Finite Fourier Transform

Digital signals

let a discrete finite sequence $\boldsymbol{s}=\lbrace s[0],…,s[N-1]\rbrace$

$$ \hat \boldsymbol{s}[\nu] = \sum_{t=0}^{N-1} s[t]e^{-i2\pi\frac{\nu}{N} t}\quad \nu\in\lbrace 0,\ldots,N-1 \rbrace $$ and $$ \boldsymbol{s}[t] = \frac{1}{N}\sum_{\nu=0}^{N-1} \hat s[\nu]e^{-i2\pi\frac{\nu}{N} t}\quad \nu\in\lbrace 0,\ldots,N-1 \rbrace $$

The time signal $\boldsymbol{s}$ and the Fourier transform $\hat \boldsymbol{s}$ are cnosidered as $N$-periodic sequences !