Sinais e sistemas
\[ E_x = \int_{-\infty}^{\infty} |x(t)|^2 \dif t \qquad P_x = \lim_{\tau \to \infty} \frac{1}{\tau} \int_{-\tau/2}^{\tau/2} |x(t)|^2 \dif t \qquad P_x = \int_{-\infty}^{\infty} S_x(f) \dif f \qquad \] \[ P_x = \frac{1}{T} \int_{-T/2}^{T/2} |x(t)|^2 \dif t \quad \text{(sinal periódico)} \]
Quantização
\[ \MSE \approx \frac{\Delta^2}{12} \qquad \SNR \approx 3 \frac{P_x}{A^2}{L^2} \]
Espaço de sinais
\[ \vec{u} \bullet \vec{v} = \sum_{i=1}^{N} u_i v_i \qquad \left\lVert \vec{u} \right\lVert^2 = \vec{u} \bullet \vec{u} \] \[ x(t) \bullet y(t) = \int_{-\infty}^{\infty} x(t) y(t) \dif t \qquad E_x = x(t) \bullet x(t) = \left\lVert x(t) \right\lVert^2 \] \[ s_{j,i} = s_j(t) \bullet \phi_i(t), \quad 1 \leq i \leq N, \ 0 \leq j < M \] \[ E_s = \frac{1}{M} \sum_{j=0}^{M-1} E_{s_j} \qquad V = \frac{1}{M} \sum_{j=0}^{M-1} V_j \] \[ \Pb = Q \left( \sqrt{\frac{d_{01}^2}{2 N_0}} \right) \quad (M = 2) \qquad \Ps \approx V Q \left( \sqrt{\frac{\dmin^2}{2 N_0}} \right) \quad (M > 2,\ \text{alta SNR}) \] \[ \Pb \approx \frac{\Ps}{k} \quad \text{(Gray, alta SNR)} \qquad \Pb = \frac{M/2}{M-1}\Ps \quad \text{(ortogonal)} \]
Transmissão digital
\[ h_\mathrm{RX}(t) = c \, h_\mathrm{TX}(\Ts-t) \quad \text{(filtro casado)} \] \[ \Rs = \frac{1}{\Ts} \qquad P = \Es\Rs \] \[ M = 2^k \qquad \Tb = \frac{\Ts}{k} \qquad \Rb = k \Rs \qquad \Eb = \frac{\Es}{k} \] \[ \Pb = Q \left( \sqrt{\frac{2 \Eb}{N_0}} \right) \quad \text{(polar)} \qquad \Pb = Q \left( \sqrt{\frac{\Eb}{N_0}} \right) \quad \text{(on-off)} \] \[ \Ps = \frac{2(M-1)}{M} Q \left( \sqrt{\frac{6 \Es}{(M^2 - 1) N_0}} \right) \quad \text{(PAM)} \] \[ \Pb = \frac{1}{2} \mathrm{e}^{-\frac{\Eb}{N_0}} \quad \text{(DBPSK, não-coerente)} \qquad \Pb = \frac{1}{2} \mathrm{e}^{-\frac{\Eb}{2 N_0}} \quad \text{(BFSK, não-coerente)} \] \[ B = (1 + \alpha) \frac{\Rs}{2} \quad \text{(PAM)} \] \[ B = (1 + \alpha) \Rs \quad \text{(ASK, PSK, QAM)} \qquad B = (M - 1)\Delta f + (1 + \alpha) R_s \quad \text{(FSK)} \qquad \] \[ \rho = \frac{R_b}{B} \]