Week of 10/29
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[http://en.wikipedia.org/wiki/Maxwell%E2%80%93Boltzmann_distribution Maxwell Boltzmann velocity distribution] | [http://en.wikipedia.org/wiki/Maxwell%E2%80%93Boltzmann_distribution Maxwell Boltzmann velocity distribution] | ||
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+ | ---- | ||
It turns out that the FWHM is nontrivial: [http://mathworld.wolfram.com/GaussianFunction.html this link shows how to compute it] | It turns out that the FWHM is nontrivial: [http://mathworld.wolfram.com/GaussianFunction.html this link shows how to compute it] | ||
the answer turns out to be <math>2 \sqrt{2 \ln 2} \sigma \approx 2.3548 \sigma </math> | the answer turns out to be <math>2 \sqrt{2 \ln 2} \sigma \approx 2.3548 \sigma </math> | ||
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+ | ---- | ||
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+ | The fundamental property of the delta function: you must remember this | ||
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+ | <math>\int _ a ^ b \delta (t - t_0) f(t) dt = f(t_0)</math> if <math>t_0 \in [a,b] \,</math>. | ||
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+ | <math>\int _ a ^ b \delta (t - t_0) f(t) dt = 0</math> if <math>t_0 \notin [a,b] \,</math> | ||
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+ | 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> | ||
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+ | [http://www-history.mcs.st-and.ac.uk/history/Biographies/Dirac.html Paul Dirac's bio] | ||
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+ | [[Image:Dirac.png]] | ||
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+ | Two nice links on Sinc functions: | ||
+ | [http://mathworld.wolfram.com/SincFunction.html from Mathworld] and | ||
+ | [http://en.wikipedia.org/wiki/Sinc_function Wikipedia] | ||
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+ | {{mathematica|filename=Spectral_interferometry.nb|title=Interfering gaussians}} | ||
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+ | The following two figures are from [http://seismo.berkeley.edu/~barbara/ Barbara Romanowicz], a seismologist at the University of California, Berkeley. | ||
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+ | [[Image:Slide0002.gif]] | ||
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+ | [[Image:Slide0030.gif]] | ||
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+ | {{PDF|filename=10-29-07.pdf|title=lecture notes from 10/29/07}} | ||
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+ | {{PDF|filename=11_1_06.pdf|title=lecture notes from 10/31/07}} | ||
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+ | [http://mathworld.wolfram.com/Convolution.html very nice graphical illustration of the convolution] | ||
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+ | {{mathematica|filename=Convolution.nb|title=Creating a realistic time series by convolving a wavelet with a series of spikes}} | ||
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+ | {{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 ... |
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 .
if
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 |