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".

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

• 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 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:

${\hat {v}}\cdot {\hat {b}}_{m}=\sum _{j=1}^{3}a_{j}{\hat {a}}_{j}\cdot {\hat {b}}_{m}=\sum _{j=1}^{3}a_{j}\underbrace {\left({\hat {a}}_{j}\cdot {\hat {b}}_{m}\right)} _{projof{\hat {a}}_{j}on{\hat {b}}_{m}}$

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

${\hat {v}}\cdot {\hat {b}}_{m}=\sum _{j=1}^{3}b_{j}{\hat {b}}_{j}\cdot {\hat {b}}_{m}=\sum _{j=1}^{3}a_{j}\underbrace {\left({\hat {a}}_{j}\cdot {\hat {b}}_{m}\right)} _{projof{\hat {a}}_{j}on{\hat {b}}_{m}}=\sum _{j=1}^{3}a_{j}k_{m}\delta mj=k_{m}\sum _{j=1}^{3}a_{j}\delta mj=\sum _{j=1}^{3}a_{m}k_{m}$ 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 ${\hat {b}}_{j}$?