Master's Thesis

Entries from Wednesday, November 15. 2006

Wednesday, November 15. 2006

The exponential in $R^d$, II

Master's Thesis

(Q15) I’m gonna show you some cool Gabor atoms on \bbR^2. First, the function u_\omega(x):=e^{\pi\i\langle x,\omega\rangle}, as described previously, indeed looks like this:

[xx yy] = meshgrid(linspace(-2,2,100)); v=[1 1.5]; e1 = exp(pi*i*( v(1)*xx + v(2)*yy )); imagesc(real(e1)) w=[-1 4]; e2 = exp(pi*i*( w(1)*xx + w(2)*yy )); imagesc(real(e2))

See how the value of the frequency changes with the length of \omega, and the direction with the orientation of \omega, just as described previously.
In the STFT, these frequencies get reduced locally by an “envelope function”. One could take the Gaussian window e^{-a^2x^2} to achieve this:

g1 = exp(-(xx.^2+yy.^2)); imagesc(g1) g2 = exp(-4*(xx.^2+yy.^2)); imagesc(g2)

And now these are the modulated Gaussians, whose set of translates across \bbR^2 forms the building blocks for Gabor analysis on \bbR^2:

imagesc(real(g1.*e1)) imagesc(real(g2.*e2))

Posted by Stephan Paukner in Master's Thesis at 18:15 | Comments (0) | Trackback (1)
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