![]() ![]() ![]() ![]() After just a few your eye will be unable to detect the difference.Would be interesting to see what the ear picks up? Signal after each successive stage will more closely resemble a sinewave. Really it only resembles one.Then there's the practical problem of keeping an integrator zeroed - it'll integrate its offset error and eventually saturate.So if you build one use a low pass filters instead of integrators. As you cascade stages of integration and attenuate the harmonics you approach just the fundamental which would be a pure sine wave.Apparently some people are happy enough with two stages to claim it's a sine wave. It attenuates not removes higher harmonics.So integrating once attenuates harmonics, integrating again attenuates them more. ![]() Fundamental plus odd harmonics.Much easier approach using FourierIntegration can be thought of as low pass filtering. It should swing between zero and some positive value, just like transformer inrush.Next look at Fourier transform of a triangle wavefor some reason system refuses to show the wikipedia link ipasted it in four times.Look for wikipedia triangle wave. It has been a long long time since I'd thought of parabolas in a non-trivial way (I basically just remembered locus and focus). Okay, I just did something I would have done the other day I was on PF if I'd have had the time, which was to wiki 'parabola'. Gets a bit difficult for higher levels of integration so you have to do some proper calculus for these.You might like to conjecture what the curve looks like after a large number of successive integrations of the original square wave form. Take integrated areas above the y=0 line as positive and below as negative.You can do square to triangular and triangular to the next one using simple geometric calculation of areas. You can see immediately what successive integrals of a square wave form look like if you remember that one definition of integration is simply the area under a curve.Just draw graphs of the integrated area under the curves as you progress along the x axis. ![]()
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