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In-Depth Information
v
R
0
Figure 1.8. Closing target with velocity
v
.
Substituting Eq. (1.26) into Eq. (1.25) and collecting terms yield
2
v
c
t
ψ
0
x
r
()
x
=
1
+
------
(1.27)
where the constant phase
is
ψ
0
2
R
0
c
2
v
c
ψ
0
=
---------
+
------
t
0
(1.28)
Define the compression or scaling factor
γ
by
2
v
c
γ
=
1
+
------
(1.29)
Note that for a receding target the scaling factor is
. Utilizing
γ
=
1
vc
(
⁄
)
Eq. (1.29) we can rewrite Eq. (1.27) as
x
r
()
x
γ
t
ψ
0
=
(
)
(1.30)
Eq. (1.30) is a time-compressed version of the return signal from a stationary
target ( ). Hence, based on the scaling property of the Fourier transform,
the spectrum of the received signal will be expanded in frequency to a factor of
.
v
=
0
γ
Consider the special case when
x
()
y
() ω
0
t
=
cos
(1.31)
where
is the radar center frequency in radians per second. The received sig-
ω
0
nal
is then given by
x
r
()
x
r
()
y
γ
t
ψ
0
=
(
)
cos
(
γω
0
t
ψ
0
)
(1.32)
The Fourier transform of Eq. (1.32) is
X
r
()
1
Y
ω
Y
ω
-----
=
---- ω
0
+
---- ω
0
+
(1.33)
2γ
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