DSP is all about when you can approximate discrete-time systems with continuous ones and vice versa. Here are some details on this time-honoured form of coarse graining.
Sampling theorems. Nyquist rates, etc. Neat tricks with SDEs; relationship to classic state filter inference. The classic connection here is difference equations versus differential equations, because of some nice Markov properties. There are other possible relations, though, with actual time-delay equations.
NB: This is a giant mess of errors and outdated stuff; I wish I had time to fix all the terrible omissions I made here.
Improving Neural Models by Compensating for Discrete Rather Than Continuous Filter Dynamics when Simulating on Digital Systems.
I assume regular discretisation here, (uniform sampling, if you are taking measurements) but there is also stuff at nonuniform sampling.
Signal processing without equal bins and thus a simple Nyquist Theorem. It turns out that this is not fatal for many systems, e.g. it’s almost simple for linear systems, although it requires a leeeettle bit of stochastic calculus. In the non-linear case, it will probably be more irritating.
Keywords: Nonuniform sampling, irregular sampling.
Signal reconstruction is when you have knowledge of the system’s bandwidth, and you wish to reconstruct the most likely actual form for a realisation.
Reviews in [BaSt10], [PiPe04], [Eng07], [Unse00].
This is sometimes extended to include signal reconstructions with uncertain sampling times.
For FFT of unevenly sampled points, you can try the Non uniform FFT (NuFFT).
TODO: Lomb–Scargle normalized periodogram and its uses.
Julius O. Smith, Free Resampling Software.
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