Graphics Programs Reference
In-Depth Information
⁄
cv
cf
r
∆
t
=
-----------
(1.17)
+
cv
⁄
cv
f
r
d
=
-------------
(1.18)
+
The reflected pulse spacing is now
and the new PRF is
, where
sd
f
r
′
cv
⁄
cv
f
r
c
f
r
′
sd
==
-----
c
∆
t
-------------
(1.19)
+
It follows that the new PRF is related to the original PRF by
cv
+
cv
-----------
f
r
′
=
f
r
(1.20)
However, since the number of cycles does not change, the frequency of the
reflected signal will go up by the same factor. Denoting the new frequency by
, it follows
f
0
′
cv
+
cv
-----------
f
0
′
=
f
0
(1.21)
where is the carrier frequency of the incident signal. The Doppler frequency
is defined as the difference
f
0
. More precisely,
f
d
f
0
′
f
0
cv
+
cv
2
v
cv
-----------
-----------
f
d
=
f
0
′
f
0
=
f
0
f
0
=
f
0
(1.22)
but since
and
, then
vc
«
c
=
λ
f
0
2
v
c
2
v
λ
------
f
d
≈
f
0
=
------
(1.23)
Eq. (1.23) indicates that the Doppler shift is proportional to the target velocity,
and, thus, one can extract
from range rate and vice versa.
f
d
The result in Eq. (1.23) can also be derived using the following approach:
Fig. 1.8
shows a closing target with velocity
. Let
refer to the range at
v
R
0
time
(time reference); then the range to the target at any time is
t
0
t
R
()
R
0
=
t
(
)
(1.24)
0
The signal received by the radar is then given by
x
r
()
xt
ψ ()
=
(
)
(1.25)
where
is the transmitted signal, and
x
()
2
---
R
0
ψ ()
=
(
vt
+
vt
0
)
(1.26)
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