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Generalizations of Fourier analysis (2021)

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Re: Generalizations of Fourier analysis (2021)

#21

Post by nextaccountic »

jdhwosnhw wrote: post #15
mananaysiempre wrote: post #9

Earlier quoted context omitted.

(Note: GGP, not GP.) I meant complex frequency and not complex amplitude though.

From a physics perspective, complex frequency results in “evanescent waves” - ie, waves that decay rather than oscillate (technically a fully complex frequency of the form a+ib will both oscillate and decay)

Yep: e^iω (with ω real) is an oscillation, but e^σ (with σ real) is an exponential decay when σ 0 and constant with σ = 1

so e^(σ + iω) = e^σ * e^iω is just an exponential growth or decay modulated by a sinusoid.. or, if σ is one, is just a pure oscillation

ω is the usual frequency, but σ + iω is the complex frequency. the fourier transform deals with function that receives ω as input, and the laplace transform deals with functions that receives σ + iω instead.

so the fourier transform is just a special case of the laplace transform with σ = 0

Re: Generalizations of Fourier analysis (2021)

#22

Post by nhatcher »

rthomas6 wrote: post #13

Ugh, I need to learn more math. How do I even start? I know multi variable calculus, I know the basics of linear algebra, and I know Fourier transforms. Yet this article is half gibberish to me.

I wish I could help you.

If you are any sort of programmer "Linear Algebra" is going to be the best bang for your buck. You can have a look a some of the books of Gilbert Strang or his online courses. Beyond that the classic books in "Abstract Algebra" are those by Serge Lang or Jacobson. Either or might be too difficult and maybe not worth it.

I always tell people to study integration correctly. By that I mean measure theory, with the Riemann integral taught in schools you can only go so far.

With those two you are in a good position to tackle functional analysis. The classic book on the subject is that of Walter Rudin but maybe it too hard for self learners. A better alternative might be the open course at MIT:

https://ocw.mit.edu/courses/18-102-introduction-to-functiona...

I also like the book by Vladimir Kadets.

Functional analysis is terribly important in practical areas like signal processing, quantum mechanics or partial differential equation in general. so it might be worth your pain.

There are many ways you can go from there.

I can't leave without recommending one more things. If you are curious of advanced mathematics as a tourist the three books by Avner Ash and Robert Gross make a great read

Re: Generalizations of Fourier analysis (2021)

#23

Post by xeonmc »

nextaccountic wrote: post #21
jdhwosnhw wrote: post #15

Earlier quoted context omitted.

From a physics perspective, complex frequency results in “evanescent waves” - ie, waves that decay rather than oscillate (technically a fully complex frequency of the form a+ib will both oscillate and decay)

Yep: e^iω (with ω real) is an oscillation, but e^σ (with σ real) is an exponential decay when σ 0 and constant with σ = 1 so e^(σ + iω) = e^σ * e^iω is just an exponential growth or decay modulated by a sinusoid.. or, if σ is one, is just a pure oscillation ω is the usual frequency, but σ + iω is the complex frequency. the fourier transform deals with function that receives ω as input, and the laplace transform deals…

> so the fourier transform is just a special case of the laplace transform with σ = 0

Another useful way of looking at it: Laplace transform is doing many extra Fourier-transform-but-with-decay giving you a map of which "global decay timescale" fits your data best -- since each "slice" is itself sufficient to fully describe the time series

They are all cases of integral transforms with different choices of the set of "primitive fingerprints" -- see chirp transform, wavelet transform, chirplet transform etc -- all taking advantage of the fact that if you choose one set of basis "brushes" that are not redundant with each other (e.g. having red-green-blue brushes is independent, as is magenta-green-yellow but having red-green-blue-yellow is not) then you will be able to describe your signal in terms of a composition of those kernels.

Re: Generalizations of Fourier analysis (2021)

#24

Post by StarlaAtNight »

I asked ChatGPT to explain the intro section of this article like I'm five:

""" Fourier analysis is like playing with music notes. We have different ways of looking at these notes, like how high or low they are, or how long they last. In the same way, we have different ways of looking at signals and how they change over time or space.

Just like how we can make different types of music by combining different notes in different ways, we can make different types of signals by combining different types of Fourier analysis.

There are four classic ways to do Fourier analysis, and each one is like a different way of looking at the signal. They're like different types of glasses we can put on to see the signal more clearly.

But there are also other ways to do Fourier analysis, like looking at signals on different types of spaces or using different types of math. It's like having even more types of glasses to look at the signal!

So when you ask a mathematician what Fourier analysis is, they might give you a different answer depending on which type of glasses they like to wear. """

Re: Generalizations of Fourier analysis (2021)

#25

Post by nextaccountic »

mananaysiempre wrote: post #7
dustingetz wrote: post #5

Earlier quoted context omitted.

what is an intuition for complex frequency?

As I meant it—of, not for. I referred to the idea that by plugging an imaginary frequency into the Fourier transform [ETA: the grown-up Fourier transform with the complex exponent, not the schoolboy cosine kludge], you get the Laplace transform, and while that changes the inverse Fourier transform in a different way, it’s not hard to work out how specifically and obtain the inverse Laplace transform. Why you’d want t…

> but are now allowed to work with exponentially increasing functions, which standard Fourier theory (even the tempered distributions version) can’t accomodate.

So you can't take the fourier transform of an exponential? But.. it seems you can? https://proofwiki.org/wiki/Fourier_Transform_of_Exponential_...

Re: Generalizations of Fourier analysis (2021)

#26

Post by nextaccountic »

xeonmc wrote: post #23
nextaccountic wrote: post #21

Earlier quoted context omitted.

Yep: e^iω (with ω real) is an oscillation, but e^σ (with σ real) is an exponential decay when σ 0 and constant with σ = 1 so e^(σ + iω) = e^σ * e^iω is just an exponential growth or decay modulated by a sinusoid.. or, if σ is one, is just a pure oscillation ω is the usual frequency, but σ + iω is the complex frequency. the fourier transform deals with function that receives ω as input, and the laplace transform deals…

> so the fourier transform is just a special case of the laplace transform with σ = 0 Another useful way of looking at it: Laplace transform is doing many extra Fourier-transform-but-with-decay giving you a map of which "global decay timescale" fits your data best -- since each "slice" is itself sufficient to fully describe the time series They are all cases of integral transforms with different choices of the set of…

So, so I'm going with a rusty knowledge of a computer engineering course from years ago,

> a map of which "global decay timescale" fits your data best

What do you mean by this?

> -- since each "slice" is itself sufficient to fully describe the time series

But each slice is multiplied by an exponent.. which, okay, becomes a convolution that lets you recover the original function

Re: Generalizations of Fourier analysis (2021)

#27

Post by xeonmc »

nextaccountic wrote: post #26
xeonmc wrote: post #23

Earlier quoted context omitted.

> so the fourier transform is just a special case of the laplace transform with σ = 0 Another useful way of looking at it: Laplace transform is doing many extra Fourier-transform-but-with-decay giving you a map of which "global decay timescale" fits your data best -- since each "slice" is itself sufficient to fully describe the time series They are all cases of integral transforms with different choices of the set of…

So, so I'm going with a rusty knowledge of a computer engineering course from years ago, > a map of which "global decay timescale" fits your data best What do you mean by this? > -- since each "slice" is itself sufficient to fully describe the time series But each slice is multiplied by an exponent.. which, okay, becomes a convolution that lets you recover the original function

The baseline decay describes the global dissipation or amplification over time.

While Laplace transform is most useful in more complicated systems, this concept is actually best illustrated in a damped/amplified harmonic oscillator model as it serves as the primitive archetype that more complex systems are composed from.

In a nutshell, the general solution is a linear combination of two exponential that are either pure-decay or oscillatory whose imaginary parts, if the signal is to have no imaginary parts, must cancel each other out so that the result is a real signal, which means that the two must be "synchronized" in time against each ither, i.e. having the same oscillation frequency but in the opposite rotation direction (so the imaginary part opposes each other out), and with the same decay progression. This means that the average of their complex frequencies must be real, i.e. +/- for underdamped and +/- for overdamped, and so you can split out the mean as a common decay function, giving you decay(t)*( a*clkwise(t) + b*ccwise(t) )

    a,b:real
    E,F:real->complex
    y:real->real
    y = aE + bF
      = sum(a,b)sum(E,F)/2 + diff(a,b)diff(E,F)/2

Re: Generalizations of Fourier analysis (2021)

#28

Post by enriquto »

nextaccountic wrote: post #25
mananaysiempre wrote: post #7

Earlier quoted context omitted.

As I meant it—of, not for. I referred to the idea that by plugging an imaginary frequency into the Fourier transform [ETA: the grown-up Fourier transform with the complex exponent, not the schoolboy cosine kludge], you get the Laplace transform, and while that changes the inverse Fourier transform in a different way, it’s not hard to work out how specifically and obtain the inverse Laplace transform. Why you’d want t…

> but are now allowed to work with exponentially increasing functions, which standard Fourier theory (even the tempered distributions version) can’t accomodate. So you can't take the fourier transform of an exponential? But.. it seems you can? https://proofwiki.org/wiki/Fourier_Transform_of_Exponential_...

exp(-|x|) is not an exponential, it's just the easy, bounded, integrable, half of it :)
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