The Fourier series and The Fourier Transforms
Fourier Series and Fourier Transform are somewhat "weird". But may be that's what we need to speak with nature.......
In 1820, the French guy who was highly interested in understanding heat conduction through solids was trying his level best to solve the unsteady heat conduction equation without any heat generation known as diffusion equation. While solving this equation he came up with something that was not anticipated at all. Now known as the “Fourier Series”, which finds applications in physics, engineering, etc. In today's series of blogs in mathematics, we are going to look at the basics of the Fourier Series and some parts of Fourier Transforms. Let’s start this interesting topic quickly.
The first question which is very basic to ask is ‘What is the Fourier series?’ some of the readers are already familiar with this but to have common ground let’s define the Fourier series.
The Fourier series is an infinite series of sine and cosine functions that are used to approximate the given function
(which shouldn’t be a trigonometric function obviously) Knowing this the second question which comes to our mind straight away is why? What’s the need of approximating any function in terms of trigonometric functions sine and cosine? Well, there are a lot of advantages but the one which is most important is, it is easy to deal with trigonometric functions. As they are continuous in nature.
Continuous? Why do we need that? ….Ya you read it correctly. I said continuous because we try to model this world which is stochastic in nature using the hypothesis of the continuum. Now should we change this approach? That is the topic of another blog, for now, we need continuous functions and our purpose of achieving that is success through the use of Fourier’s series. So now it’s clear to us why do we need it and what is an advantage of having The Fourier series.
We are going to look at an easy example of how to use the Fourier series. Consider we have a square wave and now our task is to approximate that square wave using the Fourier series. A square wave is as shown in figure 1 and its approximation using the Fourier sine series is as shown in figure 2. Remember that whenever we talk about any trigonometric function we are essentially talking about a “periodic function”. The simplest way to understand periodic functions is as follows, consider a function f(t). If we say that ‘T’ is the period of the function then, we can write f(t) = f(t+T). Meaning the output of function f repeats though the input is different.
In figure 1 we can see there are corners for a square wave but, in figure 2 where we have done the approximation using the sine signal, we don’t have sharp corners. So in the end it is still an approximation. In this way, one can use the Fourier series to find the approximate function.
Let’s look at mathematical formulas for the Fourier series now. We have a function f(x) whose period is say L. It will be defined from 0 to L as it repeats throughout the domain. In that case, we can approximate using the following formula.
Where Ak and Bk are just the coefficients that can be found using the given formula.
Ak and Bk can be thought of as a projection of f(x) on the cosine as well as the sine axis. This key idea will be helpful in understanding the Fourier transform as we go ahead into the introduction to Fourier transforms.
Complex Fourier series:
The complex functions f(x) can be approximated using the complex Fourier series. Here the function is projected using exponential function raise to ‘i’ which is Euler’s way of defining the complex functions in complex planes. So the mathematical formulation looks like something which is shown below
This can be studied in terms of sine and cosine by writing Euler’s formula for the exponential terms. These characteristics are used for the Fourier Transforms.
Fourier transform is nothing but another coordinate transform that is used to represent the data, images, solution of differential equations, etc.
Fourier, while working on an unsteady heat equation in the 2-D rectangular domain, realized that the problem will get simplified if by some way we can transform it into the Eigenvector and Eigenvalue space. So he came up with this new type of transformation. But the basis is still the sine and cosine functions. It works like an orthogonal basis we use for our 2-D x-y plane. So if you are not familiar with Eigenvectors and Eigenvalues don’t worry. Just think of sine and cosine as the basis in the world of Fourier like x and y in 2-D geometry (Famously known as Hilbert’s space in quantum mechanics).
The Fourier transform is nothing but the extension of the Fourier series where the function we approximated earlier f(x) is not periodic. Initially, we were looking at the periodic function with the period being L. Now it’s not the case anymore, it’s the case of the infinite domain. This Fourier transform is a little bit mathematically challenging but with the background understanding of what we have just done it will be easy for the reader to go to any standard book and absorb the Fourier Transform in a better way.
Something More about Joseph Fourier: Joseph Fourier was a French mathematician who started teaching mathematics at an age of 16. He was the youngest professor in Europe known at that time for his deep understanding of signals. The research carried out by Fourier found little application in the early days of the 19th century but as soon as the digital world started to get expanded the Applications of his equations and transforms were found to be tremendous. He was aware of these applications though he started with studies in heat conduction only. He could prove the effects of the atmosphere on the temperature of the earth and also worked out some of the examples from electrical engineering during the 1820's. His work was stored in Egyptian books which took almost 23 volumes to complete. He was lucky enough to write the preface of that book on his own. He has also played a significant role in the French Revolution and politics.
The Fourier series is an infinite series of sine and cosine functions that are used to approximate the given function.
If f(t) = f(t+T) then 'T' is the period of function f. These are called as Periodic functions.
The Fourier series can be used to approximate the periodic functions.
The Fourier transform can be used to approximate any function, it doesn't matter if it's periodic or not.
In this blog, we have seen some basic definitions and ideas behind the Fourier series and Fourier Transforms. I hope it inspires the readers to go ahead and study more about this topic through any standard book. We will continue to bring you interesting topics through our blogs !!!