Graphics Programs Reference
In-Depth Information
Waveform Resolution and Ambiguity
175
Let
–
j 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 f
=
(
–
f 0
)
(3.136)
Thus, Eq. (3.135) becomes
U f
U f
(
) U f
(
–
f d
)
d
=
(
–
f 0
) U f
(
–
f 0
–
f d
)
d
(3.137)
–
–
The complex frequency correlation function is then defined as
j f d t
u () 2
f () U f
χ f
=
(
) U f
(
–
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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