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Suppose we have a time bounded original signal f(t) where f is nicely behaved, for simplicity we can restrict it to a polynomial. The crux of the problem is that its analytical form is unknown, we only have the sampled values of the signal and its derivatives. The support of the function is [a,b] and we also know its derivatives on the entire support of the function.

Next suppose we want to time-stretch the signal using the following transform: $t_s \mapsto t + g(t)\cdot D$ where $g(t)$ is a smooth polynomial step function that maps $[a,b]$ to $[0, 1]$. The end result is a signal that has the original values but it is non-linearly mapped to $[a, b+D]$. The analytical form of $g(t)$ is known.

The question is: can we compute/estimate the derivative of the time-stretched signal $f'(t_s)$ as a function of the original signal's derivatives and values?

The question was prompted by a desire to avoid the numerical derivative computation of the stretched signal, since that introduces a lot of noise that is not originally present in the non-stretched derivative of the signal.

Marcus Müller
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Emanuel Hristea
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  • do we know that the polynomial is sufficiently bandwidth limited so that Fourier theory allows us to fully reconstruct it from the samples without knowing it's a polynomial? – Marcus Müller Sep 24 '23 at 20:05
  • Suppose we have a time bounded original signal f(t) where f is nicely behaved, for simplicity we can restrict it to a polynomial. that would be a simplification in many fields, but here, it makes things hard; can we make different simplifying assumptions, such as the bandlimiting above? Or a polynomial over $e^{\sqrt{-1}t}$ instead of over $t$? – Marcus Müller Sep 24 '23 at 20:06
  • But wait, all you're asking for is the derivative of $f(g(t))$, knowing the derivatives of f and g as well as g. That's trivial, just apply the chain rule of derivation; $d/dt f(g(t)) = f'(g(t))\cdot g'(t)$. Since you know the original f, and since the mean value theorem applies, you should be reasonably well able to estimate f' at points between the original sampling instants through linear approximation. Other than that, anything that gets you closer to the actual values would boil down to exactly the same numerical effort as finding the polynomial from the original samples. – Marcus Müller Sep 24 '23 at 21:06
  • @MarcusMüller Thank you for the insights. Indeed we can assume it is band-limited. Indeed we can compute the derivative using the chain rule but the estimation is the tricky bit. – Emanuel Hristea Sep 24 '23 at 21:25

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