Frequency-domain analysis | Time domain analysis | Classical control theory | Signal processing
In system analysis, among other fields of study, a linear time-invariant (LTI) system is a system that produces an output signal from any input signal subject to the constraints of linearity and time-invariance; these terms are briefly defined . These properties apply (exactly or approximately) to many important physical systems, in which case the response y(t) of the system to an arbitrary input x(t) can be found directly using convolution: y(t) = x(t) ∗ h(t) where h(t) is called the system's impulse response and ∗ represents convolution (not to be confused with multiplication, as is frequently employed by the symbol in computer languages). What's more, there are systematic methods for solving any such system (determining h(t)), whereas systems not meeting both properties are generally more difficult (or impossible) to solve analytically. A good example of an LTI system is any electrical circuit consisting of resistors, capacitors, inductors and linear amplifiers. Linear time-invariant system theory is also used in image processing, where the systems have spatial dimensions instead of, or in addition to, a temporal dimension. These systems may be referred to as linear translation-invariant to give the terminology the most general reach. In the case of generic discrete-time (i.e., sampled) systems, linear shift-invariant is the corresponding term. LTI system theory is an area of applied mathematics which has direct applications in electrical circuit analysis and design, signal processing and filter design, control theory, mechanical engineering, image processing, the design of measuring instruments of many sorts, NMR spectroscopy, and many other technical areas where systems of ordinary differential equations present themselves. (Wikipedia).
Review of Linear Time Invariant Systems
http://AllSignalProcessing.com for more great signal-processing content: ad-free videos, concept/screenshot files, quizzes, MATLAB and data files. Review: systems, linear systems, time invariant systems, impulse response and convolution, linear constant-coefficient difference equations
From playlist Introduction and Background
Introduction to Linear Time Invariant System Descriptions
http://AllSignalProcessing.com for free e-book on frequency relationships and more great signal processing content, including concept/screenshot files, quizzes, MATLAB and data files. Introduces systems and their use in signal processing; defines linearity, time invariance, and causal sys
From playlist Introduction and Background
Discrete-Time Dynamical Systems
This video shows how discrete-time dynamical systems may be induced from continuous-time systems. https://www.eigensteve.com/
From playlist Data-Driven Dynamical Systems
Intro to Linear Systems: 2 Equations, 2 Unknowns - Dr Chris Tisdell Live Stream
Free ebook http://tinyurl.com/EngMathYT Basic introduction to linear systems. We discuss the case with 2 equations and 2 unknowns. A linear system is a mathematical model of a system based on the use of a linear operator. Linear systems typically exhibit features and properties that ar
From playlist Intro to Linear Systems
Analytically Solving Systems of Linear Ordinary Differential Equations
In this video we derive the analytical solution to a system of linear ordinary differential equations. This is also referred to as the linear time invariant (LTI) system or state space system. As such, this video describes the analytical solution to a linear state space system. In addit
From playlist Ordinary Differential Equations
Lecture 5, Properties of Linear, Time-invariant Systems | MIT RES.6.007 Signals and Systems
Lecture 5, Properties of Linear, Time-invariant Systems Instructor: Alan V. Oppenheim View the complete course: http://ocw.mit.edu/RES-6.007S11 License: Creative Commons BY-NC-SA More information at http://ocw.mit.edu/terms More courses at http://ocw.mit.edu
From playlist MIT RES.6.007 Signals and Systems, 1987
Time-Independent Schrödinger Equation | Quantum Mechanics
In this video, we will talk about the time-independent Schrödinger equation in quantum mechanics. If we start with the time-dependent Schrödinger equation, we can get to the time-independent one by performing a separation of variables on the wave function, where we claim that the time depe
From playlist Quantum Mechanics, Quantum Field Theory
Lecture 6, Systems Represented by Differential Equations | MIT RES.6.007 Signals and Systems
Lecture 6, Systems Represented by Differential Equations Instructor: Alan V. Oppenheim View the complete course: http://ocw.mit.edu/RES-6.007S11 License: Creative Commons BY-NC-SA More information at http://ocw.mit.edu/terms More courses at http://ocw.mit.edu
From playlist MIT RES.6.007 Signals and Systems, 1987
Lecture 24 | The Fourier Transforms and its Applications
Lecture by Professor Brad Osgood for the Electrical Engineering course, The Fourier Transforms and its Applications (EE 261). Professor Osgood continues his lecture on linear systems. The Fourier transform is a tool for solving physical problems. In this course the emphasis is on rela
From playlist Lecture Collection | The Fourier Transforms and Its Applications
Frequency Response Descriptions for LTI Systems
http://AllSignalProcessing.com for free e-book on frequency relationships and more great signal processing content, including concept/screenshot files, quizzes, MATLAB and data files. An introduction to the description of the input output characteristics of linear time-invariant systems b
From playlist Introduction and Background
Lec 2 | MIT RES.6-008 Digital Signal Processing, 1975
Lecture 2: Discrete-time signals and systems, part 1 Instructor: Alan V. Oppenheim View the complete course: http://ocw.mit.edu/RES6-008S11 License: Creative Commons BY-NC-SA More information at http://ocw.mit.edu/terms More courses at http://ocw.mit.edu
From playlist MIT RES.6-008 Digital Signal Processing, 1975
Koopman Observable Subspaces & Finite Linear Representations of Nonlinear Dynamics for Control
This video illustrates the use of the Koopman operator to simulate and control a nonlinear dynamical system using a linear dynamical system on an observable subspace. From the Paper: Koopman observable subspaces and finite linear representations of nonlinear dynamical systems for contro
From playlist Research Abstracts from Brunton Lab
Lecture 25 | The Fourier Transforms and its Applications
Lecture by Professor Brad Osgood for the Electrical Engineering course, The Fourier Transforms and its Applications (EE 261). Professor Osgood lectures on the relationship between LTI and the Fourier transforms. The Fourier transform is a tool for solving physical problems. In this cou
From playlist Lecture Collection | The Fourier Transforms and Its Applications
Lec 14 | MIT RES.6-008 Digital Signal Processing, 1975
Lecture 14: Design of IIR digital filters, part 1 Instructor: Alan V. Oppenheim View the complete course: http://ocw.mit.edu/RES6-008S11 License: Creative Commons BY-NC-SA More information at http://ocw.mit.edu/terms More courses at http://ocw.mit.edu
From playlist MIT RES.6-008 Digital Signal Processing, 1975
Coherence and the Cross Spectrum
http://AllSignalProcessing.com for more great signal processing content, including concept/screenshot files, quizzes, MATLAB and data files. Coherence and the cross spectrum describe the relationship between two random signals in the frequency domain based on their second order statistics
From playlist Random Signal Characterization
Impulse Response Descriptions for Systems
http://AllSignalProcessing.com for more great signal-processing content: ad-free videos, concept/screenshot files, quizzes, MATLAB and data files. Introduces the impulse response and convolution sum for determining the output of a linear time-invariant system from the input. Defines finit
From playlist Introduction and Background
Lecture 23 | The Fourier Transforms and its Applications
Lecture by Professor Brad Osgood for the Electrical Engineering course, The Fourier Transforms and its Applications (EE 261). Professor Osgood lectures on linear systems, focusing on linear time and variance systems. The Fourier transform is a tool for solving physical problems. In thi
From playlist Lecture Collection | The Fourier Transforms and Its Applications