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".

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

• 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

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

Changing from one orthogonal Basis Functions to another

If you have a vector $\hat{v}=\sum _{j=1}^3{a}_j{\hat{a}}_j$ and wish to change it to $\hat{v}=\sum _{j=1}^3{b}_j{\hat{b}}_j$

• Working from the $\hat{a}$ basis set:

$\hat{v}\cdot {\hat{b}}_m=\sum _{j=1}^3{a}_j{\hat{a}}_j\cdot {\hat{b}}_m=\sum _{j=1}^3{a}_j\underset{projof{\hat{a}}_{j}on{\hat{b}}_m}{\underbrace{({\hat{a}}_j\cdot {\hat{b}}_m)}}$

• Working from the $\hat{b}$ basis set:

$\hat{v}\cdot {\hat{b}}_m=\sum _{j=1}^3{b}_j{\hat{b}}_j\cdot {\hat{b}}_m=\sum _{j=1}^3{b}_j\underset{projof{\hat{b}}_{j}on{\hat{b}}_m}{\underbrace{({\hat{b}}_j\cdot {\hat{b}}_m)}}=\sum _{j=1}^3{a}_j{k}_m\delta mj=k_m\sum _{j=1}^3{a}_j\delta mj=\sum _{j=1}^3{a}_m{k}_m$ Define $k_m={\left|{\hat{a}}_m\right|}^2$

• Why?
• How did you get the last two lines of the last page?
• What does the ${\hat{b}}_m$ represent, say compared to ${\hat{b}}_j$?
• When you do the dot product of say A \cdot B, is it always the projection of A onto B and not the opposite way around?