Teoria da probabilidade Pr [ A ∪ B ] = Pr [ A ] + Pr [ B ] − Pr [ A ∩ B ] Pr [ A ∣ B ] = Pr [ A ∩ B ] Pr [ B ] \def\arraystretch{1.25}
\Pr[A \cup B] = \Pr[A] + \Pr[B] - \Pr[A \cap B] \qquad \Pr[A \mid B] = \dfrac{\Pr[A \cap B]}{\Pr[B]} Pr [ A ∪ B ] = Pr [ A ] + Pr [ B ] − Pr [ A ∩ B ] Pr [ A ∣ B ] = Pr [ B ] Pr [ A ∩ B ] Pr [ A ∪ B ] = Pr [ A ] + Pr [ B ] , para A e B eventos disjuntos. Pr [ A ∩ B ] = Pr [ A ] Pr [ B ] , para A e B eventos independentes. \def\arraystretch{1.25}
\begin{aligned} \Pr[A \cup B] = \Pr[A] + \Pr[B], & \quad \text{para $A$ e $B$ eventos disjuntos.} \\ \Pr[A \cap B] = \Pr[A] \Pr[B], & \quad \text{para $A$ e $B$ eventos independentes.} \end{aligned} Pr [ A ∪ B ] = Pr [ A ] + Pr [ B ] , Pr [ A ∩ B ] = Pr [ A ] Pr [ B ] , para A e B eventos disjuntos. para A e B eventos independentes. Pr [ B ] = ∑ i Pr [ B ∣ A i ] Pr [ A i ] , onde { A i } s a ˜ o eventos que particionam o espa c ¸ o amostral. \def\arraystretch{1.25}
\Pr[B] = \sum_{i} \Pr[B \mid A_i] \Pr[A_i], \quad \text{onde } \{ A_i \} \text{ são eventos que particionam o espaço amostral.} Pr [ B ] = i ∑ Pr [ B ∣ A i ] Pr [ A i ] , onde { A i } s a ˜ o eventos que particionam o espa c ¸ o amostral.
Variáveis aleatórias PMF p X ( x ) Fun c ¸ a ˜ o massa de probabilidade PDF f X ( x ) Fun c ¸ a ˜ o densidade de probabilidade CDF F X ( x ) Fun c ¸ a ˜ o distribui c ¸ a ˜ o cumulativa \def\arraystretch{1.25}
\begin{alignedat}{2} \text{PMF} & \qquad & p_X(x) & \qquad \text{Função massa de probabilidade} \\ \text{PDF} & \qquad & f_X(x) & \qquad \text{Função densidade de probabilidade} \\ \text{CDF} & \qquad & F_X(x) & \qquad \text{Função distribuição cumulativa} \end{alignedat} PMF PDF CDF p X ( x ) f X ( x ) F X ( x ) Fun c ¸ a ˜ o massa de probabilidade Fun c ¸ a ˜ o densidade de probabilidade Fun c ¸ a ˜ o distribui c ¸ a ˜ o cumulativa p X ( x ) = Pr [ X = x ] Pr [ a ≤ X ≤ b ] = ∫ a − b + f X ( x ) d x F X ( x ) = Pr [ X ≤ x ] = ∫ − ∞ x + f X ( u ) d u \def\arraystretch{1.25}
p_X(x) = \Pr[X = x] \qquad \Pr[a \leq X \leq b] = \int_{a^-}^{b^+} f_X(x) \dif x \qquad F_X(x) = \Pr[X \leq x] = \int_{-\infty}^{x^+} f_X(u) \dif u p X ( x ) = Pr [ X = x ] Pr [ a ≤ X ≤ b ] = ∫ a − b + f X ( x ) d x F X ( x ) = Pr [ X ≤ x ] = ∫ − ∞ x + f X ( u ) d u f X ( x ) = ∫ − ∞ ∞ f X , Y ( x , y ) d y f X ( x ∣ Y = y ) = f X , Y ( x , y ) f Y ( y ) \def\arraystretch{1.25}
f_{X}(x) = \int_{-\infty}^\infty f_{X,Y}(x,y) \dif y \qquad \qquad f_{X}(x \mid Y=y) = \frac{f_{X,Y}(x,y)}{f_Y(y)} f X ( x ) = ∫ − ∞ ∞ f X , Y ( x , y ) d y f X ( x ∣ Y = y ) = f Y ( y ) f X , Y ( x , y ) f X ( x ) = ∑ i f X ( x ∣ A i ) Pr [ A i ] , onde { A i } s a ˜ o eventos que particionam o espa c ¸ o amostral. \def\arraystretch{1.25}
f_X(x) = \sum_{i} f_{X}(x \mid A_i) \Pr[A_i], \quad \text{onde $\{ A_i \}$ são eventos que particionam o espaço amostral.} f X ( x ) = i ∑ f X ( x ∣ A i ) Pr [ A i ] , onde { A i } s a ˜ o eventos que particionam o espa c ¸ o amostral.
Distribuição normal X ∼ N ( μ , σ 2 ) ⟺ f X ( x ) = 1 2 π σ 2 e − ( x − μ ) 2 2 σ 2 \def\arraystretch{1.25}
X \sim \mathrm{N}(\mu, \sigma^2) \quad \iff \quad f_X(x) = \dfrac{1}{\sqrt{2 \pi \sigma^2}} \ee^{-\frac{(x - \mu)^2}{2\sigma^2}} X ∼ N ( μ , σ 2 ) ⟺ f X ( x ) = 2 π σ 2 1 e − 2 σ 2 ( x − μ ) 2 Φ ( x ) = 1 2 π ∫ − ∞ x e − u 2 2 d u Q ( x ) = 1 − Φ ( x ) \def\arraystretch{1.25}
\Phi(x) = \dfrac{1}{\sqrt{2 \pi}} \int_{-\infty}^x \ee^{-\frac{u^2}{2}} \dif u \qquad \mathrm{Q}(x) = 1 - \Phi(x) Φ ( x ) = 2 π 1 ∫ − ∞ x e − 2 u 2 d u Q ( x ) = 1 − Φ ( x ) X ∼ N ( μ , σ 2 ) ⟹ Pr [ a ≤ X ≤ b ] = Φ ( b − μ σ ) − Φ ( a − μ σ ) \def\arraystretch{1.25}
X \sim \mathrm{N}(\mu, \sigma^2) \quad \implies \quad \Pr[a \leq X \leq b] = \Phi \left( \frac{b - \mu}{\sigma} \right) - \Phi \left( \frac{a - \mu}{\sigma} \right) X ∼ N ( μ , σ 2 ) ⟹ Pr [ a ≤ X ≤ b ] = Φ ( σ b − μ ) − Φ ( σ a − μ )
Valor esperado E [ g ( X ) ] = ∑ x ∈ S X g ( x ) p X ( x ) E [ g ( X ) ] = ∫ − ∞ ∞ g ( x ) f X ( x ) d x \def\arraystretch{1.25}
\EV[g(X)] = \sum_{x \in S_X} g(x) p_\mathrm{X}(x) \qquad \EV[g(X)] = \int_{-\infty}^{\infty} g(x) f_\mathrm{X}(x) \dif x E [ g ( X )] = x ∈ S X ∑ g ( x ) p X ( x ) E [ g ( X )] = ∫ − ∞ ∞ g ( x ) f X ( x ) d x E [ g ( X , Y ) ] = ∑ x ∈ S X ∑ y ∈ S Y g ( x , y ) p X , Y ( x , y ) E [ g ( X , Y ) ] = ∫ − ∞ ∞ ∫ − ∞ ∞ g ( x , y ) f X , Y ( x , y ) d x d y \def\arraystretch{1.25}
\EV[g(X, Y)] = \sum_{x \in S_X} \sum_{y \in S_Y} g(x, y) p_\mathrm{X, Y}(x, y) \qquad \EV[g(X, Y)] = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} g(x, y) f_\mathrm{X, Y}(x, y) \dif x \dif y E [ g ( X , Y )] = x ∈ S X ∑ y ∈ S Y ∑ g ( x , y ) p X , Y ( x , y ) E [ g ( X , Y )] = ∫ − ∞ ∞ ∫ − ∞ ∞ g ( x , y ) f X , Y ( x , y ) d x d y μ X = E [ X ] σ X 2 = v a r [ X ] = E [ ( X − μ X ) 2 ] = E [ X 2 ] − E [ X ] 2 \def\arraystretch{1.25}
\mu_X = \EV[X] \qquad \sigma_X^2 = \var[X] = \EV[(X - \mu_X)^2] = \EV[X^2] - \EV[X]^2 μ X = E [ X ] σ X 2 = var [ X ] = E [( X − μ X ) 2 ] = E [ X 2 ] − E [ X ] 2 c o v [ X , Y ] = E [ ( X − μ X ) ( Y − μ Y ) ] = E [ X Y ] − E [ X ] E [ Y ] ρ X , Y = c o v [ X , Y ] v a r [ X ] v a r [ Y ] \def\arraystretch{1.25}
\cov[X, Y] = \EV[(X - \mu_X)(Y - \mu_Y)] = \EV[XY] - \EV[X] \EV[Y] \qquad \rho_{X,Y} = \frac{\cov[X, Y]}{\sqrt{\var[X] \var[Y]}} cov [ X , Y ] = E [( X − μ X ) ( Y − μ Y )] = E [ X Y ] − E [ X ] E [ Y ] ρ X , Y = var [ X ] var [ Y ] cov [ X , Y ] v a r [ X + Y ] = v a r [ X ] + v a r [ Y ] + 2 c o v [ X , Y ] \def\arraystretch{1.25}
\var[X + Y] = \var[X] + \var[Y] + 2\cov[X, Y] \\ var [ X + Y ] = var [ X ] + var [ Y ] + 2 cov [ X , Y ] E [ X ] = ∑ i E [ X ∣ A i ] Pr [ A i ] , onde { A i } s a ˜ o eventos que particionam o espa c ¸ o amostral. \def\arraystretch{1.25}
\EV[X] = \sum_{i} \EV[X \mid A_i] \Pr[A_i], \quad \text{onde $\{ A_i \}$ são eventos que particionam o espaço amostral.} E [ X ] = i ∑ E [ X ∣ A i ] Pr [ A i ] , onde { A i } s a ˜ o eventos que particionam o espa c ¸ o amostral.
Vetores aleatórios μ ⃗ X ⃗ = E [ X ⃗ ] = [ E [ X 1 ] E [ X 2 ] ⋮ E [ X n ] ] C X ⃗ = E [ ( X ⃗ − μ ⃗ Y ⃗ ) ( X ⃗ − μ ⃗ Y ⃗ ) T ] = [ v a r [ X 1 ] c o v [ X 1 , X 2 ] ⋯ c o v [ X 1 , X n ] c o v [ X 2 , X 1 ] v a r [ X 2 ] ⋯ c o v [ X 2 , X n ] ⋮ ⋮ ⋱ ⋮ c o v [ X n , X 1 ] c o v [ X n , X 2 ] ⋯ v a r [ X n ] ] \def\arraystretch{1.25}
\vec{\mu}_{\vec{X}} = \EV[ \vec{X} ] = \mat{\EV[X_1] \\ \EV[X_2] \\ \vdots \\ \EV[X_n]} \qquad C_{\vec{X}} = \EV[ (\vec{X} - \vec{\mu}_{\vec{Y}}) (\vec{X} - \vec{\mu}_{\vec{Y}})^\tr] = \mat{ \var[X_1] & \cov[X_1, X_2] & \cdots & \cov[X_1, X_n] \\ \cov[X_2, X_1] & \var[X_2] & \cdots & \cov[X_2, X_n] \\ \vdots & \vdots & \ddots & \vdots \\ \cov[X_n, X_1] & \cov[X_n, X_2] & \cdots & \var[X_n] } μ X = E [ X ] = E [ X 1 ] E [ X 2 ] ⋮ E [ X n ] C X = E [( X − μ Y ) ( X − μ Y ) T ] = var [ X 1 ] cov [ X 2 , X 1 ] ⋮ cov [ X n , X 1 ] cov [ X 1 , X 2 ] var [ X 2 ] ⋮ cov [ X n , X 2 ] ⋯ ⋯ ⋱ ⋯ cov [ X 1 , X n ] cov [ X 2 , X n ] ⋮ var [ X n ] Y ⃗ = A X ⃗ + b ⃗ ⟹ μ ⃗ Y ⃗ = A μ ⃗ X ⃗ + b ⃗ , C Y ⃗ = A C X ⃗ A T \def\arraystretch{1.25}
\vec{Y} = A \vec{X} + \vec{b} \quad \implies \quad \vec{\mu}_{\vec{Y}} = A \vec{\mu}_{\vec{X}} + \vec{b}, \quad C_{\vec{Y}} = A C_{\vec{X}} A^\tr Y = A X + b ⟹ μ Y = A μ X + b , C Y = A C X A T X ⃗ ∼ N ( μ ⃗ , C ) ⟺ f X ⃗ ( x ⃗ ) = 1 ( 2 π ) n / 2 det C exp ( − 1 2 ( x ⃗ − μ ⃗ ) T C − 1 ( x ⃗ − μ ⃗ ) ) \def\arraystretch{1.25}
\vec{X} \sim \mathrm{N}(\vec{\mu}, C) \quad \iff \quad f_{\vec{X}}(\vec{x}) = \frac{1}{(2 \pi)^{n/2} \sqrt{\det C}} \exp \left( -\frac{1}{2} (\vec{x} - \vec{\mu})^\tr C^{-1} (\vec{x} - \vec{\mu}) \right) X ∼ N ( μ , C ) ⟺ f X ( x ) = ( 2 π ) n /2 det C 1 exp ( − 2 1 ( x − μ ) T C − 1 ( x − μ ) )
Processos estocásticos μ X ( t ) = E [ X ( t ) ] C X ( t 1 , t 2 ) = c o v [ X ( t 1 ) , X ( t 2 ) ] = E [ X ( t 1 ) X ( t 2 ) ] − E [ X ( t 1 ) ] E [ X ( t 2 ) ] \def\arraystretch{1.25}
\mu_X(t) = \EV[X(t)] \qquad C_X(t_1, t_2) = \cov[X(t_1), X(t_2)] = \EV[ X(t_1) X(t_2) ] - \EV[X(t_1)] \EV[X(t_2)] μ X ( t ) = E [ X ( t )] C X ( t 1 , t 2 ) = cov [ X ( t 1 ) , X ( t 2 )] = E [ X ( t 1 ) X ( t 2 )] − E [ X ( t 1 )] E [ X ( t 2 )]
Processos estocásticos estacionários no sentido amplo S X ( f ) = F { R X ( τ ) } μ Y = h ^ ( 0 ) μ X S Y ( f ) = ∣ h ^ ( f ) ∣ 2 S X ( f ) \def\arraystretch{1.25}
S_X(f) = \Fourier \{ R_X(\tau) \} \qquad \mu_Y = \hat{h}(0) \mu_X \qquad S_Y(f) = |\hat{h}(f)|^2 S_X(f) S X ( f ) = F { R X ( τ )} μ Y = h ^ ( 0 ) μ X S Y ( f ) = ∣ h ^ ( f ) ∣ 2 S X ( f ) P X = E [ X 2 ( t ) ] = R X ( 0 ) = ∫ − ∞ ∞ S X ( f ) d f \def\arraystretch{1.25}
P_X = \EV[X^2(t)] = R_X(0) = \int_{-\infty}^{\infty} S_X(f) \dif f P X = E [ X 2 ( t )] = R X ( 0 ) = ∫ − ∞ ∞ S X ( f ) d f
Séries ∑ k = 0 ∞ r k = 1 1 − r , ∑ k = 0 ∞ k r k = r ( 1 − r ) 2 , ∑ k = 0 ∞ k 2 r k = r ( 1 + r ) ( 1 − r ) 3 . ( ∣ r ∣ < 1 ) \def\arraystretch{1.25}
\sum_{k=0}^{\infty} r^k = \frac{1}{1 - r}, \qquad \qquad \sum_{k=0}^{\infty} kr^k = \frac{r}{(1 - r)^2}, \qquad \qquad \sum_{k=0}^{\infty} k^2r^k = \frac{r(1+r)}{(1 - r)^3}. \qquad \qquad (|r| < 1) k = 0 ∑ ∞ r k = 1 − r 1 , k = 0 ∑ ∞ k r k = ( 1 − r ) 2 r , k = 0 ∑ ∞ k 2 r k = ( 1 − r ) 3 r ( 1 + r ) . ( ∣ r ∣ < 1 )
Identidades trigonométricas cos ( a ± b ) = cos a cos b ∓ sin a sin b sin ( a ± b ) = sin a cos b ∓ sin b cos a \def\arraystretch{1.25}
\cos(a \pm b) = \cos a \cos b \mp \sin a \sin b \qquad \sin(a \pm b) = \sin a \cos b \mp \sin b \cos a cos ( a ± b ) = cos a cos b ∓ sin a sin b sin ( a ± b ) = sin a cos b ∓ sin b cos a cos a cos b = 1 2 [ cos ( a − b ) + cos ( a + b ) ] sin a sin b = 1 2 [ cos ( a − b ) − cos ( a + b ) ] \def\arraystretch{1.25}
\cos a \cos b = \frac{1}{2} [ \cos (a-b) + \cos (a+b) ] \qquad \sin a \sin b = \frac{1}{2} [ \cos (a-b) - \cos (a+b) ] cos a cos b = 2 1 [ cos ( a − b ) + cos ( a + b )] sin a sin b = 2 1 [ cos ( a − b ) − cos ( a + b )] cos 2 a = 1 2 ( 1 + cos 2 a ) sin 2 a = 1 2 ( 1 − cos 2 a ) \def\arraystretch{1.25}
\cos^2 a = \frac{1}{2} (1 + \cos 2a) \qquad \sin^2 a = \frac{1}{2} (1 - \cos 2a) cos 2 a = 2 1 ( 1 + cos 2 a ) sin 2 a = 2 1 ( 1 − cos 2 a )
Fórmula de Euler e j θ = cos θ + j sin θ cos θ = 1 2 ( e j θ + e − j θ ) sin θ = 1 2 j ( e j θ − e − j θ ) \def\arraystretch{1.25}
\ee^{\iu\theta} = \cos \theta + \iu \sin \theta \qquad \cos \theta = \frac{1}{2} (\mathrm{e}^{\iu\theta} + \mathrm{e}^{-\iu\theta}) \qquad \sin \theta = \frac{1}{2\iu} (\mathrm{e}^{\iu\theta} - \mathrm{e}^{-\iu\theta}) e j θ = cos θ + j sin θ cos θ = 2 1 ( e j θ + e − j θ ) sin θ = 2 j 1 ( e j θ − e − j θ )
Sinais básicos r e c t ( x ) = { 1 , se ∣ x ∣ < 1 / 2 , 0 , caso contr a ˊ rio. t r i ( x ) = { 1 − ∣ x ∣ , se ∣ x ∣ < 1 , 0 , caso contr a ˊ rio. s i n c ( x ) = sin ( π x ) π x \def\arraystretch{1.25}
\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} rect ( x ) = { 1 , 0 , se ∣ x ∣ < 1/2 , caso contr a ˊ rio. tri ( x ) = { 1 − ∣ x ∣ , 0 , se ∣ x ∣ < 1 , caso contr a ˊ rio. sinc ( x ) = π x sin ( π x ) s i g n ( x ) = { 1 , x > 0 , 0 , x = 0 , − 1 , x < 0. u ( x ) = { 1 , x > 0 , 0 , x < 0. \def\arraystretch{1.25}
\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} sign ( x ) = ⎩ ⎨ ⎧ 1 , 0 , − 1 , x > 0 , x = 0 , x < 0. u ( x ) = { 1 , 0 , x > 0 , x < 0.
Definição e propriedades da transformada de Fourier em tempo contínuo
∫ − 1 2 1 2 ∑ n = − ∞ ∞ x ^ ( f ) = F { x ( t ) } = ∫ − ∞ ∞ x ( t ) e − j 2 π f t d t x ( t ) = F − 1 { x ^ ( f ) } = ∫ − ∞ ∞ x ^ ( f ) e j 2 π t f d f \def\arraystretch{1.25}
\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 ∫ − 2 1 2 1 n = − ∞ ∑ ∞ x ^ ( f ) = F { x ( t )} = ∫ − ∞ ∞ x ( t ) e − j 2 π f t d t x ( t ) = F − 1 { x ^ ( f )} = ∫ − ∞ ∞ x ^ ( f ) e j 2 π t f d f F { a x 1 ( t ) + b x 2 ( t ) } = a F { x 1 ( t ) } + b F { x 2 ( t ) } \def\arraystretch{1.25}
\Fourier \{ a x_1(t) + b x_2(t) \} = a \Fourier \{ x_1(t) \} + b \Fourier \{ x_2(t) \} F { a x 1 ( t ) + b x 2 ( t )} = a F { x 1 ( t )} + b F { x 2 ( t )} F { x ( t − t 0 ) } = F { x ( t ) } e − j 2 π t 0 f F { x ( t ) e j 2 π f 0 t } = x ^ ( f − f 0 ) F { x ( − t ) } = x ^ ∗ ( f ) \def\arraystretch{1.25}
\Fourier \{ x(t - t_0) \} = \Fourier \{ x(t) \} \ee^{-\iu 2 \pi t_0 f} \qquad \Fourier \{ x(t) \ee^{\iu 2 \pi f_0 t} \} = \hat{x}(f - f_0) \qquad \Fourier \{ x(-t) \} = \hat{x}^*(f) F { x ( t − t 0 )} = F { x ( t )} e − j 2 π t 0 f F { x ( t ) e j 2 π f 0 t } = x ^ ( f − f 0 ) F { x ( − t )} = x ^ ∗ ( f ) F { x 1 ( t ) x 2 ( t ) } = F { x 1 ( t ) } ⋆ F { x 2 ( t ) } F { x 1 ( t ) ⋆ x 2 ( t ) } = F { x 1 ( t ) } F { x 2 ( t ) } \def\arraystretch{1.25}
\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) \} F { x 1 ( t ) x 2 ( t )} = F { x 1 ( t )} ⋆ F { x 2 ( t )} F { x 1 ( t ) ⋆ x 2 ( t )} = F { x 1 ( t )} F { x 2 ( t )} F { x ^ ( t ) } = x ( − f ) \def\arraystretch{1.25}
\Fourier \{ \hat{x}(t) \} = x(-f) F { x ^ ( t )} = x ( − f ) F { x ( a t ) } = 1 ∣ a ∣ x ^ ( f a ) \def\arraystretch{1.25}
\Fourier \{ x(at) \} = \dfrac{1}{|a|} \hat{x} \left( \dfrac{f}{a} \right) F { x ( a t )} = ∣ a ∣ 1 x ^ ( a f ) F { d n d t n x ( t ) } = ( j 2 π f ) n x ^ ( f ) F { t n x ( t ) } = ( j 2 π ) n d n d f n x ^ ( f ) \def\arraystretch{1.25}
\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) F { d t n d n x ( t ) } = ( j 2 π f ) n x ^ ( f ) F { t n x ( t )} = ( 2 π j ) n d f n d n x ^ ( f )
x ^ ( ϕ ) = F { x [ n ] } = ∑ n = − ∞ ∞ x [ n ] e − j 2 π ϕ n x [ n ] = F − 1 { x ^ ( ϕ ) } = ∫ − 1 2 1 2 x ^ ( ϕ ) e j 2 π n ϕ d ϕ \def\arraystretch{1.25}
\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 x ^ ( ϕ ) = F { x [ n ]} = n = − ∞ ∑ ∞ x [ n ] e − j 2 π ϕ n x [ n ] = F − 1 { x ^ ( ϕ )} = ∫ − 2 1 2 1 x ^ ( ϕ ) e j 2 πn ϕ d ϕ F { a x 1 [ n ] + b x 2 [ n ] } = a F { x 1 [ n ] } + b F { x 2 [ n ] } \def\arraystretch{1.25}
\Fourier \{ a x_1[n] + b x_2[n] \} = a \Fourier \{ x_1[n] \} + b \Fourier \{ x_2[n] \} F { a x 1 [ n ] + b x 2 [ n ]} = a F { x 1 [ n ]} + b F { x 2 [ n ]} F { x [ n − n 0 ] } = F { x [ n ] } e − j 2 π n 0 ϕ F { x [ n ] e j 2 π ϕ 0 n } = x ^ ( ϕ − ϕ 0 ) F { x [ − n ] } = x ^ ∗ ( ϕ ) \def\arraystretch{1.25}
\Fourier \{ x[n - n_0] \} = \Fourier \{ x[n] \} \ee^{-\iu 2 \pi n_0 \phi} \qquad \Fourier \{ x[n] \ee^{\iu 2 \pi \phi_0 n} \} = \hat{x}(\phi - \phi_0) \qquad \Fourier \{ x[-n] \} = \hat{x}^*(\phi) F { x [ n − n 0 ]} = F { x [ n ]} e − j 2 π n 0 ϕ F { x [ n ] e j 2 π ϕ 0 n } = x ^ ( ϕ − ϕ 0 ) F { x [ − n ]} = x ^ ∗ ( ϕ ) F { x 1 [ n ] x 2 [ n ] } = F { x 1 [ n ] } ⋆ F { x 2 [ n ] } F { x 1 [ n ] ⋆ x 2 [ n ] } = F { x 1 [ n ] } F { x 2 [ n ] } \def\arraystretch{1.25}
\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] \} F { x 1 [ n ] x 2 [ n ]} = F { x 1 [ n ]} ⋆ F { x 2 [ n ]} F { x 1 [ n ] ⋆ x 2 [ n ]} = F { x 1 [ n ]} F { x 2 [ n ]}
Pares transformados de Fourier em tempo contínuo
Sejam t 0 ∈ R , f 0 ∈ R e T ∈ R . \def\arraystretch{1.25}
\text{Sejam $t_0 \in \mathbb{R}$, $f_0 \in \mathbb{R}$ e $T \in \mathbb{R}$.} Sejam t 0 ∈ R , f 0 ∈ R e T ∈ R . x ( t ) x ^ ( f ) 1 δ ( f ) ( 1 ) δ ( t ) 1 ( 2 ) e j 2 π f 0 t δ ( f − f 0 ) ( 3 ) δ ( t − t 0 ) e − j 2 π t 0 f ( 4 ) cos ( 2 π f 0 t ) 1 2 δ ( f − f 0 ) + 1 2 δ ( f + f 0 ) ( 5 ) sin ( 2 π f 0 t ) 1 2 j δ ( f − f 0 ) − 1 2 j δ ( f + f 0 ) ( 6 ) r e c t ( t / t 0 ) t 0 s i n c ( t 0 f ) ( 7 ) s i n c ( t / t 0 ) t 0 r e c t ( t 0 f ) ( 8 ) t r i ( t / t 0 ) t 0 s i n c 2 ( t 0 f ) ( 9 ) s i n c 2 ( t / t 0 ) t 0 t r i ( t 0 f ) ( 10 ) s i g n ( t ) 1 j π f ( 11 ) 1 t − j π s i g n ( f ) ( 12 ) u ( t ) 1 j 2 π f + 1 2 δ ( f ) ( 13 ) J 0 ( t / t 0 ) 2 t 0 r e c t ( π t 0 f ) 1 − ( 2 π t 0 f ) 2 ( 14 ) ∑ n = − ∞ ∞ δ ( t − n T ) 1 T ∑ k = − ∞ ∞ δ ( f − k T ) ( 15 ) e − t / t 0 u ( t ) t 0 1 + j 2 π t 0 f ( 16 ) e − ∣ t / t 0 ∣ 2 t 0 1 + ( 2 π t 0 f ) 2 ( 17 ) e − 1 2 ( t / t 0 ) 2 t 0 2 π e − 1 2 ( 2 π t 0 f ) 2 ( 18 ) \def\arraystretch{1.25}
\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)} & \qquad \dfrac{1}{T} \displaystyle{\sum_{k=-\infty}^{\infty} \delta \left (f - \frac{k}{T} \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} x ( t ) 1 δ ( t ) e j 2 π f 0 t δ ( t − t 0 ) cos ( 2 π f 0 t ) sin ( 2 π f 0 t ) rect ( t / t 0 ) sinc ( t / t 0 ) tri ( t / t 0 ) sinc 2 ( t / t 0 ) sign ( t ) t 1 u ( t ) J 0 ( t / t 0 ) n = − ∞ ∑ ∞ δ ( t − n T ) e − t / t 0 u ( t ) e − ∣ t / t 0 ∣ e − 2 1 ( t / t 0 ) 2 x ^ ( f ) δ ( f ) 1 δ ( f − f 0 ) e − j 2 π t 0 f 2 1 δ ( f − f 0 ) + 2 1 δ ( f + f 0 ) 2 j 1 δ ( f − f 0 ) − 2 j 1 δ ( f + f 0 ) t 0 sinc ( t 0 f ) t 0 rect ( t 0 f ) t 0 sinc 2 ( t 0 f ) t 0 tri ( t 0 f ) j π f 1 − j π sign ( f ) j 2 π f 1 + 2 1 δ ( f ) 1 − ( 2 π t 0 f ) 2 2 t 0 rect ( π t 0 f ) T 1 k = − ∞ ∑ ∞ δ ( f − T k ) 1 + j 2 π t 0 f t 0 1 + ( 2 π t 0 f ) 2 2 t 0 t 0 2 π e − 2 1 ( 2 π t 0 f ) 2 ( 1 ) ( 2 ) ( 3 ) ( 4 ) ( 5 ) ( 6 ) ( 7 ) ( 8 ) ( 9 ) ( 10 ) ( 11 ) ( 12 ) ( 13 ) ( 14 ) ( 15 ) ( 16 ) ( 17 ) ( 18 )
Sejam n 0 ∈ Z , ϕ 0 ∈ R e a ∈ R , com ϕ 0 ∈ [ − 1 2 , 1 2 ) e ∣ a ∣ < 1 . \def\arraystretch{1.25}
\text{Sejam $n_0 \in \mathbb{Z}$, $\phi_0 \in \mathbb{R}$ e $a \in \mathbb{R}$, com $\phi_0 \in \left[ -\frac{1}{2}, \frac{1}{2} \right)$ e $|a| < 1$.} Sejam n 0 ∈ Z , ϕ 0 ∈ R e a ∈ R , com ϕ 0 ∈ [ − 2 1 , 2 1 ) e ∣ a ∣ < 1. x [ n ] x ^ ( ϕ ) 1 δ ( ϕ ) ( 1 ) δ [ n ] 1 ( 2 ) e j 2 π ϕ 0 n δ ( ϕ − ϕ 0 ) ( 3 ) δ [ n − n 0 ] e − j 2 π n 0 ϕ ( 4 ) cos ( 2 π ϕ 0 n ) 1 2 δ ( ϕ − ϕ 0 ) + 1 2 δ ( ϕ + ϕ 0 ) ( 5 ) sin ( 2 π ϕ 0 n ) 1 2 j δ ( ϕ − ϕ 0 ) − 1 2 j δ ( ϕ + ϕ 0 ) ( 6 ) u [ n ] 1 1 − e − j 2 π ϕ + 1 2 δ ( ϕ ) ( 7 ) a n u [ n ] 1 1 − a e − j 2 π ϕ ( 8 ) a ∣ n ∣ 1 − a 2 1 − 2 a cos ( 2 π ϕ ) + a 2 ( 9 ) \def\arraystretch{1.25}
\def\arraystretch{2} \begin{alignedat}{2} x[n] & \qquad \hat{x}(\phi) & \qquad \\ \hline 1 & \qquad \delta(\phi) & \qquad (1) \\ \delta[n] & \qquad 1 & \qquad (2) \\ \ee^{\iu 2\pi \phi_0 n} & \qquad \delta(\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} \delta(\phi - \phi_0) + \dfrac{1}{2} \delta(\phi + \phi_0) & \qquad (5) \\ \sin(2 \pi \phi_0 n) & \qquad \dfrac{1}{2\iu} \delta(\phi - \phi_0) - \dfrac{1}{2\iu} \delta(\phi + \phi_0) & \qquad (6) \\ \step[n] & \qquad \dfrac{1}{1 - \ee^{-\iu 2 \pi \phi}} + \dfrac{1}{2} \delta(\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} x [ n ] 1 δ [ n ] e j 2 π ϕ 0 n δ [ n − n 0 ] cos ( 2 π ϕ 0 n ) sin ( 2 π ϕ 0 n ) u [ n ] a n u [ n ] a ∣ n ∣ x ^ ( ϕ ) δ ( ϕ ) 1 δ ( ϕ − ϕ 0 ) e − j 2 π n 0 ϕ 2 1 δ ( ϕ − ϕ 0 ) + 2 1 δ ( ϕ + ϕ 0 ) 2 j 1 δ ( ϕ − ϕ 0 ) − 2 j 1 δ ( ϕ + ϕ 0 ) 1 − e − j 2 π ϕ 1 + 2 1 δ ( ϕ ) 1 − a e − j 2 π ϕ 1 1 − 2 a cos ( 2 π ϕ ) + a 2 1 − a 2 ( 1 ) ( 2 ) ( 3 ) ( 4 ) ( 5 ) ( 6 ) ( 7 ) ( 8 ) ( 9 )