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I'm trying to detect rapid changes in a one-dimensional signal say $[0,1]\ni t \mapsto f(t) \in [-1,1]$. By rapid changes, I mean corner points, edges, or sharp transitions at a point for example the signal switches from a value $-1$ to $+1$ on a point.

For this, I'm using the STFT i.e., the Short-Time Fourier transform using some window functions (see Foundations of Time-Frequency Analysis by Grochenig, Chapter 3).

My question is:

Is there any justification or estimate indicating for what classes of signals STFT can provide some kind of qualitative answer about the discontinuity detection accuracy? I see that mathematically STFT can be applied for any $L^2([0,1];[-1,1])$ class of signals which can be very wild. But how does the data derived from an STFT, i.e., the time-frequency plot indicates, mathematically which part of the signal is jittery or at which point the signal switches? Is there any formal results/theorems in this line?

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  • $\begingroup$ Have you ruled out using a time domain derivative estimator? $\endgroup$ Feb 23 at 15:59
  • $\begingroup$ @DanBoschen, no I haven't. I'm specifically investigating STFT. $\endgroup$
    – Zeno San
    Feb 23 at 16:34
  • $\begingroup$ Why are you investigating STFT? I think the STFT is a poor method for this type of thing mainly because of the time/frequency uncertainty and the whacky way transients are represented in the phase of the STFT. Hence I'm guessing it will be hard to find a mathematical justification for something that's unlikely to work well in the first place. $\endgroup$
    – Hilmar
    Feb 23 at 18:41
  • $\begingroup$ @Hilmar, I'm investigating because I'm interested in it, and what is the justification for this 'unlikeliness' and 'poor' attribution? I can see STFT working pretty well, see Fig.2.1, Chapter 2 and Fig 3.1, 3.3, 3.4 Chapter 3, of the book I mentioned above, and I'm sure there will be some mathematical justification as the mathematical theory (even for tempered distributions) is very well developed. $\endgroup$
    – Zeno San
    Feb 24 at 5:08

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