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2.2 Mathematical Modeling

Periodic functions g(t) = g(t + T) for some constant T, called period, and all t.
Frequency f = (1/T)--the number of times a repetition occures in a fixed time interval (e.g., cycles per second or Hertz). 
   -|--
  --  --
 --    --  --
--      -|--
 ---- - - - - ------------ ------------- ------------- ------- - - -
Low frequency High frequency
Phase 
|||||||||||||              |||||||||||||
 ||||||||   |||||       ||||    ||||||||   sin(2pft)
---------------------------------------|
           ||||||||||||||||||           sin(2pf t+ p/2)

Data Transmitted by varying some physical property. Possible representations for the character whose binary code is `01110010':
Binary data from frequency 0's are represented by low frequencies; 1's by high frequencies. 
|    |   |    |    |    |    |   |    |
|-   ----|-- ---- -|-   |-   ----||-  |
---  -----------------  ---  ----|--  |
- - ---------------- -  - - --------- |
| -------|----|----|  --| -------|  --|
|  --|-- |--- |- --|  |-|  --|-- |  --|
|    |   |    |    |    |    |   |    |
  0   1    1    1    0    0   1    0

Binary data from amplitude 0's and 1's are represented by voltage level 
|  |  | |  |  | |  |  |
|  --------|  | ---|  |
|  |  | |  |  | |  |  |
|  |  | |  |  | |  |  |
|  |  | |  |  | |  |  |
---|  | |  -----|  ---|
|  |  | |  |  | |  |  |
 0  1 1  1  0 0  1  0
Fourier analysis Any reasonably behaved periodic function can be represented by sums of sine and cosine functions
sum oo sum oo g(t) = an sin(2pnft)+ bncos(2pnft) n=0 n=1

where f = ----1------ period of g , and an and bn are computable from g by integration.

Example: The character 'b' The bit pattern to be submitted is `01100010'. The voltage output can be the function g(t) with the coefficients a0 = 3/8, an = (1/pn)[...], and bn = (1/pn)[...].
Example: square waves Can be approximated by
oo s(t) = sum A-sin(2pnf t) n=1,3,5,...n 1
n = 1 sin(2pft) -- --- ---- - - - --- ---- -- |-
n = 3 13sin(6pf t) --- ------ -----------
n = 1, 3 sin(2pft)+ 1sin(6pf t) 3 --| --| | || | | | --| |-|| --- --
n = 1, 3, 5 1 1 sin(2pft)+ 3sin(6pf t)+ 5sin(10pft) --| --| | || | | | | | | --| |-||

Observation The larger the bandwidth (of frequencies) the easier the interpretation is, and so also the data rate can be increased.
Cost The energy to transmit frequency nf is proportional to an 2 + bn 2 . Hence, larger bandwidth implies higher cost.

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