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HomeHomework Helpcontrol-systemsLinear Time-Invariant Systems

Linear Time-Invariant Systems

Linear Time-Invariant (LTI) systems are a class of systems in signal processing and control theory that exhibit linearity and time-invariance, allowing for the analysis and design of systems through convolution and the use of impulse responses.

intermediate
3 hours
Control Systems
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Overview

Linear Time-Invariant (LTI) systems are fundamental in engineering and mathematics, providing a framework for analyzing and designing systems that respond predictably to inputs. Their linearity and time-invariance properties simplify the analysis, allowing for the use of powerful mathematical tools ...

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Key Terms

Linearity
A property of a system where the output is directly proportional to the input.

Example: If you double the input, the output also doubles.

Time-Invariance
A property of a system where the behavior does not change over time.

Example: A system that behaves the same way today as it will tomorrow.

Impulse Response
The output of an LTI system when the input is an impulse function.

Example: The impulse response can be used to determine the output for any input.

Convolution
A mathematical operation that combines two functions to produce a third function.

Example: Convolution is used to find the output of an LTI system given its impulse response.

Superposition
The principle that the total response of a linear system is the sum of the responses to individual inputs.

Example: If a system responds to input A and input B, the response to A + B is the sum of the two responses.

Transfer Function
A mathematical representation of the relationship between the input and output of a system in the frequency domain.

Example: The transfer function can be used to analyze system stability.

Related Topics

Control Theory
The study of how to manipulate the behavior of dynamical systems.
advanced
Signal Processing
The analysis, interpretation, and manipulation of signals.
intermediate
Fourier Transform
A mathematical transform that expresses a function in terms of its frequency components.
intermediate

Key Concepts

LinearityTime-InvarianceImpulse ResponseConvolution