Fourier Series, Representing Functions, and Applications

Definition and Motivation

In Chapter 6 of Understanding Analysis by Abbott, we explored some important properties of power series, where we tried to represent a function f(x) as a limit of polynomials. In this blog, we want to look at another type of powerful and widely-used series called Fourier series or trigonometric series, by representing a function as a series of sines and cosines.

First, let’s look at the definition of a Fourier series:

Let p>0 be a fixed number and f(x) be a periodic function with period 2p, defined on (-p,p). The Fourier series of f(x) is a way of expanding the function f(x) into an infinite series involving sines and cosines:

f(x)=\dfrac{a_0}{2}+\sum_{n=1}^\infty a_n\cos(\dfrac{n\pi x}{p})+\sum_{n=1}^\infty b_n\sin(\dfrac{n\pi x}{p}),

where a_0, a_n and b_n are called the Fourier coefficients of f(x), and are given by the formulas

a_0=\dfrac{1}{p}\int_{-p}^p f(x)dx,

a_n=\dfrac{1}{p}\int_{-p}^p f(x)\cos(\dfrac{n\pi x}{p})dx, and

b_n=\dfrac{1}{p}\int_{-p}^p f(x)\sin(\dfrac{n\pi x}{p})dx.

For example, if we want to represent an electrocardiogram as a Fourier series by a linear combinations of sine and cosine functions, we may want to do something like below:

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Let’s take some time and interpret this illustration: (1)The solid blue line represents the electrocardiogram wave which we want to approximate; (2) the dotted blue line is the average value of the electrocardiogram; (3) the dotted magenta line is a sine wave; and (4) the red solid line represents the sum of line (2) and (3). If we keep adding more sinusoidal waves to the sum, we may eventually get a closer and closer approximation to the desired electrocardiogram as showing below.

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But, you may wonder, how do we determine the phase and magnitude of each of those sine and cosine functions? Next, we will see how do we achieve that.

Representation and Transformation

Consider a periodic signal x_T(t) with period T. Since the period is T, we take the fundamental frequency to be  \omega_0=2\pi/T. We can represent any such function (with some very minor restrictions) using Fourier Series. The Fourier Series is more easily understood if we first restrict ourselves to functions that are either even or odd.

So an even function, x_e(t), can be represented as a sum of cosines of various frequencies via the equation:

x_e(t)=a_0+\sum_{n=1}^\infty a_n\cos(n\omega_0 t)

Consider the following function, x_T and its corresponding values for a_n. This function has T=1 so \omega_0=2\pi/T=2\pi.

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The right column shows the sum from n=0 to n=4. But how do we find the coefficient a_n? That’s a very good question. The process of solving a_n would require more algebraic work that involves integration and some trigonometric tricks, but we won’t go into details here.

Similarly, for odd functions, we can represent them as the sum of sine functions:

x_o(t)=\sum_{n=1}^\infty b_n\sin(n\omega_0 t)

Note that there is no b_0 term since the average value of an odd function over one period is always zero. If we have a function that is neither even nor odd, or is a combination of both, we may also use Fourier series to represent it, by creating two sets of functions to represent the even and odd parts, and then use the previous method to find each part of the coefficients respectively.

However, for most of the functions which are not periodic, we want to incorporate another approach called Fourier transform. The idea is that we want to let the period get very large, i.e., T\to\infty. Though this seems straightforward in concept, it fundamentally changes the nature of the transformation.

Historical Context

Baron Jean-Baptiste-Joseph Fourier, born in poor circumstances in Auxerre, introduced the idea that

There is no function f(x), or part of a function, which cannot be expressed by a trigonometric series.

Fourier was obsessed with heat, keeping his rooms uncomfortably hot for visitors. And also because of this, Fourier came upon his idea in connection with the problem of the flow of heat in solid bodies, including the earth.

Fourier analysis plays a key role in a lot of scientific and math studies. In the study of signals, for example if you say the word “hello”, you can actually represent the audio signals by a Fourier series.

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Besides, Fourier series also has a broad application in approximation theory, control systems, and in solving higher order partial differential equations, etc.

Reference

https://www.math.purdue.edu/academic/files/courses/2014fall/MA16021/FourierSeries(nopauses).pdf

https://www.khanacademy.org/science/electrical-engineering/ee-signals/ee-fourier-series/v/ee-fourier-series-intro

http://lpsa.swarthmore.edu/Fourier/Series/WhyFS.html

Posted on by Han Gao
Categories: Fall 2016

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