--- Digital signal processing ---

 time domain :

   array of sampling times =  [ 1 dt, 2 dt, 3 dt, ... , N dt ] 
   longest time interval  = T = N dt
   shortest time interval = dt

   example: 4 minute song at 44.1 kHz.  N = ?

 frequency domain :

   what are the possible frequencies ?

   highest is [ 1, -1, 1, -1, ... , ] 
   which has period = 2 dt, so f_high = 1/(2 dt)

   lowest periodic is [ 0 , ...,  1 , ... , 0 , ... , -1 , ... 0 ]
   which has a period of T = N dt, so f_low = 1/(N dt) 

   just as multiples of shortest time gives all times, 
   so multiples of this lowest are only possible frequences.

   But that gives 1/(N*dt) * ( 1, 2, 3, 4, ..., N/2) 
   which is only half as much data as time points.
   What gives?

   Answer: each of these frequences comes in two phases,
   i.e. cosine and sine

   Caveat: actually, the high frequency only has one phase.
   But there's also a zero frequency, so we do get N frequencies.
   
   example: N = 4

   time: 
           (1 0 0 0) x amplitude_1
         + (0 1 0 0) x amplitude_2
         + (0 0 1 0) x amplitude_3
           (0 0 0 1) x amplitude_4

   freq:
           ( 1  1  1  1) x how_much_freq_0
         + ( 1 -1 -1  1) x how_much_freq_1a  (twice period of 2)
         + (-1 -1  1  1) x how_much_freq_1b  (shifted 1/4 cyle)
         + ( 1 -1  1 -1) x how_much_freq_2   (highest = half of 4)
 
   Aside - a weird way to think of this :
    We can call the other phase a "negative frequency",
    which makes sense when you do all this with complex numbers,
    and otherwise is completely confusing.  
    Then the frequency labels would be 
      N = 4 : m=(-1,0,1,2),
      N = 8 : m=(-3,-2,-1,0,1,2,3,4)
    where the frequency f = m df = m/(N dt)

 --- filter ---
   
   Just a simple multiplication in frequency domain...
 
   So digital signal processing looks like this

   1) start with [ y1, y2, y3, ...., yN ] at each time value
 
   2) transform to [a1, a2, a3, ..., aN ] at each freq value

   3) multiply (filter) by [ 1, 1, 1, 0, 0, 0, ]  (low freq filter)
      or whatever

   4) inverse transform back to time [ new_y1, new_y2, ... ]

   By the way, that transform is called 
     "FFT" Fast Fourier Transform   
   or  "DFT" Discrete Fourier Transform
   and is also what's used in jpg images for compression