====== Oct 20. Exercise Day! ======

So far, we have learned many transformations and inverse transformation. Conceptually, we are just trying to decompose a given function $f(x)$ as a linear combination of $e^{ax}$ for various $a$. Because it is an eigenfunction of $(d/dx)$: 
$$ (d/dx) e^{ax} = a e^{ax}$$


==== Ex 1 ====
Decomposition of a standing wave with Dirichlet condition. When we pluck a string on a violin, the string vibrate with the endpoints fixed. 

Suppose the string's length is $L$, and you took a snapshot of the string and obtained the vertical displacement as a function $f: [0,L] \to \R$ (yes, it is $\R$ valued, not $\C$ valued). 

Design a way to do Fourier decomposition of $f$. 

If you like, pick $L = 2$ and 
$$ f(x) = 1 - |x-1|. $$
(just a linear peak). 


==== Ex 2 ==== 
Damped oscillation, reverse engineering. 

Suppose you observe some damped oscillation that goes like 
$$ f(t) = A e^{-a t} sin(b t), \quad t > 0 $$
Can you find the second order equation that $f(t)$ satisfies? Namely, for what constant $A,B$ does $f(t)$ satisfies the equation?

$$ f"(t) + A f'(t) + B f(t) = 0. $$

If you like, pick $a=1, b= 2$. 

==== Ex 3 ====
Tuning the friction. 

Consider the pure oscillation equation: 
$$ f"(t) + \omega^2 f(t) = 0 $$
it has solution like $f(t) = a \sin(\omega t) + b \cos(\omega t). $

What happens if we add some friction term? 
$$ f"(t) + a f'(t) + \omega^2 f(t) = 0 $$
  * does it matter if $a$ is positive or negative? 
  * does it matter if $a$ is small or large? 

Make a guess first. Then solve it explicitly, for $\omega = 1$, and try various $a$. 

Hint: Is there a 'planewave' that solves the equation, i.e. somethign like $e^{ct}$ for some $c$? 

==== Ex 4: Discrete Fourier transformation ====
You received the following a periodic sequence of numbers, 

  * sequence of period 9: 0 0 1 0 0 2 0 0 3

Use discrete Fourier transformation to analyse it. 
  * You can use this input number to as values of $f(x)$ for $x=0,1, \cdots, 8$. 
  * you know $f(x)$ can be written as 
$$ f(x) = \sum_{p=0}^8 F(p) e^{2\pi i (xp/9)} $$
Can you find $F(p)$? 
  * These $f(x)$ are all real valued, what does that say about $F(p)$? (try to take complex conjugate of the equation)
  * Can you guess, which $|F(p)|$s are largest? 


If you have done the above exercise, try this one
  * sequence of period 10: 2 5 3 8 2 7 1 9 0 4