Circuit Theory/Transform Tables

Laplace Transform Appendix

The Laplace Transform is defined as such:

$F(s)=ℒ\{f(t)\}=\underset{0^{-}}{\overset{\mathrm{\infty }}{\int }}{e}^{-st}f(t)dt.$

The result of the transform of a time-domain function f(t) is F(s).

Laplace Transform Table

Time Domain	Laplace Domain
$x(t)=\frac{1}{2\pi j}\underset{\sigma -j\mathrm{\infty }}{\overset{\sigma +j\mathrm{\infty }}{\int }}X(s)e^{st}ds$	$X(s)=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}x(t)e^{-st}dt$
$\delta (t)$	$1$
$\delta (t-a)$	$e^{-as}$
$u(t)$	$\frac{1}{s}$
$u(t-a)$	$\frac{e^{-as}}{s}$
$tu(t)$	$\frac{1}{s^2}$
$t^{n}u(t)$	$\frac{n!}{s^{n+1}}$
$\frac{1}{\sqrt{\pi t}}u(t)$	$\frac{1}{\sqrt{s}}$
$e^{at}u(t)$	$\frac{1}{s-a}$
$t^n{e}^{at}u(t)$	$\frac{n!}{(s-a{)}^{n+1}}$
$\cos (\omega t)u(t)$	$\frac{s}{s^2+{\omega }^2}$
$\sin (\omega t)u(t)$	$\frac{\omega }{s^2+{\omega }^2}$
$\cosh (\omega t)u(t)$	$\frac{s}{s^2-{\omega }^2}$
$\sinh (\omega t)u(t)$	$\frac{\omega }{s^2-{\omega }^2}$
$e^{at}\cos (\omega t)u(t)$	$\frac{s-a}{(s-a{)}^2+{\omega }^2}$
$e^{at}\sin (\omega t)u(t)$	$\frac{\omega }{(s-a{)}^2+{\omega }^2}$
$\frac{1}{2{\omega }^3}(\sin \omega t-\omega t\cos \omega t)$	$\frac{1}{(s^2+{\omega }^2{)}^2}$
$\frac{t}{2\omega }\sin \omega t$	$\frac{s}{(s^2+{\omega }^2{)}^2}$
$\frac{1}{2\omega }(\sin \omega t+\omega t\cos \omega t)$	$\frac{s^2}{(s^2+{\omega }^2{)}^2}$

Laplace Transform Properties

Property	Definition
Linearity	$ℒ\{af(t)+bg(t)\}=aF(s)+bG(s)$
Differentiation	$ℒ\{f^{\prime }\}=sℒ\{f\}-f(0^{-})$ $ℒ\{f^{″}\}=s^2ℒ\{f\}-sf(0^{-})-f^{\prime }(0^{-})$ $ℒ\left\{f^{(n)}\right\}=s^nℒ\{f\}-s^{n-1}f(0^{-})-\cdots -f^{(n-1)}(0^{-})$
Frequency Division	$ℒ\{tf(t)\}=-F^{\prime }(s)$ $ℒ\{t^{n}f(t)\}=(-1{)}^n{F}^{(n)}(s)$
Frequency Integration	$ℒ\left\{\frac{f(t)}{t}\right\}=\underset{s}{\overset{\mathrm{\infty }}{\int }}F(\sigma )d\sigma$
Time Integration	$ℒ\{\underset{0}{\overset{t}{\int }}f(\tau )d\tau \}=ℒ\{u(t)*f(t)\}=\frac{1}{s}F(s)$
Scaling	$ℒ\{f(at)\}=\frac{1}{a}F\left(\frac{s}{a}\right)$
Initial value theorem	$f(0^{+})=\underset{s\to \mathrm{\infty }}{\lim }sF(s)$
Final value theorem	$f(\mathrm{\infty })=\underset{s\to 0}{\lim }sF(s)$
Frequency Shifts	$ℒ\{e^{at}f(t)\}=F(s-a)$ ${ℒ}^{-1}\{F(s-a)\}=e^{at}f(t)$
Time Shifts	$ℒ\{f(t-a)u(t-a)\}=e^{-as}F(s)$ ${ℒ}^{-1}\{e^{-as}F(s)\}=f(t-a)u(t-a)$
Convolution Theorem	$ℒ\{f(t)*g(t)\}=F(s)G(s)$

Where:

$f(t)={ℒ}^{-1}\{F(s)\}$

$g(t)={ℒ}^{-1}\{G(s)\}$

$s=\sigma +j\omega$

Fourier Transform Appendix

The Fourier Transform is defined as such:

$F(j\omega )=ℱ\{f(t)\}=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}f(t)e^{-j\omega t}dt$

The result of the transform of a time-domain function f(t) is F(jω).

Fourier Transform Table

Time Domain	Fourier Domain
$x(t)=\frac{1}{2\pi }\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}X(j\omega )e^{j\omega t}d\omega$	$X(j\omega )=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}x(t)e^{-j\omega t}dt$
$1$	$2\pi \delta (\omega )$
$-0.5+u(t)$	$\frac{1}{j\omega }$
$\delta (t)$	$1$
$\delta (t-c)$	$e^{-j\omega c}$
$u(t)$	$\pi \delta (\omega )+\frac{1}{j\omega }$
$e^{-bt}u(t)$	$\frac{1}{j\omega +b}$
$\cos {\omega }_{0}t$	$\pi [\delta (\omega +{\omega }_0)+\delta (\omega -{\omega }_0)]$
$\cos ({\omega }_{0}t+\theta )$	$\pi [e^{-j\theta }\delta (\omega +{\omega }_0)+e^{j\theta }\delta (\omega -{\omega }_0)]$
$\sin {\omega }_{0}t$	$j\pi [\delta (\omega +{\omega }_0)-\delta (\omega -{\omega }_0)]$
$\sin ({\omega }_{0}t+\theta )$	$j\pi [e^{-j\theta }\delta (\omega +{\omega }_0)-e^{j\theta }\delta (\omega -{\omega }_0)]$
$rect(\frac{t}{\tau })$	$\tau sinc\frac{\tau \omega }{2\pi }$
$\tau sinc\frac{\tau t}{2\pi }$	$2\pi p_{\tau }(\omega )$
$(1-\frac{2|t|}{\tau })p_{\tau }(t)$	$\frac{\tau }{2}sin{c}^2\frac{\tau \omega }{4\pi }$
$\frac{\tau }{2}sin{c}^2(\frac{\tau t}{4\pi })$	$2\pi (1-\frac{2|\omega |}{\tau })p_{\tau }(\omega )$

Note: $sinc(x)=\sin (x)/x$ ; $p_{\tau }(t)$ is the rectangular pulse function of width $\tau$