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Laplace Transform

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Two Famous Linear Transforms

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Pierre-Simon Laplace:

23 March 1749 – 5 March 1827) was a French scholar whose work was important to the development of engineering, mathematics, statistics, physics, astronomy, and philosophy.

Jean-Baptiste Joseph Fourier:

21 March 1768 – 16 May 1830) was a French mathematician and physicist and best known for initiating the investigation of Fourier series, which eventually developed into Fourier analysis and harmonic analysis, and their applications to problems of heat transfer and vibrations.

Laplace transform

Fourier transform

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Inner Product

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Inner Product

  • For real valued vector

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  • Orthogonal

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Inner Product

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Hermitian Transpose

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Two Linear Transformations

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Fourier Transform

Laplace Transform

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Laplace Transform

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Laplace Transform

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Laplace Transform

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Eigenfunction

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Definition: Laplace Transform

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Example 1

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Example 2

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Example 3

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Unit Impulse Signal

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Laplace Transform of Unit-Impulse Signal

  • Laplace transform of unit-impulse function

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Unit Step Signal

  • The indefinite integral of the unit-impulse is the unit-step

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  • Laplace transform of unit-step function

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Convolution Property of Laplace Transform

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Convolution Property of Laplace Transform

  • Change of variables

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Response to General Input

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Laplace Transform of a Derivative

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Laplace Transform of a Derivative

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Solving Differential Equations with Laplace Transform

  • Example of the first ODE

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  • Laplace transform

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  • This is a simple algebraic expression
  • Laplace transform converts a differential equation to an equivalent algebraic equation

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Solving Differential Equations with Laplace Transform

  • Example of the second ODE

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Laplace Transform of Integral

  • Laplace transform of integral

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Laplace Transform of Integral

  • Laplace transform of integral

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Transfer Function

  • Transfer function of an LTI system is defined as the Laplace transform of the impulse response.

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  • Describe input-output behavior of the system.

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Rational Transfer Function

  • Example

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Transfer Function

  • Transfer function of differential equation system

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Poles and Zeros

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