Formulário: Sinais e sistemas
Sinais básicos
\[ \rect(x) = \begin{cases} 1, & \text{se } |x| < 1/2, \\ 0, & \text{caso contrário.} \end{cases} \qquad \tri(x) = \begin{cases} 1 - |x|, & \text{se } |x| < 1, \\ 0, & \text{caso contrário.} \end{cases} \qquad \sinc(x) = \frac{\sin(\pi x)}{\pi x} \] \[ \sign(x) = \begin{cases} 1, & x > 0, \\ 0, & x = 0, \\ -1, & x < 0. \end{cases} \qquad \step(x) = \begin{cases} 1, & x > 0, \\ 0, & x < 0. \end{cases} \]Energia de um sinal real
\[ E_x = \int_{-\infty}^{\infty} [x(t)]^2 \dif t = \int_{-\infty}^{\infty} \Psi_x (f) \dif f \] \[ \psi_x(\tau) = \int_{-\infty}^{\infty} x(t) x(t - \tau) \dif t \qquad \Psi_x(f) = | X(f) |^2 \] \[ \Psi_x(f) = \cal{F} \{ \psi_x(\tau) \} \]Potência de um sinal real
\[ P_x = \lim_{T \to \infty} \frac{1}{T} \int_{-T/2}^{T/2} [x(t)]^2 \dif t = \int_{-\infty}^{\infty} S_x(f) \dif f \] \[ P_x = \frac{1}{T_0} \int_{-T_0/2}^{T_0/2} |x(t)|^2 \dif t \quad \text{(sinal periódico)} \] \[ R_x(\tau) = \lim_{T \to \infty} \frac{1}{T} \int_{-T/2}^{T/2} x(t) x(t - \tau) \dif t \qquad S_x(f) = \lim_{T \to \infty} \frac{1}{T} | X_T (f) |^2 \] \[ S_x(f) = \cal{F} \{ R_x(\tau) \} \]Definição e propriedades da transformada de Fourier em tempo contínuo
\[ \vphantom{\int_{-\frac{1}{2}}^{\frac{1}{2}}} \vphantom{\sum_{n=-\infty}^{\infty}} \hat{x}(f) = \Fourier \{ x(t) \} = \int_{-\infty}^{\infty} x(t) \ee^{-\iu 2 \pi f t} \dif t \qquad x(t) = \Fourier^{-1} \{ \hat{x}(f) \} = \int_{-\infty}^{\infty} \hat{x}(f) \ee^{\iu 2 \pi t f} \dif f \] \[ \Fourier \{ a x_1(t) + b x_2(t) \} = a \Fourier \{ x_1(t) \} + b \Fourier \{ x_2(t) \} \] \[ \Fourier \{ x_1(t) x_2(t) \} = \Fourier \{ x_1(t) \} \star \Fourier \{ x_2(t) \} \qquad \Fourier \{ x_1(t) \star x_2(t) \} = \Fourier \{ x_1(t) \} \Fourier \{ x_2(t) \} \] \[ \Fourier \{ x(t - t_0) \} = \hat{x}(f) \ee^{-\iu 2 \pi t_0 f} \qquad \Fourier \{ x(t) \ee^{\iu 2 \pi f_0 t} \} = \hat{x}(f - f_0) \] \[ \Fourier \{ \hat{x}(t) \} = x(-f) \] \[ \Fourier \{ x(at) \} = \dfrac{1}{|a|} \hat{x} \left( \dfrac{f}{a} \right) \] \[ \Fourier \left \{ \dfrac{\dif^n}{\dif t^n} x(t) \right \} = (\iu 2 \pi f)^n \hat{x}(f) \qquad \Fourier \{ t^n x(t) \} = \left( \dfrac{\iu}{2\pi} \right)^n \dfrac{\dif^n}{\dif f^n} \hat{x}(f) \]
\[ \hat{x}(\phi) = \Fourier \{ x[n] \} = \sum_{n=-\infty}^{\infty} x[n] \ee^{-\iu 2 \pi \phi n} \qquad x[n] = \Fourier^{-1} \{ \hat{x}(\phi) \} = \int_{-\frac{1}{2}}^{\frac{1}{2}} \hat{x}(\phi) \ee^{\iu 2 \pi n \phi} \dif \phi \] \[ \Fourier \{ a x_1[n] + b x_2[n] \} = a \Fourier \{ x_1[n] \} + b \Fourier \{ x_2[n] \} \] \[ \Fourier \{ x_1[n] x_2[n] \} = \Fourier \{ x_1[n] \} \star \Fourier \{ x_2[n] \} \qquad \Fourier \{ x_1[n] \star x_2[n] \} = \Fourier \{ x_1[n] \} \Fourier \{ x_2[n] \} \] \[ \Fourier \{ x[n - n_0] \} = \hat{x}(\phi) \ee^{-\iu 2 \pi n_0 \phi} \qquad \Fourier \{ x[n] \ee^{\iu 2 \pi \phi_0 n} \} = \hat{x}(\phi - \phi_0) \]
Pares transformados de Fourier em tempo contínuo
$\text{Sejam $t_0 \in \mathbb{R}$ e $f_0 \in \mathbb{R}$ constantes.}$ \[ \def\arraystretch{2} \begin{alignedat}{2} x(t) & \qquad \hat{x}(f) & \qquad \\ \hline 1 & \qquad \delta(f) & \qquad (1) \\ \delta(t) & \qquad 1 & \qquad (2) \\ \ee^{\iu 2\pi f_0 t} & \qquad \delta(f - f_0) & \qquad (3) \\ \delta(t - t_0) & \qquad \ee^{-\iu 2\pi t_0 f} & \qquad (4) \\ \cos(2 \pi f_0 t) & \qquad \dfrac{1}{2} \delta(f - f_0) + \dfrac{1}{2} \delta(f + f_0) & \qquad (5) \\ \sin(2 \pi f_0 t) & \qquad \dfrac{1}{2\iu} \delta(f - f_0) - \dfrac{1}{2\iu} \delta(f + f_0) & \qquad (6) \\ \rect(t/t_0) & \qquad t_0 \sinc(t_0 f) & \qquad (7) \\ \sinc(t/t_0) & \qquad t_0 \rect(t_0 f) & \qquad (8) \\ \tri(t/t_0) & \qquad t_0 \sinc^2(t_0 f) & \qquad (9) \\ \sinc^2 (t/t_0) & \qquad t_0 \tri(t_0 f) & \qquad (10) \\ \sign(t) & \qquad \dfrac{1}{\iu \pi f} & \qquad (11) \\ \dfrac{1}{t} & \qquad -\iu \pi \sign(f) & \qquad (12) \\ \step(t) & \qquad \dfrac{1}{\iu 2 \pi f} + \dfrac{1}{2} \delta(f) & \qquad (13) \\ \mathrm{J}_0(t/t_0) & \qquad \dfrac{2 t_0 \rect(\pi t_0 f)}{\sqrt{1 - (2 \pi t_0 f)^2}} & \qquad (14) \\ \displaystyle{\sum_{n=-\infty}^{\infty} \delta(t - n t_0)} & \qquad \dfrac{1}{t_0} \displaystyle{\sum_{k=-\infty}^{\infty} \delta \left (f - \frac{k}{t_0} \right)} & \qquad (15) \\ \ee^{-t/t_0} \step(t) & \qquad \dfrac{t_0}{1 + \iu 2 \pi t_0 f} & \qquad (16) \\ \ee^{-|t/t_0|} & \qquad \dfrac{2 t_0}{1 + (2 \pi t_0 f)^2} & \qquad (17) \\ \ee^{-\frac{1}{2}(t/t_0)^2} & \qquad t_0 \sqrt{2 \pi} \ee^{-\frac{1}{2} (2 \pi t_0 f)^2} & \qquad (18) \end{alignedat} \]
$\text{Sejam $n_0 \in \mathbb{Z}$, $\phi_0 \in \mathbb{R}$ e $a \in \mathbb{R}$ constantes, com $|a| < 1$.}$ \[ \def\arraystretch{2} \begin{alignedat}{2} x[n] & \qquad \hat{x}(\phi) & \qquad \\ \hline 1 & \qquad \deltaper(\phi) & \qquad (1) \\ \delta[n] & \qquad 1 & \qquad (2) \\ \ee^{\iu 2\pi \phi_0 n} & \qquad \deltaper(\phi - \phi_0) & \qquad (3) \\ \delta[n - n_0] & \qquad \ee^{-\iu 2\pi n_0 \phi} & \qquad (4) \\ \cos(2 \pi \phi_0 n) & \qquad \dfrac{1}{2} \deltaper(\phi - \phi_0) + \dfrac{1}{2} \deltaper(\phi + \phi_0) & \qquad (5) \\ \sin(2 \pi \phi_0 n) & \qquad \dfrac{1}{2\iu} \deltaper(\phi - \phi_0) - \dfrac{1}{2\iu} \deltaper(\phi + \phi_0) & \qquad (6) \\ \step[n] & \qquad \dfrac{1}{1 - \ee^{-\iu 2 \pi \phi}} + \dfrac{1}{2} \deltaper(\phi) & \qquad (7) \\ a^n \step[n] & \qquad \dfrac{1}{1 - a \ee^{-\iu 2 \pi \phi}} & \qquad (8) \\ a^{|n|} & \qquad \dfrac{1 - a^2}{1 - 2 a \cos(2 \pi \phi) + a^2} & \qquad (9) \end{alignedat} \]