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If the events are not mutually exclusive then | If the events are not mutually exclusive then | ||
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| + | {| | ||
| + | ! colspan="2" style="background: #e4bc7e; font-size: 110%;" | Discrete-time Fourier Transform Pairs and Properties | ||
| + | |- | ||
| + | ! colspan="2" style="background: #eee;" | Property of Probability Functions | ||
| + | |- | ||
| + | | align="right" style="padding-right: 1em;" | The complement of an event A (i.e. the event A not occurring) || <math>\,P(A^c) = 1 - P(A)\,</math> | ||
| + | |- | ||
| + | | align="right" style="padding-right: 1em;" | The intersection of two independent events A and B || <math>\,P(A \mbox{ and }B) = P(A \cap B) = P(A) P(B)\,</math> | ||
| + | |- | ||
| + | | align="right" style="padding-right: 1em;" | The union of two events A and B (i.e. either A or B occurring) || <math>\,P(A \mbox{ or } B) = P(A) + P(B) - P(A \mbox{ and } B)\,</math> | ||
| + | |- | ||
| + | | align="right" style="padding-right: 1em;" | The union of two mutually exclusive events A and B || <math>\,P(A \mbox{ or } B) = P(A \cup B)= P(A) + P(B)\,</math> | ||
| + | |} | ||
| + | {| | ||
| + | |- | ||
| + | ! colspan="4" style="background: #eee;" | DT Fourier Transform Pairs | ||
| + | |- | ||
| + | | align="right" style="padding-right: 1em;" | || <math>x[n]</math> || <math>\longrightarrow</math>|| <math> \mathcal{X}(\omega) </math> | ||
| + | |- | ||
| + | | align="right" style="padding-right: 1em;" | DTFT of a complex exponential || <math>e^{jw_0n}</math> || || <math>\pi\sum_{l=-\infty}^{+\infty}\delta(w-w_0-2\pi l) \ </math> | ||
| + | |- | ||
| + | |- | ||
| + | | align="right" style="padding-right: 1em;" | || <math>a^{n} u[n], |a|<1 \ </math> || ||<math>\frac{1}{1-ae^{-j\omega}} \ </math> | ||
| + | |- | ||
| + | | align="right" style="padding-right: 1em;" | || <math>\sin\left(\omega _0 n\right) u[n] \ </math> || ||<math>\frac{1}{2j}\left( \frac{1}{1-e^{-j(\omega -\omega _0)}}-\frac{1}{1-e^{-j(\omega +\omega _0)}}\right)</math> | ||
| + | |- | ||
| + | |} | ||
| + | |||
| + | {| | ||
| + | |- | ||
| + | ! colspan="4" style="background: #eee;" | DT Fourier Transform Properties | ||
| + | |- | ||
| + | | align="right" style="padding-right: 1em;" | || <math>x[n]</math> || <math>\longrightarrow</math>|| <math> \mathcal{X}(\omega) </math> | ||
| + | |- | ||
| + | | align="right" style="padding-right: 1em;" | multiplication property|| <math>x[n]y[n] \ </math> || || <math>\frac{1}{2\pi} \int_{2\pi} X(\theta)Y(\omega-\theta)d\theta</math> | ||
| + | |- | ||
| + | | align="right" style="padding-right: 1em;" | convolution property || <math>x[n]*y[n] \!</math> || ||<math> X(\omega)Y(\omega) \!</math> | ||
| + | |- | ||
| + | | align="right" style="padding-right: 1em;" | time reversal ||<math>\ x[-n] </math> || ||<math>\ X(-\omega)</math> | ||
| + | |- | ||
| + | |} | ||
| − | + | {| | |
| − | + | |- | |
| + | ! colspan="2" style="background: #eee;" | Other DT Fourier Transform Properties | ||
| + | |- | ||
| + | | align="right" style="padding-right: 1em;" | Parseval's relation || <math>\frac {1}{N} \sum_{n=-\infty}^{\infty}\left| x[n] \right|^2 = </math> | ||
| + | |} | ||
| + | ---- | ||
| + | [[Collective_Table_of_Formulas|Back to Collective Table]] | ||
| + | [[Category:Formulas]] | ||
Revision as of 08:51, 22 October 2010
If the events are not mutually exclusive then
$ \mathrm{P}\left(A \hbox{ or } B\right)=\mathrm{P}\left(A\right)+\mathrm{P}\left(B\right)-\mathrm{P}\left(A \mbox{ and } B\right) $
Conditional probability is written P(A|B), and is read "the probability of A, given B"
$ P(A \mid B) = \frac{P(A \cap B)}{P(B)}\, $
| Discrete-time Fourier Transform Pairs and Properties | |
|---|---|
| Property of Probability Functions | |
| The complement of an event A (i.e. the event A not occurring) | $ \,P(A^c) = 1 - P(A)\, $ |
| The intersection of two independent events A and B | $ \,P(A \mbox{ and }B) = P(A \cap B) = P(A) P(B)\, $ |
| The union of two events A and B (i.e. either A or B occurring) | $ \,P(A \mbox{ or } B) = P(A) + P(B) - P(A \mbox{ and } B)\, $ |
| The union of two mutually exclusive events A and B | $ \,P(A \mbox{ or } B) = P(A \cup B)= P(A) + P(B)\, $ |
| DT Fourier Transform Pairs | |||
|---|---|---|---|
| $ x[n] $ | $ \longrightarrow $ | $ \mathcal{X}(\omega) $ | |
| DTFT of a complex exponential | $ e^{jw_0n} $ | $ \pi\sum_{l=-\infty}^{+\infty}\delta(w-w_0-2\pi l) \ $ | |
| $ a^{n} u[n], |a|<1 \ $ | $ \frac{1}{1-ae^{-j\omega}} \ $ | ||
| $ \sin\left(\omega _0 n\right) u[n] \ $ | $ \frac{1}{2j}\left( \frac{1}{1-e^{-j(\omega -\omega _0)}}-\frac{1}{1-e^{-j(\omega +\omega _0)}}\right) $ | ||
| DT Fourier Transform Properties | |||
|---|---|---|---|
| $ x[n] $ | $ \longrightarrow $ | $ \mathcal{X}(\omega) $ | |
| multiplication property | $ x[n]y[n] \ $ | $ \frac{1}{2\pi} \int_{2\pi} X(\theta)Y(\omega-\theta)d\theta $ | |
| convolution property | $ x[n]*y[n] \! $ | $ X(\omega)Y(\omega) \! $ | |
| time reversal | $ \ x[-n] $ | $ \ X(-\omega) $ | |
| Other DT Fourier Transform Properties | |
|---|---|
| Parseval's relation | $ \frac {1}{N} \sum_{n=-\infty}^{\infty}\left| x[n] \right|^2 = $ |
