Poisson Formula
Spectrum of an ideally sampled signal
\(S\left(f\right)=\sum_{n=-\infty}^{\infty}{x\left(n\right)e^{-j2\pi\frac{nf}{F_e}}}\) (8)
The spectrum of an ideally sampled signal is continuous and periodic with a period of \(f_e=\frac{1}{T_e}\)
\(S\left(f\right)=\frac{1}{T_e}\sum_{n=-\infty}^{\infty}{S(f-nf_e)}\) (9)
Sampling \(\rightarrow\) Spectrum periodization
\(S\left(f\right)=\sum_{n=-\infty}^{\infty}{x\left(n\right)e^{-j2\pi\frac{nf}{F_e}}}=\frac{1}{T_e}\sum_{n=-\infty}^{\infty}{S(f-nf_e)}\) (10)