is
T
p
q
q
e
−
j
(
z
i
(
m
)
−
z
j
(
m
))
2
(
R
m
)
i
,
j
=
E
{
x
i
(
n
)
x
j
(
n
)
}
=
σ
.
(11)
1
With abuse of notation, as
m
,
n
are not related to the time variables used earlier.
