Circuit Theory/Transform Tables

Laplace Transform Appendix

The Laplace Transform is defined as such:

$F(s) = \mathcal{L} \left\{f(t)\right\} =\int_{0^-}^\infty e^{-st} f(t)\,dt.$

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

Laplace Transform Table

	Time Domain	Laplace Domain
	$x(t) = \mathcal{L}^{-1}\left\{ X(s) \right\}$	$X(s) = \mathcal{L} \left\{ x(t) \right\}$
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	$t u(t) \,$	$\frac{1}{s^2}$
7	$t^nu(t) \,$	$\frac{n!}{s^{n+1}}$
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	$\frac{1}{2\omega}(\sin \omega t+\omega t \cos \omega t)$	$\frac{s^2}{(s^2+\omega^2)^2}$

Laplace Transform Properties

Property	Definition
Linearity	$\mathcal{L}\left\{a f(t) + b g(t) \right\} = a F(s) + b G(s)$
Differentiation	$\mathcal{L}\{f'\} = s \mathcal{L}\{f\} - f(0^-)$ $\mathcal{L}\{f''\} = s^2 \mathcal{L}\{f\} - s f(0^-) - f'(0^-)$ $\mathcal{L}\left\{ f^{(n)} \right\} = s^n \mathcal{L}\{f\} - s^{n - 1} f(0^-) - \cdots - f^{(n - 1)}(0^-)$
Frequency Division	$\mathcal{L}\{ t f(t)\} = -F'(s)$ $\mathcal{L}\{ t^{n} f(t)\} = (-1)^{n} F^{(n)}(s)$
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 = σ + jω

Fourier Transform Appendix

The Fourier Transform is defined as such:

$F(j\omega) = \mathcal{F} \left\{f(t) \right\} = \int_{-\infty}^\infty f(t) e^{-j\omega t}dt$

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

Fourier Transform Table

	Time Domain	Frequency Domain
	$x(t) = \mathcal{F}^{-1}\left\{ X(\omega) \right\}$	$X(\omega) = \mathcal{F} \left\{ x(t) \right\}$
1	$X(j \omega)=\int_{-\infty}^\infty x(t) e^{-j \omega t}d t$	$x(t)=\frac{1}{2 \pi} \int_{-\infty}^{\infty} X(j \omega)e^{j \omega t}d \omega$
2	$1 \,$	$2\pi\delta(\omega) \,$
3	$-0.5 + u(t) \,$	$\frac{1}{j \omega} \,$
4	$\delta (t) \,$	$1 \,$
5	$\delta (t-c) \,$	$e^{-j \omega c} \,$
6	$u(t) \,$	$\pi \delta(\omega)+\frac{1}{j \omega} \,$
7	$e^{-bt}u(t) \, (b > 0)$	$\frac{1}{j \omega+b} \,$
8	$\cos \omega_0 t \,$	$\pi \left[ \delta(\omega+\omega_0)+\delta(\omega-\omega_0) \right] \,$
9	$\cos (\omega_0 t + \theta) \,$	$\pi \left[ e^{-j \theta}\delta(\omega+\omega_0)+e^{j \theta}\delta(\omega-\omega_0) \right] \,$
10	$\sin \omega_0 t \,$	$j \pi \left[ \delta(\omega +\omega_0)-\delta(\omega-\omega_0) \right] \,$
11	$\sin (\omega_0 t + \theta) \,$	$j \pi \left[ e^{-j \theta}\delta(\omega +\omega_0)-e^{j \theta}\delta(\omega-\omega_0) \right] \,$
12	$\mbox{rect} \left( \frac{t}{\tau} \right) \,$	$\tau \mbox{sinc} \left( \frac{\tau \omega}{2 \pi} \right) \,$
13	$\tau \mbox{sinc} \left( \frac{\tau t}{2 \pi} \right) \,$	$2 \pi p_\tau(\omega)\,$
14	$\left( 1-\frac{2 |t|}{\tau} \right) p_\tau (t) \,$	$\frac{\tau}{2} \mbox{sinc}^2 \left( \frac{\tau \omega}{4 \pi} \right) \,$
15	$\frac{\tau}{2} \mbox{sinc}^2 \left( \frac{\tau t}{4 \pi} \right) \,$	$2 \pi \left( 1-\frac{2|\omega|}{\tau} \right) p_\tau (\omega) \,$
16	$e^{-a|t|}, \Re\{a\}>0 \,$	$\frac{2a}{a^2 + \omega^2} \,$
Notes:	sinc(x) = sin(x) / x pτ(t) is the rectangular pulse function of width τ u(t) is the Heavyside step function δ(t) is the Dirac delta function