Graphics Programs Reference
In-Depth Information
It is customary to let . The output of the mixer is the product of the
received and reference signals. After low pass filtering the signal is
f
r
=
f
0
2πµ∆τ
t
πµ ∆(
2
s
0
()
a
=
cos
(
2π
f
0
∆τ
+
)
(5.31)
Substituting Eq. (5.28) into (5.31) and collecting terms yield
4π
BR
c
τ′
2
R
c
2π
BR
c
τ′
t
--------------
-------
2π
f
0
s
0
()
a
=
cos
+
--------------
(5.32)
and since
τ′
»
2
Rc
⁄
, Eq. (5.32) is approximated by
4π
BR
c
τ′
4π
R
c
t
s
0
()
a
≈
cos
--------------
+
----------
f
0
(5.33)
The instantaneous frequency is
1
2π
d
4π
BR
c
τ′
4π
R
c
2
BR
c
τ′
------
--------------
t
----------
f
0
f
inst
=
+
=
-----------
(5.34)
d
t
which clearly indicates that target range is proportional to the instantaneous
frequency. Therefore, proper sampling of the LPF output and taking the FFT of
the sampled sequence lead to the following conclusion: a peak at some fre-
quency
f
1
indicates presence of a target at range
R
1
=
f
1
c
τ′
⁄
2
B
(5.35)
Assume
I
close targets at ranges
R
1
,
R
2
, and so forth (
R
1
<<<
R
2
…
R
I
).
From superposition, the total signal is
I
∑
µ
---
t
)
2
s
r
()
=
a
i
()
cos
2π
f
0
(
t
τ
i
)
+
(
τ
i
(5.36)
i
=
1
where are proportional to the targetsÓ cross sections,
antenna gain, and range. The times represent
the two-way time delays, where coincides with the start of the receive win-
dow. Using Eq. (5.32) the overall signal at the output of the LPF can then be
described by
{
a
i
()
i
;
=
12…
I
,, ,
}
{
τ
i
=
(
2
R
i
⁄
c
)
;
i
=
12…
I
,, ,
}
τ
1
I
∑
4π
BR
i
c
τ′
2
R
i
c
2π
BR
i
c
τ′
t
----------------
--------
2π
f
0
s
o
()
=
a
i
cos
+
----------------
(5.37)
i
=
1
And hence, target returns appear as constant frequency tones that can be
resolved using the FFT. Consequently, determining the proper sampling rate
and FFT size is very critical. The rest of this section presents a methodology
for computing the proper FFT parameters required for stretch processing.
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