8.3: Common Fourier Transforms
- Page ID
- 22888
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Common CTFT Properties
| Time Domain Signal | Frequency Domain Signal | Condition |
|---|---|---|
| \(e^{-(a t)} u(t)\) | \(\frac{1}{a+j \omega}\) | \(a>0\) |
| \(e^{at}u(−t)\) | \(\frac{1}{a-j \omega}\) | \(a>0\) |
| \(e^{−(a|t|)}\) | \(\frac{2a}{a^2+\omega^2}\) | \(a>0\) |
| \(te^{−(at)}u(t)\) | \(\frac{1}{(a+j \omega)^2}\) | \(a>0\) |
| \(t^ne^{−(at)}u(t)\) | \(\frac{n !}{(a+j \omega)^{n+1}}\) | \(a>0\) |
| \(\delta(t)\) | \(1\) | |
| \(1\) | \(2 \pi \delta(\omega)\) | |
| \(e^{j \omega_0 t}\) | \(2 \pi \delta\left(\omega-\omega_{0}\right)\) | |
| \( \cos (\omega_0 t) \) | \(\pi\left(\delta\left(\omega-\omega_{0}\right)+\delta\left(\omega+\omega_{0}\right)\right)\) | |
| \(\sin (\omega_0 t)\) | \(j \pi\left(\delta\left(\omega+\omega_{0}\right)-\delta\left(\omega-\omega_{0}\right)\right)\) | |
| \(u(t)\) | \(\pi \delta(\omega)+\frac{1}{j \omega}\) | |
| sgn (\(t)\) | \(\frac{2}{j \omega}\) | |
| \(\cos \left(\omega_{0} t\right) u(t)\) | \(\frac{\pi}{2}\left(\delta\left(\omega-\omega_{0}\right)+\delta\left(\omega+\omega_{0}\right)\right)+\frac{j \omega}{\omega_{0}^{2}-\omega^{2}}\) | |
| \(\sin \left(\omega_{0} t\right) u(t)\) | \(\frac{\pi}{2 j}\left(\delta\left(\omega-\omega_{0}\right)-\delta\left(\omega+\omega_{0}\right)\right)+\frac{\omega_{0}}{\omega_{0}^{2}-\omega^{2}}\) | |
| \(e^{-(a t)} \sin \left(\omega_{0} t\right) u(t)\) | \(\frac{\omega_{0}}{(a+j \omega)^{2}+\omega_{0}^{2}}\) | \(a>0\) |
| \(e^{-(a t)} \cos \left(\omega_{0} t\right) u(t)\) | \(\frac{a+j \omega}{(a+j \omega)^{2}+\omega_{0}^{2}}\) | \(a>0\) |
| \(u(t+\tau)-u(t-\tau)\) | \(2 \tau \frac{\sin (\omega \tau)}{\omega \tau}=2 \tau \operatorname{sinc}(\omega t)\) | |
| \(\frac{\omega_{0}}{\pi} \frac{\sin \left(\omega_{0} t\right)}{\omega_{0} t}=\frac{\omega_{0}}{\pi} \operatorname{sinc}\left(\omega_{0}\right)\) | \(u\left(\omega+\omega_{0}\right)-u\left(\omega-\omega_{0}\right)\) | |
| \(\begin{array}{l} \left(\frac{t}{\tau}+1\right)\left(u\left(\frac{t}{\tau}+1\right)-u\left(\frac{t}{\tau}\right)\right) \\ \left(-\frac{t}{\tau}+1\right)\left(u\left(\frac{t}{\tau}\right)-u\left(\frac{t}{\tau}-1\right)\right)= \\ \operatorname{triag}\left(\frac{t}{2 \tau}\right) \end{array}\) |
\(\tau \operatorname{sinc}^{2}\left(\frac{\omega \tau}{2}\right)\) | |
| \(\frac{\omega_{0}}{2 \pi} \operatorname{sinc}^{2}\left(\frac{\omega_{0} t}{2}\right)\) | \(\begin{array}{l} \left(\frac{\omega}{\omega_{0}}+1\right)\left(u\left(\frac{\omega}{\omega_{0}}+1\right)-u\left(\frac{\omega}{\omega_{0}}\right)\right) + \\ \left(-\frac{\omega}{\omega_{0}}+1\right)\left(u\left(\frac{\omega}{\omega_{0}}\right)-u\left(\frac{\omega}{\omega_{0}}-1\right)\right)= \\ \operatorname{triag}\left(\frac{\omega}{2 \omega_{0}}\right) \end{array}\) |
|
| \(\sum_{n=-\infty}^{\infty} \delta(t-n T)\) | \(\omega_{0} \sum_{n=-\infty}^{\infty} \delta\left(\omega-n \omega_{0}\right)\) | \(\omega_0 = \frac{2 \pi}{T}\) |
| \(e^{-\frac{t^{2}}{2 \sigma^{2}}}\) | \(\sigma \sqrt{2 \pi} e^{-\frac{\sigma^{2} \omega^{2}}{2}}\) |
triag[n] is the triangle function for arbitrary real-valued \(n\).
\[ \operatorname{triag}[\mathrm{n}]=\left\{\begin{array}{ll}
1+n & \text { if }-1 \leq n \leq 0 \\
1-n & \text { if } 0<n \leq 1 \\
0 & \text { otherwise }
\end{array}\right. \nonumber \]