Digital Signal Processing Reference
In-Depth Information
FOV
geometric
delay
g
1
g
2
g
J
baseline
x
1
(
t
)
x
2
(
t
)
x
J
(
t
)
Fig. 2
Schematic overview of a radio interferometer
3.1
Data Acquisition
Mathematically, the correlation process is described as follows. Assume that there
are
J
array elements (telescopes). The RF signal
x
j
(
)
t
from the
j
th telescope is first
(
)
moved to baseband where it is denoted by
x
j
, then sampled and split into narrow
subbands, e.g., of 100 kHz each, such that the “narrowband condition” holds. This
condition states that the maximal geometrical delay across the array should be fairly
representable by a phase shift of the complex baseband signal, and this property is
discussed in more detail in the next subsection.
The resulting signal is called
x
j
t
(
,
)
,forthe
j
th telescope,
n
th time bin, and for
the subband frequency centered at RF frequency
f
k
.The
J
signals are stacked into a
J
n
k
×
(
,
)
.
A single correlation matrix is formed by “integrating” (summing) the crosscor-
relation products
x
1 vector
x
n
k
x
H
(
,
)
(
,
)
n
k
n
k
over
N
subsequent samples,
mN
−
1
1
N
R
m
,
k
=
∑
n
=(
m
−
1
)
N
x
H
x
(
n
,
k
)
(
n
,
k
)
,
(6)
where
m
is the index of the corresponding “short-term interval” (STI) over which is
The duration of a STI depends on the stationarity of the data, which is limited by
factors like Earth rotation and the diameter of the array. For the Westerbork array,
a typical value for the STI is 10-30 s; the total observation can last for up to 12 h.
The resulting number of samples
N
in a snapshot observation is equal to the product
of bandwidth and integration time and typically ranges from 10
3
(1 s, 1 kHz) to 10
6
(10 s, 100 kHz) in radio astronomical applications.
