Fourier Analysis

Review: Mathematical Physics

Euler-Fourier Theorem

Suppose a function f(α) has the following properties:

• periodic with period 2π , i.e f(α ± 2π)=f(α)

• finite and has single/unique value

• has finite maximum and finite minimum within a finite interval

• has finite discontinuities in a finite interval

• absolutely integrable :

: bounded

• Then this function f(α) can be expanded as a series of the form:

Orthogonality and Fourier Coefficient

• The coefficients : an dan bn can be determined by applying the orthogonality properties of the basis functions : sine and cosine functions.

m,n : integers.

• Using these orthogonality, it can be shown that:

• For bn=1,2,3,…. While an=0,1,2,3,….

Other Intervals

• Let f(x) be a periodic function with period of λ : f(x± λ)=f(x)

• Hence substitute α = 2π x/λ → because when x=x ± λ then α ± 2π,

• Define k= 2π/λ, hence α = kx, which enable us to rewrite f(x) as:

where

Similarly, if f(t) is periodic with period T : f(t+T)=f(t), we substitute. α = 2π t/T and define ω= 2π/T, then

General Interval (L)

• Suppose a periodic function f(x) has period L : f(x± L)=f(x),

• In this case, substitute α = 2π x/L which is equivalent to α ± 2π,

• Now f(x) can be expanded as Fourier series :

The coefficients can be shown as :

Example: Fourier Series

Given a periodic square wave :

• Hence :

Thus :

0 2

h

-h

Term by term contribution on a Fourier Series

Fourier Series : Complex Variable

• Fourier Series in complex representation is written using fundamental identities:

• So another way to express f(x) under Fourier expansion would be:

• Where (n=0, ±1, ±2 ….)

With 𝑘 =2𝜋

𝐿

the orthogonality is : −𝐿/2𝐿/2

exp(𝑖𝑛𝑘𝑥) exp −𝑖𝑚𝑘𝑥 𝑑𝑥 = 𝐿𝛿𝑛,𝑚

The coefficient will be :

Fourier Series Spectrum

• Coefficient/ amplitude related to each term on the series is termed the distribution/ spectrum of the series.

• Example:

Fourier Series of Even Function

•When the function to be expanded is an even function : 𝑓 −𝑥 = 𝑓(𝑥) then a simpler computation can be derived for Fourier series.

•Suppose the length of interval is L : -L/2 to L/2.

•The Fourier Series is given by

•𝑓 𝑥 =1

2𝑎0 + 𝑛=1(𝑎𝑛 cos 𝑛𝑘𝑥 + 𝑏𝑛 sin(𝑛𝑘𝑥)) where as

usual 𝑘 =2𝜋

𝐿.

•The coefficients are given by:

𝑎𝑛 =2

𝐿 −𝐿/2𝐿/2

𝑓 𝑥 cos 𝑛𝑘𝑥 𝑑𝑥 & 𝑏𝑛 =2

𝐿 −𝐿/2𝐿/2

𝑓 𝑥 sin 𝑛𝑘𝑥 𝑑𝑥

Fourier Series of Even Function

•Now, 𝑓 −𝑥 = 𝑓 𝑥 and we know that cos −𝑥 = cos(𝑥)andsin −𝑥 = −sin 𝑥 . So:

𝑎𝑛 =2

𝐿 −𝐿

2

0𝑓 𝑥 cos 𝑛𝑘𝑥 𝑑𝑥 + 0

𝐿

2 𝑓 𝑥 cos 𝑛𝑘𝑥 𝑑𝑥

For the first integral substitute 𝑥 = −𝑥′ so −𝐿

2→𝐿

2

and the integral becomes:

𝑎𝑛 =4

𝐿 0

𝐿2𝑓 𝑥 cos 𝑛𝑘𝑥 𝑑𝑥

Whereas the 𝑏𝑛 =2

𝐿 −𝐿/2𝐿/2

𝑓 𝑥 sin 𝑛𝑘𝑥 𝑑𝑥 = 0!

Notice that in computing 𝑎𝑛 we only need half of the interval!

Fourier Series of Odd Function

• If the function is an odd function : 𝑓 −𝑥 = −𝑓(𝑥) then a similar result is obtained:

𝑏𝑛 =4

𝐿 0

𝐿2𝑓 𝑥 cos 𝑛𝑘𝑥 𝑑𝑥

Whereas the 𝑎𝑛 =2

𝐿 −𝐿/2𝐿/2

𝑓 𝑥 cos 𝑛𝑘𝑥 𝑑𝑥 = 0!

Again we only need half of the interval!

Fourier Series of a Wave

• A wave function f(x-vt) is periodic with spatial and temporal period of λ and T respectively, thus :

f(x-vt)= f(x-v(t±T)) = f((x-vt) ±λ) with λ=vT• Define x’=x-vt so that : f(x’)=f(x’±λ)

• So we can Implement Fourier series:

• Or simpler in complex notation:

Fourier Transform Analysis

• If the function of interest f(t) for example is not periodic then we modify the Fourier series analysis into Fourier integral or Fourier transform analysis.

• The non periodic function may be thought as a function with period infinity.

• Fourier series in complex notation of a function f(t) with period T :

• If f(t) is not periodic or its period’s interval is ∞, then T→ ∞, hence the freq. ν=1/T → 0 so 1/T → dv, with nν becomes a continuous variable nv→ v and Σ → ʃ

Fourier Transform Analysis

• Thus :

Define the following integral :

Then :

The pair of functions f(t) and G(v) are related by integral transformation.

G(v) is a Fourier transform of f(t), and

f(t) is the inverse Fourier Transform of G (v).

This transformations allow us to switch our working domain from v to t and vice-versa.

Fourier Transform

Alternative form of Fourier transform using variables : 𝜔 = 2𝜋𝜈 and t are:

Or, sometimes the pair of transformation are written as:

Fourier Transform

In special cases in which the function’s parity is known (even/odd), a simpler more direct expression of FT/ IFT can be used :

Even function Odd function