Difference between revisions of "Table of Formulas and Properties ECE301Fall2008mboutin" - Rhea

Latest revision as of 06:20, 5 February 2009

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General Purpose Formulas
Series
Finite Geometric Series Formula_ECE301Fall2008mboutin	$\sum_{k=0}^n x^k = \left\{ \begin{array}{ll} \frac{1-x^{n+1}}{1-x}&, \text{ if } x\neq 1\\ n+1 &, \text{ else}\end{array}\right.$
Infinite Geometric Series Formula_ECE301Fall2008mboutin	$\sum_{k=0}^\infty x^k = \left\{ \begin{array}{ll} \frac{1}{1-x}&, \text{ if } \|x\|\leq 1\\ \text{diverges} &, \text{ else }\end{array}\right.$
Euler's Formula
Complex exponential in terms of sinusoidal signals_ECE301Fall2008mboutin	$e^{jw_0t}=cosw_0t+jsinw_0t$
Cosine function in terms of complex exponential_ECE301Fall2008mboutin	$cos\theta=\frac{e^{j\theta}+e^{-j\theta}}{2}$
Sine function in terms of complex exponential_ECE301Fall2008mboutin	$sin\theta=\frac{e^{j\theta}-e^{-j\theta}}{2j}$
Other
sinc function_ECE301Fall2008mboutin	$sinc(\theta)=\frac{sin(\pi\theta)}{\pi\theta}$
Discrete-Time Domain
Useful Formulas
DT Fourier Transform_ECE301Fall2008mboutin	$\,\mathcal{X}(\omega)=\mathcal{F}(x[n])=\sum_{n=-\infty}^{\infty}x[n]e^{-j\omega n}\,$
DT Inverse Fourier Transform_ECE301Fall2008mboutin	$\,x[n]=\mathcal{F}^{-1}(\mathcal{X}(\omega))=\frac{1}{2\pi} \int_{0}^{2\pi}\mathcal{X}(\omega)e^{j\omega n} d \omega\,$
DT Fourier Transform Pairs
DT Fourier Transform Pair_ECE301Fall2008mboutin	$e^{jw_0n} \longrightarrow 2\pi\sum_{l=-\infty}^{+\infty}\delta(w-w_0-2\pi l) \$
DT Fourier an_ECE301Fall2008mboutin	$a^{n} u[n], \|a\|<1 \longrightarrow \frac{1}{1-ae^{-j\omega}} \$

DT Fourier Transform Pair 3_ECE301Fall2008mboutin $sin[\omega _0 n]u[n]\rightarrow \frac{1}{2j}[\frac{1}{1-e^{-j(\omega -\omega _0)}}-\frac{1}{1-e^{-j(\omega +\omega _0)}}]$

DT Fourier Transform Multiplication_ECE301Fall2008mboutin $x[n]y[n]\longleftrightarrow \frac{1}{2\pi} \int_{2\pi} X(e^{j\theta})Y(e^{j(\omega-\theta)})d\theta$
DT Fourier Transform Convolution_ECE301Fall2008mboutin $x[n]*y[n] = X(e^{jw})Y(e^{jw}) \!$
DT Fourier Transform Time Reversal_ECE301Fall2008mboutin $\ x[-n] \longleftrightarrow X(e^{-j \omega})$
DT Fourier Transform Duality_ECE301Fall2008mboutin $F(x(t)) = X(w) \longleftrightarrow F(X(t)) = 2\pi x(-w)$

Parsevel Relationship for DT signals

Parsevel Relationship for DT signals_ECE301Fall2008mboutin $\frac {1}{N} \sum_{n=<N>}^{}|x[n]|^2 = \sum_{k=<N>}{}|a_k|^2$

Continuous-time domain

Useful Formulas

CT Fourier Transform_ECE301Fall2008mboutin $\ \mathcal{X}(\omega)=\mathcal{F}(x(t))=\int_{-\infty}^{\infty}x(t)e^{-j\omega t} dt$
CT Inverse Fourier Transform_ECE301Fall2008mboutin $\ x(t) =\mathcal{F}^{-1}(\mathcal{X}(\omega))=\frac{1}{2 \pi}\int_{-\infty}^{\infty}\mathcal{X}(\omega)e^{j\omega t} d \omega$
CT Laplace Transform_ECE301Fall2008mboutin $\mathcal{L}(x(t)) = X(s) = \int_{-\infty}^\infty x(t) e^{-st} dt$
CT_Fourrier_Coefficients_ECE301Fall2008mboutin $ak = \frac{1}{T} \int_{0}^{T}x(t)e^{-jk} \frac{2\pi}{T} dt$
Time_Shifting_ECE301Fall2008mboutin $F(x(t-t_0)) = e^{-jwt_0}F(x(t))\,$
CT Energy of a Signal_ECE301Fall2008mboutin $E\infty = \int_{-\infty}^{+\infty}|x(t)|^2 dt\!$
CT Time-averaged Power of a Signal over an infinite interval_ECE301Fall2008mboutin $P_\infty = \lim_{T \to \infty} \frac{1}{2T} \int_{-T}^{T} \left | x\ (t) \right |^2 \, dt$

CT Fourier Transform Pairs

(1) $x(t)= \sum^{\infty}_{k=-\infty} a_{k}e^{jkw_{0}t} \longrightarrow {\mathcal X}(\omega)= 2\pi\sum^{\infty}_{k=-\infty}a_{k}\delta(w-kw_{0})\,$
(2) $x(t)=\sum^{\infty}_{n=-\infty} \delta(t-nT) \longrightarrow {\mathcal X}(\omega)= \frac{2\pi}{T}\sum^{\infty}_{k=-\infty}\delta(w-\frac{2\pi k}{T})\,$
(3) $x(t)=\cos(\omega_0 t) \longrightarrow {\mathcal X}(\omega)=\pi \left[\delta (\omega - \omega_0) + \delta (\omega + \omega_0)\right]$
(4) $x(t)=sin(\omega_0 t) \longrightarrow {\mathcal X}(\omega)=\frac{\pi}{j} \left[\delta (\omega - \omega_0) - \delta (\omega + \omega_0)\right]$
(5) $x(t)=\delta (t)\longrightarrow {\mathcal X}(\omega)=1 \!$
(6) $x(t)= u(t)\longrightarrow {\mathcal X}(\omega)= \frac{1}{jw} + \pi \delta (w) \!$
(7) $x(t)=\delta (t-t_0)\longrightarrow {\mathcal X}(\omega)= e^{jwt_0} \!$
(8) $x(t)=e^{-at}u(t),\text{ where }a\text{ is real,}a>0 \longrightarrow {\mathcal X}(\omega)=\frac{1}{a+j\omega}$
(9) $x(t)=e^{j\omega_0 t} \longrightarrow {\mathcal X}(\omega)= 2\pi \delta (\omega - \omega_0)$
(10) $x(t)=te^{-at}u(t), \text{ where }a\text{ is real,} a>0 \longrightarrow {\mathcal X}(\omega)=(\frac{1}{a+j\omega})^2$
(11) $x(t)=\left\{\begin{array}{ll}1, & \text{ if }|t|<T,\\ 0, & \text{else.}\end{array} \right. \longrightarrow {\mathcal X}(\omega)=\frac{2 \sin \left( T \omega \right)}{\omega}$
(12) $x(t)=\frac{2 \sin \left( W t \right)}{\pi t } \longrightarrow \mathcal{X}(\omega)=\left\{\begin{array}{ll}1, & \text{ if }|\omega| <W,\\ 0, & \text{else.}\end{array} \right. \$

CT Fourier Transform Properties

CT_Fourier_Int/Diff_ECE301Fall2008mboutin $\; \; \; (1)\frac{dx(t)}{dt} \rightarrow j\omega \Chi (\omega)\; \; \; \; \; \; (2) \int_{-\infty}^{t}x(\tau)d\tau \rightarrow \frac{1}{j\omega}\Chi (\omega) + \pi \Chi (0) \delta (\omega)$

CT Time and Frequency Scaling_ECE301Fall2008mboutin : $x(at) \leftarrow \rightarrow \frac{1}{|a|}X(\frac{j\omega }{a})\,$

CT Differentiation in Frequency_ECE301Fall2008mboutin $tx(t)\rightarrow j\frac{d}{d\omega}X(j\omega)$

CT Convolution_ECE301Fall2008mboutin : $F(x_1(t)*x_2(t)) = X_1(\omega)X_2(\omega) \!$

CT Frequency Shifting_ECE301Fall2008mboutin : $F(e^{jw0t}x(t)) = X(j(w - w0)) \!$

$F(x(t)y(t))=\frac{1}{2\pi}X(j\omega)*Y(j\omega)=\frac{1}{2\pi}\int_{-\infty}^{\infty}X(j\theta)Y(j(\omega-\theta))d\theta$

CT Time Reversal_ECE301Fall2008mboutin $x(-t) \leftarrow \rightarrow X(-j\omega )\,$

CT Multiplication Property Mimis Version_ECE301Fall2008mboutin $F(x_1(t)x_2(t)) = \frac {1} {2\pi} X_1(\omega)*X_2(\omega)$

CT Duality Property_ECE301Fall2008mboutin : $F(x(t)) = X(w) = 2\pi x(-w) \!$

F(x(t)) = X(w) = 2\pi x(-w) \!

CT Conjugate Symmetry_ECE301Fall2008mboutin ==Conjugate Symmetry==

if

\ F(x(t)) = X(w)

then,

\ F(x(t)^*) = X^*(-w)

Parsevel Relationship for CT signals

Parsevel Relationship for CT signals_ECE301Fall2008mboutin $\frac {1}{T} \int_{0}^{T}|x(t)|^2 dt = \sum_{k=-\infty}^{\infty}|a_k|^2$ ???--Mboutin 10:12, 22 October 2008 (UTC)

Parseval's Relation for Aperiodic Signals_ECE301Fall2008mboutin $\int_{-\infty}^{\infty} |x(t)|^2 dt = \frac{1}{2\pi} \int_{-\infty}^{\infty} |\mathcal{X}(w)|^2 dw$

put a property here following syntax described at top of page.

Some Laplace Transform Pairs

Laplace Transform Pairs
Transform Pair	Signal	Transform	ROC
1	$\,\!\delta(t)$	$$ 1 $$	$All\,\, s$
2	$\,\! u(t)$	$\frac{1}{s}$	$\mathcal{R} \mathfrak{e} \lbrace s \rbrace > 0$
3	$\,\! -u(-t)$	$\frac{1}{s}$	$\mathcal{R} \mathfrak{e} \lbrace s \rbrace < 0$
4	$\frac{t^{n-1}}{(n-1)!}u(t)$	$\frac{1}{s^{n}}$	$\mathcal{R} \mathfrak{e} \lbrace s \rbrace > 0$
5	$-\frac{t^{n-1}}{(n-1)!}u(-t)$	$\frac{1}{s^{n}}$	$\mathcal{R} \mathfrak{e} \lbrace s \rbrace < 0$
6	$\,\!e^{-\alpha t}u(t)$	$\frac{1}{s+\alpha}$	$\mathcal{R} \mathfrak{e} \lbrace s \rbrace > -\alpha$
7	$\,\! -e^{-\alpha t}u(-t)$	$\frac{1}{s+\alpha}$	$\mathcal{R} \mathfrak{e} \lbrace s \rbrace < -\alpha$
8	$\frac{t^{n-1}}{(n-1)!}e^{-\alpha t}u(t)$	$\frac{1}{(s+\alpha )^{n}}$	$\mathcal{R} \mathfrak{e} \lbrace s \rbrace > -\alpha$
9	$-\frac{t^{n-1}}{(n-1)!}e^{-\alpha t}u(-t)$	$\frac{1}{(s+\alpha )^{n}}$	$\mathcal{R} \mathfrak{e} \lbrace s \rbrace < -\alpha$
10	$\,\!\delta (t - T)$	$\,\! e^{-sT}$	$All\,\, s$
11	$\,\![cos( \omega_0 t)]u(t)$	$\frac{s}{s^2+\omega_0^{2}}$	$\mathcal{R} \mathfrak{e} \lbrace s \rbrace > 0$
12	$\,\![sin( \omega_0 t)]u(t)$	$\frac{\omega_0}{s^2+\omega_0^{2}}$	$\mathcal{R} \mathfrak{e} \lbrace s \rbrace > 0$
13	$\,\![e^{-\alpha t}cos( \omega_0 t)]u(t)$	$\frac{s+\alpha}{(s+\alpha)^{2}+\omega_0^{2}}$	$\mathcal{R} \mathfrak{e} \lbrace s \rbrace > -\alpha$
14	$\,\![e^{-\alpha t}sin( \omega_0 t)]u(t)$	$\frac{\omega_0}{(s+\alpha)^{2}+\omega_0^{2}}$	$\mathcal{R} \mathfrak{e} \lbrace s \rbrace > -\alpha$
15	$u_n(t) = \frac{d^{n}\delta (t)}{dt^{n}}$	$\,\!s^{n}$	$All\,\, s$
16	$u_{-n}(t) = \underbrace{u(t) \dots u(t)}_{n\,\,times}$	$\frac{1}{s^{n}}$	$\mathcal{R} \mathfrak{e} \lbrace s \rbrace > 0$

(1) $\delta(t) \leftrightarrow 1$ , for all s
(2) $\ u(t) \leftrightarrow \frac{1}{s}$ , for Re{s} > 0
(3) $\ -u(-t) \leftrightarrow \frac{1}{s}$ , for Re{s} < 0
(4) $\frac{t^{n - 1}}{(n - 1)!}u(t) \leftrightarrow \frac{1}{s^{n}}$ , for Re{s} > 0
(5) $- \frac{t^{n - 1}}{(n - 1)!}u(-t) \leftrightarrow \frac{1}{s^{n}}$ , for Re{s} < 0
(6) $\ e^{\alpha t }u(t) \leftrightarrow \frac{1}{s + \alpha}$ , for Re{s} > $\ - \alpha$
(7) $\ -e^{\alpha t }u(-t) \leftrightarrow \frac{1}{s + \alpha}$ , for Re{s} < $\ - \alpha$

@@ Line 112: / Line 112: @@
 == Some Laplace Transform Pairs ==
-{{:LaplaceTransforms}}
+{{:LaplaceTransforms_ECE301Fall2008mboutin}}
 *[[Laplace Pairs 1_ECE301Fall2008mboutin| (1)]]{{:Laplace Pairs 1_ECE301Fall2008mboutin}}
 *[[Laplace Pairs 2_ECE301Fall2008mboutin| (2)]]{{:Laplace Pairs 2_ECE301Fall2008mboutin}}