Fourier Series - Project Introduction

There are many ways to approximate badly-behaved functions (functions with discontinuities, corners, or functions with no antiderivatives) with functions that are more nicely behaved. Well-behaved functions are always differentiable and integrable everywhere, so we can use all the techniques of calculus to our advantage. If we use polynomials to approximate a function, we have what is known as a Taylor series. BC Calculus students study these approximations as a part of the BC course. Fourier series use sums of simple sines and cosines to approximate a function. Sines and cosines are easy to manipulate and their derivatives and integrals are well known, so they make a good choice for approximations, particularly for periodic functions.

Jean Baptiste Joseph Fourier (1768 - 1830)

Since sines and cosines are periodic, Fourier series are often used to represent periodic functions such as sound waves. You may have noticed that your voice on the phone (or on your school's intercom) doesn't sound like your voice in person. That's because a Fourier series has been used to eliminate some frequencies in your voice an amplify others.

This project will explain how Fourier series are created, compare them to Taylor series (this section can be skipped if you haven't studied Taylor series), move from the theory of continuous functions to the real-world of discrete data, and demonstrate how a Fourier series can be manipulated to remove unwanted frequencies in old recordings.

The project has six major sections:

• What is a Fourier series? - This section introduces the computational formulas for a Fourier series. Fourier series are defined by integration on the interval . We consider even function initially, then move to odd functions and finally to an arbitrary function on .
• Least Squares, Fourier, and Taylor Series - This section compares the errors on the interval in the Fourier series with those is a Taylor series of the same order. The Fourier series is the series that satisfies the least squares criterion, that is, has the minimum sum of squared errors over the interval. (This section can be skipped if students have not studies Taylor series).
• Changing the domain of the approximation - The Fourier series is defined initially on the interval , but more commonly we are interested in a different domain. This section explains how to modify the series to accommodate an arbitrary domain and period.
• Discrete Fourier Series - A continuous function can be sampled at regular intervals to reduce the continuous signal to a discrete signal. A common example is the conversion of the continuous signal of a sound wave to a sequence of data points. The data points can then be turned back into the continuous signal (this is the basic principle of a CD player) using techniques built on those described here.
• Using Audacity to Capture Sound - This section explains how to use the free software Audacity to produce sound waves that can be analyzed using Fourier series. Several student projects are suggested, including analyzing ides and recorded vs. live voices.
• Fourier Filter to Clean Up Old Recordings - Old recordings often have a fuzzy buzzing sound when played due to age and damage. It is possible to remove these unwanted noises by creating a Fourier series approximation, removing the terms related to those unwanted sounds, and playing back the filtered series. This section describes how sound waves can be "cleaned up" to remove high frequency or high energy components.

If you choose to have your students work through this investigation and find errors or have questions, please e-mail: "teague at ncssm dot edu." One of us in the department will get back to you with support.

Note: Solutions are password-protected. Please feel free to email one of us in the NCSSM math dept to receive the login credentials.

Soultions: Part 1, Part 2, Part 3, Part 4