Jean Baptiste Joseph Fourier And the Fourier Series

Highlights Joseph Fourier invented the Fourier series in his experiments on heat diffusion through solid objects. Served under Napoleon in Egypt, in which time he collected and published a great deal of information on ancient Egyptian artifacts. Fourier series is used today in a huge number of disciplines to analyze numerical information. Similar formulas are also used as the basis of nearly all computer multimedia compression.

Biography Born in Auxerre France, March 1768, the son of a tailor. Orphaned by the age of 9 Showed talent in math very early in life, and wished to become an officer in the French army after finishing college. Was denied the opportunity to become an officer because he was not born nobility. Was told that it would make no difference even if he was a, “second Newton.” At the age of 19, entered a Benedictine abbey to train for priesthood.

Biography Left the abbey when he was 21. His ambition to make an impact on the world of mathematics was too great. In a letter to one of his professors at his former college he wrote: “ Yesterday was my 21’st birthday, at that age Newton and Pascal had already acquired many claims to immortality” He returned to teaching at his college, the Ecole Royale Militaire . Played a part in the French Revolution, but was repulsed by the persecution of innocents. He was jailed for a short time, until Robespierre was overthrown.

Biography Fourier accompanied Napoleon on his invasion of Egypt, to usurp the land in the name of France from it’s Turkish rulers. It was Napoleon’s intention to begin an institute of learning in Cairo, and he took with him a large number of scientists and scholars. At the Institute de Egypt, Fourier undertook Egyptological work, and oversaw a number of Napoleon’s projects for three years until he returned to France in 1801.

Biography By 1807, Fourier had completed his work On the Propagation of Heat in Solid Bodies. Where he described his method of representing arbitrary functions as trigonometric series’. His contemporaries were reluctant to accept his claim that any function could be represented as an infinite series of trigonometric functions. Even after submitting clarifications over the course of the next few years, he was only able to gain mixed acceptance.

It was Fourier’s theory that any function defined on the interval of can be expressed as an infinite sum of sine and cosine waves. The amplitudes of the waves, a and b for function f, are found with these integrals.

The Fourier Series for: In this case, all a’s are zero, except for the first. .

It is now a matter of simple arithmetic to find the values for and It is now a matter of simple arithmetic to find the values for and . Only the odd values of n produce nonzero b’s since . Graphing these seven terms with the Fourier series: We begin to see an approximation of f(x)’s square wave appearing, even after just the first seven terms.

Numerical Data can be processed with the Discrete Fourier Transform N= the number of samples This uses Euler’s identity to express both the sin and cosine elements of the series as a complex power of e.

The essential quality of the Fourier transform is that it breaks allows us to view a function as the magnitudes of the frequencies that make it up. Things that use the Fourier Transform(or a similar function): Audio filters Audio Visualization i.e. voiceprint’s , spectrum analyzers Graphical manipulation and compression Signal analysis Many many more.

A voiceprint of music.

A two dimensional Fourier transform can be done on graphical images. The transformed data can be reduced in size by either cutting out frequencies, or quantizing the data(reducing the precision).

The ringing around borders is a common artifact in highly compressed graphics, and results from the loss of higher frequency data during compression. The first Fourier integral of the presentation showed the one-dimensional equivalent of this phenomenon in the sinusoid distortion of what should have been a perfect square wave.

MP3, JPEG, and MPEG MP3 compresses audio information by not storing inaudible frequencies JPEG(joint photographic experts group) image compression breaks an image into 8x8 blocks, does a Discrete cosine transform, and quantizes the data to fit the required quality level. Classical methods are then used to eliminate redundancy. MPEG(moving picture experts group) video compression(a variety of which is used to store the data on all DVD’s) is much like JPEG compression, except that it also attempts to find redundant data between frames,(still backgrounds, simple moving objects, etc.) and re-use the information from previous (or even future!) frames.