10/02 - Fourier Series

Fourier Series (as compared to vectors)

If a function is periodic, $x(t)=x(t+T)\,\!$, and it meets the Dirichlet conditions, then we can write it as $x(t)=\sum _{n=-\infty }^{\infty }\alpha _{n}e^{j2\pi nt/T}$

• Dirichlet conditions
  • x(t) must have a finite number of extrema in any given interval
  • x(t) must have a finite number of discontinuities in any given interval
  • x(t) must be absolutely integrable over a period
  • x(t) must be bounded

Like vectors we can change to a new basis function by taking the inner product of $x(t)\,\!$ with the mth basis function.