Fourier Series

Fourier Series - Definition

Cosine & Sine Form

The Fourier series of a periodic function 𝑓(𝑥) of period 𝑃 is:

• $f(x)=\frac{A_0}{2}+{\sum }_{n=1}^{\infty }[A_{n}cos(2\pi \frac{n}{P}x)+B_{n}sin(2\pi \frac{n}{P}x)]$

where:

• 𝑃 - is the interval length of the repeating period
• $A_n=\frac{2}{P}{\int }_0^{P}f(x)cos(2\pi \frac{n}{P}x)dx\text{for }n\ge 1$
• $B_n=\frac{2}{P}{\int }_0^{P}f(x)sin(2\pi \frac{n}{P}x)dx\text{for }n\ge 1$

Intuition of 𝐴_𝑛 and 𝐵_𝑛

Click here to expand... _𝑛 and 𝐵_𝑛 act as inner product of functions:

• $A_n=\frac{2}{P}{\int }_0^{P}f(x)cos(2\pi \frac{n}{P}x)dx=\frac{1}{||cos(2\pi \frac{n}{P}x)|{|}^2}⟨f(x),cos(2\pi \frac{n}{P}x)⟩$
• $B_n=\frac{2}{P}{\int }_0^{P}f(x)sin(2\pi \frac{n}{P}x)dx=\frac{1}{||sin(2\pi \frac{n}{P}x)|{|}^2}⟨f(x),sin(2\pi \frac{n}{P}x)⟩$

In other words how much of 𝑓(𝑥) is in 𝑐𝑜𝑠(2𝜋(𝑛/𝑃)𝑥) and 𝑠𝑖𝑛(2𝜋(𝑛/𝑃)𝑥) for each 𝑛 from 1 to 𝑃.

Thus, the Fourier Series transforms a periodic function from the 𝑥 domain into the frequency domain.

𝐴

Complex Form Fourier Series

The complex Fourier Series of a periodic function 𝑓(𝑥) of period 𝑃 is:

• $f(x)={\sum }_{n=-\infty }^{\infty }{C}_n{e}^{i2\pi \frac{n}{P}x}$
• $f(x)={\sum }_{n=-\infty }^{\infty }({𝛼}_n+i{𝛽}_n)(cos(2\pi \frac{n}{P}x)+isin(2\pi \frac{n}{P}x))$

where:

• 𝐶_𝑛 is defined by the inner product of functions:
  • $C_n=⟨f(x),e^{i2\pi \frac{n}{P}x}⟩$
• 𝑒^𝑖𝑥 is Euler’s Formula

For more details, see: Complex Fourier Series

Fourier Series - Derivation

• Sal Khan’s Series - Derivation of Fourier Series

Fourier Series - Examples

Fourier Series of a Square Wave

Find the Fourier series of the square wave, for which the function over one period is

The function is odd and has an average value of zero, with period 𝑇=1. Therefore, all 𝑎_𝑘 vanish; one must only compute the integrals to find the 𝑏_𝑘

where in the last line the fact that 𝑘 is a positive integer was used. Therefore, the Fourier series for the square wave is

Subpages

• Complex Fourier Series

Resources

• Steve Brunton’s Fourier Series