Graphics Programs Reference
In-Depth Information
Waveform Resolution and Ambiguity
175
Let
∞
∫
j
2π
ft
Ψ ()
=
ψ()
e
d
(3.133)
∞
Due to the Doppler shift associated with the target, the received signal spec-
trum will be shifted by . In other words the received spectrum can be repre-
sented by . In order to distinguish between the two targets located at
the same range but having different velocities, one may use the integral square
error. More precisely,
f
d
Ψ
f
(
)
d
∞
∫
2
2
ε
f
=
Ψ () Ψ
f
(
)
d
(3.134)
d
∞
Using similar analysis as that which led to Eq. (3.125), one should minimize
∞
∫
Ψ
∗
()Ψ
f
2
Re
(
)
d
(3.135)
d
∞
By using the analytic signal in Eq. (3.118) it can be shown that
Ψ ()
U
2π
f
=
(
2π
f
0
)
(3.136)
Thus, Eq. (3.135) becomes
∞
∫
∞
∫
U
∗
2π
f
U
∗
2π
f
(
)
U
2π
f
(
2π
f
d
)
d
=
(
2π
f
0
)
U
2π
f
(
2π
f
0
2π
f
d
)
d
(3.137)
∞
∞
The complex frequency correlation function is then defined as
∞
∫
∞
∫
j
2π
f
d
t
u
()
2
f
()
U
∗
2π
f
χ
f
=
(
)
U
2π
f
(
2π
f
d
)
d
=
d
(3.138)
∞
∞
and the Doppler resolution constant
is
∆
f
d
∞
∫
∞
∫
f
()
2
u
()
4
χ
f
d
f
d
d
1
τ'
∞
∞
∆
f
d
=
---------------------------------
=
---
--------------------------
---
--
=
---
(3.139)
χ
f
2
2
∞
∫
()
u
()
2
d
∞
where
is pulsewidth.
τ'
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