⛓️ Fourier Series

= a sum that represents any function as a sum of sine and cosine waves


Periodic functions

Let be T-periodic function:

  • and : Fourier coefficients
    • used for the unique mathematical description of periodic functions
Even/Odd functions
  • even function (symmetry to y-axis): only cosine waves
  • odd function (symmetry to origin): only sine waves
Complex numbers

Each sine + cosine pair can be interpreted as a complex number:

So the Fourier series can be written as:

Non-Periodic functions

To extend the Fourier series to non-periodic signals, its period needs to be extended to becomes a continuous variable → becomes a function of k → summation needs to be an integral