10/01 - Vectors & Functions

Vectors & Functions

• How to related the vector v to the sampling?

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, $\delta$

• $\delta (x)=\{\begin{array}{ll}+\mathrm{\infty },& x=0\\ 0,& x\ne 0\end{array}$
• $\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\delta (x)dx=1.$

By using the Dirac Delta function the summation becomes an integral

$f(t)=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}f(u)\cdot \delta (t-u)du$

• explain b_j