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Don’t forget that Heisenberg’s uncertainty principle can be derived as a property (theorem?) of the Fourier transform [1]. I found the Laplace transform intrigu
by elcritch 7y ago
Don’t forget that Heisenberg’s uncertainty principle can be derived as a property (theorem?) of the Fourier transform [1]. I found the Laplace transform intriguing as well [2].
The Hilbert transform sounds intriguing.
1: http://www-users.math.umn.edu/~garrett/m/fun/uncertainty.pdf http://www-users.math.umn.edu/~garrett/m/fun/uncertainty.pdf
2: https://en.m.wikipedia.org/wiki/Laplace_transform https://en.m.wikipedia.org/wiki/Laplace_transform
edit: fixed link
- pitaj 7y agoLaplace transforms, being essentially a more powerful superset of Fourier transforms, are awesome. They also make calculus way easier, differential equations specifically. As an electrical engineer, I love me some Laplace transforms.
- posix_me_less 7y ago> a more powerful superset What do you mean? How is integral over half R more powerful than integral over whole R?
- danaos 7y agoParent is right. The Laplace transform is a generalization of Fourier's. There are two types of Laplace tranforms, the two-sided and the one-sided. The two-sided is defined in all of R. The s variable is a complex frequency in the form s = σ + jw. Setting σ = 0 in yields the Fourier tranform. The single-sided LT transform is equivalent to the one-sided if the signal is causal. A causal signal f(t) is zero for all t < 0. All of the signals in the real world are causal. This is convenient as the exponential e^(-st) = e^(-(σ + jw)t) gets really big for values of σ close to minus infinity something the renders the whole integral non convergent. We can avoid that if we make sure the signal is causal.