⛓️ Fourier Series §

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

Formal §

Periodic functions §

Let $f$ be T-periodic function:

$f(t)=\frac{a[0]}{2}+{\sum }_{k\ge 1}a[k]\cos (\frac{2\pi }{T}kt)+b[k]\sin (\frac{2\pi }{T}kt)$

• $a[k]\in R$ and $b[k]\in R$: 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:

$e^{i\phi }=\cos (\phi )\sin (\phi )$

So the Fourier series can be written as:

$f(t){\sum }_{k=-\infty }^{\infty }c[k]e^{2\pi ikt/T},c[k]=\frac{1}{T}{\int }_0^{T}f(t)e^{-2\pi ikt/T}dt$

Non-Periodic functions §

To extend the Fourier series to non-periodic signals, its period $T$ needs to be extended to $\infty$ → $k$ becomes a continuous variable → $c[k]$ becomes a function of k → summation needs to be an integral

$f(t)={\int }_{-\infty }^{\infty }F(k)e^{2\pi ikt}dk$

with:

• $F(k)$: fourier transform