Introduction
If this series (1) converges, the inverse Fourier transform is defined by :
\(x\left(n\right)=\frac{1}{F_e}\int_{-\frac{F_e}{2}}^{+\frac{F_e}{2}}{X(f)}e^{j2\pi\frac{nf}{F_e}}\) (6)
Noticed :
We verify that X\(\left(f\right)\ \\) is a periodic function of period \(F_e\) (due to sampling). If we replace f by \({f+kF}_e\) :
\(e^{-j2\pi\frac{n(f+k.F_e)}{F_e}}=e^{-j2\pi\frac{nf}{F_e}}+e^{-j2\pi\frac{nk.F_e}{F_e}}=e^{-j2\pi\frac{nf}{F_e}}\) (7)