DSP is all about when you can approximate discrete systems with continuous ones and vice versa. Sampling theorems. Nyquist rates, Compressive sampling, nonuniform signal sampling, stochastic signal sampling etc.
There are a few ways to frame this. Traditionally we talk about Shannon sampling theorems, Nyquist rates and so on. To be frank, I haven’t actually read Shannon, because the setup is not useful for the types of problems I face in my work, although I’m sure it boils down to some very similar results.
The received-wisdom version of the Shannon theorem is that you can reconstruct a signal if you know it has frequencies in it that are “too high”. Specifically, if you sample a continuous time signal at intervals of \(T\) seconds, then you had better have no frequencies of period shorter than \(2T\). I’m playing fast-and-loose with definitions here - the spectrum here is the continuous Fourier spectrogram. If you do much non-trivial signal processing, (in my case I constantly need to do things like multiplying signals) it rapidly becomes impossible to maintain bounds on the support of the spectrogram (TODO explain this with diagrams)
This doesn’t tell us much about more bizarre nonuniform sampling regimes, mild violations of frequency constraints, or whether other sets of (perhaps more domain-appropriate) constraints on our signals will lead to a sensible reconstruction theory.
Let’s talk about the modern, abstract and fashionable Hilbert-space framing of this problem This way is general, and based on projections between Hilbert spaces. Nice works in this tradition are, e.g. (Vetterli, Marziliano, and Blu 2002) that observes that you don’t care about Fourier spectrogram support, but rather the rate of degrees of freedom to construct a coherent sampling theory. Also accessible is (Unser 2000), which constructs the problem of discretising signals as a minimal-loss projection/reconstruction problem.
More recently you have fancy persons such as Adcock and Hansen unifying compressed sensing and signal sampling (Adcock et al. 2014; Adcock and Hansen 2016) with more or less the same framework, so I’ll dive into their methods here.
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Adcock, Ben, Anders Hansen, Bogdan Roman, and Gerd Teschke. 2014. “Generalized Sampling: Stable Reconstructions, Inverse Problems and Compressed Sensing over the Continuum.” In Advances in Imaging and Electron Physics, edited by Peter W. Hawkes, 182:187–279. Elsevier. https://doi.org/10.1016/B978-0-12-800146-2.00004-7.
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