Engineering Tables/DFT Transform Table

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Time-Domain
x[n]
Frequency Domain
X[k]
Notes
x_n\equiv\frac{1}{N}\sum_{k=0}^{N-1}X_k \cdot e^{i 2 \pi kn/N} X_k\equiv\sum_{n=0}^{N-1}x_n \cdot e^{-i 2 \pi kn/N} DFT Definition
x_n \cdot e^{i 2 \pi kn/N} \, X_{n-k}\, Shift theorem
x_{n-k}\, X_k \cdot e^{-i 2 \pi kn/N}
x_n \in \mathbf{R} X_k=X_{N-k}^*\, Real DFT
a^n\, \frac{1-a^N}{1-a \cdot e^{-i 2 \pi k/N} }  
{N-1 \choose n}\, \left(1+e^{-i 2 \pi k/N} \right)^{N-1}\,