# Link between time and complex amplitude argument, DFT

I have some discrete points, with time $t_n$ and value $u(t_n)$. I perform discrete fourier transormation using values $u(t_n)$. Now I have some complex values, as I understood absolute value is amplitude and argument is frequency, right? well my teacher said i need to compute something using $t_n$'s, $u(t_n)$'s, $A_n$'s and $\omega_n$'s (assuming $A_n$ is absolute value, $\omega_n$ is argument). It's somehow related to statistics variance and uncertainity principle. As you see, I don't know about Fourier transformations :D. Now I've made a program that performs DFT and builds a plot using $\omega_n$'s as $Ox$ axis and $A_n$'s as $Oy$ axis. It looks like this: screenshot

Value i need to compute is $\Delta T \Delta \omega$

$T_1=\frac{\sum_{n=1}^N u(t_n)*t_n} {\sum_{n=1}^N u(t_n)}$ $T_2=\frac{\sum_{n=1}^N u(t_n)*t_n^2} {\sum_{n=1}^N u(t_n)}$ $\Delta T = \sigma^2 = 2\sqrt{T_2-T_1^2}$

and same for $\Delta \omega$:

$W_1=\frac{\sum_{n=1}^N A_n*\omega_n} {\sum_{n=1}^N A_n}$ $W_2=\frac{\sum_{n=1}^N A_n*\omega_n^2} {\sum_{n=1}^N A_n}$ $\Delta \omega = \sigma^2 = 2\sqrt{W_2-W_1^2}$

What my teacher wants me to compute and which values should I expect as corresponding to theory? Can you give me step by step tutorial from source values to result? May be there's something wrong with the task.

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Why not ask your teacher? There's a lot you can do (e.g. spectrograms), and it's not entirely clear what you (or your teacher, as it seems) are expecting to see from what you've written. –  Ｊ. Ｍ. Oct 3 '10 at 14:15
My bad, added screenshot. According to pictures in my notebook made when teacher announced this task, plot is right. Unfortunately i cant contact with teacher. Fixed question contents, added formulas from notebook. –  Anton Oct 3 '10 at 15:34