Linear Systems and Unit Impulse Response

Overview

• A linear system transforms an input function into an output function.
• Different linear systems produce different outputs for the same input.
• Predicting the output for any possible input is key for a given linear system.
• Input functions can be any size and shape.

Linearity Property

• If the input is the sum of two different input functions, then the output is the sum of the two associated output functions.
• Knowing the system's response to a unit impulse allows us to predict the response to any input.

Unit Impulse and Impulse Response

• *Unit Impulse:
  • A function with an area equal to one and a width approaching zero.
  • As width becomes smaller, height becomes larger to maintain the area of one.
• Unit Impulse Response:
  • The system's output when the input is a unit impulse.
  • Key to understanding and predicting the system's response to any input.

Infinite Sum Representation

• Any input can be viewed as an infinite sum of impulse functions.
• Correspondingly, the output is an infinite sum of output functions.
• Each output function is scaled by the height of the associated input pulse.

Time and Tau Variables

• Time: Fixed at the moment of interest.
• Tau: A variable to calculate the area of each red rectangle corresponding to input functions.
  • Each red rectangle is associated with the input function’s value at “Time minus Tau”.
  • It also relates to the unit impulse response at time equals Tau.

Height and Area Calculation

• Height of each rectangle depends on the input function at “Time minus Tau” and the unit impulse response at Tau.
• Width is denoted by “d tau”, which approaches zero.
• Area of each rectangle: height x width.
• The total area of all rectangles at a moment in time gives the total output.
  • As “d tau” approaches zero, each rectangle’s area approaches zero, but summing them gives the convolution integral.

Convolution

• The process of summing the areas of these rectangles is called convolution.
• Convolution equation represents the total output at a given moment in time.
• Essential for analyzing continuous linear systems.

Laplace Transform

• Used to compute output functions alternatively.
• Transfer Function:
  • Laplace Transform of the unit impulse response.
  • Important for studying frequency response and stability of the system.

Summary

• Understanding the unit impulse and convolution is crucial to predicting a linear system's behavior.
• The transfer function provides insights into frequency response and system stability.
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