Signals and systems/GF Fourier

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Fourier series

The Fourier series is used to analyze arbitrary periodic functions by showing them as a composite of sines and cosines.

A function is considered periodic if for .

The exponential form of the Fourier series is defined as

Determining the coefficient

  • The definition of the Fourier series

  • Integrating both sides for one period. The range of integration is arbitrary, but using scales nicely when extending the Fourier series to a non-periodic function

  • Multiply by the complex conjugate

    • Using L'Hopitals to evaluate the case. Note that n & m are integers

Notation

Linear Time Invariant Systems

Must meet the following criteria

  • Time independance
  • Linearity
    • Superposition (additivity)
    • Scaling (homogeneity)

Complex Conjugate

Changing Basis Functions

Identities