Fourier Analysis

Usefulsness

• 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)

One formula to rule them all

Let $s(t)$ be a signal (continuous, discrete, periodic…). Let $e_{\nu}(t)$ be the (continuous, discrete, periodic…) complex sine at frequency $\nu$, then $$ \hat s(\nu) = \langle s(t), e_{\nu}(t)\rangle $$ and $$ s(t) = \langle s(\nu), \overline{e_{\nu}(t)}\rangle $$ with appropriate inner products !