[SS] Properties of (unilateral) Laplace Transform

Unilateral Laplace Transform의 주요 성질.

Linearity

$$a{ x }_{ 1 }\left( t \right) +b{ x }_{ 2 }\left( t \right) \longleftrightarrow a{ X }_{ 1 }\left( s \right) +{ X }_{ 2 }\left( s \right)$$

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Time Shifting

$$x\left( t-{ t }_{ 0 } \right) u\left( t-{ t }_{ 0 } \right) \longleftrightarrow { e }^{ -s{ t }_{ 0 } }X\left( s \right)$$

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$s$-domain Shifting (or Complex Shifting)

$$x( t ) e^{ s_0 t} \longleftrightarrow X\left( s-{ s }_{ 0 } \right)$$

다음과 같은 형태로도 많이 사용됨.

$$x( t ) e^{ -s_0 t} \longleftrightarrow X( s+s_0 )$$

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Scaling

$${ x }\left( t \right) \longleftrightarrow { X }\left( s \right) ,\quad Re\left[ s \right] >b\\ { x }\left( at \right) \longleftrightarrow { \frac { 1 }{ a } X }\left( \frac { s }{ a } \right) ,\quad Re\left[ s \right] >ab$$

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Convolution

$${ x }\left( t \right) \longleftrightarrow { X }\left( s \right) ,\quad { h }\left( t \right) \longleftrightarrow { H }\left( s \right) \\ { h }\left( t \right) \ast { x }\left( t \right) \longleftrightarrow { H }\left( s \right) { X }\left( s \right)$$

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다음 성질들은 미분방정식에서 Laplace Transform이 빛을 발하는 강점인 성질들임.

Differentiation (on time domain)

$$x(t)\longleftrightarrow X(s)\\ \dfrac{d x(t)}{dt} \longleftrightarrow sX(s)-x(0^-)$$

위는 1st order derivative이고, $n$-th order derivatives는 다음과 같음.

일반화된 form으로는 다음과 같음. $$x(t)\longleftrightarrow X(s)\\ \dfrac{d^n x(t)}{dt^n} \longleftrightarrow s^nX(s)-s^{n-1}x(0^-)-s^{n-2}\overset{(1)}{x}(0^-)-\cdots-s\overset{(n-2)}{x}(0^-)-s^0\overset{(n-1)}x(0^-)$$ where

• $\overset{(n-1)}x(t)= \dfrac{d^{(n-1)}x(t)}{dt^{(n-1)}}$

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(Time) Integration

$$x(t)\longleftrightarrow X(s)\\ y(t)=\int^t_0 x(\tau) d\tau \longleftrightarrow Y(s)=\dfrac{X(s)}{s}$$

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Differentiation (on $s$-domain) (or $t$-multiplication)

$$(-t)^nx(t) \longleftrightarrow \dfrac{d^nX(s)}{ds^n}$$

다음과 같은 형태가 보다 많이 쓰이긴 함.

$$(t)^nx(t) \longleftrightarrow -\dfrac{d^nX(s)}{ds^n}$$

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Integration (on $s$-domain) (or $t$-division)

$$\dfrac{x(t)}{t} \longleftrightarrow \int^\infty_s X(u) du$$

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다음의 정리들은 회로해석에서 중요함.
Laplace trasnform으로 대수적 연산으로 미분방정식을 풀고
이를 역변환하기 전에
$s$-domain에서 무한대와 0을 취하는 극한을 구하여
해당 회로(system)의 값을 확인함으로서 역변환 전의 체크를 수행할 수 있음.

Initial Value Theorem

$$\underset{t\to0^+}{\lim}x(t) = \underset{s\to\infty}{\lim}sF(s)$$

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Final Value Theorem

$$\underset{t\to \infty}{\lim}x(t) = \underset{s\to 0}{\lim}sF(s)$$

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그 외의 정리들

Time Reflection

$$x (t) \longleftrightarrow X(s) \\x(-t)=X(-s)$$