0. Laplace Transform
0-0. Bi-lateral Laplace Transform
0-1. Uni-lateral Laplace Transform
2022.10.24 - [.../Signals and Systems] - [SS] Laplace Transform Table
[SS] Laplace Transform Table
Signal Laplace Transform RoC ... 1 $u(t)$ $\frac{1}{s}$ $\text{Re}(s)>0$ 2 $u(t)-u(t-a)$ $\frac{1-e^{-as}}{s}$ $\text{Re}(s)>0$ 3 $\delta(t)$ 1 all complex plane 4 $\delta(t-a)$ $e^{-as}$ all complex plane 5 $e^{-at}u(t)$ $\frac{1}{s+a}$ $\text{Re}(s)>-a$
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1. Properties of Laplace Transform (Uni-lateral)
2022.10.25 - [.../Signals and Systems] - [SS] Properties of (unilateral) Laplace Transform
[SS] Properties of (unilateral) Laplace Transform
Unilateral Laplace Transform의 주요 성질. Linearity $$a{ x }_{ 1 }\left( t \right) +b{ x }_{ 2 }\left( t \right) \longleftrightarrow a{ X }_{ 1 }\left( s \right) +{ X }_{ 2 }\left( s \right)$$ Time Shifting $$x\left( t-{ t }_{ 0 } \right) u\left( t-{
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2. Inverse Lapalce Transform
2-0. Partial Fraction Decomposition
2021.11.11 - [.../Signals and Systems] - [SS] Partial Fraction Decomposition (부분분수분해)
[SS] Partial Fraction Decomposition (부분분수분해)
Partial fraction decomposition은 이항분리 라는 이름으로도 불림. 여러 방법이 있지만, Heaviside라는 분이 제안한 Cover-up 기법이 가장 효과적인 기법으로 알려져 있음. 아주 간단한 경우에는 통분 후 등식
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https://github.com/dsaint31x/SS/blob/master/SS_04_PartialFractionDecomposition.ipynb
3. Differential Equation (미분방정식)
[SS] Differential Equation and Response (zero-input, zero-state, natural, forced) w/ Laplace Transform
Laplace Transform을 이용한 Differential Equation을 풀기. 문제 다음 미분방정식 의 시스템이 있다고 하자. $$ \frac{d^2 y(t)}{dt^2} + 3\frac{dy(t)}{dt}+2y(t)=\frac{dx(t)}{dt} $$ 아래와 같은 입력과 초기조건에서 zero-state
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4. Transfer Function (전달함수)
2024.12.15 - [.../Signals and Systems] - [SS] 예제: 미분방정식
[SS] 예제: 미분방정식
input signal: $x(t) = 10 e^{-3t} u(t)$initial condition: $y(0^-) = 0, y^\prime(i^-) = 0$8. differential equation9. frequency response10. transfer function11. impulse response12. zero input response and zero state response 8-11: https://youtu.be/3pgvBgIf99
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5. 시스템 구현도
2023.11.16 - [.../Signals and Systems] - [SS] System Representation w/ Laplace Transform
[SS] System Representation w/ Laplace Transform
Laplace transform 의 인수분해(factorization) 등을 이용하여 다양한 형태의 구현도(Implementation) 가 가능해짐. Differential Equation and Transfer Function $$\dfrac{d^2y(t)}{dt^2}+a_1\dfrac{dy(t)}{dt}+a_0y(t)=b_2\dfrac{d^2x(t)}{dt^2}+
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5-0. 시스템 연결
2023.11.16 - [.../Signals and Systems] - [SS] Cascade Connection : Transfer Function
[SS] Cascade Connection : Transfer Function
Transfer function은 impulse response의 Laplace transform이고, time-domain에서의 convolution은 s-domain에서 multiplication이므로 다음이 성립. $$H(s)=H_1(s)H_2(s)\cdots H_n(s)\\ h(t)=h_1(t) * h_2(t)* \cdots * h_n(t)$$ cascade connection의
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2023.11.16 - [.../Signals and Systems] - [SS] Parallel Connection : Transfer Function
[SS] Parallel Connection : Transfer Function
Cascade connection이 곱셈인 반면, Parallel transform은 덧셈임. $$H(s)=H_1(s)+H_2(s)+\cdots + H_n(s)\\h(t)=h_1(t)+h_2(t)+\cdots +h_n(t)$$ Example $$\begin{aligned}H(s)&=\frac{2}{s^2+3s+2} \\ &=\frac{A_1}{s+1}\frac{A_2}{s+2}\end{aligned} \\ \quad \
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6. Poles and Zeros
[SS] Poles and Zeros : Impulse Response and Frequency Response
Pre-requiredments Region of Convergence (ROC) Laplace Transform의 ROC는 다음과 같은 특징을 가짐. 유한한 구간내에 존재하며 발산하지 않는 signal : ROC가 모든 s 평면에서 수렴. right-sided signal (어떤 값 이상의 구
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