Fourier Series

*Periodic Square Wave Curve Decomposed Into a Series of Sine Wave Curves – Medium*

A Fourier series is an expansion of a periodic function f(x) in terms of an infinite sum of sines and cosines. Fourier series make use of the orthogonality relationships of the sine and cosine functions. The computation and study of Fourier series is known as harmonic analysis and is extremely useful as a way to break up an arbitrary periodic function into a set of simple terms that can be plugged in, solved individually, and then recombined to obtain the solution to the original problem or an approximation to it to whatever accuracy is desired or practical. Examples of successive approximations to common functions using Fourier series are illustrated below.¹

A Fourier Series has many applications in mathematical analysis as it is defined as the sum of multiple sines and cosines. Thus, it can be easily differentiated and integrated, which usually analyses the functions such as saw waves which are periodic signals in experimentation. It also provides an analytical approach to solve the discontinuity problem. In calculus, this helps in solving complex differential equations.²

Take Away

Simply put, a Fourier Series allows us to model any arbitrary periodic signal with a combination of sines and cosines.

References

¹ “Fourier Series — From Wolfram MathWorld”. 2022. mathworld.wolfram.com. https://mathworld.wolfram.com/FourierSeries.html.

² “Fourier Series – Definition, Formula, Applications And Examples”. 2022. BYJUS. https://byjus.com/maths/fourier-series/.

Additional Reading

“Fourier Analysis – Wikipedia”. 2022. en.wikipedia.org. https://en.wikipedia.org/wiki/Fourier_analysis.

In mathematics, Fourier analysis is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions. Fourier analysis grew from the study of Fourier series, and is named after Joseph Fourier, who showed that representing a function as a sum of trigonometric functions greatly simplifies the study of heat transfer.

Videos

Fourier Series introduction

Introduction to Fourier Series

Signal and System: Introduction to Fourier Series Topics Discussed:

1. What is the Fourier Series?
2. Use of Fourier Series.
3. Difference between Fourier Series and Fourier Transform.
4. Revision of periodic signals.
5. Harmonics.
6. Different types of Fourier Series Expansion.

But what is the Fourier Transform? A visual introduction.