10/01 - Vectors & Functions: Difference between revisions

Revision as of 16:12, 6 November 2008

We could sample a continuous function every T seconds, creating a "bar graph".

$f(t)=\sum _{i=0}^{N-1}f(i\cdot T)\cdot p(t-i\cdot T)$

$f(i\cdot T)$ are the coefficients
$p(t-i\cdot T)$ are the basis functions, where $p(t)\,\!$ is a rectangle 1 unit high and T units wide

In an effort to make this more exact, will will continue to shrink the rectangle down to the Dirac Delta function, $\delta \,\!$

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 ==Vectors & Functions==
 *I'm not sure what my moodle log on is :(
+*How to related the vector v to the sampling?
 We could sample a continuous function every T seconds, creating a "bar graph".
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 <math> f(t)= \sum_{i=0}^{N-1} f(i \cdot T) \cdot p(t - i \cdot T)</math>
 *<math> f (i \cdot T) </math> are the coefficients
-*<math> p(t - i \cdot T) </math> are the basis functions
+*<math> p(t - i \cdot T) </math> are the basis functions, where <math> p(t) \,\! </math> is a rectangle 1 unit high and T units wide
+In an effort to make this more exact, will will continue to shrink the rectangle down to the Dirac Delta function, <math> \delta \,\!</math>
+*<math>\delta(x) = \begin{cases} +\infty, & x = 0 \\ 0, & x \ne 0 \end{cases}</math>
+*<math>\int_{-\infty}^\infty \delta(x) \, dx = 1.</math>