Hey hey,
this is a maths question, perhaps someone can point me in the right direction or point out my error of thinking.
I tried to use quartic functions to create waveforms. Formulae of the kind:
a x^4 +- b x^2 + c
are symmetric in respect to the y-axis. With the right coefficients the resultant shape has three extrema. Example:
1/4 x^4 - 2 x^2 + 4
has two minima at -2 and 2 at y=0 and one maximum in between at 0 with y=4.
The upside-down version of the function:
-1/4 x^4 + 2 x^2 - 4
has maxima at -2 and 2 of y=0 and a minimum at 0 y=-4.
These two shapes, in the set boundaries of -2 and 2, could be concatenated to create one symmetric wave. At -2 and 2 their rate of change is 0, so that should avoid undue aliasing or other issues.
My question: is there a way to increase the steepness of the "bump" between such outer maxima/minima and construct the whole function in such a way that all important values are known: i.e. the x coordinates of the extrema.
I tried to create different functions, but they all sound the same. I've used two approaches: create the first derivative of degree three in factorial form:
(x + a) * (x + 0) * (x -a)
And coming from a more visual point of view: x^4 has a minimum at 0. A "bump" cna be introduced by subtracting a quadratic function with a factor. The minima can be calculated with:
x^4 = a * x^2
with a good factor a this is easy enough to solve.
Though all experiments sound the same. I suppose that this is due to the "bump" having the same steepness. Or could it be that I just need to increase the factor a quite a lot?
Or is this futile with just quartic functions?
Best wishes,
Jeanette
