# A mathematical justification of discontinuity detection using STFT

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?

• Have you ruled out using a time domain derivative estimator? Feb 23 at 15:59
• @DanBoschen, no I haven't. I'm specifically investigating STFT. Feb 23 at 16:34
• 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. Feb 23 at 18:41
• @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. Feb 24 at 5:08