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I'm requesting resourses to dive deep and get a good grip on complex signals, Fourier analysis and connections to information theory and information encoded by complex signals.

My background: 3rd year physics undergraduate, in the process of learning quantum mechanics, have some intuitive understanding of Fourier transforms.

My motivation: I am trying to understand and reason about information encoded by wave functions. Specifically, I am trying to draw connections to Nyquist-Shannon sampling theorem.

Here's some problems I encountered:

  • a lot of resources related to signal processing spend a lot of time talking about real-valued functions and I need a more general understanding of how things work for complex-valued functions without putting a lot of emphasis on the special case of real functions;
  • a lot of resources related to signal processing quickly go from idealized signals to those constrained by the limitations of the real world(noise and stuff) and I need a better understanding of these idealized signals without delving deeper into things a real engineer would need;

Additionally, I'd also love to see some quantum information resources that go into information encoded in continuous systems. A lot of what I've seen is centered on discrete cases like spin because that's what is actually feasible to use in quantum computing.

I'm trying to get the knowledge I need piece by piece, but I thought it doesn't hurt to try and you people about some recommendations :)

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    $\begingroup$ I probably won't have an answer for you in any case, but just to be clear, do you seek a better understanding of; (A) how to find the Fourier series (FT) for a given periodic function? (B) how to interpret the FT for a given periodic function? (C) techniques for obtaining the discrete Fourier transform (DFT) of some sampled signal, or (D) how to interpret the DFT of a signal? $\endgroup$ Dec 10, 2023 at 18:30
  • $\begingroup$ I am interested in a more general case of aperiodic functions and their Fourier transforms. (C) and (D) would also be helpful, though! $\endgroup$
    – sensorer
    Dec 10, 2023 at 18:50
  • $\begingroup$ Just out of curiosity, can you give more context for how you want to apply the Nyquist theorem to quantum mechanics? The Nyquist theorem is useful when you sample a continuous signal at a finite sampling rate (so instead of measuring the full function $f(t)$, you only measure $f(n/F_s)$ at some discrete times $n/F_s$ where $n$ is an integer and $F_s$ is the sampling rate). In quantum mechanics, especially undergrad quantum mechanics, normally one calculates the wavefunction (or at least eigenfunctions) at all values of space and time and doesn't worry about sampling/discretization issues. $\endgroup$
    – Andrew
    Dec 10, 2023 at 20:22
  • $\begingroup$ I have this thing when if I do not understand something I go through gedanken experiments to see how the thing I do not understand could work in principle. A few months ago I tried to explain to myself how a continuous wavefunction could exist in discrete space. Replacing differentials with finite differences seemed boring and unsatisfying, but then I remembered about Nyquist theorem and the fact that one can reconstruct continuous functions from discrete samples. I've been obsessed with that idea for the last 4 months. It made me stumble onto many unexpected results and I want to go further $\endgroup$
    – sensorer
    Dec 10, 2023 at 20:38
  • $\begingroup$ There's probably not much to it(that said, delusional as I am, I have already convinced myself that I've stumbled upon a deep law of nature). But it is a great problem to work on, in a sense that it gives me lots of motivation to master QM and go into GR and QFT in the future. Which means that on the other side I'll have a better understanding of how our universe works either way $\endgroup$
    – sensorer
    Dec 10, 2023 at 20:46

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Recommendations are no longer needed because Achim Kempf and Yufang Hao seem to have already discovered everything there is to know here.

For anyone curious, here's a link: https://uwspace.uwaterloo.ca/bitstream/handle/10012/6311/Hao_Yufang.pdf

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