Fourier Series and Laplace Transform
First published: 2011
Brief summary
MIT OCW material connecting waves, Fourier analysis and normal modes.
Article
Fourier series represent periodic functions as sums of sinusoidal components with different frequencies, amplitudes and phases. This allows a complicated repeating signal to be decomposed into harmonically related components.
For linear differential equations, the response to a periodic input can be found by solving for the response to each sinusoidal component and then adding the results through the superposition principle.
The Laplace transform represents a function in a new variable and converts many differential equations into algebraic equations. Initial conditions can be incorporated directly, and the transformed solution can then be converted back to the original time domain.
The unit also introduces convolution and the delta function. Convolution describes how a system combines its impulse response with an input, while the delta function provides an idealised representation of a sudden impulse.
Source details and credits
- Source / publisher: MIT OCW
- Source type: Course material
- URL type: WWW
- Credits: MIT OCW
- URL: https://ocw.mit.edu/courses/18-03sc-differential-equations-fall-2011/pages/unit-iii-fourier-series-and-laplace-transform/