Amostragem e quantização
\[ f_\text{Nyq} = 2 B_x \qquad T_\text{Nyq} = \frac{1}{f_\text{Nyq}} \qquad x(t) = \sum_{n \in \mathbb{Z}} \mathbf{x}[n] \sinc \left( \frac{t - n \Ta}{\Ta} \right) \] \[ \MSE \approx \frac{\Delta^2}{12} \qquad \SNR \approx 3 \frac{P_x}{A^2}{L^2} \qquad L = 2^k \qquad \Rb = \fa k \]
Modulação por amplitude de pulso
\[ x(t) = \sum_{n \in \mathbb{Z}} \mathbf{u}[n]p(t - n \Ts) \qquad \Rs = \frac{1}{\Ts} \] \[ S_x(f) = \frac{1}{\Ts} |P(f)|^2 S_\mathbf{u}(\Ts f), \quad S_\mathbf{u}(\phi) = \Fourier \{ C_\mathbf{u}[\ell] + \mu_\mathbf{u}^2 \} \] \[ q(t) = p(-t) \quad \text{(filtro casado)} \] \[ \sum_{k \in \mathbb{Z}} H(f - k \Rs) = \Ts \quad \text{(critério de Nyquist)} \] \[ \Delta = W - \frac{\Rs}{2} \qquad \alpha = \frac{\Delta}{\Rs / 2} \qquad W = \frac{\Rs}{2}(1 + \alpha) \]
Transmissão digital em banda base
\[ \Es = \frac{1}{M} \sum_{j=0}^{M-1} E_{s_j} \qquad P_x = \Es\Rs = \Eb\Rb \qquad \rho = \frac{\Rb}{W} \] \[ \Pb = Q \left( \sqrt{\frac{\Eb - \mu_\mathrm{b}^2}{N_0 / 2}} \right) \quad \text{(PAM binário)} \] \[ \Pb = Q \left( \sqrt{\frac{2 \Eb}{N_0}} \right) \quad \text{(binário, polar)} \qquad \Pb = Q \left( \sqrt{\frac{\Eb}{N_0}} \right) \quad \text{(binário, unipolar)} \] \[ \Ps = \frac{2(M-1)}{M} Q \left( \sqrt{\frac{6 (\Es - \mu_\mathrm{s}^2)}{(M^2 - 1) N_0}} \right) \quad \text{($M$-PAM uniforme)} \]
Bits vs símbolos
\[ M = 2^k \qquad \Tb = \frac{\Ts}{k} \qquad \Rb = k \Rs \qquad \Eb = \frac{\Es}{k} \] \[ \Pb \approx \frac{\Ps}{k} \quad \text{(Gray, alta SNR)} \qquad \Pb = \frac{M/2}{M-1}\Ps \quad \text{(ortogonal)} \]
Transmissão digital em banda passante
\[ \Pb = Q \left( \sqrt{\frac{d_{01}^2}{2 N_0}} \right) \quad (M = 2) \qquad \Ps \approx V Q \left( \sqrt{\frac{\dmin^2}{2 N_0}} \right) \quad (M > 2,\ \text{alta SNR}), \] \[ \Pb = \frac{1}{2} \mathrm{e}^{-\frac{\Eb}{N_0}} \quad \text{(DBPSK, não-coerente)} \qquad \Pb = \frac{1}{2} \mathrm{e}^{-\frac{\Eb}{2 N_0}} \quad \text{(BFSK, não-coerente)} \] \[ W = (1 + \alpha) \Rs \quad \text{(ASK, PSK, QAM)} \qquad W = (M - 1)\Delta f + (1 + \alpha) \Rs \quad \text{(FSK)} \qquad \]
Espaço de sinais
\[ \vec{u} \bullet \vec{v} = \sum_{i=1}^{N} u_i v_i \qquad \left\lVert \vec{u} \right\lVert^2 = \vec{u} \bullet \vec{u} \] \[ x(t) \bullet y(t) = \int_{-\infty}^{\infty} x(t) y(t) \dif t \qquad E_x = x(t) \bullet x(t) = \left\lVert x(t) \right\lVert^2 \] \[ s_{j,i} = s_j(t) \bullet \phi_i(t), \quad 0 \leq i < N, \ 0 \leq j < M \]