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Signals and Systems – Table of Fourier Transform Pairs
Fourier Transform
Fourier transform is a transformation technique that transforms signals from the continuous-time domain to the corresponding frequency domain and vice-versa.
The Fourier transform of a continuous-time function $x(t)$ is defined as,
$$\mathrm{X(\omega)=\int_{-\infty}^{\infty} x(t)e^{-j\omega t}dt… (1)}$$
Inverse Fourier Transform
The inverse Fourier transform of a continuous-time function is defined as,
$$\mathrm{x(t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}X(\omega)\:e^{j\omega t}d\omega… (2)}$$
Equations (1) and (2) for $X(\omega)$ and $x(t)$ are known as Fourier transform pair and can be represented as −
$$\mathrm{X(\omega)=F[x(t)]}$$
And
$$\mathrm{x(t)=F^{-1}[X(\omega)]}$$
Table of Fourier Transform Pairs
| Function,x(t) | Fourier Transform, X(ω) |
|---|---|
| $\delta(t)$ | 1 |
| $\delta(t-t_{0})$ | $e^{-j \omega t_{0}}$ |
| 1 | $2\pi \delta(\omega)$ |
| u(t) | $\pi\delta(\omega)+\frac{1}{j\omega}$ |
| $\sum_{n=−\infty}^{\infty}\delta(t-nT)$ | $\omega_{0}\sum_{n=−\infty}^{\infty}\delta(\omega-n\omega_{0});\:\:\left(\omega_{0}=\frac{2\pi}{T} \right)$ |
| sgn(t) | $\frac{2}{j\omega}$ |
| $ e^{j\omega_{0}t}$ | $ 2\pi\delta(\omega-\omega_{0})$ |
| $ cos\:\omega_{0}t$ | $\pi[\delta(\omega-\omega_{0})+\delta(\omega+\omega_{0})]$ |
| $sin\:\omega_{0}t$ | $-j\pi[\delta(\omega-\omega_{0})-\delta(\omega+\omega_{0})]$ |
| $e^{-at}u(t);\:\:\:a >0$ | $\frac{1}{a+j\omega}$ |
| $t\:e^{at}u(t);\:\:\:a >0$ | $\frac{1}{(a+j\omega)^{2}}$ |
| $e^{-|at|};\:\:a >0$ | $\frac{2a}{a^{2}+\omega^{2}}$ |
| $e^{-|t|}$ | $\frac{2}{1+\omega^{2}}$ |
| $\frac{1}{\pi t}$ | $-j\:sgn(\omega)$ |
| $\frac{1}{a^{2}+t^{2}}$ | $\frac{\pi}{a}e^{-a|\omega|}$ |
| $\Pi (\frac{t}{τ})$ | $τ\:sin c(\frac{\omega τ}{2})$ |
| $\Delta(\frac{t}{τ})$ | $\frac{τ}{2}sin C^{2}(\frac{\omega τ}{4})$ |
| $\frac{sin\:at}{\pi t}$ | $P_{a}(\omega)=\begin{cases}1 & for\:|\omega|\:< a\0 & for\:|\omega|\: > a \end{cases}$ |
| $cos\:\omega_{0}t\:u(t)$ | $\frac{\pi}{2}[\delta(\omega-\omega_{0})+\delta(\omega+\omega_{0})]+\left [ \frac{j\omega}{(j\omega)^{2}+\omega_{0}^{2}} \right ]$ |
| $sin\:\omega_{0}t\:u(t)$ | $-j\frac{\pi}{2}[\delta(\omega-\omega_{0})-\delta(\omega+\omega_{0})]+\left [\frac{\omega_{0}}{(j\omega)^{2}+\omega_{0}^{2}} \right ]$ |
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