Fourier Analysis from heat to JPEG
School of Mathematics, UNSW
Fourier Analysis attempts to understand complex phenomena in terms of
simpler phemonena. In some senses, it is a very
old idea. For instance, early astronomers who worked in terms of
geocentric models of the universe observed that the planets moved
around the heavens in a somewhat irregular fashion. Ptolemy
of Alexandria is credited with the idea of resolving these complex
into a sum of circular motions; I consider this attempt to be an early
example of harmonic analysis. Another instructive example is musical; a
violin and a flute sound different even when they are playing the
``same'' note. This is now explained by saying that a flute produces a
``pure'' (or harmonic-free) tone, while a violin produces a ``complex''
tone rich in ``harmonics'', which are resonances at frequencies which
are multiples of the basic frequency. The
characteristic timbre of an instrument is determined by the harmonics
present in its sound, and by their relative intensities.
Fourier Analysis is named after Jean Baptiste
who had an interesting life in revolutionary
France. He was a passionate mathematician, interested in
understanding nature through Mathematics.
His prize-winning essay Théorie analytique de la
chaleur was published in 1822, some 15 years after it was
first circulated. He was elected Fellow of the Royal Society in
Fourier Analysis is the mathematical idea that one can
understand a complex signal
as the sum of simple signals. To illustrate that this has some
real physical meaning, let us consider some acoustic signals. First of
all, a pure
tone can be written
as a cosinusoidal signal of amplitude a,
frequency w , and phase q:
x(t) = a cos(w t
tones can be represented by a Fourier series,
a sum of pure tones whose frequencies are integer multiples (harmonics)
of a fundamental frequency, w :
x(t) = a1 cos(w t
+ a2 cos(2w t
+ a3 cos(3w t
+ ... .
All the information about the tone is encapsulated in the amplitudes an, the frequencies nw and the
phases qn. Very few
coefficients and phases are needed to describe sine waves or triangle
waves, but many are needed to describe square waves and sawtooth
waves. In white noise,
the frequencies are random and not multiples of a basic
There are mathematical formulae to describe the coefficients and the
phases, and it is possible to see how well the "partial sums"
Fourier series represent the original signal. Mathematicians play with
powerful symbolic computation packages, such
as MAPLE, to view signals in more
mathematical language and manipulate them. We tend to treat
anything which can be graphed in the same way as a signal and to
Fourier analyse it.
It is also possible to treat higher-dimensional
phenomena in the same way. Fourier analysis is very useful
in physics, because it is very suitable for
discussing wave phenomena such as electromagnetic waves.
Fourier analysis was invented (or discovered) in the 19th Century, but
it is still undergoing development. This can be
compared to the internal combustion engine powered vehicle; it too was
discovered in the 19th Century, but has been developed greatly
since then, to become more powerful and cleaner. Many users of
like drivers: they are perfectly happy to leave the inner workings of
their vehicles to mechanics. But mathematicians are like
mechanics: they like to see how the machine works and to improve it
In recent decades, mathematicians started asking questions about
Fourier analysis in strange environments like spheres. Here,
simple sinusoidal waves are no longer available, and it becomes
necessary to break things up into other pieces. My own research
is concerned with Fourier analysis in "manifolds" such as spheres and
"negatively curved spaces". The basic questions are to find
the "simple" building blocks and decompose and synthesise complex
phenomena in terms of these simple phenomena.
As part of this investigation, the mathematicians R.R. Coifman and G.
Weiss talked about
breaking up functions into "atoms". More or less at the same
time, the physicist Grossmann and the geophysicist Morlet looked as
special kinds of signals which were the physical versions of these
atoms. This has led to a new kind of analysis. The
basic objects are no longer sinusoidal functions, but much more
irregular objects, called wavelets. This
has been one of the most significant mathematical discoveries of
the 20th Century, and Coifman has done
very well out of it.
It turns out that complex signals can often be written better using
wavelets than using sines and cosines. Anything which can be
written as a sum of a few sine functions must repeat periodically to
infinity, and be very smooth, but real-world signals are not like
this-they are spatially
localised, and often irregular. The sum of a few wavelets has some
approximating a localised signal.
Let us consider a piece of music
by Bach. On a CD, this requires tens of megabytes of space. The
original score contains less than a megabyte of information, but on
the other hand this does not define the final sound completely.
The mp3 file contains about four megabytes, and is probably quite close
to the minimum required to define the sound. This is achieved
with careful encoding. By the way, this was downloaded from Scientific
are running a competition to see whether listeners can distinguish a
violin from a modern copy. At least one wavelet site claims to know the
Digital photography produces files of the order of one megabyte in
size, usually in jpg format. The same photograph, in "bitmap"
format, that is, where the colour of each pixel on the screen is
defined, is about ten times as big. The next version of jpg,
which achieves more compression, is based on wavelets. As we live
in an era in which we want to transmit and store ever more data,
compression becomes ever more important.
The interesting challenge now is to understand how we hear and how we
see. This may enable us to compress even further. Putting
"wavelets and hearing" into google finds over three thousand
references, and I understand from a Swiss colleague who spent six
months in Cambridge last year that wavelet analysis and Fourier
analysis are casting new light on our understanding of the cochlear and
how it works. It will be interesting to see whether wavelet and
Fourier ideas will end up playing a role in vision.
Acknowledgement: Many parts of this material include fragments taken from
the quoted web pages, and other web-published material has informed this
discussion. In particular, Amara Graps' pages on
wavelets have been very useful.