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	${\frac {1}{2\omega }}(\sin \omega t+\omega t\cos \omega t)$	${\frac {s^{2}}{(s^{2}+\omega ^{2})^{2}}}$