Fourier Analysis from heat to JPEG

Michael Cowling

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 motions 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 Joseph Fourier, 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 1823.

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 + q).

More complicated 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 + q1) + a2 cos(2w t + q2) + a3 cos(3w t + q3) + ... .

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 frequency.

There are mathematical formulae to describe the coefficients and the phases, and it is possible to see how well the "partial sums" of the 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 mathematics are 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 constantly.

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 chance of 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 American; they are running a competition to see whether listeners can distinguish a Stradivarius violin from a modern copy. At least one wavelet site claims to know the answer.

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.