14 | | [[Image(http://www.pas.rochester.edu/~afrank/Fig1.jpg)]] |
| 18 | [[Image(http://www.pas.rochester.edu/~afrank/Fig1.jpg, 250px)]] |
| 19 | |
| 20 | Use the derivative property of [[latex($ \Xi[f] $)]] |
| 21 | |
| 22 | [[latex($ \Xi[f ] = \frac{1}{2\pi i k} \Xi[ \frac{df}{dx} ] $)]] |
| 23 | |
| 24 | Since derivative of f is the direct delta function |
| 25 | |
| 26 | [[latex($ \frac{df}{dx} = \delta(x-a) $)]] |
| 27 | |
| 28 | and |
| 29 | |
| 30 | [[latex($ \Xi[ \delta(x-a) ] = e^{-i2\pi k a} $)]] |
| 31 | |
| 32 | Thus we have |
| 33 | |
| 34 | [[latex($ \Xi[f ] = F(k) = \frac{1}{2\pi i k} e^{-i2\pi k a} $)]] |
| 35 | |
| 36 | The next step is to define the Energy Spectral Density (ESD) |
| 37 | |
| 38 | We use Parcivals Thm which tell us "energy" under the curve f(x) |
| 39 | |
| 40 | [[latex($ \int^{+\infty}_{-\infty} | f(x)|^2 dx = \int^{+\infty}_{-\infty} | F(k)|^2 dk $)]] |
| 41 | |
| 42 | Thus the ESD which we write as E(k) is |
| 43 | |
| 44 | [[latex($ E(k) = | F(k)|^2 $)]] |
| 45 | |
| 46 | which for the sgn function is |
| 47 | |
| 48 | [[latex($ 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} $)]] |
| 51 | |
| 52 | So for a step function |
| 53 | |
| 54 | [[latex($ E(k) \propto k^{-2} $)]] |
| 55 | |
| 56 | |
| 57 | |
| 58 | |
| 59 | |