Why the Fourier Transform

Lecture 5: Digital Signal What?

20/10/2004
Summary:
  • Signal domains and their Fourier transforms
    • Continuous infinite (FT)
    • Continuous finite (FS)
    • Discrete infinite (DTFT)
    • Discrete finite (DFT)
  • Energy for complex-valued signals
  • Dot product for complex valued signals
  • Eigenvectors of any LTI system
    • Exponentials
    • Exponentials with imaginary rates (spirals!)
    • Proof - Cork-Screw Effect
      • A screw translated is a screw rotated
      • A spiral later is like the spiral rotated
      • Rotation in complex numbers is scaling
      • Thus, a spiral later is the spiral scaled
      • Input scales, output scales by same amount
      • Same scale, same rotation
      • Output later is output rotated
    • Eigenvalue corresponding to this eigenvector is also Fourier transform
    • Only like spirals interact
    • Convolution in signal domain is multiplication in Fourier domain