Digital Signal Processing Reference
In-Depth Information
DFS
←→
X
symmetries:
x
[
−
n
]
[
−
k
]
DFS
←→
X
∗
[
−
x
∗
[
n
]
k
]
l
g
r
,
y
i
d
.
,
©
,
L
s
x
(
)
DFS
←→
W
kn
0
N
X
shifts:
n
−
n
0
[
k
]
X
(
k
0
)
DFS
←→
W
−nk
0
N
x
[
n
]
k
−
N
−
1
x
N
−
1
X
1
N
2
2
Parseval:
[
n
]
=
[
k
]
n
=
0
k
=
0
Discrete Fourier Transform (DFT)
used for:
finite support signals (
x
[
n
]
∈
N
)
N
−
1
W
n
N
,
[
]=
[
]
=
analysis formula:
X
k
x
n
k
0,...,
N
−
1
n
=
0
N
−
1
1
N
W
−nk
N
synthesis formula:
x
[
n
]=
X
[
k
]
,
n
=
0,...,
N
−
1
k
=
0
DFT
←→
symmetries:
x
[
−
n
mod
N
]
X
[
−
k
mod
N
]
DFT
←→
x
∗
[
X
∗
[
−
n
]
k
mod
N
]
x
(
mod
N
DFT
←→
W
kn
N
X
shifts:
n
−
n
0
)
[
k
]
X
(
mod
N
DFT
←→
W
−nk
0
N
x
[
n
]
k
−
k
0
)
N
−
1
x
]
N
−
1
X
]
1
N
2
2
Parseval:
[
n
=
[
k
n
=
0
k
=
0
Some DFT pairs for length-
N
signals
(
n
,
k
=
0,1,...,
N
−
1)
e
−j
2
N
k
x
[
n
]=
δ
[
n
−
k
]
X
[
k
]=
x
[
n
]=
1
X
[
k
]=
N
δ
[
k
]
e
j
2
N
L
x
[
n
]=
X
[
k
]=
N
δ
[
k
−
L
]
cos
2
2
e
j
φ
δ
[
)]
N
Ln
N
e
−j
φ
δ
[
x
[
n
]=
+
φ
X
[
k
]=
k
−
L
]+
k
−
N
+
L
sin
2
e
j
φ
δ
[
]
N
Ln
]=
−
jN
2
e
−j
φ
δ
[
x
[
n
]=
+
φ
X
[
k
k
−
L
]
−
k
−
N
+
L
1for
n
sin
(
π/
Mk
≤
M
−
1
N
)
e
−j
N
(
M−
1
)
k
x
[
n
]=
X
[
k
]=
sin
(
π/
k
0for
M
≤
n
≤
N
−
1
N
)
