$$F_k[n] = \frac{1}{N}\sum_{m=0}^{N-1} f[n+m]\, e^{-j2\pi km/N} \quad\Longrightarrow\quad \mathbf{F}[n] = \mathbf{W}\,\mathbf{f}_{n:n+N}, \qquad W_{k,m} = \tfrac{1}{N} e^{-j2\pi km/N}$$
$$F_k[n] = \frac{1}{N}\sum_{m=0}^{N-1} f[n+m]\, e^{-j2\pi km/N} \quad\Longrightarrow\quad \mathbf{F}[n] = \mathbf{W}\,\mathbf{f}_{n:n+N}, \qquad W_{k,m} = \tfrac{1}{N} e^{-j2\pi km/N}$$