Fourier Analysis — Pt 1. Fourier Series
I’ve been looking into Fourier theory for my project for a while. I feel like it’s a good time to write some blogs about it. Although it might look very difficult and complex at first glance, Fourier analysis is actually not as mysterious as it seems.
Here I would like to discuss Fourier analysis from the very basic Fourier Series, then work my way to Fourier Transforms, and to 2D Fourier Transform eventually. I hope to make everything as clear as possible so I’ll be providing mathematical proofs in support of Fourier theory. This is going to be pretty hardcore math proofs and require knowledge of calculus especially in integral as well as trigonometry. But I find it helps a lot in order to establish a solid understanding of Fourier theory. Hope this is helpful to anyone who’s curious about Fourier Analysis.
Complicated time waveforms can be expressed as a sum of sine and cosine wave
From the study of vector space we know that every vector can be expressed in only one way as a linear combination of the basis vectors. Similarly, we can express a periodic function f as a linear combination of the Fourier basis. But first, we need to prove that the set of Fourier basis is orthogonal.
Fourier Basis
The essential idea of Fourier theory is that we can represent any periodic function with simple sine and cosine waves. If m, n are any two different integers, then the function set {1, sin(mx), cos(nx), …} is an orthogonal set of functions on [-π, π]. In Fourier theory, we can establish such a set of sinusoids of different frequency as an orthogonal basis for continuous signals f.
Prove {1, sin(mx), cos(nx), …} is an orthogonal set of functions on [-π, π] when m≠n.
To prove that, we need to prove that for any sinusoids w, v in the set, w∘v=0
(1) sin(kπ)=0 is true for all integers k
(2) Therefore, we only need to prove that cos(mθ)∘ cos(nθ)=0
Now that we’ve proved the set {1, sin(mx), cos(nx), …} is an orthogonal set, we can use it as our Fourier basis.
Fourier Series (sinusoids)
Now that we have our orthogonal basis, we can make use of it by constructing linear combination of those basis to represent the complex function f.
Now we need to calculate the coefficients:
(1) For a_0, integral both sides of the equation:
(2) For a_n(n≥1), multiply both sides by cos(mx) then integral:
(3) Similarly for b_n (n≥1), multiply both sides by sin(mx) then integral:
Now you have it! The Fourier Series.
Note that we can also replace x with 2πx/T and with the help of Trigonometric Identities we can convert it into:
Here we have a periodic function with period T. Coefficient A_n denotes the amplitudes and ϕ_n is the phases.
Complex Fourier Series
For sinusoid Fourier series, we have coefficients a_0, a_n, and b_n in different formulas respectively. But wouldn’t it be nice if we have just one formula for all the coefficients? That’s where the complex Fourier series comes around. The idea of complex Fourier series is to express the same sinusoid Fourier series formula with complex exponentials.
In order to do that, let’s look at Euler’s Equation first.
We can represent sinθ and cosθ using e^{iθ}:
Here’s a table that demonstrates how much simpler it will be if using e^{iθ}
Now that we get the complex representation of sinθ and cosθ, we can substitute sinθ and cosθ (θ=nx) with it in the Fourier series.
We can also calculate the complex representation of the coefficient U_n by multiplying both sides with e^{-ikx} and then integral over x
Similarly, we can replace x with 2πx/T and we then can get a more general complex Fourier series that is defined on [-T/2,T/2] with period T.
Here in this blog we discussed Fourier series in terms of sinusoids and complex Fourier Series. In theory, we can use Fourier series to represent any periodic function. However, Fourier series does not extend Fourier theory beyond periodic functions and discrete frequency spectra. What should we do then? I’ll continue this in my next blog about Fourier Transform.
