Sinais e Sistemas Formulario completo

Prof. Dr. Marcelo de Oliveira Rosa

26 de setembro de 2013

1 Eletricidade

Descritivo básico de equações em circuitos elétricos (para facilitar modelagem).

Corrente no indutor

\begin{aligned}i_{l}(t) & = \frac{1}{L} ∫_{−∞}^{t} v_{l}(τ)dτ \\ & = i_{l}(t_{0}) + \frac{1}{L} ∫_{t_{0}}^{t} v_{l}(τ)dτ \end{aligned}  (1)

Tensão no indutor

v_{l}(t) = L\frac{d}{dt}i_{l}(t)  (2)

Corrente no capacitor

i_{c}(t) = C \frac{d}{dt}v_{c}(t)  (3)

Tensão no capacitor

\begin{aligned}v_{c}(t) & = \frac{1}{C} ∫_{−∞}^{t} i_{c}(τ)dτ \\ & = v_{c}(t_{0}) + \frac{1}{C} ∫_{t_{0}}^{t} i_{c}(τ)dτ \end{aligned}  (4)

2 Domínio do tempo

E_{x} = ∫_{−∞}^{+∞} |x(t)|^{2} dt  (5)

P_{x} = lim_{T →+∞} \frac{1}{T} ∫_{t_{0}}^{t_{0}+T} |x(t)|^{2} dt  (6)

y(t) = ∫_{−∞}^{+∞} x(τ)h(t − τ)dτ  (7)

\begin{aligned}y(t) & = x(t) ∗ h(t) ⇔ \\ \frac{d}{dt}y(t) & = \frac{d}{dt}x(t) ∗ h(t) \\ & = x(t) ∗ \frac{d}{dt}h(t) \end{aligned}  (8)

\begin{aligned}y(t) & = x(t) ∗ h(t) ⇔ \\ y(at) & = |a| x(at) ∗ h(at) \end{aligned}  (9)

2.1 EDO

Seja n a ordem da maior diferencial de y(t) e m a ordem da maior diferencial de x(t) de uma EDO linear com coeficientes constantes.

y_{p}(t) = \begin{cases}0 & n > m \\ Kδ(t) & n = m \\ \begin{matrix}K_{m−n} \frac{d^{m−n}}{dt^{m−n}} δ(t)+ \\ K_{m−n−1} \frac{d^{m−n−1}}{dt^{m−n−1}} δ(t)+ \\ ⋯ + \\ K_{0}δ(t) \end{matrix} & n < m \end{cases} (10)

3 Domínio de Fourier

e^{±ȷx} = cos(x) ± ȷ sen(x)  (11)

\begin{aligned}cos(x) & = \frac{e^{+ȷx} + e^{−ȷx}}{2} \\ sen(x) & = \frac{e^{+ȷx} − e^{−ȷx}}{2ȷ} \end{aligned}  (12)

cos(a ± b) = cos(a) cos(b) ∓ sen(a) sen(b)
sen(a ± b) = sen(a) cos(b) ± sen(b) cos(a)

(13)

X[k] = \frac{1}{T} ∫_{t_{0}}^{t_{0}+T} x(t)e^{−ȷkΩt}dt  (14)

X_{c}[k] = \frac{2}{T} ∫_{t_{0}}^{t_{0}+T} x(t) cos(ȷkΩt)dt
X_{c}[k] = X[k] + X[−k]
X_{s}[k] = \frac{2}{T} ∫_{t_{0}}^{t_{0}+T} x(t) sen(ȷkΩt)dt
X_{s}[k] = ȷ(X[k] − X[−k])

(15)

x(t) = ∑_{k=−∞}^{+∞} X[k]e^{+ȷkΩt}  (16)

x(t) = X[0]

+ ∑_{k=1}^{+∞} {X_{c}[k] cos(kΩt)

+ X_{s}[k] sen(kΩt)}

(17)

SF {x(−t)} = X[−k]  (18)
\begin{aligned}SF \{x(at)\} & = X[k], T_{SF} = aT_{x} \\ & = X[k/a], T_{SF} = T_{x} \end{aligned}  (19)
SF \{\frac{d}{dt}x(t)\} = ȷkΩX[k]  (20)
SF \{∫_{−∞}^{t} x(τ)dτ\} = \frac{X[k]}{ȷkΩ}  (21)
SF {x^{∗}(t)} = X^{∗}[−k]  (22)
SF \{δ_{T_{0}}(t)\} = \frac{Ω_{0}}{2π} δ_{m}[k], T = mT_{0} (23)
SF \{\frac{1}{a} rect (\frac{t}{a}) ∗ δ_{T_{0}}\} = \frac{Ω_{0}}{2π} sinc (ak \frac{Ω_{0}}{2π} ) (24)
SF \{\frac{1}{a} tri (\frac{t}{a}) ∗ δ_{T_{0}}\} = \frac{Ω_{0}}{2π} sinc^{2} (ak \frac{Ω_{0}}{2π} ) (25)
X(ȷΩ) = ∫_{−∞}^{+∞} x(t)e^{−ȷΩt}dt  (26)
x(t) = \frac{1}{2π} ∫_{−∞}^{+∞} X(ȷΩ)e^{+ȷΩt}dΩ  (27)
TF \{x(at)\} = \frac{1}{|a|}X (\frac{ȷΩ}{a})  (28)
TF \{\frac{1}{a}x(t/a)\} = X(ȷaΩ)  (29)
TF {x^{∗}(t)} = X^{∗}(−ȷΩ)  (30)
TF \{\frac{d}{dt}x(t)\} = ȷΩX(ȷΩ)  (31)
TF \{∫_{−∞}^{t} x(τ)dτ\} = \frac{X(ȷΩ)}{ȷΩ} + πX(0)δ(Ω) (32)
\begin{aligned}x(t) & ⇔ X(ȷΩ) \\ TF \{X(ȷt)\} & = 2πx(−Ω) \end{aligned}  (33)

TF { ∑_{k=∞}^{+∞} X[k]e^{ȷkΩ_{a}t}} = 2π ∑_{k=−∞}^{+∞} X[k]δ(Ω − kΩ_{a})

(34)

TF {δ_{T_{0}}(t)} = Ω_{0}δ_{Ω_{0}}(Ω)  (35)
TF \{rect(t)\} = sinc (\frac{Ω}{2π} )  (36)
TF \{tri(t)\} = sinc^{2} (\frac{Ω}{2π} )  (37)
TF \{e^{−at}u(t)\} = \frac{1}{a + ȷΩ}  (38)
TF {1} = 2πδ(Ω)  (39)
TF {δ(t)} = 1  (40)
TF \{u(t)\} = \frac{1}{ȷΩ} + πδ(Ω)  (41)
TF {cos(Ω_{0}t)} = π [δ(Ω + Ω_{0}) + δ(Ω − Ω_{0})] (42)

TF {sen(Ω_{0}t)} = ȷπ [δ(Ω + Ω_{0}) − δ(Ω − Ω_{0})]

(43)

4 Domínio de Laplace

X(s) = ∫_{−∞}^{+∞} x(t)e^{−st}dt  (44)
x(t) = \frac{1}{ȷ2π} ∫_{σ+j∞}^{σ−j∞} X(s)e^{+st}ds  (45)
TL \{x(at)\} = \frac{1}{|a|}X (\frac{s}{a})  (46)
TL \{\frac{1}{a}x(t/a)\} = X(as)  (47)
TL {x^{∗}(t)} = X^{∗}(−s)  (48)
TL \{\frac{d}{dt}x(t)\} = sX(s) − x(0^{−})  (49)

\begin{aligned}TL \{\frac{d^{N}}{dt^{N}} x(t)\} & = s^{N}X(s) \\ & − ∑_{n=1}^{N} s^{N−n} [\frac{d^{n−1}}{dt^{n−1}} x(t)]_{t=0^{−}} \end{aligned}

(50)

TL \{∫_{−∞}^{t} x(τ)dτ\} = \frac{X(ȷΩ)}{ȷΩ}  (51)

TL \{−tx(t)\} = \frac{d}{ds}X(s)  (52)

TL \{e^{−at}u(t)\} = \frac{1}{s + a}, ℜ(s) > −a  (53)

TL \{−e^{−at}u(−t)\} = \frac{1}{s + a}, ℜ(s) < −a (54)

TL {δ(t)} = 1  (55)

TL \{u(t)\} = \frac{1}{s}, ℜ(s) > 0  (56)

TL \{−u(−t)\} = \frac{1}{s}, ℜ(s) < 0  (57)

TL \{t^{n}u(t)\} = \frac{n!}{s^{n+1}} , ℜ(s) > 0  (58)

TL \{−t^{n}u(−t)\} = \frac{n!}{s^{n+1}} , ℜ(s) < 0  (59)

TL \{cos(Ω_{0}t)u(t)\} = \frac{s}{s^{2} + Ω_{0}^{2}} , ℜ(s) > 0 (60)

TL \{sen(Ω_{0}t)u(t)\} = \frac{Ω_{0}}{s^{2} + Ω_{0}^{2}} , ℜ(s) > 0 (61)