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)
2
The zero time cut along the Doppler frequency axis has a
(
sin
x
⁄
x
shape.
It extends from
∞
to
∞
. The first null occurs at
f
d
=
±
1 τ'
⁄
. Hence, it is
possible to detect two targets that are shifted by
1 τ'
⁄
, without any ambiguity.
We conclude that a single pulse range and Doppler resolutions are limited by
the pulsewidth . Fine range resolution requires that a very short pulse be
used. Unfortunately, using very short pulses requires very large operating
bandwidths, and may limit the radar average transmitted power to impractical
values.
τ'
4.2.2. LFM Ambiguity Function
Consider the LFM complex envelope signal defined by
1
τ'
t
τ'
--
e
j
πµ
t
2
s
()
=
-------
Rect
(4.13)
In order to compute the ambiguity function for the LFM complex envelope, we
will first consider the case when
0 ττ′
≤≤
. In this case the integration limits
are from
τ′
⁄
2
to
(
τ′
⁄
2
)
τ
. Substituting Eq. (4.13) into Eq. (4.9) yields
∞
∫
1
τ′
t
τ′
t
τ′
τ
e
j
πµ
t
2
)
2
Rect
e
j
2π
f
d
t
j
πµ
t
τ
(
----
---
----------
χτ
f
d
(
;
)
=
Rect
e
dt
(4.14)
∞
It follows that
τ'
---
τ
πµτ
2
e
e
j
2πµτ +
d
(
)
t
∫
---------------
χτ
f
d
(
;
)
=
dt
(4.15)
τ'
2
τ'
------
Finishing the integration process in Eq. (4.15) yields
τ
τ'
sin
πτ' µτ
(
+
f
d
) 1
---
τ
τ'
e
j
πτ
f
d
χτ
f
d
(
;
)
=
1
---
-------------------------------------------------------------
0 ττ′
≤≤
(4.16)
τ
τ'
πτ' µτ
(
+
f
d
) 1
---
Similar analysis for the case when can be carried out, where in
this case the integration limits are from to . The same result
can be obtained by using the symmetry property of the ambiguity function
(
τ′
≤≤
τ
0
(
τ′
⁄
2
)
τ
τ'2
⁄
χτ
f
d
(
,
)
=
χ τ
f
d
(
,
)
). It follows that an expression for
χτ
f
d
(
;
)
that is
valid for any
τ
is given by
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