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.)
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[[Image:Squarewave.png]]
 
[[Image:Squarewave.png]]
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I imported the data and plotted it along with its periodogram.
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[[Image:Sqwavexmgr.png]]
 
[[Image:Sqwavexmgr.png]]
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Notice that only the odd harmonics are present.
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Here is a mathematica notebook that simulates this.
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{{mathematica|filename=fouriertransformexamples.nb|title=lots of fourier transform examples}}
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----
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Sampling theorem.  See 10/31/07 lecture notes
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We end up with
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<math>
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f(t) = \sum _ {n = - \infty} ^ \infty f(n/2f_s) \frac{\sin(\pi (2 f_s t -n))}{\pi (2 f_s t -n)}
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</math>
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This amounts to taking samples of the data every <math>1/2f_s</math> and multiplying them by a sinc function and adding up the results.
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{{mathematica|filename=Sincinterpolation.nb|title=Sinc function interpolation via the samping theorem}}
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Mathematica can  find the fourier transform of a box function of width h centered on zero times  <math>sin(2 \pi x)</math>:
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<math>-\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}
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</math>
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{{mathematica|filename=Sinpiece.nb|title=FT of finite sines}}
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{{PDF|filename=11-7-07.pdf|title=11-7-07 notes}}
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{{mathematica|filename=Timingfft.nb|title=timing the fft}}
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{{mathematica|filename=Nyquist.nb|title=Nyquist exercise}}
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{{mathematica|filename=Sounds.nb|title=cool sound examples with Mathematica}}
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{{PDF|filename=11_9_07.pdf|title=11-9-07 notes}}
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[[Image:3pointnyquist.png]]

Latest revision as of 21:01, 6 December 2007

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.)

Squarewave.png



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


Sqwavexmgr.png


Notice that only the odd harmonics are present.

Here is a mathematica notebook that simulates this.

Mathematica.png 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 / 2fs and multiplying them by a sinc function and adding up the results.

Mathematica.png 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 sin(2πx):

-\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}


Mathematica.png Download FT of finite sines


Pdf.png Download 11-7-07 notes
Mathematica.png Download timing the fft


Mathematica.png Download Nyquist exercise


Mathematica.png Download cool sound examples with Mathematica


Pdf.png Download 11-9-07 notes

3pointnyquist.png

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