10/01 - Vectors & Functions: Difference between revisions

Revision as of 12:49, 9 November 2008

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

$f(t)=\sum _{i=0}^{N-1}\underbrace {f(iT)} _{coefficients}\cdot \underbrace {p(t-iT)} _{basisfunctions}$

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

By using the Dirac Delta function the summation becomes an integral

$f(t)=\int _{-\infty }^{\infty }f(u)\cdot \delta (t-u)\,du$

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 We could sample a continuous function every T seconds, creating a "bar graph".
-<math> f(t)= \sum_{i=0}^{N-1} f(i T) \cdot p(t - i T)</math>
+<math> f(t)= \sum_{i=0}^{N-1} \underbrace{f(i T)}_{coefficients} \cdot \underbrace{p(t - i T)}_{basis functions}</math>
-*<math> f (i T) \,\!</math> are the coefficients
+*Where <math> p(t) \,\! </math> is a rectangle 1 unit high and T units wide
-*<math> p(t - i 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>