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I need to design a function with an specific shape. I cannot come up with some equation that fits the kind of function that I am looking for.

My requirements are:

  1. The function is defined from 0 to N, where N is a natural number (it is a sample).after N, the signal returns 0.

  2. the signal has a plateau or maximum where the signal starts. After some time, the amplitude decays rapidly until it touches the bottom (zero).

  3. we can control the speed of decay by a coefficient k, that allows the function to be reduced more or less rapidly.

Here is an example of something similar but I am looking for a mirrored version:

What kind of function could fit these features?

Thanks!

GGChe
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    Try $0.125-h(x)$. In general try manipulating the sigmoid function. It doesn't get to exactly zero though. – Benjamin Wang Oct 25 '23 at 21:45
  • Great! thank you very much for your quick response. that is exactly what I am looking for. I modified the sigmoid function a little bit and it fits what I am looking for. I will set it manually to zero after it decays below the curve. – GGChe Oct 25 '23 at 22:04
  • You could also try $\exp\left(k-\frac{k}{1-\left(\frac{x}{N}\right)^2}\right)$, which smoothly approaches $0$ as $x\to N^-$. Such a function would look like this. – Kyan Cheung Oct 26 '23 at 00:50
  • that would also work! thanks. I will test it out – GGChe Oct 26 '23 at 18:17

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