There is plenty of literature indicating that the overshoot at a jump discontinuity from the GIBBS effect is 1.08949. However, this has primarily been demonstrated through the STEP input function. This is fine since the jump discontinuity occurs once and the amplitude of the function remains constant after the discontinuity has been encountered.
But what happens if the function amplitude changes after the discontinuity such as in the case of the triangle function? I have demonstrated that the rate at which the overshoot approaches the theoretical value of 1.08949 as a function of k terms in the series is different depending on the function behavior after the discontinuity. So I question the theoretical value of 1.08949 for all functions in general. I have yet to find a good explanation on this matter.
What I have attached is a plot for abs(Ppeak-1) vs k terms in the Fourier series approximation of: 1) rectangle function; 2) triangle function; 3) Friedlander function. The amplitude for the jump discontinuity is UNITY for all 3 cases & all 3 cases has the same period or function width.
The plots for the latter 2 show a dip opposed to the gradual asymptotic approach to 1.08949 value of the rectangle function. The reason for this is at the lower k values the peak value of the series approximation is less than UNITY. This is evident in the Output vs time plots. As the peak value approaches UNITY for higher k values the dip occurs due to abs(Ppeak-1). After the dip the overshoot approaches the 1.08949 value as k increases; however, it is done at different rates.
So this distinction in behavior leads me to question the 1.08949 value as being universal for all functions? Are there any good demonstrations for the GIBBS effect on different functions to show that this theoretical value is universally applicable?
I also included the m-files that generated the data for the plots in case one wants to examine the code.
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