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When a square wave doesn't jump, its oscillators aren't aligned:

But if they are in sync, the wave will jump to its extrema:

However this is just an example of square waves. The tool Understanding Fourier Approximations & Oscillators only have square, sawtooth or triangle wave to examine, but as how I see when they are in sync, the wave is always at the maximum or minimum. This matches the intuition when the "arm" is longest. However in the square or sawtooth wave, one position in sync yields both maximum and minimum as the same time, not one as usual. While this can be explained as they have to be close around the jump, this show me that a more rigorous mathematical description is needed. What would happen for a more usual waveform?

Do you know what special things could be when all components are in sync?

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  • $\begingroup$ what is a "more usual periodic waveform that jumps" than a rectangular wave? I'm a bit confused as to what you're asking us. It's the DFT - it does what the DFT does. There's nothing special about any signal you put into that or get out of that, unless you "attribute" specialness to it from a human point of view. $\endgroup$ – Marcus Müller Mar 20 '18 at 17:12
  • $\begingroup$ I would expect that phase-aligned harmonics imply large changes in amplitude in a short period of time; IOW a large signal slope. $\endgroup$ – MBaz Mar 20 '18 at 18:41
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    $\begingroup$ @MarcusMüller I'm looking for more rigorous math to describe the extrema. It seems that in the point of view of signal processing, those points are no special, right? $\endgroup$ – Ooker Mar 21 '18 at 0:56
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sine waves all have zero crossings at the same phase of $0$ or $\pi$ and have a peak (or valley) with a phase of $\pm\tfrac{\pi}{2}$.

cosine waves all have zero crossings at the same phase of $\pm\tfrac{\pi}{2}$ and have a peak (or valley) with a phase of $0$ or $\pi$.

so if it's a $\sin(\cdot)$ series (with no cosine terms), expect a big zero-crossing when all of the phases line up at $0$ or $\pi$.

or expect a $\cos(\cdot)$ series (with no sine terms) to have a major zero-crossing transition to occur when the phases are $\pm\tfrac{\pi}{2}$.

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