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! colspan="2" style="background: #bbb; font-size: 110%;" | General Purpose Formulas
 
! 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;" | Euler's formula || <math>e^{jw_0t}=\cos w_0t+j\sin w_0t</math>
 
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| align="right" style="padding-right: 1em;" | Cosine function in terms of complex exponentials|| <math>\cos\theta=\frac{e^{j\theta}+e^{-j\theta}}{2}</math>
 
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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
 
! colspan="2" style="background: #eee;" | Function (or Signal) Metrics

Latest revision as of 07:23, 29 October 2009

Some General Purpose Formulas and Definitions

General Purpose Formulas
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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Basic linear algebra uncovers and clarifies very important geometry and algebra.

Dr. Paul Garrett