Discrete-Time Fourier Transform Properties with Proofs
| Property Name | Property | Proof |
|---|---|---|
| Periodicity | $ \chi(\omega + 2\pi) = \chi(\omega) $ | Example |
| Linearity | $ ax_{1}[n] + bx_{2}[n] → a\chi_{1}(\omega) + b\chi_{2}(\omega) $ | Example |
| Time Shifting & Frequency Shifting | 1) x[n - no] → e-jωnoX(ω) 2) e-jωonx[n] → X[ω - ωo] |
Example |
| Conjugate & Conjugate Symmetry | x[n] → X*(-ω) | Example |
| Parversal Relation | $ \sum_{n=-\infty}^{\infty }\left | x[n] \right |^{2} = \frac{1}{2\pi }\int_{0}^{2\pi}\left | \chi (\omega) \right |^{2}d\omega $ | Example |
| Convolution | $ x[n]*y[n] \rightarrow \chi(\omega)\gamma (\omega) $ | Example |
| Multiplication | Example | Example |
| Duality | Example | Example |
| Differentiation in Frequency | Example | Example |
