Digital Signal Processing Reference
In-Depth Information
Table 11.2. DTFTs and DTFSs for elementary DT sequences
Note that the DTFS does not exist for aperiodic sequences
k
=−∞
x
[
k
]e
−
j
Ω
k
∞
1
K
0
x
[
k
]e
−
j
n
Ω
0
k
Sequence:
x
[
k
]
DTFS:
D
n
=
DTFT:
X
(
Ω
)
=
k
=
K
0
m
=−∞
δ
(
Ω
−
2
m
π
)
∞
(1)
x
[
k
]
=
1
D
n
=
1
X
(
Ω
)
=
2
π
(2)
x
[
k
]
= δ
[
k
]
does not exist
X
(
Ω
)
=
1
X
(
Ω
)
=
e
−
j
Ω
k
0
(3)
x
[
k
]
= δ
[
k
−
k
0
]
does not exist
m
=−∞
δ
(
k
−
mK
0
)
∞
m
=−∞
δ
∞
1
K
0
X
(
Ω
)
=
2
π
K
0
−
2
m
π
K
0
(4)
x
[
k
]
=
D
n
=
for all
n
Ω
m
=−∞
δ
(
Ω
−
2
m
π
)
+
∞
1
1
−
e
−
j
Ω
x
[
k
]
=
u
[
k
]
X
(
Ω
)
= π
(5)
does not exist
1
(6)
x
[
k
]
=
p
k
u
[
k
] with
p
<
1
does not exist
X
(
Ω
)
=
1
−
p
e
−
j
Ω
1
(7) First-order time-rising
decaying exponential
x
[
k
]
=
(
k
+
1)
p
k
u
[
k
]
,
with
p
<
1.
does not exist
X
(
Ω
)
=
(1
−
p
e
−
j
Ω
)
2
1
n
=
p
rK
0
0 elsewhere
for
−∞ <
r
< ∞
m
=−∞
δ
(
Ω
−
Ω
0
−
2
m
π
)
∞
=
X
(
Ω
)
=
2
π
(8) Complex exponential
(periodic)
x
[
k
]
=
e
j
k
Ω
0
K
0
D
n
=
2
π
p
/
Ω
0
m
=−∞
δ
(
Ω
−
Ω
0
−
2
m
π
)
∞
X
(
Ω
)
=
2
π
(9) Complex exponential
(aperiodic)
x
[
k
]
=
e
j
k
Ω
0
,
2
π/
Ω
0
does not exist
=
rational
m
=−∞
δ
(
Ω
+
Ω
0
−
2
m
π
)
+ π
∞
(10) Cosine (periodic)
x
[
k
]
=
cos(
Ω
0
k
)
K
0
1
X
(
Ω
)
= π
n
=
p
rK
0
2
D
n
=
m
=−∞
δ
(
Ω
−
Ω
0
−
2
m
π
)
∞
=
2
π
p
/
Ω
0
0 elsewhere
for
−∞ <
r
< ∞
m
=−∞
δ
(
Ω
+
Ω
0
−
2
m
π
)
+ π
∞
(11) Cosine (aperiodic)
x
[
k
]
=
cos(
Ω
0
k
)
,
2
π/
Ω
0
does not exist
X
(
Ω
)
= π
m
=−∞
δ
(
Ω
−
Ω
0
−
2
m
π
)
∞
=
rational
m
=−∞
δ
(
Ω
+
Ω
0
−
2
m
π
)
−
j
π
∞
(12) Sine (periodic)
x
[
k
]
=
sin(
Ω
0
k
)
K
0
1
X
(
Ω
)
=
j
π
=
p
rK
0
2
j
n
D
n
=
m
=−∞
δ
(
Ω
−
Ω
0
−
2
m
π
)
∞
=
2
π
p
/
Ω
0
0 elsewhere
for
−∞ <
r
< ∞
m
=−∞
δ
(
Ω
+
Ω
0
−
2
m
π
)
−
j
π
∞
(13) Sine (aperiodic)
x
[
k
]
=
sin(
Ω
0
k
),
2
π/
Ω
0
does not exist
X
(
Ω
)
=
j
π
m
=−∞
δ
(
Ω
−
Ω
0
−
2
m
π
)
(
cont.
)
∞
=
rational
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