CT Fourier Transform Pairs and Properties
Using $ \omega $ in radians to parametrize the Fourier transforms.
| CT Fourier transform and its Inverse | |
|---|---|
| CT Fourier Transform | $ \mathcal{X}(\omega)=\mathcal{F}(x(t))=\int_{-\infty}^{\infty} x(t) e^{-j\omega t} dt $ |
| Inverse DT Fourier Transform | $ \, x(t)=\mathcal{F}^{-1}(\mathcal{X}(\omega))=\frac{1}{2\pi} \int_{-\infty}^{\infty}\mathcal{X}(\omega)e^{j\omega t} d \omega\, $ |
| CT Fourier Transform Pairs | |||
|---|---|---|---|
| $ x(t) $ | $ \longrightarrow $ | $ \mathcal{X}(\omega) $ | |
| CTFT of a complex exponential | $ e^{jw_0t} $ | ||
| CT Fourier Transform Properties | |||
|---|---|---|---|
| $ x(t) $ | $ \longrightarrow $ | $ \mathcal{X}(\omega) $ | |
| multiplication property | $ x(t)y(t) \ $ | $ \frac{1}{2\pi} X(\omega)*Y(\omega) =\frac{1}{2\pi} \int_{-\infty}^{\infty} X(\theta)Y(\omega-\theta)d\theta $ | |
| convolution property | $ x(t)*y(t) \! $ | $ X(\omega)Y(\omega) \! $ | |
| time reversal | $ \ x(-t) $ | $ \ X(-\omega) $ | |
| Other CT Fourier Transform Properties | |
|---|---|
| Parseval's relation | |
