Fourier reminder

Section: Time-Frequency

From time representation to frequency

  • Spectral analysis : frequency content of a function
  • Think about musical notes !
  • Measure the similute (correlation, angle) between pure (complex) sine and a signal
  • Sines are eigen signal of time invariant linear systems (filters)

Fourier analysis computes the correlation between the signal $s$ and a pure sine at various frequencies $\nu$: $$ \hat s(\nu) = \langle s(t), e_{\nu}(t)\rangle $$

Limitation of Fourier analysis

  • We obtain a pure frequency content from a pure temporal content
  • What is the difference between a sum of sine, and a succession of sine ?