# The Fourier Series

## Contents

People are generally comfortable thinking about functions of time. For example a signal might be described as x(t), where "t" is time.  This is referred to as the "time domain."  However, it is often useful to think of signals and systems in the "frequency domain" where frequency, instead of time, is the independent variable, e.g., X(f) where "f" is frequency.  This brings us to the concept of Fourier Analysis.  The next several paragraphs try to describe why Fourier Analysis is important.

## Motivation

What follows are some examples that show how a function can be made up of a sum of sinusoidal functions.  These are given to show that it is plausible that general waveforms can be considered as a sum of sines and cosines.  The next section will describe exactly how we determine how we determine which sinusoids need to added together to form a particular function. Included are

Also included are a few examples that show, in a very basic way, a couple of applications of Fourier Theory, thought the number of applications and the ways that Fourier Theory is used are many.

### Examples: Functions as Sums of Sinusoids

##### Example: The Square Wave as a Sum of Sinusoids Average + 1st harmonic up to 3rd harmonic ...5th harmonic ...7th ...21st

If you click the leftmost button, you will see four functions: 1) a square wave (solid blue line), 2) the average value of the square wave, 3) a sine wave with a period of two seconds (dash-dot blue), or frequency = ½ Hz (solid magenta), and 4) the sum of the last three functions (solid red).  This last function is the a very crude approximation to the square wave.   Note: the sine wave is the same frequency as the square wave; we call this the 1st (or fundamental) harmonic.

If you click the second button another (smaller) sine wave is added to the picture with a frequency of 3/2 Hz  (this is three times as fast as the square wave (and the original sine wave); we call this the 3rd harmonic).  If you look at the sum of the average plus the two sine waves, you see that we get an even better approximation to the original square wave.

Notes:

• As you add sinusoids waves of increasingly higher frequency, the approximation gets better and better.
• The addition of higher frequencies better approximates the rapid changes, or details, (i.e., the discontinuity) of the original function (in this case, the square wave).
• On either side of the discontinuity there is a small overshoot (called "Gibb's phenomenon" or "Gibb's overshoot").  This overshoot is always present in the Fourier representation of a signal with a discontinuity and has the same height for any number of harmonics greater than 1.  The magnitude of the overshoot varies with the number of harmonics used but quickly converges to an amplitude of about 9% of the magnitude of the discontinuity for a square wave.  In this case the discontinuity has an amplitude of 1 unit, so the Gibb's overshoot is about 0.09.  Though the height of the overshoot is finite (at about 9%) as we add more harmonics, note that the width decreases, so the area of the overshoot (and hence the energy) decreases.  As the area goes to zero, its effect in most systems of interest to engineers also goes to zero.
• Because of the symmetry of the waveform, only odd harmonics (1, 3, 5, ...) are needed to approximate the function.  The reasons for this are discussed elsewhere.
• The rightmost button shows the sum of all harmonics up to the 21st harmonic, but not all of the individual sinusoids are explicitly shown on the plot.  In particular harmonics between 7 and 21 are not shown.
##### Example: The Triangle Wave as a Sum of Sinusoids Average + 1st harmonic up to 3rd harmonic ...5th harmonic ...7th

Note:

• As you add sine waves of increasingly higher frequency, the approximation gets better and better.
• The addition of higher frequencies better approximates the details, (i.e., the change in slope) in the original function.
• The amplitudes of the harmonics for this example drop off much more rapidly than those for the square wave (we prove this later).  As an example, compare the amplitude of the 3rd harmonic in both cases.  Conceptually, this occurs because the triangle wave looks much more like the 1st harmonic, so the contribution of the higher harmonics is less.  Even with only the 1st through 7th harmonics we have a very good approximation to the original function.
• There is no discontinuity, so no Gibb's overshoot.
• As before, only odd harmonics (1, 3, 5, ...) are needed to approximate the function.
##### Example: The Sawtooth as a Sum of Sinusoids Average + 1st harmonic up to 3rd harmonic ...5th harmonic ...7th ...21st

Notes:

• As you add sine waves of increasingly higher frequency, the approximation gets better and better.
• The addition of higher frequencies better approximates the rapid changes, or details, (i.e., the discontinuity) in the original function.
• Because of the discontinuity we see Gibb's phenomenon.
• As before, only odd harmonics (1, 3, 5, ...) are needed to approximate the function.
• The rightmost button shows the sum of all harmonics up to the 21st harmonic, but not all of the individual sinusoids are explicitly included in the plot.  In particular harmonics between 7 and 21 are not shown.
##### Example: The Electrocardiogram as a Sum of Sinusoids Average + 1st harmonic up to 2nd harmonic ...3rd harmonic ...4th ...10th ...20th

Notes:

• As you add sine waves of increasingly higher frequency, the approximation gets better and better.
• The addition of higher frequencies better approximates the rapid changes, or details, in the original function.
• Because the waveform lacks symmetry, both even and odd harmonics are needed to approximate the function.
• The two rightmost button shows the sum of all harmonics up to the 10th and 20th harmonics, but not all of the individual sinusoids are explicitly included in the plot.

### A Fourier Approach to Audio Signals

Fourier analysis plays a key role in the study of signals.  For example consider the function of time shown at the left below (the vertical axis is arbitrary).  If you play the signal (controls are below the image) you will hear the word "hello."  The signal is quite complicated as you can see by a detail of the image (from 0.30 to 0.35 seconds is shown)  shown in the upper right.  It is possible, using Fourier Analysis to find the magnitude of sinusoidal functions in the signal.  This is shown in the lower right where the amplitude of the signal as a function of frequency ("frequency domain") is shown.  Again, the units of the vertical axis can be considered to be arbitrary - the reason for the large scale (and correspondingly small signal) is to match the scale for the example after this one.

##### Example: Saying "Hello" Now consider  the signal shown below.  The time domain representation appears to be even more complicated than the one above.  If you play it, you will find that there is a lot of noise in the signal. Looking at the detail, you see that the signal is quite complicated to describe as a function of time. However, if you look at the frequency domain representation (lower right) the problem becomes immediately apparent. There are large signals near 300 and 700 Hz. The distortion is easily seen in both frequency and time, but only the frequency domain clearly shows the nature of the distortion.

##### Example: Saying "Hello" with Noise Added ## Fourier Theory Applied to Systems Analysis Now consider the example of a car moving a long a road.  This model of a car is very simple - a mass (m), connected to the road through a suspension system (spring (k) and dashpot, or friction (b)).  The road surface is irregular, but we can find the frequencies of the sinusoids that add up to describe the surface using Fourier Analysis.  We can also find the frequencies at which the suspension might be resonant using sinusoidal steady state analysis, or the Bode plot.  If certain frequencies are generally present, we would design are suspension so that it damps those frequencies.

The same principle could apply to the design of buildings.  If we are trying to design a building that is resistant to earthquakes, we could find the range of frequencies that are present in the shaking of the ground during an earthquake, and ensure that are building is insensitive to those frequencies.

## For Fun - A Short Biography of Fourier

A short biography of Fourier is available for your amusement and edification. According to wikipedia, he also discovered the greenhouse effect.

## Moving forward

The following are devoted to the development of Fourier Series.

References