Revision as of 06:11, 29 October 2009

Some General Purpose Formulas and Definitions

General Purpose Formulas
Euler's Formula and Related Equalities
Euler's formula	$e^{jw_0t}=\cos w_0t+j\sin w_0t$
Cosine function in terms of complex exponentials	$\cos\theta=\frac{e^{j\theta}+e^{-j\theta}}{2}$
Sine function in terms of complex exponentials	$\sin\theta=\frac{e^{j\theta}-e^{-j\theta}}{2j}$
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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 ! colspan="2" style="background: #bbb; font-size: 110%;" | General Purpose Formulas
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-! colspan="2" style="background: #eee;" | Series
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-| align="right" style="padding-right: 1em;" | Finite Geometric Series Formula || <math>\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. </math>
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-| align="right" style="padding-right: 1em;" | Infinite Geometric Series Formula || <math>\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. </math>
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 ! colspan="2" style="background: #eee;" | Euler's Formula and Related Equalities
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 | align="right" style="padding-right: 1em;" | Sine function in terms of complex exponentials||<math>\sin\theta=\frac{e^{j\theta}-e^{-j\theta}}{2j}</math>
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-! colspan="2" style="background: #eee;" | Definition of some Basic Functions (what engineers call "Signals")
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-| align="right" style="padding-right: 1em;" | sinc function || <math>sinc(\theta)=\frac{sin(\pi\theta)}{\pi\theta}</math>
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-| align="right" style="padding-right: 1em;" | CT unit step function || <math> u(t)=\left\{ \begin{array}{ll}1, & t\geq 0 \\ 0, & \text{ else}\end{array}\right., \text{ for }t\in {\mathbb R}</math>
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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>
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 ! colspan="2" style="background: #eee;" | Function (or Signal) Metrics

Difference between revisions of "SomeGeneralPurposeFormulas1" - Rhea

Revision as of 06:11, 29 October 2009

Some General Purpose Formulas and Definitions

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