Changes between Version 4 and Version 5 of u/afrank


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Timestamp:
12/22/14 20:55:51 (10 years ago)
Author:
Adam Frank
Comment:

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  • u/afrank

    v4 v5  
    66The basic equation for the fourier transform.
    77
    8  [[latex($ F(k) = \int^{+\infty}_{-\infty} f(x) e^{(-i 2 \pi k x)} dx   $)]]
     8 [[latex($ \Xi[f] = F(k) = \int^{+\infty}_{-\infty} f(x) e^{(-i 2 \pi k x)} dx   $)]]
     9
     10with the inverse
     11
     12 [[latex($ \Xi[F]^{-1} = f(x) = \int^{+\infty}_{-\infty} F(k)) e^{(i 2 \pi k x)} dk   $)]]
    913
    1014Here we want to solve the signup function which is runs from -1 to 1 with a step at x=a
     
    1216 [[latex($ f(x) = sgn(x-a)   $)]]
    1317
    14 [[Image(http://www.pas.rochester.edu/~afrank/Fig1.jpg)]]
     18[[Image(http://www.pas.rochester.edu/~afrank/Fig1.jpg, 250px)]]
     19
     20Use the derivative property of  [[latex($ \Xi[f]   $)]]
     21
     22 [[latex($ \Xi[f ] = \frac{1}{2\pi i k} \Xi[ \frac{df}{dx} ]  $)]]
     23
     24Since derivative of f is the direct delta function
     25
     26 [[latex($ \frac{df}{dx} = \delta(x-a)   $)]]
     27
     28and
     29
     30 [[latex($ \Xi[ \delta(x-a) ] =  e^{-i2\pi k a} $)]]
     31
     32Thus we have
     33
     34 [[latex($ \Xi[f ] = F(k) = \frac{1}{2\pi i k} e^{-i2\pi k a}  $)]]
     35
     36The next step is to define the Energy Spectral Density (ESD)
     37
     38We use Parcivals Thm which tell us "energy" under the curve f(x)
     39
     40 [[latex($ \int^{+\infty}_{-\infty} | f(x)|^2 dx  = \int^{+\infty}_{-\infty} | F(k)|^2 dk $)]]
     41
     42Thus the ESD which we write as E(k) is
     43
     44[[latex($  E(k) =  | F(k)|^2  $)]]
     45
     46which for the sgn function is
     47
     48 [[latex($ E(k) = (\frac{1}{2\pi i k} e^{-i2\pi k a})^*  (\frac{1}{2\pi i k} e^{-i2\pi k a}) $)]]
     49
     50 [[latex($ E(k) = \frac{1}{4\pi^2 k^2}   $)]]
     51
     52So for a step function
     53
     54 [[latex($ E(k) \propto k^{-2}   $)]]
     55
     56
     57
     58
     59
    1560
    1661== Description of Problem ==