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, $\delta \,\!$

• $\delta (x)={\begin{cases}+\infty ,&x=0\\0,&x\neq 0\end{cases}}$
• $\int _{-\infty }^{\infty }\delta (x)\,dx=1.$

By using the Dirac Delta function the summation becomes an integral

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

Changing Basis Functions

• explain b_j