11: Laplace Transform and Continuous Time System Design
- Page ID
- 22913
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- 11.1: Laplace Transform
- This page discusses the Laplace transform, which generalizes the Continuous-Time Fourier Transform and is preferred for its convergence and notation. It utilizes a complex parameter and distinguishes between bilateral and unilateral transforms while drawing parallels to Fourier transforms. It notes visualization challenges and suggests the poles and zeros method for clarity.
- 11.2: Common Laplace Transforms
- This page provides a table that lists various signals along with their Laplace transforms and regions of convergence. It covers essential signals such as the Dirac delta function, unit step function, polynomials, exponential functions, and trigonometric functions. Each entry specifies the conditions for the transforms' validity based on the real part of \(s\).
- 11.3: Properties of the Laplace Transform
- This page details various properties of Laplace transforms, including linearity, shifting, scaling, conjugation, convolution, differentiation, and time integration. Each property is accompanied by its corresponding signal, transform, and required region of convergence, illustrating how signals are transformed in the Laplace domain.
- 11.4: Inverse Laplace Transform
- This page outlines four key methods for finding the inverse Laplace transform \(h(t)\) from \(H(s)\): Inspection, Partial-Fraction Expansion, Power Series Expansion, and Contour Integration. Each method is briefly described, with examples included to demonstrate their applications. The discussion concludes that these techniques are essential for filter design in engineering.
- 11.5: Poles and Zeros in the S-Plane
- This page discusses poles and zeros in the context of the Laplace transform and transfer functions, highlighting how they impact system behavior when plotted on the S-plane. It emphasizes their significance in analyzing system stability and filter design, providing qualitative insights into dynamics. Pole-zero plots are crucial in control theory and understanding frequency response.
- 11.6: Region of Convergence for the Laplace Transform
- This page explains the region of convergence (ROC) in Laplace transforms for continuous-time LTI systems, detailing how certain signals yield converging outputs while others do not. It differentiates convergence conditions for causal and anti-causal signals based on the real part of the complex variable \(s\). The ROC is depicted in the s-plane, with the overall ROC for multiple poles determined by the intersection of their individual regions.
- 11.7: Rational Functions and the Laplace Transform
- This page introduces rational functions as the ratio of two polynomials, highlighting the significance of their roots for understanding properties like discontinuities and domain. Discontinuities arise when the denominator is zero, leading to vertical asymptotes in graphs. It covers x-intercepts and y-intercepts based on the roots of the numerator and the value of \(x\) at zero.
- 11.8: Differential Equations
- This page discusses the role of differential equations in describing system dynamics and the use of linear constant coefficient ordinary differential equations (LCCDE). It explains the application of the unilateral Laplace transform to solve nonhomogeneous equations and derive the transfer function \(H(s)\) for frequency response analysis.
- 11.9: Continuous Time Filter Design
- This page discusses the design and utility of analog filters, particularly Continuous-Time filters, in signal processing. It covers the estimation of frequency response using pole/zero plots and compares different filter types, including Butterworth, Chebyshev, and elliptic filters. Chebyshev filters use Taylor series for effective passband performance, while elliptic filters combine approximations for both passband and stopband efficiency.