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| + | [[Category:ECE 301]] | ||
| + | [[Category:Fall 2008]] | ||
| + | [[Category:mboutin]] | ||
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== Phasors == | == Phasors == | ||
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<math>\cos{\theta}=\frac{e^{j\theta}+e^{-j\theta}}{2}</math> | <math>\cos{\theta}=\frac{e^{j\theta}+e^{-j\theta}}{2}</math> | ||
| + | |||
| + | <math>\sin{\theta}=\frac{e^{j\theta}-e^{-j\theta}}{2j}</math> | ||
| + | |||
| + | <math>|e^{j\theta}|=1</math> | ||
| + | |||
| + | |||
| + | == Energy == | ||
| + | |||
| + | ==== Discrete ==== | ||
| + | |||
| + | |||
| + | <math>E_\infty = \sum_{n=-\infty}^\infty |x[n]|^2</math> | ||
| + | |||
| + | |||
| + | ==== Continuous ==== | ||
| + | |||
| + | |||
| + | <math>E_\infty = \int_{-\infty}^\infty |x(t)|^2\,dt)</math> | ||
| + | |||
| + | |||
| + | == Power == | ||
| + | |||
| + | ==== Discrete ==== | ||
| + | |||
| + | |||
| + | <math>P_\infty = \lim_{N \to \infty} \left (\frac{1}{2N + 1} \sum_{n=-N}^{+N} |x[n]|^2 \right)</math> | ||
| + | |||
| + | ==== Continuous ==== | ||
| + | |||
| + | |||
| + | <math>P_\infty = \lim_{T \to \infty} \left (\frac{1}{2T} \int_{-T}^T |x(t)|^2\,dt \right)</math> | ||
| + | |||
| + | == Geometric Series == | ||
Latest revision as of 17:51, 5 November 2008
Contents
Phasors
$ x(t)=Ae^{j\theta+\phi} $
Where A is the radius of the phasor and $ \phi $ if the offset.
Useful Phasors Facts
$ e^{j\theta} = \cos{\theta}+j\sin{\theta} $
$ Ae^{j[\theta+\phi]}=Ae^{j\theta}e^{j\phi} $
$ \cos{\theta}=\frac{e^{j\theta}+e^{-j\theta}}{2} $
$ \sin{\theta}=\frac{e^{j\theta}-e^{-j\theta}}{2j} $
$ |e^{j\theta}|=1 $
Energy
Discrete
$ E_\infty = \sum_{n=-\infty}^\infty |x[n]|^2 $
Continuous
$ E_\infty = \int_{-\infty}^\infty |x(t)|^2\,dt) $
Power
Discrete
$ P_\infty = \lim_{N \to \infty} \left (\frac{1}{2N + 1} \sum_{n=-N}^{+N} |x[n]|^2 \right) $
Continuous
$ P_\infty = \lim_{T \to \infty} \left (\frac{1}{2T} \int_{-T}^T |x(t)|^2\,dt \right) $
