Remark: All the proof can be adapted from the continuous Fourier Transform and are omitted.
Fourier transform Let $s = \lbrace s[t]\rbrace_{t\in\mathbb{Z}}\in \ell^1(\mathbb{Z})\cap \ell^2(\mathbb{Z})$ a $\mathbb{R}$ or $\mathbb{C}$ valued sequence . The discrete time Fourier transform of $\lbrace s[t]\rbrace_{t\in\mathbb{Z}}$ is: $$ \hat s(\nu) = \sum_{t=-\infty}^{+\infty} s[t] e^{-i2\pi n\nu}\ . $$
Plancherel-Parseval Let $s = \lbrace s[t]\rbrace_{t\in\mathbb{Z}} \in \ell^1(\mathbb{R})\cap \ell^2(\mathbb{R})$ and $\hat s$ its Fourier transform. Then $$ \Vert s\Vert_2^2 = \Vert\hat s\Vert_2^2 $$ ie $$ \sum_{n=-\infty}^{\infty} |s[t]|^2 = \int_{-\frac{1}{2}}^{\frac{1}{2}} |\hat s(\nu)|^2 \mathrm{d} \nu $$
Inversion of the Fourier transform Let $s = \lbrace s[t]\rbrace_{t\in\mathbb{Z}}$ a stable signal of finite energy (denoted by $s\in \ell^1(\mathbb{Z})\cap \ell^2(\mathbb{Z})$). Let $\hat s$ its Fourier transform. Then $$ s[n] = \int_{-\frac{1}{2}}^{\frac{1}{2}} \hat s(\nu) e^{i2\pi n\nu} \mathrm{d} \nu\ . $$
Let $u,v\in \ell^1(\mathbb{Z})\cap \ell^2(\mathbb{Z})$ and $\hat u, \hat v$ their Fourier transform
- Convolution: $$\widehat{u* v}(\nu) = \hat u(\nu) \hat v(\nu)$$
- Multiplication: $$\widehat{u . v}(\nu) = (\hat u * \hat v ) (\nu)$$
- Translation: let $v_a[t] = u[t-a]$ $$\hat v_a(\nu) = e^{-i2\pi a\nu} \hat u(\nu)$$
- Modulation: let $v_{\theta}[t] = e^{i2\pi \theta t}u[t]$ $$\hat{v}_\theta(\nu) = \hat u(\nu - \theta)$$
- Dérivée fréquentielle: let $v_p[t] = (-i2\pi t)^p u[t]$ $$\hat v_p(\nu) = \hat u^{(p)}(\nu)$$
- Complexe conjuguée $$\widehat{\bar u}(\nu) = \overline{ \hat{ u}(-\nu)}$$
- Symétrie hermitienne $$\forall t \in\mathbb{Z}\quad u[t]\in\mathbb{R} \Rightarrow \hat u(-\nu) = \overline{\hat{ u}(\nu)}$$