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Engineering Tables/DFT Transform Table
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Engineering Tables
Time-Domain
x[n]
Frequency Domain
X[k]
Notes
x
n
≡
1
N
∑
k
=
0
N
−
1
X
k
⋅
e
i
2
π
k
n
/
N
{\displaystyle x_{n}\equiv {\frac {1}{N}}\sum _{k=0}^{N-1}X_{k}\cdot e^{i2\pi kn/N}}
X
k
≡
∑
n
=
0
N
−
1
x
n
⋅
e
−
i
2
π
k
n
/
N
{\displaystyle X_{k}\equiv \sum _{n=0}^{N-1}x_{n}\cdot e^{-i2\pi kn/N}}
DFT Definition
x
n
⋅
e
i
2
π
k
n
/
N
{\displaystyle x_{n}\cdot e^{i2\pi kn/N}\,}
X
n
−
k
{\displaystyle X_{n-k}\,}
Shift theorem
x
n
−
k
{\displaystyle x_{n-k}\,}
X
k
⋅
e
−
i
2
π
k
n
/
N
{\displaystyle X_{k}\cdot e^{-i2\pi kn/N}}
x
n
∈
R
{\displaystyle x_{n}\in \mathbf {R} }
X
k
=
X
N
−
k
∗
{\displaystyle X_{k}=X_{N-k}^{*}\,}
Real DFT
a
n
{\displaystyle a^{n}\,}
1
−
a
N
1
−
a
⋅
e
−
i
2
π
k
/
N
{\displaystyle {\frac {1-a^{N}}{1-a\cdot e^{-i2\pi k/N}}}}
(
N
−
1
n
)
{\displaystyle {N-1 \choose n}\,}
(
1
+
e
−
i
2
π
k
/
N
)
N
−
1
{\displaystyle \left(1+e^{-i2\pi k/N}\right)^{N-1}\,}