Week of 11/5

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Here is the display of my oscilloscope when the input is a 1000 pulse per second output of a time-code generator. (The time-code generator is a device that locks to the 10 MHz output of an atomic clock and produces 1 Hz or 1 KHz pulse trains as well as human readable time synchronized to the atomic standard.)

I imported the data and plotted it along with its periodogram.

Notice that only the odd harmonics are present.

Here is a mathematica notebook that simulates this.

Download lots of fourier transform examples

Sampling theorem. See 10/31/07 lecture notes

We end up with $f(t) = \sum _ {n = - \infty} ^ \infty f(n/2f_s) \frac{\sin(\pi (2 f_s t -n))}{\pi (2 f_s t -n)}$

This amounts to taking samples of the data every $1 / 2 f s$ and multiplying them by a sinc function and adding up the results.

Download Sinc function interpolation via the samping theorem

Mathematica can find the fourier transform of a box function of width h centered on zero times $s i n (2π x)$ :

$-\frac{i \sqrt{\frac{2}{\pi }} \left(2 \pi \cos (h \pi ) \sin \left(\frac{h k}{2}\right)-k \cos \left(\frac{h k}{2}\right) \sin (h \pi )\right)}{4 \pi ^2-k^2}$