Fourier series

Approximates an original function by sines. If the function is "well behaved enough", the approximation is to arbitrary precision.

Fourier's original motivation, and a key application, is solving partial differential equations with the Fourier series.

Can only be used to approximate for periodic functions (obviously from its definition!). The Fourier transform however overcomes that restriction:

The Fourier series behaves really nicely in

L^{2}

, where it always exists and converges pointwise to the function: Carleson's theorem.

Video 1. But what is a Fourier series? by 3Blue1Brown (2019) Source. Amazing 2D visualization of the decomposition of complex functions.

Applications of the Fourier series

Solving partial differential equations with the Fourier series

See: math.stackexchange.com/questions/579453/real-world-application-of-fourier-series/3729366#3729366 from heat equation solution with Fourier series.

Separation of variables of certain equations like the heat equation and wave equation are solved immediately by calculating the Fourier series of initial conditions!

Other basis besides the Fourier series show up for other equations, e.g.:

Discrete Fourier transform (DFT)

Input: a sequence of

N

complex numbers

x_{k}

Output: another sequence of

N

complex numbers

X_{k}

such that:

x_{n} = \frac{1}{N} \sum_{k = 0}^{N - 1} X_{k} e^{i 2 π \frac{k n}{N}}

(1)

Intuitively, this means that we are braking up the complex signal into

N

sinusoidal frequencies:

$X_{0}$ : is kind of magic and ends up being a constant added to the signal because $e^{i 2 π \frac{k n}{N}} = e^{0} = 1$
$X_{1}$ : sinusoidal that completes one cycle over the signal. The larger the $N$ , the larger the resolution of that sinusoidal. But it completes one cycle regardless.
$X_{2}$ : sinusoidal that completes two cycles over the signal
...
$X_{N - 1}$ : sinusoidal that completes $N - 1$ cycles over the signal

and is the amplitude of each sine.

We use Zero-based numbering in our definitions because it just makes every formula simpler.

Motivation: similar to the Fourier transform:

compression: a sine would use N points in the time domain, but in the frequency domain just one, so we can throw the rest away. A sum of two sines, only two. So if your signal has periodicity, in general you can compress it with the transform
noise removal: many systems add noise only at certain frequencies, which are hopefully different from the main frequencies of the actual signal. By doing the transform, we can remove those frequencies to attain a better signal-to-noise

In particular, the discrete Fourier transform is used in signal processing after a analog-to-digital converter. Digital signal processing historically likely grew more and more over analog processing as digital processors got faster and faster as it gives more flexibility in algorithm design.

Sample software implementations:

numpy.fft, notably see the example: numpy/fft.py

Discrete Fourier transform of a real signal

See sections: "Example 1 - N even", "Example 2 - N odd" and "Representation in terms of sines and cosines" of www.statlect.com/matrix-algebra/discrete-Fourier-transform-of-a-real-signal

The transform still has complex numbers.

Summary:

$X_{0}$ is real
$X_{1} = \overset{ˉ}{X_{N - 1}}$
$X_{2} = \overset{ˉ}{X_{N - 2}}$
$X_{k} = \overset{ˉ}{X_{N - k}}$

Therefore, we only need about half of

X_{k}

to represent the signal, as the other half can be derived by conjugation.

"Representation in terms of sines and cosines" from www.statlect.com/matrix-algebra/discrete-Fourier-transform-of-a-real-signal then gives explicit formulas in terms of

X_{k}

NumPy for example has "Real FFTs" for this: numpy.org/doc/1.24/reference/routines.fft.html#real-ffts

Normalized DFT

There are actually two possible definitions for the DFT:

1/N, given as "the default" in many sources:
$x_{n} = \frac{1}{N} \sum_{k = 0}^{N - 1} X_{k} e^{i 2 π \frac{k n}{N}}$
(1)
$1 / N$ , known as the "normalized DFT" by some sources: www.dsprelated.com/freebooks/mdft/Normalized_DFT.html, definition which we adopt:
$x_{n} = \frac{1}{N} \sum_{k = 0}^{N - 1} X_{k} e^{i 2 π \frac{k n}{N}}$
(2)

The

1 / N

is nicer mathematically as the inverse becomse more symmetric, and power is conserved between time and frequency domains.

Fast Fourier transform

An efficient algorithm to calculate the discrete Fourier transform.

Fourier transform

Continuous version of the Fourier series.

Can be used to represent functions that are not periodic: math.stackexchange.com/questions/221137/what-is-the-difference-between-fourier-series-and-fourier-transformation while the Fourier series is only for periodic functions.

Of course, every function defined on a finite line segment (i.e. a compact space).

Therefore, the Fourier transform can be seen as a generalization of the Fourier series that can also decompose functions defined on the entire real line.

As a more concrete example, just like the Fourier series is how you solve the heat equation on a line segment with Dirichlet boundary conditions as shown at: Section "Solving partial differential equations with the Fourier series", the Fourier transform is what you need to solve the problem when the domain is the entire real line.

Multidimensional Fourier transform

Lecture notes:

www.robots.ox.ac.uk/~az/lectures/ia/lect2.pdf Lecture 2: 2D Fourier transforms and applications by A. Zisserman (2014)

Video 1. How the 2D FFT works by Mike X Cohen (2017) Source. Animations showing how the 2D Fourier transform looks like for simple inpuf functions.

Fourier inversion theorem

A set of theorems that prove under different conditions that the Fourier transform has an inverse for a given space, examples:

Carleson's theorem for $L^{2}$

Video 1. The Laplace Transform: A Generalized Fourier Transform by Steve Brunton (2020) Source. Explains how the Laplace transform works for functions that do not go to zero on infinity, which is a requirement for the Fourier transform. No applications in that video yet unfortunately.

Fourier series

Applications of the Fourier series

Solving partial differential equations with the Fourier series

Discrete Fourier transform (DFT)

Discrete Fourier transform of a real signal

Normalized DFT

Fast Fourier transform

Fourier transform

Multidimensional Fourier transform

Fourier inversion theorem

Laplace transform

History of the Fourier series

 Ancestors

 Incoming links

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