Applications of the Fourier Series

     Applications of the Fourier Series

The Fourier Series, the founding principle behind the _eld of Fourier Analysis, is an in_nite

expansion of a function in terms of sines and cosines. In physics and engineering, expanding functions

in terms of sines and cosines is useful because it allows one to more easily manipulate functions that

are, for example, discontinuous or simply di_cult to represent analytically. In particular, the _elds

of electronics, quantum mechanics, and electrodynamics all make heavy use of the Fourier Series.

Introduction

The Fourier Series, the founding principle behind the

_eld of Fourier Analysis, is an in_nite expansion of a func-

tion in terms of sines and cosines or imaginary exponen-

tials. The series is de_ned in its imaginary exponential

form as follows:

where the An's are given by the expression

Thus, the Fourier Series is an in_nite superposition

of imaginary exponentials with frequency terms that in-

crease as n increases. Since sines and cosines (and in turn,

imaginary exponentials) form an orthogonal set1, this se-

ries converges for any moderately well-behaved function

f(x). Examples of the Fourier Series for di_erent wave-

forms are given in _gure I.

II. THE FAST FOURIER TRANSFORM

The Fourier Series is only capable of analyzing the fre-

quency components of certain, discreet frequencies (in-

tegers) of a given function. In order to study the case

where the frequency components of the sine and cosine

terms are continuous, the concept of the Fourier Trans-

form must be introduced. The imaginary exponential

form of the Fourier Transform is de_ned as follows:

This equation is called the Discreet Fourier Transform

(DFT) of the function h(t). If we denote Hn as

the Fourier Transform, H(!), may then be approxi-

mated using the expression

Comparing equation (6) with the Fourier Series given

in equation (1), it is clear that this is a form of the Fourier

Series with non-integer frequency components.

Currently, the most common and e_cient method of

numerically calculating the DFT is by using a class of al-

gorithms called \Fast Fourier Transforms" (FFTs). The

_rst known discovery of the FFT was by Gauss in 1805;

however, the _rst modern \rediscovery" of the FFT was

done in 1942 by Danielson and Lanczos4. They were

able to show one may divide any DFT into a sum of two

DFT's which each correspond to N

2 􀀀 1 points.

The proof of Danielson and Lanczos's assertion is the

following4:

First, de_ne W as the complex number

Here, Hen

denotes the even terms of the sum (the ones

corresponding to the index 2k) and HO

n denotes the odd

terms (the ones corresponding to the index 2k + 1).

The most useful part of this formula is that it can be

used recursively, since each of these Hen

and HO

n terms

may be independently expanded using the same algo-

rithm, each time reducing the number of calculations by

a factor of 2. In fact, this class of FFT algorithm shrinks

the compution time from O(N2) operations to the much

more manageable O(N log2 N) operations. There are

many di_erent FFT algorithms; the one presented here is

simply the most common one, known as a Cooley-Tukey

FFT algorithm. There are other algorithms which can

decrease computation time by 20 or 30 percent (so-called

base-4 FFTs or base-8 FFTs)4. Most importantly, both

classes of FFT algorithms are fast enough to embed into

modern digital oscilloscopes and other such electronic

equipment. Thus, FFTs have many modern applications,

such as Spectrum Analyzers, Digital Signal Processors

(DSPs), and the numerical computation arbitrary-size

multiplication operations.

III. THE SPECTRUM ANALYZER

An important instrument to any experimentalist is the

spectrum analyzer. This instrument reads a signal (usu-

ally a voltage) and provides the operator with the Fourier

coe_cients which correspond to each of the sine and co-

sine terms of the Fourier expansion of the signal. Sup-

pose an instrument takes a time-domain signal, such as

the amplitude of the output voltage of an instrument.

Let us call this signal V(t). Then the DFT of V(t) is

We see that this equation is of the same form of

equation (6), which means that the previously described

methods of the FFT apply to the function. Thus, any

digital oscilloscope that is su_ciently fast and equipped

with a FFT algorithm is capable of providing the user

with the frequency components of the source signal.

Oscilloscopes which are equipped with the ability to

FFT their inputs are termed \Digital Spectral Analyz-

ers". Although they were once a separate piece of equip-

ment for experimentalists, improvements in digital elec-

tronics has made it practical to merge the role of oscil-

loscopes with that of the Spectral Analyzer; it is quite

common now that FFT algorithms come built into oscil-

loscopes.

Spectrum Analyzers have many uses in the laboratory,

but one of the most common uses is for signal noise stud-

ies. As shown above, the FFT of the signal gives the

amplitudes of the various oscillatory components of the

input. After normalization, this allows for the experi-

mentalist to determine what frequencies dominate their

signal. For example, if you have a DC signal, you would

expect the FFT to show only very low frequency oscil-

lations (i.e., the largest amplitudes should correspond to

f _ 0). However, if you see a sharp peak of amplitudes

around 60 Hz, you would know that something is feed-

ing noise into your signal with a frequency of 60 Hz (for

example, an AC leakage from your power source).

IV. DIGITAL SIGNAL PROCESSING

We have already seen how the Fourier Series allows

experimentalists to identify sources of noise. It may also

be used to eliminate sources of noise by introducing the

idea of the Inverse Fast Fourier Transform (IFFT).

In general, the goal of an Inverse Fourier Transform is to

take the An (the ones that appear in eq (5) and use them

to reconstruct the original function, f(t).

Analytically, this is done by multiplying each An by

e2_ik n

N then taking the sum over all n. However, this

is an ine_cient algorithm to use when the calculation

must be done numerically. Just as there is a fast numer-

ical algorithm for approximating the Fourier coe_cients

(the FFT), there is another e_cient algorithm, called the

IFFT, which is capable of calculating the Inverse Fourier

Transform much faster than the brute-force method.

In 1988, it was shown by Duhamel, Piron, and Etcheto7

that the IFFT is simply

In other words, you can calculate the IFFT directly

from the FFT; you simply ip the real and imaginary

parts of the coe_cients calculated by the original FFT.

Thus, the IFFT algorithms are essentially the same as

the FFT algorithms; all one must do is ip the numbers

around at the beginning of the calculation.

Since the IFFT inherits all of the speed bene_ts of the

FFT, it is also quite practical to use it in real time in

the laboratory. One of the most common applications of

the IFFT in the laboratory is to provide Digital Signal

Processing (DSP). In general, the idea of DSP is to use

con_gurable digital electronics to clean up, transform,

or amplify a signal by _rst FFT'ing the signal, removing,

shifting or damping the unwanted frequency components,

and then transforming the signal back using the IFFT on

the _ltered signal.

There are many advantages to doing DSP as opposed

to doing analog signal processing. To begin with, prac-

tically speaking, you can have a much more complicated

_ltering function (the function that transforms the coe_-

cients of the DFT) with DSP than analog signal process-

ing. While it is fairly easy to make a single band pass,

low pass, or high pass _lter with capacitors, resistors,

and inductors, it is relatively di_cult and time consum-

ing to implement anything more complicated than these

three simple _lters. Furthermore, even if a more compli-

cated _lter was implemented with analog electronics, it

is di_cult to make even small modi_cations to the _lter

(there are exceptions to this, such as FPGA's, but those

are also more di_cult to implement than simple software

solution). DSP is not limited by either of these e_ects

since the processing is (usually) done in software, which

can be programmed to do whatever the user desires.

Probably the most important advantage that DSP has

over analog signal processing is the fact that the pro-

cessing may be done after the signal has been taken.

V. ANALYTICAL APPLICATIONS

The Fourier Series also has many applications in math-

ematical analysis. Since it is a sum of multiple sines and

cosines, it is easily di_erentiated and integrated, which

often simpli_es analysis of functions such as saw waves

which are common signals in experimentation.

A. Discontinuous Functions

The Fourier Series also o_ers a simpli_ed analytical ap-

proach to dealing with discontinuous functions. Dirich-

let's Theorem states the following9:

If f(x) is a periodic of period 2_, and if be-

tween 􀀀_ and _ it is single-valued, has a _nite

number of maximum and minimum values,

and a _nite number of discontinuities, and

if

R _

􀀀_ jf(x)jdx is _nite, then the Fourier se-

ries converges to f(x) at all the points where

f(x) is continuous; at jumps, the Fourier se-

ries converges to the midpoint of the jump

(This includes jumps that occur at __ for

the periodic function).

In other words, nearly every function encountered in

physics, both continuous and discontinuous, may be rep-

resented in terms of the Fourier Series. This gives the

Fourier Series a distinct advantage over the Taylor Se-

ries expansion of a function, since the Taylor Series

places much more stringent limits on convergence than

the Fourier Series does (continuity is a requirement, for

example).

B. Convolutions

The Convolution Theorem states the following:

where F[f] denotes the Fourier Transform of the

function f. Since the Fourier Transform may be approx-

imated by a Fourier Series, FFT algorithms may be ap-

plied to the numerical calculation of the convolution; in

fact, the FFT method is the preferred method of cal-

culating convolutions which prevents the need for direct

integration5.

C. Generalized Fourier Series

The concept of the Fourier Series may be generalized

to any complete orthogonal system of functions. An \or-

thogonal system" satis_es the following relation.

In a generalized Fourier Series, we use these func-

tions _(x) as the expansion functions instead of sines and

cosines (or imaginary exponentials). Then our expansion

takes on the following form

One may then _nd the coe_cients an in an analogous

way that one _nds the coe_cients in the Fourier Series:

where cn is the normalization constant given by the

orthogonality relationship de_ned in (13). Equating the

_rst and last parts leaves us with

This result is analogous to the result that was presented

in equation (2), and can be used to derive these expres-

sions. An example of another complete orthogonal sys-

tem which can be used as the basis element for a General-

ized Fourier Series is the set of Spherical Harmonics. The

Spherical Harmonics provide an series that is analogous

to the Fourier Series, called the Laplace series, which is

given by the expression

Functional expansions of this form are termed \Gener-

alized Fourier Series" since they utilize the orthogonality

relationships of functional systems in the same way that

the Fourier Series does.

VI. CONCLUSION

The Fourier Series is useful in many applications rang-

ing from experimental instruments to rigorous mathe-

matical analysis techniques. Thanks to modern develop-

ments in digital electronics, coupled with numerical al-

gorithms such as the FFT, the Fourier Series has become

one of the most widely used and useful mathematicaltools available to any scientist.