Week of 10/29
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The fundamental property of the delta function: you must remember this | The fundamental property of the delta function: you must remember this | ||
| − | <math>\int _ a ^ b \delta (t - t_0) f(t) dt = f(t_0)</math> if <math>t_0 \ | + | <math>\int _ a ^ b \delta (t - t_0) f(t) dt = f(t_0)</math> if <math>t_0 \in [a,b] \,</math>. |
| − | <math>\int _ a ^ b \delta (t - t_0) f(t) dt = 0</math> if <math>t_0 \ | + | <math>\int _ a ^ b \delta (t - t_0) f(t) dt = 0</math> if <math>t_0 \notin [a,b] \,</math> |
| + | Remember, in the former case, | ||
| + | you evaluate <math>f \,</math> at the point where the argument of the delta function is zero. In this case when<math> t = t_0. \,</math> | ||
[http://www-history.mcs.st-and.ac.uk/history/Biographies/Dirac.html Paul Dirac's bio] | [http://www-history.mcs.st-and.ac.uk/history/Biographies/Dirac.html Paul Dirac's bio] | ||
| Line 42: | Line 44: | ||
[http://mathworld.wolfram.com/SincFunction.html from Mathworld] and | [http://mathworld.wolfram.com/SincFunction.html from Mathworld] and | ||
[http://en.wikipedia.org/wiki/Sinc_function Wikipedia] | [http://en.wikipedia.org/wiki/Sinc_function Wikipedia] | ||
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| + | |||
| + | {{mathematica|filename=Spectral_interferometry.nb|title=Interfering gaussians}} | ||
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[[Image:Slide0030.gif]] | [[Image:Slide0030.gif]] | ||
| − | + | {{PDF|filename=10-29-07.pdf|title=lecture notes from 10/29/07}} | |
{{PDF|filename=11_1_06.pdf|title=lecture notes from 10/31/07}} | {{PDF|filename=11_1_06.pdf|title=lecture notes from 10/31/07}} | ||
| − | {{mathematica|filename= | + | [http://mathworld.wolfram.com/Convolution.html very nice graphical illustration of the convolution] |
| + | |||
| + | {{mathematica|filename=Convolution.nb|title=Creating a realistic time series by convolving a wavelet with a series of spikes}} | ||
| + | |||
| + | {{PDF|filename=11-2-07.pdf|title=Optional digression on sampling theorem}} | ||
Latest revision as of 16:30, 5 November 2007
 Download FT, wavepackets, bandwidth ...
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A few examples of the importance of Gaussian distributions
Maxwell Boltzmann velocity distribution
It turns out that the FWHM is nontrivial: this link shows how to compute it
the answer turns out to be 
The fundamental property of the delta function: you must remember this
if ![t_0 \in [a,b] \,](/csm/wiki/images/math/f/5/d/f5da11ff3894d1eac9f725c7419c854b.png)
.
if ![t_0 \notin [a,b] \,](/csm/wiki/images/math/c/2/0/c205147a5cfbdbd0af90f97c8de32a3e.png)
Remember, in the former case,
you evaluate 
at the point where the argument of the delta function is zero. In this case when
Two nice links on Sinc functions: from Mathworld and Wikipedia
 Download Interfering gaussians
|
The following two figures are from Barbara Romanowicz, a seismologist at the University of California, Berkeley.
| Download lecture notes from 10/29/07
|
| Download lecture notes from 10/31/07
|
very nice graphical illustration of the convolution
| Download Creating a realistic time series by convolving a wavelet with a series of spikes
|
| Download Optional digression on sampling theorem
|
