Noise Source Derivation

TODO...

$\Huge S_n(f) = 2kTR$

       +--o a                +------+-o a         
       |                     |      |      ^    
       |                     |      |      |      
      .+.                   .+.     |           
    R | |                 R | |     |      |    
      '+'                   '+'    _|_          
       |                     |     ---  C  |  Vo 
    + ,+.                 + ,+.     |            
  Vg (   )              Vg (   )    |      |     
    - `+'                 - `+'     |            
       |                     |      |      |      
       |                     |      |      v     
       +--o b                +------+-o b        

$\Huge H(f) = \frac{1/j2\pi fC}{R+1/j2\pi fC} = \frac{1}{1+j2\pi fRC}$

$\Huge S_0(f) = |\frac{1}{1+j2\pi fRC}|^2 2kTR = \frac{2kTR}{1 + 4\pi^2f^2R^2C^2}$

$\Huge \overline{v_o^2} = \int_{-\infty}^{+\infty}{\frac{2kTR}{1 + 4\pi^2f^2R^2C^2}df} = \frac{kT}{C}$