Fourier Transform Catalog
#Math
| Time Domain | Frequency Domain | Note |
|---|---|---|
| $\displaystyle \text{rect}(t /T)$ | $\displaystyle T \text{ sinc}(\omega T /2\pi)$ | |
| $\displaystyle e^{-at}u(t)$ | $\displaystyle \frac{1}{a+j\omega}$ | |
| $\displaystyle e^{a\lvert t\rvert}$ | $\displaystyle \frac{2a}{a^{2}+\omega ^{2}}$ | |
| $\displaystyle \text{sinc}(t /2\pi)$ | $\displaystyle 2\pi ,\text{rect}(\omega)$ | |
| $\displaystyle \Delta(t)$ | $\displaystyle T^{2}\text{ sinc}^{2}(\omega T /2\pi)$ | |
| $\displaystyle \text{sinc}^{2}(t)$ | $\displaystyle \Delta(\omega /2\pi)$ | |
| $\displaystyle \delta(t)$ | $\displaystyle 1$ | |
| $\displaystyle \delta(t-\tau)$ | $\displaystyle e^{-j\omega \tau}$ | |
| $\displaystyle 1$ | $\displaystyle 2\pi \delta(\omega)$ | |
| $\displaystyle u(t)$ | $\displaystyle \pi \delta(\omega)+\frac{1}{j\omega}$ | |
| $\displaystyle e^{j{\omega}_{0}t}$ | $\displaystyle 2\pi \delta(\omega-{\omega}_{0})$ | |
| $\displaystyle \cos({\omega}_{0}t)$ | $\displaystyle \pi(\delta(\omega-{\omega}{0})+\delta(\omega+{\omega}{0}))$ | |
| $\displaystyle \sin({\omega}_{0}t)$ | $\displaystyle j\pi(\delta(\omega+{\omega}{0})-\delta(\omega-{\omega}{0}))$ | |
| $\displaystyle \delta_{T}(t)$ | $\displaystyle {\omega}_{0}\delta {\omega{0}}(\omega)$ | $\displaystyle \delta_{T}(t)$ is the impulse train |
| $\displaystyle \text{square wave}$ | $\displaystyle T\text{ sinc}(\omega T /2\pi) \delta_{T}(t)$ |