Revision as of 09:07, 28 October 2009

Some General Purpose Formulas and Definitions

General Purpose Formulas
Series
Finite Geometric Series Formula	$\sum_{k=0}^n x^k = \left\{ \begin{array}{ll} \frac{1-x^{n+1}}{1-x}&, \text{ if } x\neq 1\\ n+1 &, \text{ else}\end{array}\right.$
Infinite Geometric Series Formula	$\sum_{k=0}^n x^k = \left\{ \begin{array}{ll} \frac{1}{1-x}&, \text{ if } \|x\|\leq 1\\ \text{diverges} &, \text{ else }\end{array}\right.$
Euler's Formula and Related Equalities
Euler's formula	$e^{jw_0t}=cosw_0t+jsinw_0t$
Cosine function in terms of complex exponentials	$cos\theta=\frac{e^{j\theta}+e^{-j\theta}}{2}$
Sine function in terms of complex exponentials	$sin\theta=\frac{e^{j\theta}-e^{-j\theta}}{2j}$
Definition of some Basic Functions (what engineers call "Signals")
sinc function	$sinc(\theta)=\frac{sin(\pi\theta)}{\pi\theta}$
CT unit step function	$u(t)=\left\{ \begin{array}{ll}1, & t\geq 0 \\ 0, & \text{ else}\end{array}\right., \text{ for }t\in {\mathbb R}$
DT unit step function	$u[n]=\left\{ \begin{array}{ll}1, & n\geq 0 \\ 0, & \text{ else}\end{array}\right., \text{ for }n\in {\mathbb Z}$
Function (or Signal) Metrics
CT signal energy	$E_\infty=\int_{-\infty}^\infty \| x(t) \|^2 dt$
DT signal energy	$E_\infty=\sum_{n=-\infty}^\infty \| x[n] \|^2$

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 | align="right" style="padding-right: 1em;" | DT unit step function || <math>u[n]=\left\{ \begin{array}{ll}1, & n\geq 0 \\ 0, & \text{ else}\end{array}\right., \text{ for }n\in {\mathbb Z} </math>
+|-
+|-
+! colspan="2" style="background: #eee;" | Function (or Signal) Metrics
+|-
+| align="right" style="padding-right: 1em;" | CT signal energy || <math>E_\infty=\int_{-\infty}^\infty | x(t) |^2 dt </math>
+|-
+| align="right" style="padding-right: 1em;" | DT signal energy || <math>E_\infty=\sum_{n=-\infty}^\infty | x[n] |^2 </math>
 |-
 |}