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

Latest revision as of 07:20, 5 February 2009

Please follow the following model to add your formulas:

{{:name of the page with your formula}}

in the place where you want your formula to appear in this table. (Look at the syntax of the geometric series below for an example.) This will allow other people to refer to your formula later on (by refering to the corresponding page) while still being able to view all formulas on this page.

• 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 Properties

• 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

• (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 $