# Electronics Communication/Signal Processing/Signal Transformation/Laplace Transform
## Laplace Transform

Laplace Transform is a process to transfor function in Frequency domain to time domain
    $F(s)={\mathcal {L}}\left\{f(t)\right\}=\int _{0^{-}}^{\infty }e^{-st}f(t)\,dt.$

## Inverse Laplace Transform

    ${\mathcal {L}}^{-1}\left\{F(s)\right\}={1 \over {2\pi i}}\int _{c-i\infty }^{c+i\infty }e^{ft}F(s)\,ds=f(t)$

## Laplace Transform Properties

| Property | Definition |
| Linearity | ${\mathcal {L}}\left\{af(t)+bg(t)\right\}=aF(s)+bG(s)$ |
| Differentiation | ${\mathcal {L}}\{f'\}=s{\mathcal {L}}\{f\}-f(0^{-})$ ${\mathcal {L}}\{f''\}=s^{2}{\mathcal {L}}\{f\}-sf(0^{-})-f'(0^{-})$ ${\mathcal {L}}\left\{f^{(n)}\right\}=s^{n}{\mathcal {L}}\{f\}-s^{n-1}f(0^{-})-\cdots -f^{(n-1)}(0^{-})$ |
| Frequency Division | ${\mathcal {L}}\{tf(t)\}=-F'(s)$ ${\mathcal {L}}\{t^{n}f(t)\}=(-1)^{n}F^{(n)}(s)$ |
| Frequency Integration | ${\mathcal {L}}\left\{{\frac {f(t)}{t}}\right\}=\int _{s}^{\infty }F(\sigma )\,d\sigma$ |
| Time Integration | ${\mathcal {L}}\left\{\int _{0}^{t}f(\tau )\,d\tau \right\}={\mathcal {L}}\left\{u(t)*f(t)\right\}={1 \over s}F(s)$ |
| Scaling | ${\mathcal {L}}\left\{f(at)\right\}={1 \over a}F\left({s \over a}\right)$ |
| Initial value theorem | $f(0^{+})=\lim _{s\to \infty }{sF(s)}$ |
| Final value theorem | $f(\infty )=\lim _{s\to 0}{sF(s)}$ |
| Frequency Shifts | ${\mathcal {L}}\left\{e^{at}f(t)\right\}=F(s-a)$ ${\mathcal {L}}^{-1}\left\{F(s-a)\right\}=e^{at}f(t)$ |
| Time Shifts | ${\mathcal {L}}\left\{f(t-a)u(t-a)\right\}=e^{-as}F(s)$ ${\mathcal {L}}^{-1}\left\{e^{-as}F(s)\right\}=f(t-a)u(t-a)$ |
| Convolution Theorem | ${\mathcal {L}}\{f(t)*g(t)\}=F(s)G(s)$ |
Where:
    $f(t)={\mathcal {L}}^{-1}\{F(s)\}$
    $g(t)={\mathcal {L}}^{-1}\{G(s)\}$
    $s=\sigma +j\omega$

## Table of Laplace Transforms

| No. | Time Domain $x(t)={\mathcal {L}}^{-1}\left\{X(s)\right\}$ | Laplace Domain $X(s)={\mathcal {L}}\left\{x(t)\right\}$ |
| 1 | ${\frac {1}{2\pi j}}\int _{\sigma -j\infty }^{\sigma +j\infty }X(s)e^{st}ds$ | $\int _{-\infty }^{\infty }x(t)e^{-st}dt$ |
| 2 | $\delta (t)\,$ | $1\,$ |
| 3 | $\delta (t-a)\,$ | $e^{-as}\,$ |
| 4 | $u(t)\,$ | ${\frac {1}{s}}$ |
| 5 | $u(t-a)\,$ | ${\frac {e^{-as}}{s}}$ |
| 6 | $tu(t)\,$ | ${\frac {1}{s^{2}}}$ |
| 7 | $t^{n}u(t)\,$ | ${\frac {n!}{s^{n+1}}}$ |
| 8 | ${\frac {1}{\sqrt {\pi t}}}u(t)$ | ${\frac {1}{\sqrt {s}}}$ |
| 9 | $e^{at}u(t)\,$ | ${\frac {1}{s-a}}$ |
| 10 | $t^{n}e^{at}u(t)\,$ | ${\frac {n!}{(s-a)^{n+1}}}$ |
| 11 | $\cos(\omega t)u(t)\,$ | ${\frac {s}{s^{2}+\omega ^{2}}}$ |
| 12 | $\sin(\omega t)u(t)\,$ | ${\frac {\omega }{s^{2}+\omega ^{2}}}$ |
| 13 | $\cosh(\omega t)u(t)\,$ | ${\frac {s}{s^{2}-\omega ^{2}}}$ |
| 14 | $\sinh(\omega t)u(t)\,$ | ${\frac {\omega }{s^{2}-\omega ^{2}}}$ |
| 15 | $e^{at}\cos(\omega t)u(t)\,$ | ${\frac {s-a}{(s-a)^{2}+\omega ^{2}}}$ |
| 16 | $e^{at}\sin(\omega t)u(t)\,$ | ${\frac {\omega }{(s-a)^{2}+\omega ^{2}}}$ |
| 17 | ${\frac {1}{2\omega ^{3}}}(\sin \omega t-\omega t\cos \omega t)$ | ${\frac {1}{(s^{2}+\omega ^{2})^{2}}}$ |
| 18 | ${\frac {t}{2\omega }}\sin \omega t$ | ${\frac {s}{(s^{2}+\omega ^{2})^{2}}}$ |
| 19 | ${\frac {1}{2\omega }}(\sin \omega t+\omega t\cos \omega t)$ | ${\frac {s^{2}}{(s^{2}+\omega ^{2})^{2}}}$ |