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0
∑
N
∑
1
q
1
----
e
j
2π
f
d
iT
χτ
f
d
(
;
)
=
χ
1
(
τ
qT
;
f
d
)
(4.30)
q
=
(
N
1
)
i
=
0
N
∑
1
N
∑
1
q
1
----
e
j
2π
f
d
qT
χ
1
e
j
2π
f
d
jT
+
(
τ
qT
;
f
d
)
q
=
1
j
=
0
Setting
z
=
exp
(
j
2π
f
d
T
)
, and using the relation
N
∑
1
q
z
Nq
1
1
z
j
=
----------------------
(4.31)
z
j
=
0
yield
N
∑
1
q
sin
[
π
f
d
(
N
1
qT
)
]
e
j
2π
f
d
iT
[
j
π
f
d
N
(
1
qT
)
]
=
e
------------------------------------------------------
(4.32)
sin
(
π
f
d
T
)
i
=
0
Using Eq. (4.32) in Eq. (4.30) yields two complementary sums for positive and
negative
q
. Both sums can be combined as
N
∑
1
sin
[
π
f
d
(
NqT
)
]
1
----
[
j
π
f
d
N
(
1
+
q
)
T
]
---------------------------------------------
χτ
f
d
(
;
)
=
χ
1
(
τ
qT
;
f
d
)
e
(4.33)
sin
(
π
f
d
T
)
q
=
(
N
1
)
Finally, the ambiguity function associated with the coherent pulse train is com-
puted as the modulus square of Eq. (4.33). For
τ′
<
T
⁄
2
, the ambiguity func-
tion reduces to
N
∑
1
1
----
sin
[
π
f
d
(
NqT
)
]
χτ
f
d
(
;
)
=
χ
1
(
τ
qT
;
f
d
)
---------------------------------------------
(4.34)
sin
(
π
f
d
T
)
q
=
(
N
1
)
Thus, the ambiguity function for a coherent pulse train is the superposition
of the individual pulseÓs ambiguity functions. The ambiguity function cuts
along the time delay and Doppler axes are, respectively, given by
∑
N
1
2
q
-----
τ
τ′
qT
)
2
χτ;
(
=
1
1
------------------
;
τ
qT
<
τ′
(4.35)
q
=
(
N
1
)
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