We have infinite binary sequences of type $$\langle g_n \rangle_{j=4}=\{0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,1,...\} \,;\, n\to\infty$$ where $j$ indicates the length of a period.
we try to express them with a Fourier in a shape similar the spectrum:
$$G(x)=\sum_{k=1}^\infty a_k\,e^{(i\,2\pi\,k\,x / j)}$$
None of us is good in Fourier at all. May someone help us with the right expression.
Many thanks
The inverse Fourier in its most elegant form is for such sequences:
$$G(x)=\sum_{k=-j}^j \frac{2 \pi}{j}\,e^{(i\,2\pi\,(k-1)\,x / j)}$$
while $i=\sqrt{-1}$ and angular frequency:
$$\omega_j=\frac{2 \pi}{j}$$