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3: Time Domain Analysis of Continuous Time Systems

Last updated

Feb 23, 2021
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- 2.2: Linear Time Invariant Systems
- 3.1: Continuous Time Systems

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3.1: Continuous Time Systems
This page focuses on continuous time systems, specifically linear time invariant (LTI) systems. It defines linearity through additivity and homogeneity, and explains time invariance as consistent system responses to time shifts. The benefits of analyzing LTI systems with convolution and frequency domain techniques are noted, along with examples such as modeling a series RLC circuit with linear constant coefficient ordinary differential equations.
3.2: Continuous Time Impulse Response
This page explains that the output of a Linear Time-Invariant (LTI) system depends on its impulse response and input. The impulse response is the system's output from a unit impulse. By using the sifting property, inputs can be represented as shifted impulses, allowing output computation through convolution. Impulse responses can be found by solving differential equations or using impulse-like signals for measurement.
3.3: Continuous Time Convolution
This page discusses convolution as a key principle in electrical engineering for determining the output of linear time-invariant systems using input signals and impulse responses. It covers mathematical definitions, properties like commutativity, and the sifting property of the unit impulse function. The text illustrates continuous and circular convolution with graphical methods, emphasizing their importance in analyzing system responses.
3.4: Properties of Continuous Time Convolution
This page explores the essential properties of continuous time convolution in signal processing, including associativity, commutativity, distributivity, and more. It provides mathematical proofs for each property, highlighting key relationships such as the derivative of a convolution and the impact of the Dirac delta function. Additionally, it discusses the duration of convoluted signals and notes exceptions in circular convolution.
3.5: Eigenfunctions of Continuous Time LTI Systems
This page explains the role of complex exponentials as eigenfunctions in linear time invariant (LTI) systems, noting that inputting a continuous time signal like $e^{st}$ results in an output that is a scaled complex exponential. The discussion, although informal, highlights how these exponentials simplify calculations for LTI systems and relate to the continuous time Fourier transform and series for signal analysis.
3.6: BIBO Stability of Continuous Time Systems
This page discusses BIBO stability, which is a property of systems where bounded inputs lead to bounded outputs. It emphasizes that a bounded signal stays within a finite limit. For continuous-time LTI systems, BIBO stability is attained if the impulse response is absolutely integrable. In the Laplace domain, stability is represented by the pole-zero plot, requiring poles to be on the left side of the imaginary axis. BIBO stability is essential for the effectiveness of systems.
3.7: Linear Constant Coefficient Differential Equations
This page discusses the significance of differential equations in modeling continuous time systems within signals and systems. It distinguishes between ordinary differential equations (ODEs) and partial differential equations, focusing on linear constant coefficient ODEs. These equations arise in contexts like radioactive decay and electrical circuits and can be solved uniquely with the right initial conditions, emphasizing their importance in electrical engineering and practical applications.
3.8: Solving Linear Constant Coefficient Differential Equations
This page covers the solving of linear constant coefficient ordinary differential equations (ODEs), emphasizing initial value problems. It differentiates between initial and boundary value problems, requiring $N$ initial conditions for unique solutions in $N$ th-order ODEs. The general solution comprises a homogeneous solution from the characteristic polynomial and a particular solution based on the forcing function.

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