本文僅供學習使用
本文參考：
B站：DR_CAN
Dr. CAN學習筆記-數學基礎Ch0-5Laplace Transform of Convolution卷積的拉普拉斯變換
Laplace Transform : X ( s ) = L [ x ( t ) ] = ∫ 0 ∞ x ( t ) e ? s t d t X\left( s \right) =\mathcal{L} \left[ x\left( t \right) \right] =\int_0^{\infty}{x\left( t \right) e^{-st}}\mathrm{d}t X(s)=L[x(t)]=∫0∞?x(t)e?stdt
Convolution : x ( t ) ? g ( t ) = ∫ 0 t x ( τ ) g ( t ? τ ) d τ x\left( t \right) *g\left( t \right) =\int_0^t{x\left( \tau \right) g\left( t-\tau \right)}\mathrm{d}\tau x(t)?g(t)=∫0t?x(τ)g(t?τ)dτ
證明： L [ x ( t ) ? g ( t ) ] = X ( s ) G ( s ) \mathcal{L} \left[ x\left( t \right) *g\left( t \right) \right] =X\left( s \right) G\left( s \right) L[x(t)?g(t)]=X(s)G(s)
L [ x ( t ) ? g ( t ) ] = ∫ 0 ∞ ∫ 0 t x ( τ ) g ( t ? τ ) d τ e ? s t d t = ∫ 0 ∞ ∫ τ ∞ x ( τ ) g ( t ? τ ) e ? s t d t d τ \mathcal{L} \left[ x\left( t \right) *g\left( t \right) \right] =\int_0^{\infty}{\int_0^t{x\left( \tau \right) g\left( t-\tau \right) \mathrm{d}\tau}e^{-st}}\mathrm{d}t=\int_0^{\infty}{\int_{\tau}^{\infty}{x\left( \tau \right) g\left( t-\tau \right)}e^{-st}}\mathrm{d}t\mathrm{d}\tau L[x(t)?g(t)]=∫0∞?∫0t?x(τ)g(t?τ)dτe?stdt=∫0∞?∫τ∞?x(τ)g(t?τ)e?stdtdτ
>令： u = t ? τ , t = u + τ , d t = d u + d τ , t ∈ [ τ , + ∞ ) ? u ∈ [ 0 , + ∞ ) u=t-\tau ,t=u+\tau ,\mathrm{d}t=\mathrm{d}u+\mathrm{d}\tau ,t\in \left[ \tau ,+\infty \right) \Rightarrow u\in \left[ 0,+\infty \right) u=t?τ,t=u+τ,dt=du+dτ,t∈[τ,+∞)?u∈[0,+∞)
L [ x ( t ) ? g ( t ) ] = ∫ 0 ∞ ∫ 0 ∞ x ( τ ) g ( u ) e ? s ( u + τ ) d u d τ = ∫ 0 ∞ x ( τ ) e ? s τ d τ ∫ 0 ∞ g ( u ) e ? s u d u = X ( s ) G ( s ) \mathcal{L} \left[ x\left( t \right) *g\left( t \right) \right] =\int_0^{\infty}{\int_0^{\infty}{x\left( \tau \right) g\left( u \right)}e^{-s\left( u+\tau \right)}}\mathrm{d}u\mathrm{d}\tau =\int_0^{\infty}{x\left( \tau \right)}e^{-s\tau}\mathrm{d}\tau \int_0^{\infty}{g\left( u \right)}e^{-su}\mathrm{d}u=X\left( s \right) G\left( s \right) L[x(t)?g(t)]=∫0∞?∫0∞?x(τ)g(u)e?s(u+τ)dudτ=∫0∞?x(τ)e?sτdτ∫0∞?g(u)e?sudu=X(s)G(s)
L [ x ( t ) ? g ( t ) ] = L [ x ( t ) ] L [ g ( t ) ] = X ( s ) G ( s ) \mathcal{L} \left[ x\left( t \right) *g\left( t \right) \right] =\mathcal{L} \left[ x\left( t \right) \right] \mathcal{L} \left[ g\left( t \right) \right] =X\left( s \right) G\left( s \right) L[x(t)?g(t)]=L[x(t)]L[g(t)]=X(s)G(s)
malloc,calloc,realloc函數的介紹使用及柔性數組的介紹)
