I have to approximate the red function with Fourier harmonics.

***enter image description here***

I am calculating complex Ck the following way.

enter image description here

I am not sure what to put for period - T (6 or 2pi or what?). What integral limits to use (2/6*2pi and 4/6*2pi ??)? The way I showed on the picture gives me invalid results... so how to fix it?


If you already know the explicit form of the function $f(x)$, then it's possible to find the periode by solving this equation $f(x) = f(x+P)$.

If you have a graphical representation of the function (finite), maybe by collecting the numerical values or other means then you have to follow the next two steps:

Step one:

Trivial extension

Consider the general finite signal $f \in [[a, b) \rightarrow \mathbb{C}]$ with $p= b-a \in \mathbb{R}$ then its trivial extension is:

$$f_{triv} =\begin{cases}f(x)&x\in [a, b)\\ 0& otherwise \end{cases} $$

Step two:

Periodic extension $$f_{per} = \sum_{m \in \mathbb{Z}} f_{triv}(x-mp)$$

and now you can talk about the periodic function and its Fourier representation.

The periode is indeed six.


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